tracking the norm with
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Tracking the Norm with ` 2 Constant Update Time Chi-Ning Chou - PowerPoint PPT Presentation

Nov 28, 2022 •332 likes •1.55k views

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  13. <latexit sha1_base64="Bok7UKzayt/MUBbkMQ2wte4LM+g=">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</latexit> <latexit sha1_base64="fQ9XwkiByEOGBnsGfGq1nvKe7f8=">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</latexit> <latexit sha1_base64="G/KNIkeIBrEkZfB98ZOk1fm8DGg=">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</latexit> <latexit sha1_base64="Qt/7gi9eWTWoK9/0n10gwW7xL/g=">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</latexit> <latexit sha1_base64="r8yjykUf+rGX+xatj5cCNF7h5k8=">ACMXicbVDLSgNBEJz1Gd9Rj14Gg+Bwq4Kegx48ahgVMguYXbSGwfnscz0Rpclf+JVf8CvyU28+hNOYg6+Cgaq7rpnkpzKRyG4SiYmZ2bX1isLS2vrK6tb9Q3t6dKSyHNjfS2NuUOZBCQxsFSrjNLTCVSrhJ78/G/s0ArBNGX2GZQ6JYX4tMcIZe6tbrGAvdUckBFWOik269ETbDCehfEk1Jg0x0d0MluKe4YUCjVwy5zpRmGNSMYuCSxgux4WDnPF71oeOp5opcEk1OX1I97zSo5mx/mkE/X7RMWUc6VKfadieOd+e2PxP69TYHaVELnBYLmX4uyQlI0dJwD7QkLHGXpCeNW+Fspv2OWcfRp/dhSmkL3kaX+JxoeuFGK6V4V50aWwypGeESXVZNq6MOLfkf1l1wfNqOj5uHlcaPVmsZYIztkl+yTiJyQFjknF6RNOBmQJ/JMXoLXYBS8Be9frTPBdGab/EDw8Qm6Naqj</latexit> Streaming Algorithms • Input : A stream of inputs from the alphabet set. Example: . a 1 , a 2 , . . . , a m ∈ [ n ] • Output : Some statistics of the inputs. Example: # of distinct elements in the input stream. • Frequency Vector : For each , define t ∈ [ m ] , i ∈ [ n ] f ( t ) = | { t 0 ∈ [ t ] : a t 0 = i } | . i Example: norm = # of distinct elements; norm = t . ` 0 ` 1 • Applications : Database optimization, network tra ffi c etc. • Goal : Randomized algorithms using sublinear space. � 3

  14. <latexit sha1_base64="uQ3EyafsaxwtH4v0Zgfy4fLE6Y=">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</latexit> <latexit sha1_base64="r8yjykUf+rGX+xatj5cCNF7h5k8=">ACMXicbVDLSgNBEJz1Gd9Rj14Gg+Bwq4Kegx48ahgVMguYXbSGwfnscz0Rpclf+JVf8CvyU28+hNOYg6+Cgaq7rpnkpzKRyG4SiYmZ2bX1isLS2vrK6tb9Q3t6dKSyHNjfS2NuUOZBCQxsFSrjNLTCVSrhJ78/G/s0ArBNGX2GZQ6JYX4tMcIZe6tbrGAvdUckBFWOik269ETbDCehfEk1Jg0x0d0MluKe4YUCjVwy5zpRmGNSMYuCSxgux4WDnPF71oeOp5opcEk1OX1I97zSo5mx/mkE/X7RMWUc6VKfadieOd+e2PxP69TYHaVELnBYLmX4uyQlI0dJwD7QkLHGXpCeNW+Fspv2OWcfRp/dhSmkL3kaX+JxoeuFGK6V4V50aWwypGeESXVZNq6MOLfkf1l1wfNqOj5uHlcaPVmsZYIztkl+yTiJyQFjknF6RNOBmQJ/JMXoLXYBS8Be9frTPBdGab/EDw8Qm6Naqj</latexit> <latexit sha1_base64="Bok7UKzayt/MUBbkMQ2wte4LM+g=">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</latexit> <latexit sha1_base64="G/KNIkeIBrEkZfB98ZOk1fm8DGg=">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</latexit> <latexit sha1_base64="Qt/7gi9eWTWoK9/0n10gwW7xL/g=">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</latexit> <latexit sha1_base64="fQ9XwkiByEOGBnsGfGq1nvKe7f8=">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</latexit> Streaming Algorithms • Input : A stream of inputs from the alphabet set. Example: . a 1 , a 2 , . . . , a m ∈ [ n ] Deterministic algorithm: space. Θ (min { m, n } ) • Output : Some statistics of the inputs. Example: # of distinct elements in the input stream. • Frequency Vector : For each , define t ∈ [ m ] , i ∈ [ n ] f ( t ) = | { t 0 ∈ [ t ] : a t 0 = i } | . i Example: norm = # of distinct elements; norm = t . ` 0 ` 1 • Applications : Database optimization, network tra ffi c etc. • Goal : Randomized algorithms using sublinear space. � 3

  15. <latexit sha1_base64="uQ3EyafsaxwtH4v0Zgfy4fLE6Y=">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</latexit> <latexit sha1_base64="r8yjykUf+rGX+xatj5cCNF7h5k8=">ACMXicbVDLSgNBEJz1Gd9Rj14Gg+Bwq4Kegx48ahgVMguYXbSGwfnscz0Rpclf+JVf8CvyU28+hNOYg6+Cgaq7rpnkpzKRyG4SiYmZ2bX1isLS2vrK6tb9Q3t6dKSyHNjfS2NuUOZBCQxsFSrjNLTCVSrhJ78/G/s0ArBNGX2GZQ6JYX4tMcIZe6tbrGAvdUckBFWOik269ETbDCehfEk1Jg0x0d0MluKe4YUCjVwy5zpRmGNSMYuCSxgux4WDnPF71oeOp5opcEk1OX1I97zSo5mx/mkE/X7RMWUc6VKfadieOd+e2PxP69TYHaVELnBYLmX4uyQlI0dJwD7QkLHGXpCeNW+Fspv2OWcfRp/dhSmkL3kaX+JxoeuFGK6V4V50aWwypGeESXVZNq6MOLfkf1l1wfNqOj5uHlcaPVmsZYIztkl+yTiJyQFjknF6RNOBmQJ/JMXoLXYBS8Be9frTPBdGab/EDw8Qm6Naqj</latexit> <latexit sha1_base64="Bok7UKzayt/MUBbkMQ2wte4LM+g=">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</latexit> <latexit sha1_base64="G/KNIkeIBrEkZfB98ZOk1fm8DGg=">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</latexit> <latexit sha1_base64="Qt/7gi9eWTWoK9/0n10gwW7xL/g=">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</latexit> <latexit sha1_base64="fQ9XwkiByEOGBnsGfGq1nvKe7f8=">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</latexit> Streaming Algorithms • Input : A stream of inputs from the alphabet set. Example: . a 1 , a 2 , . . . , a m ∈ [ n ] Deterministic algorithm: space. Θ (min { m, n } ) • Output : Some statistics of the inputs. Example: # of distinct elements in the input stream. • Frequency Vector : For each , define t ∈ [ m ] , i ∈ [ n ] f ( t ) = | { t 0 ∈ [ t ] : a t 0 = i } | . i Example: norm = # of distinct elements; norm = t . ` 0 ` 1 • Applications : Database optimization, network tra ffi c etc. • Goal : Randomized algorithms using sublinear space. Faster time!? � 3

  16. <latexit sha1_base64="uduSAHc0LNEyQ/YF/UPTbTsE68=">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</latexit> Estimation ` 2 � 4

  17. <latexit sha1_base64="uduSAHc0LNEyQ/YF/UPTbTsE68=">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</latexit> <latexit sha1_base64="N0sSOIwTN4Wmria72AQEY1gd0Og=">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</latexit> Estimation ` 2 • Goal : Estimating the norm of the frequency vector in ` 2 sublinear space. � 4

  18. <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="uduSAHc0LNEyQ/YF/UPTbTsE68=">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</latexit> <latexit sha1_base64="SKZeXgsCq5/7d14G1DzuCv0kNI8=">ACfXicdVDdbtMwGHXD3zb+OrjkxqKaNKQRpenW7gpV4obLItFtUp1FjuOk1vwT7C9AlObleAvegFt4AXC7IjE0jmTr+Jzv82efrJLCQR96wV37t67/2Bnd+/ho8dPnvb3n505U1vG58xIYy8y6rgUms9BgOQXleVUZKfZ1dv1/75J26dMPoDNBVPFC21KASj4KW0T8jMEskLWGz2FXGiVDRVr8mquGwP1auOrNL4MiZWlEtYvSG8ckIafaud+Ds+kpxLoJgc4TDtD6IwmpwM4xMchaPTUTyJPRmPR+PjCA/DaIMB2mKW7vd2SW5YrbgGJqlzi2FUQdJSC4J3u2R2vGKsita8oWnmiruknYTQ4cPvJLjwli/NOCN+ndHS5Vzjcp8paKwdP96a/E2b1FDcZq0Qlc1cM2uBxW1xGDwOlOcC8sZyMYTyqzwb8VsS1l4JO/MaUxtS6BZv4nmn9mRimq85ZURjZdS4B/AVe0m1Pnw/uTEP4/OYvD4SiM3x8PptNtjDvoBXqJDtEQTdAUvUMzNEcMfUXf0Q/0s/crOAiOgvC6NOhte56jGwgmvwF9zMZl</latexit> <latexit sha1_base64="7nCA/8Ujab8yoBjsWdZAY4VRanU=">ACJ3icbVDLSgNBEJz1/Tbq0ctiEDyFXRX0GPDiUcFoIBtC76Q3Ds5jmelVlyU/4V/wK/xJnr0T5zEHwVDNRUdPdleZSOIqi92BqemZ2bn5hcWl5ZXVtvbaxelMYTm2uJHGtlNwKIXGFgmS2M4tgkolXqU3JyP/6hatE0ZfUJljV8FAi0xwIC+1EycGCnqV6tHjWiM8C+J6TOJjrbQSLSd/wQqEmLsG5Thzl1K3AkuASh0tJ4TAHfgMD7HiqQaHrVuOFh+GuV/phZqx/msKx+r2jAuVcqVJfqYCu3W9vJP7ndQrKjruV0HlBqPnXoKyQIZlwdH3YFxY5ydIT4Fb4XUN+DRY4+Yx+TClNoQcEqb9E4x03SoHuV0luZDmsEsJ7clk1/g19ePHvqP6Sy/1GfNDYPz+sN5uTGBfYNtheyxmR6zJTtkZazHOJHtgj+wpeA5egtfg7at0Kpj0bLEfCD4+AWFtp4E=</latexit> <latexit sha1_base64="N0sSOIwTN4Wmria72AQEY1gd0Og=">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</latexit> Estimation ` 2 • Goal : Estimating the norm of the frequency vector in ` 2 sublinear space. • -One-shot estimation : Output s.t. ( ✏ , � ) σ m h� � i � � m � k f ( m ) k 2 � > ✏ k f ( m ) k 2  � . Pr � � 2 2 � 4

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( ✏ , � ) σ m h� � i � � m � k f ( m ) k 2 � > ✏ k f ( m ) k 2  � . Pr � � 2 2 • -Weak tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 � 4

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( ✏ , � ) σ m h� � i � � m � k f ( m ) k 2 � > ✏ k f ( m ) k 2  � . Pr � � 2 2 • -Weak tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 • -Strong tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( t ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 � 4

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sha1_base64="SKZeXgsCq5/7d14G1DzuCv0kNI8=">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</latexit> <latexit sha1_base64="N0sSOIwTN4Wmria72AQEY1gd0Og=">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</latexit> Estimation ` 2 • Goal : Estimating the norm of the frequency vector in ` 2 sublinear space. • -One-shot estimation : Output s.t. ( ✏ , � ) σ m h� � i � � m � k f ( m ) k 2 � > ✏ k f ( m ) k 2  � . Pr � � 2 2 • -Weak tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 • -Strong tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( t ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 Strong tracking => Weak tracking => One-shot � 4

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sha1_base64="SKZeXgsCq5/7d14G1DzuCv0kNI8=">ACfXicdVDdbtMwGHXD3zb+OrjkxqKaNKQRpenW7gpV4obLItFtUp1FjuOk1vwT7C9AlObleAvegFt4AXC7IjE0jmTr+Jzv82efrJLCQR96wV37t67/2Bnd+/ho8dPnvb3n505U1vG58xIYy8y6rgUms9BgOQXleVUZKfZ1dv1/75J26dMPoDNBVPFC21KASj4KW0T8jMEskLWGz2FXGiVDRVr8mquGwP1auOrNL4MiZWlEtYvSG8ckIafaud+Ds+kpxLoJgc4TDtD6IwmpwM4xMchaPTUTyJPRmPR+PjCA/DaIMB2mKW7vd2SW5YrbgGJqlzi2FUQdJSC4J3u2R2vGKsita8oWnmiruknYTQ4cPvJLjwli/NOCN+ndHS5Vzjcp8paKwdP96a/E2b1FDcZq0Qlc1cM2uBxW1xGDwOlOcC8sZyMYTyqzwb8VsS1l4JO/MaUxtS6BZv4nmn9mRimq85ZURjZdS4B/AVe0m1Pnw/uTEP4/OYvD4SiM3x8PptNtjDvoBXqJDtEQTdAUvUMzNEcMfUXf0Q/0s/crOAiOgvC6NOhte56jGwgmvwF9zMZl</latexit> <latexit sha1_base64="N0sSOIwTN4Wmria72AQEY1gd0Og=">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</latexit> Estimation ` 2 • Goal : Estimating the norm of the frequency vector in ` 2 sublinear space. • -One-shot estimation : Output s.t. ( ✏ , � ) σ m h� � i � � m � k f ( m ) k 2 � > ✏ k f ( m ) k 2  � . Pr � � 2 2 • -Weak tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 • -Strong tracking : Output s.t. ( ✏ , � ) σ 1 , σ 2 , . . . , σ m � � h i � � t � k f ( t ) k 2 � > ✏ k f ( t ) k 2 9 t ∈ [ m ]  � . Pr � � 2 2 Strong tracking => Weak tracking => One-shot � 4

  23. Linear Sketch � 5

  24. 
 Linear Sketch • Linear sketch is a special class of streaming algorithms. 
 
 � 5

  25. <latexit sha1_base64="XGrsU9Prt6lEANDymq/YHLa61es=">ACInicbVDLSgNBEJz1mcRXokcvi0HwFHZV0IOHgBePEU0UskFmJ71xcB7LTK+6LPkEr/oDfo038ST4MU4eBzUWDNRUdPdFaeCWwyCT29ufmFxablUrqysrq1vVGubHaszw6DNtNDmOqYWBFfQRo4CrlMDVMYCruK705F/dQ/Gcq0uMU+hJ+lA8YQzik6iFr8ploPGsEY/iwJp6ROpmjd1Lxy1Ncsk6CQCWptNwxS7BXUIGcChpUos5BSdkcH0HVUQm2V4x3Hfq7Tun7iTbuKfTH6s+Ogkprcxm7Sknx1v71RuJ/XjfD5LhXcJVmCIpNBiWZ8FH7o8P9PjfAUOSOUGa429Vnt9RQhi6eX1NynakB0thdouCBaSmp6hdRqkU+LCKER7RJMf4NXjh36hmSWe/ER409s8P682TaYwlsk12yB4JyRFpkjPSIm3CyIA8kWfy4r16b9679zEpnfOmPVvkF7yvb+WLpSc=</latexit> 
 <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> Linear Sketch • Linear sketch is a special class of streaming algorithms. 
 n 
 } } k ⌧ n Sketching matrix Π � 5

  26. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="XGrsU9Prt6lEANDymq/YHLa61es=">ACInicbVDLSgNBEJz1mcRXokcvi0HwFHZV0IOHgBePEU0UskFmJ71xcB7LTK+6LPkEr/oDfo038ST4MU4eBzUWDNRUdPdFaeCWwyCT29ufmFxablUrqysrq1vVGubHaszw6DNtNDmOqYWBFfQRo4CrlMDVMYCruK705F/dQ/Gcq0uMU+hJ+lA8YQzik6iFr8ploPGsEY/iwJp6ROpmjd1Lxy1Ncsk6CQCWptNwxS7BXUIGcChpUos5BSdkcH0HVUQm2V4x3Hfq7Tun7iTbuKfTH6s+Ogkprcxm7Sknx1v71RuJ/XjfD5LhXcJVmCIpNBiWZ8FH7o8P9PjfAUOSOUGa429Vnt9RQhi6eX1NynakB0thdouCBaSmp6hdRqkU+LCKER7RJMf4NXjh36hmSWe/ER409s8P682TaYwlsk12yB4JyRFpkjPSIm3CyIA8kWfy4r16b9679zEpnfOmPVvkF7yvb+WLpSc=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="1N3lREL/14B0c0pxvPYRSXJ/1zQ=">ACKnicbVDLSgNBEJz1/Tbq0ctgEPQSdlXQgwfBi8cI5gHZKLOT3jg4j2WmV12WfIZX/QG/xpt49UOcxBzUWDBQXdVN91SeEwDN+DqemZ2bn5hcWl5ZXVtfXKxmbTmdxyaHAjW0nzIEUGhoUEI7s8BUIqGV3J0P/dY9WCeMvsIig65ifS1SwRl6qRPXBU2vyz3cH9xUqmEtHIFOkmhMqmSM+s1GsBj3DM8VaOSOdeJwgy7JbMouITBUpw7yBi/Y3oeKqZAtctRzcP6K5XejQ1j+NdKT+nCiZcq5Qie9UDG/dX28o/ud1ckxPuqXQWY6g+feiNJcUDR0GQHvCAkdZeMK4Ff5Wym+ZRx9TL+2FCbXfWSJ/4mGB26UYrpXxpmRxaCMER7RpeWoGoYX/Y1qkjQPatFh7eDyqHp2Oo5xgWyTHbJHInJMzsgFqZMG4cSQJ/JMXoLX4C14Dz6+W6eC8cwW+YXg8wuvkKgY</latexit> 
 Linear Sketch • Linear sketch is a special class of streaming algorithms. 
 n 
 } } k ⌧ n Sketching vector Π f ( t ) Sketching matrix Π � 5

  27. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="1N3lREL/14B0c0pxvPYRSXJ/1zQ=">ACKnicbVDLSgNBEJz1/Tbq0ctgEPQSdlXQgwfBi8cI5gHZKLOT3jg4j2WmV12WfIZX/QG/xpt49UOcxBzUWDBQXdVN91SeEwDN+DqemZ2bn5hcWl5ZXVtfXKxmbTmdxyaHAjW0nzIEUGhoUEI7s8BUIqGV3J0P/dY9WCeMvsIig65ifS1SwRl6qRPXBU2vyz3cH9xUqmEtHIFOkmhMqmSM+s1GsBj3DM8VaOSOdeJwgy7JbMouITBUpw7yBi/Y3oeKqZAtctRzcP6K5XejQ1j+NdKT+nCiZcq5Qie9UDG/dX28o/ud1ckxPuqXQWY6g+feiNJcUDR0GQHvCAkdZeMK4Ff5Wym+ZRx9TL+2FCbXfWSJ/4mGB26UYrpXxpmRxaCMER7RpeWoGoYX/Y1qkjQPatFh7eDyqHp2Oo5xgWyTHbJHInJMzsgFqZMG4cSQJ/JMXoLX4C14Dz6+W6eC8cwW+YXg8wuvkKgY</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> 
 <latexit sha1_base64="wGt8SuM6EbjbBjIMYiQ5PWRpQKs=">ACM3icbZDJSgNBEIZ73HejHr0MBsFTmHFBDx4ELx4VjAqZEHo6NbGxl7G7RjM08ype9QV8GPEmXn0HOzEHt4KGv/6qoq/NBfcYhS9BGPjE5NT0zOzc/MLi0vLtZXVC6sLw6DJtNDmKqUWBFfQRI4CrnIDVKYCLtOb40H98g6M5VqdY5lDW9Ke4hlnFL3Vqa0maHMxZVL7K1Bt1dVnVo9akTDCP+KeCTqZBSnZVgNulqVkhQyAS1thVHObYdNciZgGouKSzklN3QHrS8VFSCbvh8VW46Z1umGnjn8Jw6H6fcFRaW8rUd0qK1/Z3bWD+V2sVmB20HVd5gaDY16KsECHqcEAi7HIDEXpBWG+1tDdk09DfS8fmwpdaF6SFP/EwX3TEtJVdcluRal54bQR5u5YTaAF/9G9VdcbDfincb2W796HCEcYaskw2yRWKyT47ICTklTcJInzyQR/IUPAevwVvw/tU6Foxm1siPCD4+Ac5lrEU=</latexit> <latexit sha1_base64="XGrsU9Prt6lEANDymq/YHLa61es=">ACInicbVDLSgNBEJz1mcRXokcvi0HwFHZV0IOHgBePEU0UskFmJ71xcB7LTK+6LPkEr/oDfo038ST4MU4eBzUWDNRUdPdFaeCWwyCT29ufmFxablUrqysrq1vVGubHaszw6DNtNDmOqYWBFfQRo4CrlMDVMYCruK705F/dQ/Gcq0uMU+hJ+lA8YQzik6iFr8ploPGsEY/iwJp6ROpmjd1Lxy1Ncsk6CQCWptNwxS7BXUIGcChpUos5BSdkcH0HVUQm2V4x3Hfq7Tun7iTbuKfTH6s+Ogkprcxm7Sknx1v71RuJ/XjfD5LhXcJVmCIpNBiWZ8FH7o8P9PjfAUOSOUGa429Vnt9RQhi6eX1NynakB0thdouCBaSmp6hdRqkU+LCKER7RJMf4NXjh36hmSWe/ER409s8P682TaYwlsk12yB4JyRFpkjPSIm3CyIA8kWfy4r16b9679zEpnfOmPVvkF7yvb+WLpSc=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> Linear Sketch AMS Sketch [Alon-Matias-Szegedy 96] • Linear sketch is a special class of streaming algorithms. 
 n 
 } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π � 5

  28. <latexit sha1_base64="wGt8SuM6EbjbBjIMYiQ5PWRpQKs=">ACM3icbZDJSgNBEIZ73HejHr0MBsFTmHFBDx4ELx4VjAqZEHo6NbGxl7G7RjM08ype9QV8GPEmXn0HOzEHt4KGv/6qoq/NBfcYhS9BGPjE5NT0zOzc/MLi0vLtZXVC6sLw6DJtNDmKqUWBFfQRI4CrnIDVKYCLtOb40H98g6M5VqdY5lDW9Ke4hlnFL3Vqa0maHMxZVL7K1Bt1dVnVo9akTDCP+KeCTqZBSnZVgNulqVkhQyAS1thVHObYdNciZgGouKSzklN3QHrS8VFSCbvh8VW46Z1umGnjn8Jw6H6fcFRaW8rUd0qK1/Z3bWD+V2sVmB20HVd5gaDY16KsECHqcEAi7HIDEXpBWG+1tDdk09DfS8fmwpdaF6SFP/EwX3TEtJVdcluRal54bQR5u5YTaAF/9G9VdcbDfincb2W796HCEcYaskw2yRWKyT47ICTklTcJInzyQR/IUPAevwVvw/tU6Foxm1siPCD4+Ac5lrEU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> 
 <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="XGrsU9Prt6lEANDymq/YHLa61es=">ACInicbVDLSgNBEJz1mcRXokcvi0HwFHZV0IOHgBePEU0UskFmJ71xcB7LTK+6LPkEr/oDfo038ST4MU4eBzUWDNRUdPdFaeCWwyCT29ufmFxablUrqysrq1vVGubHaszw6DNtNDmOqYWBFfQRo4CrlMDVMYCruK705F/dQ/Gcq0uMU+hJ+lA8YQzik6iFr8ploPGsEY/iwJp6ROpmjd1Lxy1Ncsk6CQCWptNwxS7BXUIGcChpUos5BSdkcH0HVUQm2V4x3Hfq7Tun7iTbuKfTH6s+Ogkprcxm7Sknx1v71RuJ/XjfD5LhXcJVmCIpNBiWZ8FH7o8P9PjfAUOSOUGa429Vnt9RQhi6eX1NynakB0thdouCBaSmp6hdRqkU+LCKER7RJMf4NXjh36hmSWe/ER409s8P682TaYwlsk12yB4JyRFpkjPSIm3CyIA8kWfy4r16b9679zEpnfOmPVvkF7yvb+WLpSc=</latexit> <latexit sha1_base64="1N3lREL/14B0c0pxvPYRSXJ/1zQ=">ACKnicbVDLSgNBEJz1/Tbq0ctgEPQSdlXQgwfBi8cI5gHZKLOT3jg4j2WmV12WfIZX/QG/xpt49UOcxBzUWDBQXdVN91SeEwDN+DqemZ2bn5hcWl5ZXVtfXKxmbTmdxyaHAjW0nzIEUGhoUEI7s8BUIqGV3J0P/dY9WCeMvsIig65ifS1SwRl6qRPXBU2vyz3cH9xUqmEtHIFOkmhMqmSM+s1GsBj3DM8VaOSOdeJwgy7JbMouITBUpw7yBi/Y3oeKqZAtctRzcP6K5XejQ1j+NdKT+nCiZcq5Qie9UDG/dX28o/ud1ckxPuqXQWY6g+feiNJcUDR0GQHvCAkdZeMK4Ff5Wym+ZRx9TL+2FCbXfWSJ/4mGB26UYrpXxpmRxaCMER7RpeWoGoYX/Y1qkjQPatFh7eDyqHp2Oo5xgWyTHbJHInJMzsgFqZMG4cSQJ/JMXoLX4C14Dz6+W6eC8cwW+YXg8wuvkKgY</latexit> <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> Linear Sketch AMS Sketch [Alon-Matias-Szegedy 96] • Linear sketch is a special class of streaming algorithms. 
 n 
 } -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π 2 � 5

  29. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="1N3lREL/14B0c0pxvPYRSXJ/1zQ=">ACKnicbVDLSgNBEJz1/Tbq0ctgEPQSdlXQgwfBi8cI5gHZKLOT3jg4j2WmV12WfIZX/QG/xpt49UOcxBzUWDBQXdVN91SeEwDN+DqemZ2bn5hcWl5ZXVtfXKxmbTmdxyaHAjW0nzIEUGhoUEI7s8BUIqGV3J0P/dY9WCeMvsIig65ifS1SwRl6qRPXBU2vyz3cH9xUqmEtHIFOkmhMqmSM+s1GsBj3DM8VaOSOdeJwgy7JbMouITBUpw7yBi/Y3oeKqZAtctRzcP6K5XejQ1j+NdKT+nCiZcq5Qie9UDG/dX28o/ud1ckxPuqXQWY6g+feiNJcUDR0GQHvCAkdZeMK4Ff5Wym+ZRx9TL+2FCbXfWSJ/4mGB26UYrpXxpmRxaCMER7RpeWoGoYX/Y1qkjQPatFh7eDyqHp2Oo5xgWyTHbJHInJMzsgFqZMG4cSQJ/JMXoLX4C14Dz6+W6eC8cwW+YXg8wuvkKgY</latexit> <latexit sha1_base64="wGt8SuM6EbjbBjIMYiQ5PWRpQKs=">ACM3icbZDJSgNBEIZ73HejHr0MBsFTmHFBDx4ELx4VjAqZEHo6NbGxl7G7RjM08ype9QV8GPEmXn0HOzEHt4KGv/6qoq/NBfcYhS9BGPjE5NT0zOzc/MLi0vLtZXVC6sLw6DJtNDmKqUWBFfQRI4CrnIDVKYCLtOb40H98g6M5VqdY5lDW9Ke4hlnFL3Vqa0maHMxZVL7K1Bt1dVnVo9akTDCP+KeCTqZBSnZVgNulqVkhQyAS1thVHObYdNciZgGouKSzklN3QHrS8VFSCbvh8VW46Z1umGnjn8Jw6H6fcFRaW8rUd0qK1/Z3bWD+V2sVmB20HVd5gaDY16KsECHqcEAi7HIDEXpBWG+1tDdk09DfS8fmwpdaF6SFP/EwX3TEtJVdcluRal54bQR5u5YTaAF/9G9VdcbDfincb2W796HCEcYaskw2yRWKyT47ICTklTcJInzyQR/IUPAevwVvw/tU6Foxm1siPCD4+Ac5lrEU=</latexit> <latexit sha1_base64="XGrsU9Prt6lEANDymq/YHLa61es=">ACInicbVDLSgNBEJz1mcRXokcvi0HwFHZV0IOHgBePEU0UskFmJ71xcB7LTK+6LPkEr/oDfo038ST4MU4eBzUWDNRUdPdFaeCWwyCT29ufmFxablUrqysrq1vVGubHaszw6DNtNDmOqYWBFfQRo4CrlMDVMYCruK705F/dQ/Gcq0uMU+hJ+lA8YQzik6iFr8ploPGsEY/iwJp6ROpmjd1Lxy1Ncsk6CQCWptNwxS7BXUIGcChpUos5BSdkcH0HVUQm2V4x3Hfq7Tun7iTbuKfTH6s+Ogkprcxm7Sknx1v71RuJ/XjfD5LhXcJVmCIpNBiWZ8FH7o8P9PjfAUOSOUGa429Vnt9RQhi6eX1NynakB0thdouCBaSmp6hdRqkU+LCKER7RJMf4NXjh36hmSWe/ER409s8P682TaYwlsk12yB4JyRFpkjPSIm3CyIA8kWfy4r16b9679zEpnfOmPVvkF7yvb+WLpSc=</latexit> 
 Linear Sketch AMS Sketch [Alon-Matias-Szegedy 96] • Linear sketch is a special class of streaming algorithms. 
 n 
 } -1 -2 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -2 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 0 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 0 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -2 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π 2 8 � 5

  30. <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> 
 <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="XGrsU9Prt6lEANDymq/YHLa61es=">ACInicbVDLSgNBEJz1mcRXokcvi0HwFHZV0IOHgBePEU0UskFmJ71xcB7LTK+6LPkEr/oDfo038ST4MU4eBzUWDNRUdPdFaeCWwyCT29ufmFxablUrqysrq1vVGubHaszw6DNtNDmOqYWBFfQRo4CrlMDVMYCruK705F/dQ/Gcq0uMU+hJ+lA8YQzik6iFr8ploPGsEY/iwJp6ROpmjd1Lxy1Ncsk6CQCWptNwxS7BXUIGcChpUos5BSdkcH0HVUQm2V4x3Hfq7Tun7iTbuKfTH6s+Ogkprcxm7Sknx1v71RuJ/XjfD5LhXcJVmCIpNBiWZ8FH7o8P9PjfAUOSOUGa429Vnt9RQhi6eX1NynakB0thdouCBaSmp6hdRqkU+LCKER7RJMf4NXjh36hmSWe/ER409s8P682TaYwlsk12yB4JyRFpkjPSIm3CyIA8kWfy4r16b9679zEpnfOmPVvkF7yvb+WLpSc=</latexit> <latexit sha1_base64="1N3lREL/14B0c0pxvPYRSXJ/1zQ=">ACKnicbVDLSgNBEJz1/Tbq0ctgEPQSdlXQgwfBi8cI5gHZKLOT3jg4j2WmV12WfIZX/QG/xpt49UOcxBzUWDBQXdVN91SeEwDN+DqemZ2bn5hcWl5ZXVtfXKxmbTmdxyaHAjW0nzIEUGhoUEI7s8BUIqGV3J0P/dY9WCeMvsIig65ifS1SwRl6qRPXBU2vyz3cH9xUqmEtHIFOkmhMqmSM+s1GsBj3DM8VaOSOdeJwgy7JbMouITBUpw7yBi/Y3oeKqZAtctRzcP6K5XejQ1j+NdKT+nCiZcq5Qie9UDG/dX28o/ud1ckxPuqXQWY6g+feiNJcUDR0GQHvCAkdZeMK4Ff5Wym+ZRx9TL+2FCbXfWSJ/4mGB26UYrpXxpmRxaCMER7RpeWoGoYX/Y1qkjQPatFh7eDyqHp2Oo5xgWyTHbJHInJMzsgFqZMG4cSQJ/JMXoLX4C14Dz6+W6eC8cwW+YXg8wuvkKgY</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="wGt8SuM6EbjbBjIMYiQ5PWRpQKs=">ACM3icbZDJSgNBEIZ73HejHr0MBsFTmHFBDx4ELx4VjAqZEHo6NbGxl7G7RjM08ype9QV8GPEmXn0HOzEHt4KGv/6qoq/NBfcYhS9BGPjE5NT0zOzc/MLi0vLtZXVC6sLw6DJtNDmKqUWBFfQRI4CrnIDVKYCLtOb40H98g6M5VqdY5lDW9Ke4hlnFL3Vqa0maHMxZVL7K1Bt1dVnVo9akTDCP+KeCTqZBSnZVgNulqVkhQyAS1thVHObYdNciZgGouKSzklN3QHrS8VFSCbvh8VW46Z1umGnjn8Jw6H6fcFRaW8rUd0qK1/Z3bWD+V2sVmB20HVd5gaDY16KsECHqcEAi7HIDEXpBWG+1tDdk09DfS8fmwpdaF6SFP/EwX3TEtJVdcluRal54bQR5u5YTaAF/9G9VdcbDfincb2W796HCEcYaskw2yRWKyT47ICTklTcJInzyQR/IUPAevwVvw/tU6Foxm1siPCD4+Ac5lrEU=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> Linear Sketch AMS Sketch [Alon-Matias-Szegedy 96] • Linear sketch is a special class of streaming algorithms. 
 n 
 } -3 -2 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -2 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 0 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 0 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -2 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π 2 8 4 � 5

  31. 
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 n 
 } -2 -4 -3 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -2 -2 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 0 2 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 1 1 2 0 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -2 -2 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π 2 8 4 2 � 5

  32. <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> 
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 n 
 } -3 -3 -2 -4 -1 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -2 -2 -1 -1 -3 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 0 3 1 2 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 1 0 1 2 3 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 -1 -1 -2 -2 -3 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π 2 8 4 2 5 � 5

  33. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> 
 
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 n 
 } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π � 6

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 <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> Linear Sketch AMS Sketch [Alon-Matias-Szegedy 96] • Linear sketch is a special class of streaming algorithms. 
 n 
 } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π • Space complexity : , truly random 
 O ( kn ) { , pseudo-random O ( k log n ) � 6

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 n 
 } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π • Space complexity : , truly random 
 Can be even better O ( kn ) { , pseudo-random O ( k log n ) � 6

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sha1_base64="dneH1QbJhQ/5SxpSQ3Bo5DBSeBs=">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</latexit> 
 
 <latexit sha1_base64="vh2ZQZxMjvriLVuRH7sEDLb45e0=">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</latexit> Linear Sketch AMS Sketch [Alon-Matias-Szegedy 96] • Linear sketch is a special class of streaming algorithms. 
 n 
 } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 } -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 1 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 k ⌧ n √ 5 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 Sketching vector Π f ( t ) Sketching matrix Π • Space complexity : , truly random 
 Can be even better O ( kn ) { , pseudo-random O ( k log n ) • AMS sketch : for one-shot [Alon-Matias-Szegedy 96] k = O ( ✏ − 2 ) and for weak tracking [Braverman-Chestnut-Ivkin-Nelson-Wang- Woodru ff 17]. � 6

  37. Update Time � 7

  38. Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. � 7

  39. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="nVvwCFKHmHQ7LMYJDaTjUx5d1o=">ACWnicbVBNb9NAEN2Yr6YFmgI3LhYRUnsgsgMScECKxIVjkZq2Uhyi8XqcrLIf1u64xVr5t/BruMIZiR/Dxs2BfjxpbfvzezMvrySwlGS/OlF9+4/ePhop7+79/jJ0/3BwbNTZ2rLcqNPY8B4dSaJySInlUVQucSzfP15459doHXC6BNqKpwrWGpRCg4UpMXgY3ayQoJDn3VveYtFu26Pt0hZ1g5IY3+5t+M2/ZoMRgmo6RDfJukWzJkWxwvDnr9rDC8VqiJS3BuliYVzT1YElxiu5vVDivga1jiLFANCt3cdwu08eugFHFpbDia4k79v8ODcq5ReahUQCt309uId3mzmsoPcy90VRNqfjWorGVMJt4EFhfCIifZBALcirBrzFdgVOI9dqUxtR6SZCHn2i85EYp0IXPKiOb1meE38mVvru1Ibz0ZlS3yel4lL4djb+G04m2xh32Ev2ih2ylL1nE/aFHbMp4+wH+8l+sd+9v1EU9aO9q9Kot+15zq4hevEPEpu4rA=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. • E.g., AMS sketch has update time complexity. 
 Θ ( k ) = Θ ( ✏ − 2 ) n } } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 
 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 Π = k ⌧ n 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 � 7

  40. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="nVvwCFKHmHQ7LMYJDaTjUx5d1o=">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</latexit> <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. • E.g., AMS sketch has update time complexity. 
 Θ ( k ) = Θ ( ✏ − 2 ) n } } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 
 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 Π = k ⌧ n 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 • Application: Packet passing problem [Krishnamurthy-Sen- Zhang-Chen 03] � 7

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 Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. • E.g., AMS sketch has update time complexity. 
 Θ ( k ) = Θ ( ✏ − 2 ) n } } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 
 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 Π = k ⌧ n 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 • Application: Packet passing problem [Krishnamurthy-Sen- Zhang-Chen 03] � 7

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 Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. • E.g., AMS sketch has update time complexity. 
 Θ ( k ) = Θ ( ✏ − 2 ) n } } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 
 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 Π = k ⌧ n 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 • Application: Packet passing problem [Krishnamurthy-Sen- Zhang-Chen 03] Rate: 7 . 75 × 10 6 (< 130 nanoseconds per packet) � 7

  43. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="nVvwCFKHmHQ7LMYJDaTjUx5d1o=">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</latexit> <latexit sha1_base64="0oudeB/Kgn7rfKOyftVMBp6Z1pM=">ACJ3icbVDLSgNBEJz1Gd+vo5fFIHgKuyroMeDFYwQTA9kgs5PeZHAey0yvuiz5Ca/6A36N9Gjf+Jk4NJLBioqeqmuytOBbcYBN/ewuLS8spqZW19Y3Nre2d3b79ldWYNJkW2rRjakFwBU3kKCdGqAyFnAXP1yN/LtHMJZrdYt5Cl1J+4onF0UjuC1HKh1f1uNagFJfx5Ek5IlUzQuN/z1qKeZpkEhUxQazthkGK3oAY5EzBcjzILKWUPtA8dRxWVYLtFufDQP3ZKz0+0cU+hX6p/Owoqrc1l7ColxYGd9Ubif14nw+SyW3CVZgiKjQclmfBR+6Pr/R43wFDkjlBmuNvVZwNqKEOX0dSUXGeqjzR2lyh4YlpKqnpFlGqRD4sI4RltUpS/oQsvnI1qnrROa+FZ7fTmvFqvT2KskENyRE5ISC5InVyTBmkSRgR5Ia/kzXv3PrxP72tcuBNeg7IFLyfX5PSp54=</latexit> <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. • E.g., AMS sketch has update time complexity. 
 Θ ( k ) = Θ ( ✏ − 2 ) n } } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 
 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 Π = k ⌧ n 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 • When is small, AMS sketch is slow. 🐍 ✏ � 8

  44. <latexit sha1_base64="uRayg+l3yjZbEw3Vdy/1XBlXzWc=">ACIHicbVDLSgNBEJyNr8Rn1KOXxSB4CrtR0GPAi0cDJhGSILOT3mTIPJaZXnVZ8gVe9Qf8Gm/iUb/GySYHXwUDNVXdHdFieAWg+DKy0tr6yulSvrG5tb2zvV3b2O1alh0GZaHMTUQuCK2gjRwE3iQEqIwHdaHIx87t3YCzX6hqzBAaSjhSPOaPopJa6rdaCelDA/0vCBamRBa5ud71Kf6hZKkEhE9TaXhgkOMipQc4ETNf7qYWEsgkdQc9RSXYQV5sOvWPnDL0Y23cU+gX6veOnEprMxm5SklxbH97M/E/r5difD7IuUpSBMXmg+JU+Kj92dn+kBtgKDJHKDPc7eqzMTWUoQvnx5RMp2qENHKXKLhnWkqhnk/0SKb5n2EB7RxXvymLrzwd1R/SadRD0/qjdZprdlcxFgmB+SQHJOQnJEmuSRXpE0YAfJInsiz9+K9em/e+7y05C169skPeJ9fecqkcA=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="G/yUW7iRijcLc4lzxRXATB8=">ACIXicbVDLSgNBEJyNr8Rnokcvi0HwFHajoMeAF49RjAayQWYnvXFwHstMr7os+QOv+gN+jTfxJv6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vlCura+sbm1vV2vaV1Zlh0GFaNONqQXBFXSQo4BuaoDKWMB1fHc69q/vwViu1SXmKfQlHSqecEbRSRfR6KZaDxrBP5fEs5InczQvql5lWigWSZBIRPU2l4YpNgvqEHOBIxWo8xCStkdHULPUl2H4xWXk7ztl4CfauKfQn6g/Owoqrc1l7ColxVv72xuL/3m9DJOTfsFVmiEoNh2UZMJH7Y/v9gfcAEORO0KZ4W5Xn91SQxm6dOam5DpTQ6Sxu0TBA9NSUjUolSLfFRECI9ok2LyG4cX/o7qL7lqNsLDRvP8qN5qzWIsk12yRw5ISI5Ji5yRNukQRhLyRJ7Ji/fqvXnv3se0tOTNenbIHLyvb1jlpOU=</latexit> <latexit sha1_base64="0oudeB/Kgn7rfKOyftVMBp6Z1pM=">ACJ3icbVDLSgNBEJz1Gd+vo5fFIHgKuyroMeDFYwQTA9kgs5PeZHAey0yvuiz5Ca/6A36N9Gjf+Jk4NJLBioqeqmuytOBbcYBN/ewuLS8spqZW19Y3Nre2d3b79ldWYNJkW2rRjakFwBU3kKCdGqAyFnAXP1yN/LtHMJZrdYt5Cl1J+4onF0UjuC1HKh1f1uNagFJfx5Ek5IlUzQuN/z1qKeZpkEhUxQazthkGK3oAY5EzBcjzILKWUPtA8dRxWVYLtFufDQP3ZKz0+0cU+hX6p/Owoqrc1l7ColxYGd9Ubif14nw+SyW3CVZgiKjQclmfBR+6Pr/R43wFDkjlBmuNvVZwNqKEOX0dSUXGeqjzR2lyh4YlpKqnpFlGqRD4sI4RltUpS/oQsvnI1qnrROa+FZ7fTmvFqvT2KskENyRE5ISC5InVyTBmkSRgR5Ia/kzXv3PrxP72tcuBNeg7IFLyfX5PSp54=</latexit> <latexit sha1_base64="u3fhbGlX2uVrFJ5cRsuSKojXzt0=">ACNHicbVDLSiQxFE35GB/jK+lm2AjzKqpcgbGZYMblwq2Cl2NpFK32tB5FMktQj1LW71B/wXwZ249RtMl73wdSFwcs69yQnK6VwGMcP0czs3PyPhcWl5Z8rv36vrq1vnDhTWQ59bqSxZxlzIWGPgqUcFZaYCqTcJqN9yf6SVYJ4w+xrqEoWIjLQrBGQbqfG3Tp+0SbyFvxk0qJQ1sJ+7GbdGvIJmCDpnW4fl6tJTmhlcKNHLJnBskcYlDzywKLqFZTisHJeNjNoJBgJopcEPfGjd0JzA5LYwNRyNt2fcTninapWFTsXwn3WJuR32qDCYm/ohS4rBM3fjIpKUjR0EgXNhQWOsg6AcSvCWym/YJZxDIF9cKlNpUfIsvATDVfcKMV07tPSyLrxKcI1usK3tyaEl3yO6is42e0mf7u7R/86vd40xkWyRbJH5KQ/6RHDsgh6RNOanJDbsldB89Rk/R81vrTDSd2SQfKnp5BbLTrLo=</latexit> <latexit sha1_base64="nVvwCFKHmHQ7LMYJDaTjUx5d1o=">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</latexit> <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 Update Time • Update time complexity for a linear sketch algorithm is the number of field operations needed in each update. • E.g., AMS sketch has update time complexity. 
 Θ ( k ) = Θ ( ✏ − 2 ) n } } 1 -1 1 -1 1 1 -1 -1 -1 1 1 -1 -1 1 1 
 -1 -1 1 1 -1 1 -1 -1 1 -1 -1 -1 -1 -1 1 Π = k ⌧ n 1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 1 1 1 -1 1 -1 1 1 1 -1 -1 -1 -1 1 -1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 • When is small, AMS sketch is slow. 🐍 ✏ Q: Is AMS Sketch optimal in update time complexity? � 8

  45. Faster One-Shot Estimation � 9

  46. <latexit sha1_base64="wVCOUCs2BWrGCSD575rTjH2/MPI=">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</latexit> Faster One-Shot Estimation • [Dasgupta-Kumar-Sarlós 10] and [Kane-Nelson 14] showed that sparse JL achieves one-shot with update time. O ( ✏ − 1 ) � 9

  47. <latexit sha1_base64="wVCOUCs2BWrGCSD575rTjH2/MPI=">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</latexit> <latexit sha1_base64="5f7RMjb7n3fhmLAUznT6Y/m0FBQ=">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</latexit> Faster One-Shot Estimation • [Dasgupta-Kumar-Sarlós 10] and [Kane-Nelson 14] showed that sparse JL achieves one-shot with update time. O ( ✏ − 1 ) • [Thorup-Zhang 12] showed that CountSketch (proposed by [Charikar-Chen-Farach-Colton 02] ) achieves one-shot with O (1) update time. � 9

  48. 
 <latexit sha1_base64="wVCOUCs2BWrGCSD575rTjH2/MPI=">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</latexit> <latexit sha1_base64="H5zZDIUgEGguhvnAgb7A4drEA3o=">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</latexit> 
 <latexit sha1_base64="5f7RMjb7n3fhmLAUznT6Y/m0FBQ=">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</latexit> Faster One-Shot Estimation • [Dasgupta-Kumar-Sarlós 10] and [Kane-Nelson 14] showed that sparse JL achieves one-shot with update time. O ( ✏ − 1 ) • [Thorup-Zhang 12] showed that CountSketch (proposed by [Charikar-Chen-Farach-Colton 02] ) achieves one-shot with O (1) update time. • Application: Packet passing problem [Krishnamurthy-Sen- Zhang-Chen 03] 
 Rate: 7 . 75 × 10 6 (< 130 nanoseconds per packet) � 9

  49. 
 
 <latexit sha1_base64="5f7RMjb7n3fhmLAUznT6Y/m0FBQ=">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</latexit> <latexit sha1_base64="H5zZDIUgEGguhvnAgb7A4drEA3o=">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</latexit> <latexit sha1_base64="wVCOUCs2BWrGCSD575rTjH2/MPI=">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</latexit> Faster One-Shot Estimation • [Dasgupta-Kumar-Sarlós 10] and [Kane-Nelson 14] showed that sparse JL achieves one-shot with update time. O ( ✏ − 1 ) • [Thorup-Zhang 12] showed that CountSketch (proposed by [Charikar-Chen-Farach-Colton 02] ) achieves one-shot with O (1) update time. • Application: Packet passing problem [Krishnamurthy-Sen- Zhang-Chen 03] 
 Rate: 7 . 75 × 10 6 (< 130 nanoseconds per packet) [Thorup-Zhang 12] showed that CountSketch improves AMS sketch from 182 nanoseconds to 30 nanoseconds! � 9

  50. 
 <latexit sha1_base64="H5zZDIUgEGguhvnAgb7A4drEA3o=">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</latexit> <latexit sha1_base64="wVCOUCs2BWrGCSD575rTjH2/MPI=">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</latexit> <latexit sha1_base64="5f7RMjb7n3fhmLAUznT6Y/m0FBQ=">ACMnicbVDLTtAFB0D5ZFSmsCSjUVUKWwiGyrBMlI37AoSAaQ4isbj6zBiHtbMNdQa+VPYwg/0Z9pd1W0/ohOTBSRcaQz59x7z8xJC8EtRtGvYGV17cP6xuZW6+P2p53P7c7uldWlYTBkWmhzk1ILgisYIkcBN4UBKlMB1+ndt5l+fQ/Gcq0usSpgLOlU8Zwzip6atDvfey5p1jgDWR3Xh5N2N+pHTYXLIJ6DLpnX+aQTbCWZqUEhUxQa0dxVODYUYOcCahbSWmhoOyOTmHkoaIS7Ng1pnX4xTNZmGvj8KwYV9POCqtrWTqOyXFW7uozcj3tFGJ+enYcVWUCIq9GOWlCFGHsyDCjBtgKCoPKDPcvzVkt9RQhj6uNy6VLtUaep/ouCBaSmpylxSaFHVLkH4gTZ3za324cWLUS2Dq6N+fNw/uvjaHQzmMW6SfXJAeiQmJ2RAzsg5GRJGHsgjeSLPwc/gd/An+PvSuhLMZ/bImwr+/Qf3QqtK</latexit> 
 Faster One-Shot Estimation Only for one-shot • [Dasgupta-Kumar-Sarlós 10] and [Kane-Nelson 14] showed that sparse JL achieves one-shot with update time. O ( ✏ − 1 ) • [Thorup-Zhang 12] showed that CountSketch (proposed by [Charikar-Chen-Farach-Colton 02] ) achieves one-shot with O (1) update time. • Application: Packet passing problem [Krishnamurthy-Sen- Zhang-Chen 03] 
 Rate: 7 . 75 × 10 6 (< 130 nanoseconds per packet) [Thorup-Zhang 12] showed that CountSketch improves AMS sketch from 182 nanoseconds to 30 nanoseconds! � 9

  51. What About Faster Linear Sketch for Weak Tracking? � 10

  52. <latexit sha1_base64="bWHrbm3wLTPuOtRa+5Ku/IndOg=">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</latexit> <latexit sha1_base64="/bvdKTMIWq2qfqI5jIW6l7Ey7zg=">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</latexit> What About Faster Linear Sketch for Weak Tracking? Known • time for one-shot O (1) • time for weak tracking O ( ✏ − 2 ) � 10

  53. <latexit sha1_base64="bWHrbm3wLTPuOtRa+5Ku/IndOg=">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</latexit> <latexit sha1_base64="/bvdKTMIWq2qfqI5jIW6l7Ey7zg=">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</latexit> <latexit sha1_base64="/bvdKTMIWq2qfqI5jIW6l7Ey7zg=">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</latexit> What About Faster Linear Sketch for Weak Tracking? Known Unknown • time for one-shot O (1) • time for weak tracking O (1) • time for weak tracking O ( ✏ − 2 ) � 10

  54. CountSketch Provides Weak Tracking � 11

  55. <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">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</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � � 11

  56. <latexit sha1_base64="Tuf5nd3z6evy9Sol5klnqaR94gA=">ACMnicdVDLbhMxFPW09E1LQpdsLKJK7SaCa0Cu0hs2LVIpK2UiSKP505q1Y+RfYcysuZT2NIf4Gdgh9j2I+pMg0QRvZKl43Puvc+WSmFwzj+Ea2sPltb39jc2t5vrv3otN9e5MZTmMuZHGXmbMgRQaxihQwmVpgalMwkV2/X6hX3wG64TRn7AuYarYXItCcIaBmnW6p4c+bd4C3mTNEezTi/ux23RAbD+LgFJ8N3wFNlKPLOts1o20tzwSoFGLplzkyQuceqZRcElNtp5aBk/JrNYRKgZgrc1LemDT0ITE4LY8PRSFv27wnPlHO1ykKnYnjl/tUW5P+0SYXF26kXuqwQNH8wKipJ0dBFEDQXFjKOgDGrQhvpfyKWcYxPXIpTaVniPLwk803HCjFNO5T0sj68anCF/QFb69NSG8PwnRp8H5oJ+86Q8+HvdGo2WMm+QVeU0OSUKGZEQ+kDMyJpzckK/kG7mNvkc/o1/R74fWlWg5s08eVXR3Dz3eq3U=</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">ACPXicdVDLahRBFK2Oxjw1E10mi8IhEG7klkdDeQTZYRMklgehiq29PitSjqbqtNkVv/Bq3+gP5Dj8gO3GbrTWdERKJFwrOPe+6mSlFA7j+Ge09OTp8rOV1bX1jc3nL7Y62y/PnKkshxE30tiLjDmQsMIBUq4KC0wlUk4z6O5vr5J7BOGH2KdQkTxWZaFIzDNS0s7v03aKt5A3KZROSKObtzTuJW+mnW7ci9sIedwfxIcteDf4MOjTZCF1ySJOptvRWpobXinQyCVzbpzEJU48syi4hGY9rRyUjF+xGYwD1EyBm/j2gIbuBSanhbHhaQte7/DM+VcrbJQqRheun+1OfmYNq6weD/xQpcVguZ3i4pKUjR07gnNhQWOsg6AcSvCrZRfMs4BucebKlNpWfIsvATDZ+5UYrp3KelkXjU4Qv6ArfZk0w769D9P/grN9LDnr9j4fd4XBh4yrZIa/JPknIgAzJMTkhI8LJV/KNfCc/ouvoJvoV/b4rXYoWPa/Ig4hu/wCix6+U</latexit> <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">ACTnicdVDLTtAFB2ntAT6ILRLNiOiSlSikW2oArtI3XSZSgSQ4hCNJ9dhxDysmeu2lvFX8DVs2x/otj/SHSoTE9RStUca6cw5986de9JcCodh+CNoPVp5/GS1vb+9NnzFxudzZfHzhSWw4gbaexpyhxIoWGEAiWc5haYSiWcpBfvF/7J7BOGH2EZQ4TxeZaZIz9NK08za5TIaCZmfVTvSmTi6n8Vm8m8wMut3fjrp3p1u2AsbUE/ifrjfkHf9w35Mo6XVJUsMp5vBmn+MFwo0csmcG0dhjpOKWRcQr2eFA5yxi/YHMaeaqbATapmr5q+9sqMZsb6o5E26p8dFVPOlSr1lYrhufvbW4j/8sYFZgeTSui8QND8blBWSIqGLkKiM2GBoyw9YdwK/1fKz5lH2UD6aUptBzZKnfRMNnbpRielYluZFlXSUIX9BlVXOrfXj3CdH/k+O4F+314o/73cFgGWObJFtskMi0icD8oEMyYhwckWuyVfyLfge/Axugl93pa1g2fOKPECrfQuXDLRJ</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">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</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � Corollary (informal) There is an time algorithm provides -weak tracking. ( ✏ , 0 . 1) O (1) � 11

  57. <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">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</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) • The first analysis for weak tracking with constant update time. � 11

  58. <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">ACPnicdVDLahsxFNWkjz6iNMuS0DUFNJFzXiS4mRnyKa7JlAnAY9rNJo7jogeg3SnySBm1a/pNv2B/kZ/oLuQbZeVJw4kpT0gODrnXt2rk5VSOIzjn9HSg4ePHi+vrK49efrs+Xpn48WRM5XlMOJGnuSMQdSaBihQAknpQWmMgnH2dn+3D/+AtYJoz9hXcJEsZkWheAMgzTtbH7c8mn7jLeQNymUTkijP/t3SdO8nXa6cS9uQNJBvFOS94P9gYJ7S+sLlngYLoRra54ZUCjVwy58b9uMSJZxYFl9CspZWDkvEzNoNxoJopcBPfLtDQN0HJaWFsOBpq97t8Ew5V6sVCqGp+5vby7+yxtXWOxOvNBlhaD5zaCikhQNnYdCc2GBo6wDYdyKsCvlp8wyjiG6e1NqU+kZsiz8RM5N0oxnfu0NLJufIpwga7w7a0J4d0mRP9PjpJef7uXHO50h8NFjCvkFXlNtkifDMiQfCAHZEQ4+Uq+kUvyPfoR/Yquoub0qVo0fOS3EP0+w/WC7DH</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> <latexit sha1_base64="k1Ih9tKzvFWp9c6P9peCRGru6c=">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</latexit> <latexit sha1_base64="SdQSfMYR2LBhN8aBF704nH+LNeg=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) • The first analysis for weak tracking with constant update time. • Using the median trick, there is a streaming algorithm O (log δ − 1 ) provides -weak tracking with update time. ( ✏ , � ) � 11

  59. <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">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</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> <latexit sha1_base64="k1Ih9tKzvFWp9c6P9peCRGru6c=">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</latexit> <latexit sha1_base64="SdQSfMYR2LBhN8aBF704nH+LNeg=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) • The first analysis for weak tracking with constant update time. • Using the median trick, there is a streaming algorithm O (log δ − 1 ) provides -weak tracking with update time. ( ✏ , � ) • The packet passing problem now has tracking guarantee. � 11

  60. <latexit sha1_base64="SdQSfMYR2LBhN8aBF704nH+LNeg=">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</latexit> <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">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</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> <latexit sha1_base64="k1Ih9tKzvFWp9c6P9peCRGru6c=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) • The first analysis for weak tracking with constant update time. • Using the median trick, there is a streaming algorithm O (log δ − 1 ) provides -weak tracking with update time. ( ✏ , � ) • The packet passing problem now has tracking guarantee. The rest of the talk will focus on the proof sketch . � 11

  61. <latexit sha1_base64="SdQSfMYR2LBhN8aBF704nH+LNeg=">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</latexit> <latexit sha1_base64="UEXIcjqRF7cALG2OVhbwdKla4=">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</latexit> <latexit sha1_base64="EleagbtL+HnutN8aR1soCBR7h18=">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</latexit> <latexit sha1_base64="k1Ih9tKzvFWp9c6P9peCRGru6c=">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</latexit> CountSketch Provides Weak Tracking Theorem (informal) CountSketch with rows provides -weak tracking. O ( ✏ − 2 ) ( ✏ , 0 . 1) • The first analysis for weak tracking with constant update time. • Using the median trick, there is a streaming algorithm O (log δ − 1 ) provides -weak tracking with update time. ( ✏ , � ) • The packet passing problem now has tracking guarantee. The rest of the talk will focus on the proof sketch . � 11

  62. CountSketch [Charikar-Chen-Farach-Colton 02] � 12

  63. <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 
 CountSketch [Charikar-Chen-Farach-Colton 02] • Idea : Exactly one non-zero entry in each column. 
 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 -1 0 0 -1 0 0 0 0 0 0 0 Π = 0 -1 -1 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 � 12

  64. <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 
 <latexit sha1_base64="MdyWNj/nYyOpqkzs3/uYx9fFrOo=">ACMXicdVBNT9tAEF3T0vLVNpQjl1UjJHqoZTuQcIzUS29QiQBSnEbrzTish/W7hiwLP+TXts/0F/DXHlT3QTglSq9krvXlvRjP7skIKh1F0G6y8eLn6vXa+sbm1pu371rb78+cKS2HATfS2IuMOZBCwAFSrgoLDCVSTjPZp/n/vkVWCeMPsWqgJFiUy1ywRl6adxqHe+nUDghjf5Wf0qaj+NWOwqj3mGcHNIo7Bx1kl7iSbfb6R5ENA6jBdpkiZPxdrCeTgwvFWjkjk3jKMCRzWzKLiEZiMtHRSMz9gUhp5qpsCN6sXpDd3zyoTmxvqnkS7UPydqpyrVOY7FcNL97c3F/lDUvMj0a10EWJoPnjoryUFA2d50AnwgJHWXnCuBX+VsovmWUcfVrPtlSm1FNkmf+JhmtulGJ6UqeFkVTpwg36PJ6UTU+vKeE6P/JWRLGnTD5etDu95cxrpFd8oHsk5j0SJ98ISdkQDi5It/JD/Iz+BXcBnfB/WPrSrCc2SHPEDz8Btw/qro=</latexit> CountSketch [Charikar-Chen-Farach-Colton 02] • Idea : Exactly one non-zero entry in each column. 
 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 -1 0 0 -1 0 0 0 0 0 0 0 Π = 0 -1 -1 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 [Thorup-Zhang 12] showed that CountSketch with rows O ( ✏ − 2 ) achieve one-shot estimation. � 12

  65. <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> 
 
 <latexit sha1_base64="MdyWNj/nYyOpqkzs3/uYx9fFrOo=">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</latexit> CountSketch [Charikar-Chen-Farach-Colton 02] • Idea : Exactly one non-zero entry in each column. 
 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 -1 0 0 -1 0 0 0 0 0 0 0 Π = 0 -1 -1 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 [Thorup-Zhang 12] showed that CountSketch with rows O ( ✏ − 2 ) achieve one-shot estimation. • Analysis : � 12

  66. <latexit sha1_base64="vdJD4MYAbwShZLgdTRiyT9BuBs=">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</latexit> 
 
 <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> <latexit sha1_base64="MdyWNj/nYyOpqkzs3/uYx9fFrOo=">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</latexit> CountSketch [Charikar-Chen-Farach-Colton 02] • Idea : Exactly one non-zero entry in each column. 
 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 -1 0 0 -1 0 0 0 0 0 0 0 Π = 0 -1 -1 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 [Thorup-Zhang 12] showed that CountSketch with rows O ( ✏ − 2 ) achieve one-shot estimation. • Analysis : - Obs: ( Π > Π ) ij ⇥ ⇤ = 1 i = j . E � 12

  67. <latexit sha1_base64="vdJD4MYAbwShZLgdTRiyT9BuBs=">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</latexit> <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> <latexit sha1_base64="MdyWNj/nYyOpqkzs3/uYx9fFrOo=">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</latexit> 
 
 <latexit sha1_base64="0c5/yJuTyWL3i/DCinCyt/1ahag=">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</latexit> CountSketch [Charikar-Chen-Farach-Colton 02] • Idea : Exactly one non-zero entry in each column. 
 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 -1 0 0 -1 0 0 0 0 0 0 0 Π = 0 -1 -1 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 [Thorup-Zhang 12] showed that CountSketch with rows O ( ✏ − 2 ) achieve one-shot estimation. • Analysis : - Obs: ( Π > Π ) ij ⇥ ⇤ = 1 i = j . E 2 3 - Expectation: h i ( Π > Π ) ij f ( m ) f ( m ) 4 X 5 = k f ( m ) k 2 k Π f ( m ) k 2 = E 2 . E 2 i j i,j 2 [ n ] � 12

  68. <latexit sha1_base64="vdJD4MYAbwShZLgdTRiyT9BuBs=">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</latexit> <latexit sha1_base64="XvZ2buJhId9GrMTnC1Uj/N36Nwg=">ACI3icdVDLSgNBEJz1/X4evQwGwVPYRCV6EAQvHiMYFbJBZie9yeA8lpledVnyC171B/wab+LFg/iZI2gogUDNVXdHfFqRQOw/AtGBufmJyanpmdm19YXFpeWV07dyazHFrcSGMvY+ZACg0tFCjhMrXAVCzhIr4+HvoXN2CdMPoM8xQ6ivW0SARnOJSipji8WqmE1bAE9aTeCHdLstc4aNRpbWRVyAjNq9VgNuoaninQyCVzrl0LU+wUzKLgEgZzUeYgZfya9aDtqWYKXKcolx3QLa90aWKsfxpqX7vKJhyLlexr1QM+63NxT/8toZJvudQug0Q9D8c1CSYqGDi+nXWGBo8w9YdwKvyvlfWYZR5/Pjym5yXQPWewv0XDLjVJMd4soNTIfFBHCHbqkKH8DH95XQvR/cl6v1naq9dPdytHJKMYZskE2yTapkQY5IiekSVqEkz65Jw/kMXgKnoOX4PWzdCwY9ayTHwjePwC/xKWl</latexit> <latexit sha1_base64="MdyWNj/nYyOpqkzs3/uYx9fFrOo=">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</latexit> 
 
 <latexit sha1_base64="0c5/yJuTyWL3i/DCinCyt/1ahag=">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</latexit> CountSketch [Charikar-Chen-Farach-Colton 02] • Idea : Exactly one non-zero entry in each column. 
 1 1 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 1 0 -1 0 0 -1 0 0 0 0 0 0 0 Π = 0 -1 -1 -1 0 1 0 0 0 0 0 0 0 0 0 -1 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 [Thorup-Zhang 12] showed that CountSketch with rows O ( ✏ − 2 ) achieve one-shot estimation. • Analysis : - Obs: ( Π > Π ) ij ⇥ ⇤ = 1 i = j . E 2 3 - Expectation: h i ( Π > Π ) ij f ( m ) f ( m ) 4 X 5 = k f ( m ) k 2 k Π f ( m ) k 2 = E 2 . E 2 i j i,j 2 [ n ] - Apply Chebyshev’s inequality. � 12

  69. Intuition for Weak Tracking � 13

  70. 
 
 Intuition for Weak Tracking • First attempt : Apply union bound on one-shot analysis. 
 � 13

  71. 
 
 <latexit sha1_base64="SYfJsj4PAR5+RHO0f8ExeK4RzeU=">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</latexit> Intuition for Weak Tracking • First attempt : Apply union bound on one-shot analysis. 
 ✓ ◆ ✏ , � -one-shot m � 13

  72. 
 
 <latexit sha1_base64="SYfJsj4PAR5+RHO0f8ExeK4RzeU=">ACVHicdVDdahQxGM1MrbZV2629Ca4CBVkmd1WVu8WvPGygtsWNsuSyXyzG5qfIflGHcI8iE/jbfsCgu/SC7PTFazogcDJOd/HSU5eKekxy34m6daD7YePdnb3Hj95un/QO3x27m3tBEyFVdZd5tyDkgamKFHBZeWA61zBRX71fu1fAbnpTWfsKlgrvnSyFIKjlFa9E6YghKPGVReKmtes9JxEVgBCnkbAusSgoOi1W3LnFyu8NWi18GWQcayWicnXbkzfjdeESHG6tPNjhbHCa7rLCi1mBQKO79bJhVOA/coRQK2j1We6i4uOJLmEVquAY/D12S19GpaCldfEYpJ3650bg2vtG53FSc1z5v721+C9vVmP5dh6kqWoEI+6CylpRtHRdFS2kA4GqiYQLJ+NbqVjx2A/GQu+lNLY2S+R5/ImBL8JqzU0RWGV0waG8BV9GbpbG8v73RD9PzkfDYng9H0/5ksqlxhzwnL8gxGZIxmZAP5IxMiSDfyHdyTW6SH8ltupVu342myWbniNxDuv8Lck+4Cg=</latexit> <latexit sha1_base64="Blc298wygoDucQu3sCvsZoZsWZo=">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</latexit> Intuition for Weak Tracking • First attempt : Apply union bound on one-shot analysis. 
 ✓ ◆ ✏ , � -weak tracking -one-shot ( ✏ , � ) m � 13

  73. <latexit sha1_base64="r7GVqf+Zy12af+Hmex90QVb+5vk=">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</latexit> 
 
 <latexit sha1_base64="vuiumjrqXRQdtKkSR75/ZCQ7huU=">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</latexit> <latexit sha1_base64="Blc298wygoDucQu3sCvsZoZsWZo=">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</latexit> <latexit sha1_base64="SYfJsj4PAR5+RHO0f8ExeK4RzeU=">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</latexit> Intuition for Weak Tracking • First attempt : Apply union bound on one-shot analysis. 
 ✓ ◆ ✏ , � -weak tracking -one-shot ( ✏ , � ) m - Using rows (or rows after ✏ − 2 � − 1 log m ✏ − 2 � − 1 m � � � � O O median trick). � 13

  74. 
 
 <latexit sha1_base64="Blc298wygoDucQu3sCvsZoZsWZo=">ACPXicdVBNSxBEO0xJlHz4Zoc46HJEjAQltmNYc1tIZcFVwVdpalpqdmt7E/hu6a6DsxV+Tq/6B/I78gNxCrnaO6gwTxoePVeFdX10kJT3H8M1p5tPr4ydO19Y1nz1+83GxtvTrytnQCh8Iq605S8KikwSFJUnhSOASdKjxOT78s/ONv6Ly05pCqAscapkbmUgAFadLaThTmtJNg4aWy5kOSoSJInJzO6P2k1Y47cQMeSK8f7zbkU/9zv8e7S6vNltifbEXrSWZFqdGQUOD9qBsXNK7BkRQK5xtJ6bEAcQpTHAVqQKMf180Zc/4uKBnPrQvPEG/UuxM1aO8rnYZODTz/3oL8SFvVFK+N6lKUpCI24W5aXiZPkiE5Jh4JUFQgIJ8NfuZiBA0EhuXtbKluaKUEaLjF4JqzWYLI6Kayq5nVCeE4+r5tqHsK7TYj/nxz1Ot2Pnd7BbnswWMa4xt6wt2yHdVmfDdhXts+GTLAL9p1dsqvoR/Qr+h39uWldiZYzr9k9RH+vAZunsCY=</latexit> <latexit sha1_base64="SYfJsj4PAR5+RHO0f8ExeK4RzeU=">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</latexit> <latexit sha1_base64="vuiumjrqXRQdtKkSR75/ZCQ7huU=">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</latexit> <latexit sha1_base64="r7GVqf+Zy12af+Hmex90QVb+5vk=">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</latexit> Intuition for Weak Tracking • First attempt : Apply union bound on one-shot analysis. 
 ✓ ◆ ✏ , � -weak tracking -one-shot ( ✏ , � ) m - Using rows (or rows after ✏ − 2 � − 1 log m ✏ − 2 � − 1 m � � � � O O median trick). • Idea : Using chaining argument [Braverman-Chestnut-Ivkin-Nelson- Wang-Woodru ff 17] to get a fancier (and tighter) union bound. � 13

  75. <latexit sha1_base64="Blc298wygoDucQu3sCvsZoZsWZo=">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</latexit> <latexit sha1_base64="vuiumjrqXRQdtKkSR75/ZCQ7huU=">ACWnicdVBNb9QwFPSGr24LdAvcuFiskMqBVRKlt5W4sKNIrFtpc2ycpyXrFV/RPYLbWTlt/BruMIZiR+DN10kimAkS+OZ9zT25LUDuP4xyC6dfvO3Xs7w929+w8e7o8OHp0601gOc26ksec5cyCFhjkKlHBeW2Aql3CWX7zd+GefwTph9Edsa1gqVmlRCs4wSKvR8ftMQomHGdROSKM/+ZdplxUgkQWadD7rM7yFosukqajqMiuqNb5YjcbxJO5BA0mn8VFPXk+PpylNtaYbHGyOhgMs8LwRoFGLplziySucemZRcEldLtZ46Bm/IJVsAhUMwVu6fv8j4PSkFLY8PRSHv1zw3PlHOtysOkYrh2f3sb8V/eosHyzdILXTcIml8HlY2kaOimMFoICxlGwjVoS3Ur5mlnEMtd5IaU2jK2R5+ImGS26UYrwW1kG3pEuEJX+v7WhfJ+N0T/T07TSfJqkn4Gs9m2xp3yFPyjByShEzJjLwjJ2ROPlCvpJv5PvgZxRFw2jvejQabHcekxuInvwC/he4qQ=</latexit> <latexit sha1_base64="SYfJsj4PAR5+RHO0f8ExeK4RzeU=">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</latexit> <latexit sha1_base64="r7GVqf+Zy12af+Hmex90QVb+5vk=">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</latexit> 
 
 Intuition for Weak Tracking • First attempt : Apply union bound on one-shot analysis. 
 ✓ ◆ ✏ , � -weak tracking -one-shot ( ✏ , � ) m - Using rows (or rows after ✏ − 2 � − 1 log m ✏ − 2 � − 1 m � � � � O O median trick). • Idea : Using chaining argument [Braverman-Chestnut-Ivkin-Nelson- Wang-Woodru ff 17] to get a fancier (and tighter) union bound. We can get rid of the m dependency! � 13

  76. Step 1: Extracting the Correlation � 14

  77. <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">ACMnicdVDLahtBEJy187Ccl2QfcxkiAg4EsVIcZN8EueQoQ2QbtEL0zvbKg+exzPTaXhZ9iq/OD+RnkpvJNR/h0VqBOCQFA9V3XRPpYWSnuL4e7Sx+ejxk6dbre1nz1+8fNXu7Bx7WzqBE2GVdacpeFTS4IQkKTwtHIJOFZ6k59W/skFOi+t+UJVgTMNCyNzKYCNG939hIsvFTWvE8yVATv5u1u3Isb8EAGw3i/IR+Hh8MB76+tLltjPO9ErSzotRoSCjwftqPC5rV4EgKhcvtpPRYgDiHBU4DNaDRz+rm9iV/G5SM59aFZ4g36p8TNWjvK52GTg105v/2VuK/vGlJ+cGslqYoCY24X5SXipPlqyB4Jh0KUlUgIJwMt3JxBg4EhbgebKlsaRYEafiJwUthtQaT1UlhVbWsE8Ir8ndVMsQ3u+E+P/J8aDX/9AbHO13R6N1jFvsNXvD9lifDdmIfWZjNmGCXbJrdsO+Rt+iH9Ft9PO+dSNaz+yB4h+3QHQmqs3</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � � 14

  78. <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">ACTnicdVDLTtAFB2ntAT6ILRLNiOiSlSikW2oArtI3XSZSgSQ4hCNJ9dhxDysmeu2lvFX8DVs2x/otj/SHSoTE9RStUca6cw5986de9JcCodh+CNoPVp5/GS1vb+9NnzFxudzZfHzhSWw4gbaexpyhxIoWGEAiWc5haYSiWcpBfvF/7J7BOGH2EZQ4TxeZaZIz9NK08za5TIaCZmfVTvSmTi6n8Vm8m8wMut3fjrp3p1u2AsbUE/ifrjfkHf9w35Mo6XVJUsMp5vBmn+MFwo0csmcG0dhjpOKWRcQr2eFA5yxi/YHMaeaqbATapmr5q+9sqMZsb6o5E26p8dFVPOlSr1lYrhufvbW4j/8sYFZgeTSui8QND8blBWSIqGLkKiM2GBoyw9YdwK/1fKz5lH2UD6aUptBzZKnfRMNnbpRielYluZFlXSUIX9BlVXOrfXj3CdH/k+O4F+314o/73cFgGWObJFtskMi0icD8oEMyYhwckWuyVfyLfge/Axugl93pa1g2fOKPECrfQuXDLRJ</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">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</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where � 14

  79. <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="vGhVD3Tq6L4oy1weAxZcmIto3wU=">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</latexit> <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">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</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where - and σ ∈ { − 1 , 1 } n � 14

  80. <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> <latexit sha1_base64="vGhVD3Tq6L4oy1weAxZcmIto3wU=">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</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="bFSX0M5nNZxdAOcO/APzmIf1XE=">ACJnicbVDLSgNBEJyNz/iMevSyGAS9hN0o6DHgxWMEkwhJlNlJbzJkHstMr7os+Qiv+gN+jTcRb36Kk8fBqAUDNVXdHdFieAWg+DTKywsLi2vrBbX1jc2t7ZLO7tNq1PDoMG0OYmohYEV9BAjgJuEgNURgJa0fBi7LfuwViu1TVmCXQl7Ssec0bRSa34Nj/C49FdqRxUgn8vySckTKZoX634xU7Pc1SCQqZoNa2wyDBbk4NciZgtNZJLSUDWkf2o4qKsF28m+I/QKT0/1sY9hf5E/dmRU2ltJiNXKSkO7G9vLP7ntVOMz7s5V0mKoNh0UJwKH7U/Pt7vcQMReYIZYa7X02oIYydBHNTcl0qvpI3eJgempaSql3cSLbJR3kF4RBvnk984vPB3VH9Js1oJTyrVq9NyrTaLcZXskwNyREJyRmrktRJgzAyJE/kmbx4r96b9+59TEsL3qxnj8zB+/oG+W+mvw=</latexit> <latexit sha1_base64="OljLna5zmG2xqnZR60ne95BUtA=">ACInicbVDLSgNBEJyNz/iMevSyGARPYTcKegx48RjRaCAbZHbSmwyZxzLTqy5LPsGr/oBf408CX6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vrJbX1jc2t7YrO7s3VmeGQYtpoU07phYEV9BCjgLaqQEqYwG38fB87N/eg7Fcq2vMU+hK2lc84Yyik6iJr+rVINaMIH/l4QzUiUzNO92vHLU0yToJAJam0nDFLsFtQgZwJGa1FmIaVsSPvQcVRCbZbTHYd+YdO6fmJNu4p9Cfqz46CSmtzGbtKSXFgf3tj8T+vk2Fy1i24SjMExaDkz4qP3x4X6PG2AockcoM9zt6rMBNZShi2duSq4z1Ucau0sUPDAtJVW9Ikq1yEdFhPCINikmv5EL/wd1V9yU6+Fx7X65Um10ZjFuEr2yQE5IiE5JQ1yQZqkRjpkyfyTF68V+/Ne/c+pqUlb9azR+bgfX0D5r+lKw=</latexit> <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">AClnicbVFdi9NAFJ3Er92ul19EXwJFmEFLUkV9EUpiuhjBbu70GnDZHLTHXY+QuZGDdP8UPHPOM1GsLt7YeDcz/OzJmslMJiHP8Owlu379y9t7c/OLj/4OHh8OjRiTV1xWHOjTVWcYsSKFhjgIlnJUVMJVJOM0uPm3rpz+gsLo79iUsFRsrUhOENPpcOabuhMRMXKHeOLlm7SyWryim528veOdkKugrylVqwVa1cUTUlRyBzcxzZ1/1oyWUNLAVn7st/R3jCdDkfxO4iug6SHoxIH7P0KNinueG1Ao1cMmsXSVzi0rEKBZfQDmhtoWT8gq1h4aFmCuzSdcJt9NwzeVSYyh+NUcf+P+GYsrZRme9UDM/t1dqWvKm2qLF4t3RClzWC5pdCRS0jNHW7CgXFXCUjQeMV8LfNeLnrGIc/ZfsqDSm1mtkmX+Jhp/cKMV07mhpZOMNRPiFtnBdtjUvuWrVdXAyGSevx5Nvb0bTaW/jHnlKnpFjkpC3ZEq+khmZE07+BEwCA7CJ+GH8HP45bI1DPqZx2QnwtlfrB7Ojg=</latexit> <latexit sha1_base64="eHlWyTrAx78SgjirHmxzONPdmY=">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</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where - and σ ∈ { − 1 , 1 } n - depends on and . ˜ f ( t ) B η ,f ( t ) Π � 14

  81. <latexit sha1_base64="eHlWyTrAx78SgjirHmxzONPdmY=">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</latexit> <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">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</latexit> <latexit sha1_base64="vGhVD3Tq6L4oy1weAxZcmIto3wU=">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</latexit> <latexit sha1_base64="bFSX0M5nNZxdAOcO/APzmIf1XE=">ACJnicbVDLSgNBEJyNz/iMevSyGAS9hN0o6DHgxWMEkwhJlNlJbzJkHstMr7os+Qiv+gN+jTcRb36Kk8fBqAUDNVXdHdFieAWg+DTKywsLi2vrBbX1jc2t7ZLO7tNq1PDoMG0OYmohYEV9BAjgJuEgNURgJa0fBi7LfuwViu1TVmCXQl7Ssec0bRSa34Nj/C49FdqRxUgn8vySckTKZoX634xU7Pc1SCQqZoNa2wyDBbk4NciZgtNZJLSUDWkf2o4qKsF28m+I/QKT0/1sY9hf5E/dmRU2ltJiNXKSkO7G9vLP7ntVOMz7s5V0mKoNh0UJwKH7U/Pt7vcQMReYIZYa7X02oIYydBHNTcl0qvpI3eJgempaSql3cSLbJR3kF4RBvnk984vPB3VH9Js1oJTyrVq9NyrTaLcZXskwNyREJyRmrktRJgzAyJE/kmbx4r96b9+59TEsL3qxnj8zB+/oG+W+mvw=</latexit> <latexit sha1_base64="OljLna5zmG2xqnZR60ne95BUtA=">ACInicbVDLSgNBEJyNz/iMevSyGARPYTcKegx48RjRaCAbZHbSmwyZxzLTqy5LPsGr/oBf408CX6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vrJbX1jc2t7YrO7s3VmeGQYtpoU07phYEV9BCjgLaqQEqYwG38fB87N/eg7Fcq2vMU+hK2lc84Yyik6iJr+rVINaMIH/l4QzUiUzNO92vHLU0yToJAJam0nDFLsFtQgZwJGa1FmIaVsSPvQcVRCbZbTHYd+YdO6fmJNu4p9Cfqz46CSmtzGbtKSXFgf3tj8T+vk2Fy1i24SjMExaDkz4qP3x4X6PG2AockcoM9zt6rMBNZShi2duSq4z1Ucau0sUPDAtJVW9Ikq1yEdFhPCINikmv5EL/wd1V9yU6+Fx7X65Um10ZjFuEr2yQE5IiE5JQ1yQZqkRjpkyfyTF68V+/Ne/c+pqUlb9azR+bgfX0D5r+lKw=</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 Highly correlated 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where - and σ ∈ { − 1 , 1 } n - depends on and . ˜ f ( t ) B η ,f ( t ) Π � 14

  82. <latexit sha1_base64="eHlWyTrAx78SgjirHmxzONPdmY=">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</latexit> <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">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</latexit> <latexit sha1_base64="bFSX0M5nNZxdAOcO/APzmIf1XE=">ACJnicbVDLSgNBEJyNz/iMevSyGAS9hN0o6DHgxWMEkwhJlNlJbzJkHstMr7os+Qiv+gN+jTcRb36Kk8fBqAUDNVXdHdFieAWg+DTKywsLi2vrBbX1jc2t7ZLO7tNq1PDoMG0OYmohYEV9BAjgJuEgNURgJa0fBi7LfuwViu1TVmCXQl7Ssec0bRSa34Nj/C49FdqRxUgn8vySckTKZoX634xU7Pc1SCQqZoNa2wyDBbk4NciZgtNZJLSUDWkf2o4qKsF28m+I/QKT0/1sY9hf5E/dmRU2ltJiNXKSkO7G9vLP7ntVOMz7s5V0mKoNh0UJwKH7U/Pt7vcQMReYIZYa7X02oIYydBHNTcl0qvpI3eJgempaSql3cSLbJR3kF4RBvnk984vPB3VH9Js1oJTyrVq9NyrTaLcZXskwNyREJyRmrktRJgzAyJE/kmbx4r96b9+59TEsL3qxnj8zB+/oG+W+mvw=</latexit> <latexit sha1_base64="vGhVD3Tq6L4oy1weAxZcmIto3wU=">ACQXicbVDLbhQxEPQkPJLwSuDIxWKFxAFWMwkSHFfiwjFIbBJpvUQ9np6NFT9Gdg8wsubK1+Sa/ABfwSdwQ1y54J3sgS0ZKlcVe1uV9loFSjPf2Rr67du37m7sbl17/6Dh4+2dx4fBNd6iVPptPNHJQTUyuKUFGk8ajyCKTUelqfvlvrhZ/RBOfuRugbnBhZW1UoCJep4m0cxPBI9Vr0IamGgF8qK+Kp4WYj+U7KM8nE+FL8JihUYsVXtH+9km6JysjVoSWoIYVbkDc0jeFJSY78l2oANyFNY4CxBCwbDPA5b9Px5YipeO5+OJT6w/3ZEMCF0pkxOA3QSrmtL8n/arKX67Twq27SEVl4OqlvNyfFlLrxSHiXpLgGQXqVduTwBD5JSelemdK61C4Iy/cTiF+mMAVtF0Tjd9VEQfqVQx+HWp/CK61HdBAe742JvPvh9WgyWcW4wZ6yZ+wFK9gbNmHv2T6bMsm+sTN2zi6y79nP7Ff2+9K6lq16nrArlf35CzQyseY=</latexit> <latexit sha1_base64="OljLna5zmG2xqnZR60ne95BUtA=">ACInicbVDLSgNBEJyNz/iMevSyGARPYTcKegx48RjRaCAbZHbSmwyZxzLTqy5LPsGr/oBf408CX6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vrJbX1jc2t7YrO7s3VmeGQYtpoU07phYEV9BCjgLaqQEqYwG38fB87N/eg7Fcq2vMU+hK2lc84Yyik6iJr+rVINaMIH/l4QzUiUzNO92vHLU0yToJAJam0nDFLsFtQgZwJGa1FmIaVsSPvQcVRCbZbTHYd+YdO6fmJNu4p9Cfqz46CSmtzGbtKSXFgf3tj8T+vk2Fy1i24SjMExaDkz4qP3x4X6PG2AockcoM9zt6rMBNZShi2duSq4z1Ucau0sUPDAtJVW9Ikq1yEdFhPCINikmv5EL/wd1V9yU6+Fx7X65Um10ZjFuEr2yQE5IiE5JQ1yQZqkRjpkyfyTF68V+/Ne/c+pqUlb9azR+bgfX0D5r+lKw=</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 Highly correlated 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where - and σ ∈ { − 1 , 1 } n - depends on and . ˜ f ( t ) B η ,f ( t ) Π • The bad event becomes: � 14

  83. <latexit sha1_base64="eHlWyTrAx78SgjirHmxzONPdmY=">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</latexit> <latexit sha1_base64="bFSX0M5nNZxdAOcO/APzmIf1XE=">ACJnicbVDLSgNBEJyNz/iMevSyGAS9hN0o6DHgxWMEkwhJlNlJbzJkHstMr7os+Qiv+gN+jTcRb36Kk8fBqAUDNVXdHdFieAWg+DTKywsLi2vrBbX1jc2t7ZLO7tNq1PDoMG0OYmohYEV9BAjgJuEgNURgJa0fBi7LfuwViu1TVmCXQl7Ssec0bRSa34Nj/C49FdqRxUgn8vySckTKZoX634xU7Pc1SCQqZoNa2wyDBbk4NciZgtNZJLSUDWkf2o4qKsF28m+I/QKT0/1sY9hf5E/dmRU2ltJiNXKSkO7G9vLP7ntVOMz7s5V0mKoNh0UJwKH7U/Pt7vcQMReYIZYa7X02oIYydBHNTcl0qvpI3eJgempaSql3cSLbJR3kF4RBvnk984vPB3VH9Js1oJTyrVq9NyrTaLcZXskwNyREJyRmrktRJgzAyJE/kmbx4r96b9+59TEsL3qxnj8zB+/oG+W+mvw=</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="vGhVD3Tq6L4oy1weAxZcmIto3wU=">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</latexit> <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">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</latexit> <latexit sha1_base64="8XpxfwNUJHgm3UNg92ap3vW7hE=">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</latexit> <latexit sha1_base64="OljLna5zmG2xqnZR60ne95BUtA=">ACInicbVDLSgNBEJyNz/iMevSyGARPYTcKegx48RjRaCAbZHbSmwyZxzLTqy5LPsGr/oBf408CX6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vrJbX1jc2t7YrO7s3VmeGQYtpoU07phYEV9BCjgLaqQEqYwG38fB87N/eg7Fcq2vMU+hK2lc84Yyik6iJr+rVINaMIH/l4QzUiUzNO92vHLU0yToJAJam0nDFLsFtQgZwJGa1FmIaVsSPvQcVRCbZbTHYd+YdO6fmJNu4p9Cfqz46CSmtzGbtKSXFgf3tj8T+vk2Fy1i24SjMExaDkz4qP3x4X6PG2AockcoM9zt6rMBNZShi2duSq4z1Ucau0sUPDAtJVW9Ikq1yEdFhPCINikmv5EL/wd1V9yU6+Fx7X65Um10ZjFuEr2yQE5IiE5JQ1yQZqkRjpkyfyTF68V+/Ne/c+pqUlb9azR+bgfX0D5r+lKw=</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">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</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 Highly correlated 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where - and σ ∈ { − 1 , 1 } n - depends on and . ˜ f ( t ) B η ,f ( t ) Π • The bad event becomes: � � � � > ˜ � > ✏ k f ( m ) k 2 B η ,f ( t ) � sup � � 2 t 2 [ m ] � 14

  84. <latexit sha1_base64="3ytESJ+XZCnBGY7Q2xAYl48kTjU=">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</latexit> <latexit sha1_base64="vGhVD3Tq6L4oy1weAxZcmIto3wU=">ACQXicbVDLbhQxEPQkPJLwSuDIxWKFxAFWMwkSHFfiwjFIbBJpvUQ9np6NFT9Gdg8wsubK1+Sa/ABfwSdwQ1y54J3sgS0ZKlcVe1uV9loFSjPf2Rr67du37m7sbl17/6Dh4+2dx4fBNd6iVPptPNHJQTUyuKUFGk8ajyCKTUelqfvlvrhZ/RBOfuRugbnBhZW1UoCJep4m0cxPBI9Vr0IamGgF8qK+Kp4WYj+U7KM8nE+FL8JihUYsVXtH+9km6JysjVoSWoIYVbkDc0jeFJSY78l2oANyFNY4CxBCwbDPA5b9Px5YipeO5+OJT6w/3ZEMCF0pkxOA3QSrmtL8n/arKX67Twq27SEVl4OqlvNyfFlLrxSHiXpLgGQXqVduTwBD5JSelemdK61C4Iy/cTiF+mMAVtF0Tjd9VEQfqVQx+HWp/CK61HdBAe742JvPvh9WgyWcW4wZ6yZ+wFK9gbNmHv2T6bMsm+sTN2zi6y79nP7Ff2+9K6lq16nrArlf35CzQyseY=</latexit> <latexit sha1_base64="8XpxfwNUJHgm3UNg92ap3vW7hE=">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</latexit> <latexit sha1_base64="OljLna5zmG2xqnZR60ne95BUtA=">ACInicbVDLSgNBEJyNz/iMevSyGARPYTcKegx48RjRaCAbZHbSmwyZxzLTqy5LPsGr/oBf408CX6Mk8fBqAUDNVXdHfFqeAWg+DTKy0sLi2vrJbX1jc2t7YrO7s3VmeGQYtpoU07phYEV9BCjgLaqQEqYwG38fB87N/eg7Fcq2vMU+hK2lc84Yyik6iJr+rVINaMIH/l4QzUiUzNO92vHLU0yToJAJam0nDFLsFtQgZwJGa1FmIaVsSPvQcVRCbZbTHYd+YdO6fmJNu4p9Cfqz46CSmtzGbtKSXFgf3tj8T+vk2Fy1i24SjMExaDkz4qP3x4X6PG2AockcoM9zt6rMBNZShi2duSq4z1Ucau0sUPDAtJVW9Ikq1yEdFhPCINikmv5EL/wd1V9yU6+Fx7X65Um10ZjFuEr2yQE5IiE5JQ1yQZqkRjpkyfyTF68V+/Ne/c+pqUlb9azR+bgfX0D5r+lKw=</latexit> <latexit sha1_base64="bFSX0M5nNZxdAOcO/APzmIf1XE=">ACJnicbVDLSgNBEJyNz/iMevSyGAS9hN0o6DHgxWMEkwhJlNlJbzJkHstMr7os+Qiv+gN+jTcRb36Kk8fBqAUDNVXdHdFieAWg+DTKywsLi2vrBbX1jc2t7ZLO7tNq1PDoMG0OYmohYEV9BAjgJuEgNURgJa0fBi7LfuwViu1TVmCXQl7Ssec0bRSa34Nj/C49FdqRxUgn8vySckTKZoX634xU7Pc1SCQqZoNa2wyDBbk4NciZgtNZJLSUDWkf2o4qKsF28m+I/QKT0/1sY9hf5E/dmRU2ltJiNXKSkO7G9vLP7ntVOMz7s5V0mKoNh0UJwKH7U/Pt7vcQMReYIZYa7X02oIYydBHNTcl0qvpI3eJgempaSql3cSLbJR3kF4RBvnk984vPB3VH9Js1oJTyrVq9NyrTaLcZXskwNyREJyRmrktRJgzAyJE/kmbx4r96b9+59TEsL3qxnj8zB+/oG+W+mvw=</latexit> <latexit sha1_base64="OCi27wfCkln/1H3SgiSpv6C+3G8=">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</latexit> <latexit sha1_base64="weFIyoDiZhW+8OrWc2jbD0y6rUY=">ACTnicdVDLTtAFB2ntAT6ILRLNiOiSlSikW2oArtI3XSZSgSQ4hCNJ9dhxDysmeu2lvFX8DVs2x/otj/SHSoTE9RStUca6cw5986de9JcCodh+CNoPVp5/GS1vb+9NnzFxudzZfHzhSWw4gbaexpyhxIoWGEAiWc5haYSiWcpBfvF/7J7BOGH2EZQ4TxeZaZIz9NK08za5TIaCZmfVTvSmTi6n8Vm8m8wMut3fjrp3p1u2AsbUE/ifrjfkHf9w35Mo6XVJUsMp5vBmn+MFwo0csmcG0dhjpOKWRcQr2eFA5yxi/YHMaeaqbATapmr5q+9sqMZsb6o5E26p8dFVPOlSr1lYrhufvbW4j/8sYFZgeTSui8QND8blBWSIqGLkKiM2GBoyw9YdwK/1fKz5lH2UD6aUptBzZKnfRMNnbpRielYluZFlXSUIX9BlVXOrfXj3CdH/k+O4F+314o/73cFgGWObJFtskMi0icD8oEMyYhwckWuyVfyLfge/Axugl93pa1g2fOKPECrfQuXDLRJ</latexit> <latexit sha1_base64="K8HnmoE+FDf/Yxu0h+Glc6AEMcM=">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</latexit> <latexit sha1_base64="eHlWyTrAx78SgjirHmxzONPdmY=">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</latexit> Step 1: Extracting the Correlation -Weak tracking : Output s.t. k Π f (1) k 2 2 , . . . , k Π f ( m ) k 2 ( ✏ , � ) 2 � � h i � k Π f ( t ) k 2 2 � k f ( t ) k 2 2 > ✏ k f ( m ) k 2 9 t ∈ [ m ]  � Pr � � 2 � • Rewrite the error as: 
 Highly correlated 2 = σ > ˜ k Π f ( t ) k 2 2 � k f ( t ) k 2 B η ,f ( t ) σ where - and σ ∈ { − 1 , 1 } n Di ff erent from - depends on and . [BCINWW17] ˜ f ( t ) B η ,f ( t ) Π • The bad event becomes: � � � � > ˜ � > ✏ k f ( m ) k 2 B η ,f ( t ) � sup � � 2 t 2 [ m ] � 14

  85. <latexit sha1_base64="1gFP0dAu4z9SeWLx5/ENfex+q/Q=">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</latexit> Step 2: -Net ✏ � 15

  86. <latexit sha1_base64="a5tWmMv3ACZjNHLPSCPkeTud+g=">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</latexit> <latexit sha1_base64="1gFP0dAu4z9SeWLx5/ENfex+q/Q=">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</latexit> Step 2: -Net ✏ Goal : " # � � � � > ˜ � > ✏ k f ( m ) k 2  0 . 1 . B η ,f ( t ) � Pr sup � � 2 t 2 [ m ] � 15

  87. <latexit sha1_base64="v+OUs0BkjnFgLfk5vMGMlETbs6o=">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</latexit> <latexit 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sha1_base64="ciXFfdcOxnvhyP3C6eZmvZPRhLU=">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</latexit> <latexit sha1_base64="1gFP0dAu4z9SeWLx5/ENfex+q/Q=">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</latexit> Step 2: -Net ✏ Goal : " # � � � � > ˜ � > ✏ k f ( m ) k 2  0 . 1 . B η ,f ( t ) � Pr sup � � 2 t 2 [ m ] ˜ B 4 ˜ … B m − 2 ˜ ˜ B 1 B m ˜ ˜ B 6 B 3 ˜ B m − 1 ˜ B 5 ˜ ˜ B 2 B 7 � 15

  88. <latexit sha1_base64="GTUuUNPFwSzMKRB0ke4cRvYAg=">ACLHicbVDJSgNBEO1xN25Rj14Gg+AhBkV9Bjw4jFCYgLJEHo6NbGxl7G7Rh2GfIdX/QG/xouIV7/DznJwe9Dw6lU9qvrFqeAWg+DNm5tfWFxaXlktra1vbG6Vt3eurM4MgxbTQptOTC0IrqCFHAV0UgNUxgLa8c35uN+A2O5Vk3MU4gkHSqecEbRSVGzH1Sb/bDaG2i0/XIlqAUT+H9JOCMVMkOjv+2tOiPLJChkglrbDYMUo4Ia5EzAqNTLKSU3dAhdB1VIKNisnVI/AKQM/0cY9hf5E/e4oqLQ2l7GblBSv7e/eWPyv180wOYsKrtIMQbHpoiQTPmp/HIE/4AYitwRygx3t/rsmhrK0AX1Y0uMzVEGrufKLhnWkqBkUv1SIfFT2EB7RJMalGLrzwd1R/ydVRLTyuHV2eVOr1WYwrZI/sk0MSklNSJxekQVqEkVvySJ7Is/fivXrv3sd0dM6beXbJD3ifX9vUqK8=</latexit> <latexit 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B η ,f ( t ) � Pr sup � � 2 t 2 [ m ] ˜ B 4 ˜ … B m − 2 ˜ ˜ B 1 B m ˜ ˜ B 6 B 3 ˜ B m − 1 ˜ B 5 ˜ ˜ B 2 B 7 • A sequence of nets such that T 0 , T 1 , . . . � 15

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sha1_base64="tX4zxUynHQBQzWYXhIYkW/ja6s4=">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</latexit> <latexit 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B η ,f ( t ) � Pr sup � � 2 t 2 [ m ] ˜ B 4 ˜ … B m − 2 ˜ ˜ B 1 B m ˜ ˜ B 6 B 3 ˜ B m − 1 ˜ B 5 ˜ ˜ B 2 B 7 • A sequence of nets such that T 0 , T 1 , . . . � 15

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B η ,f ( t ) � Pr sup � � 2 t 2 [ m ] ˜ B 4 ˜ … B m − 2 ˜ ˜ B 1 B m ˜ ˜ B 6 B 3 ˜ B m − 1 ˜ B 5 ˜ ˜ B 2 B 7 • A sequence of nets such that T 0 , T 1 , . . . - The coarser the net is, the smaller it is. � 15

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