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Simultaneous Nearest Neighbor Search Piotr Indyk Robert Kleinberg MIT Cornell Sepideh Mahabadi Yang Yuan MIT Cornell Nearest Neighbor Dataset of points in a metric space (, ) 6/17/2016 2 Nearest Neighbor

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SLIDE 1

Simultaneous Nearest Neighbor Search

Yang Yuan Cornell Robert Kleinberg Cornell Piotr Indyk MIT Sepideh Mahabadi MIT

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Nearest Neighbor

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)

6/17/2016 2

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SLIDE 3

Nearest Neighbor

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • A query point comes online π‘Ÿ

6/17/2016 3

π‘Ÿ

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SLIDE 4

Nearest Neighbor

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • A query point comes online π‘Ÿ
  • Goal:
  • Find the nearest data-set point π‘žβˆ—

6/17/2016 4

π‘Ÿ π‘žβˆ—

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SLIDE 5

Nearest Neighbor

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • A query point comes online π‘Ÿ
  • Goal:
  • Find the nearest data-set point π‘žβˆ—
  • Do it in sub-linear time

6/17/2016 5

π‘Ÿ π‘žβˆ—

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SLIDE 6

Approximate Nearest Neighbor

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • A query point comes online π‘Ÿ
  • Goal:
  • Find the nearest data-set point π‘žβˆ—
  • Do it in sub-linear time
  • Approximate Nearest Neighbor

6/17/2016 6

π‘Ÿ π‘žβˆ— π‘ž

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SLIDE 7

What if

We have multiple queries We need the results of the queries to be related.

6/17/2016 7

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What if

We have multiple queries We need the results of the queries to be related. Example:

  • Noisy image
  • For each pixel find the true color
  • Neighboring pixels have similar color

6/17/2016 8

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SLIDE 9

Simultaneous NN Problem

6/17/2016 Sepideh Mahabadi 9

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The SNN problem

(Felzenszwalb’15)

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)

6/17/2016 10

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The SNN problem

(Felzenszwalb’15)

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • Query comes online and contains
  • 𝑙 query points 𝑅 = (π‘Ÿ1, … , π‘Ÿπ‘™)

6/17/2016 11

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3

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The SNN problem

(Felzenszwalb’15)

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • Query comes online and contains
  • 𝑙 query points 𝑅 = (π‘Ÿ1, … , π‘Ÿπ‘™)
  • A compatibility graph 𝐻 = 𝑅, 𝐹𝐻

6/17/2016 12

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3

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SLIDE 13

The SNN problem

(Felzenszwalb’15)

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • Query comes online and contains
  • 𝑙 query points 𝑅 = (π‘Ÿ1, … , π‘Ÿπ‘™)
  • A compatibility graph 𝐻 = 𝑅, 𝐹𝐻

6/17/2016 13

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3

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The SNN problem

(Felzenszwalb’15)

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • Query comes online and contains
  • 𝑙 query points 𝑅 = (π‘Ÿ1, … , π‘Ÿπ‘™)
  • A compatibility graph 𝐻 = 𝑅, 𝐹𝐻

6/17/2016 14

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3

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SLIDE 15

The SNN problem

(Felzenszwalb’15)

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • Query comes online and contains
  • 𝑙 query points 𝑅 = (π‘Ÿ1, … , π‘Ÿπ‘™)
  • A compatibility graph 𝐻 = 𝑅, 𝐹𝐻
  • Goal is to report (π‘ž1, … , π‘žπ‘™) , π‘žπ‘— ∈ 𝑄 , that minimizes

𝒋=𝟐

𝒍

𝒆𝒀(𝒓𝒋, 𝒒𝒋) + 𝒓𝒋,π’“π’Œ βˆˆπ‘­π‘― 𝒆𝒀(𝒒𝒋, π’’π’Œ)

6/17/2016 15

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3

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The Generalized SNN

  • Dataset of π‘œ points 𝑄 in a metric space (π‘Œ, π‘’π‘Œ)
  • Query comes online and contains
  • 𝑙 query points 𝑅 = (π‘Ÿ1, … , π‘Ÿπ‘™)
  • A compatibility graph 𝐻 = 𝑅, 𝐹𝐻
  • Goal is to report (π‘ž1, … , π‘žπ‘™) , π‘žπ‘— ∈ 𝑄 , that minimizes

𝒋=𝟐

𝒍

𝝀𝒋𝑒𝒁(𝒓𝒋, 𝒒𝒋) + 𝒓𝒋,π’“π’Œ βˆˆπ‘­π‘― 𝝁𝒋,π’Œπ’†π’€(𝒒𝒋, π’’π’Œ)

6/17/2016 16

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3

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Independent NN Algorithm

6/17/2016 Sepideh Mahabadi 17

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Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅

6/17/2016 18

π‘Ÿ1 π‘Ÿ3 π‘Ÿ2

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Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅
  • Independently find a (approximate) nearest neighbor

𝒒𝒋 (Searching step)

6/17/2016 19

π‘Ÿ1 π‘Ÿ2 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3

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Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅
  • Independently find a (approximate) nearest neighbor

𝒒𝒋 (Searching step)

  • Replace the label set 𝑄 with the reduced set

𝑸 = { π’’πŸ, … , 𝒒𝒍} (Pruning step)

6/17/2016 20

π‘Ÿ1 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3 π‘Ÿ2

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Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅
  • Independently find a (approximate) nearest neighbor

𝒒𝒋 (Searching step)

  • Replace the label set 𝑄 with the reduced set

𝑸 = { π’’πŸ, … , 𝒒𝒍} (Pruning step)

  • Solve the problem for

𝑄

6/17/2016 21

π‘Ÿ1 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3 π‘Ÿ2

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SLIDE 22

Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅
  • Independently find a (approximate) nearest neighbor

𝒒𝒋 (Searching step)

  • Replace the label set 𝑄 with the reduced set

𝑸 = { π’’πŸ, … , 𝒒𝒍} (Pruning step)

  • Solve the problem for

𝑄

  • Reduces the size of labels from π‘œ down to 𝑙

6/17/2016 22

π‘Ÿ1 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3 π‘Ÿ2

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SLIDE 23

Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅
  • Independently find a (approximate) nearest neighbor

𝒒𝒋 (Searching step)

  • Replace the label set 𝑄 with the reduced set

𝑸 = { π’’πŸ, … , 𝒒𝒍} (Pruning step)

  • Solve the problem for

𝑄

  • Reduces the size of labels from π‘œ down to 𝑙
  • The optimal value increases by a factor 𝜷
  • pruning gap

6/17/2016 23

π‘Ÿ1 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3 π‘Ÿ2

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SLIDE 24

Independent NN Algorithm

INN Algorithm

  • For each query point π‘Ÿπ‘— ∈ 𝑅
  • Independently find a (approximate) nearest neighbor

𝒒𝒋 (Searching step)

  • Replace the label set 𝑄 with the reduced set

𝑸 = { π’’πŸ, … , 𝒒𝒍} (Pruning step)

  • Solve the problem for

𝑄

  • Reduces the size of labels from π‘œ down to 𝑙
  • The optimal value increases by a factor 𝜷
  • pruning gap
  • Any metric labeling 𝛾-approximate algorithm can be used on the

reduced set , giving us (𝛽 β‹… 𝛾)-approximate algorithm.

6/17/2016 24

π‘Ÿ1 π‘Ÿ3 π‘ž1 π‘ž2 π‘ž3 π‘Ÿ2

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Results

6/17/2016 Sepideh Mahabadi 25

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Results

  • Prove bounds for the pruning gap

6/17/2016 26

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Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

6/17/2016 27

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Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

  • 𝜷 = 𝛁

𝐦𝐩𝐑 𝒍

6/17/2016 28

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Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

  • 𝜷 = 𝛁

𝐦𝐩𝐑 𝒍

  • For 𝑠-sparse graph: 𝜷 = 𝑷 𝒔

6/17/2016 29

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Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

  • 𝜷 = 𝛁

𝐦𝐩𝐑 𝒍

  • For 𝑠-sparse graph: 𝜷 = 𝑷 𝒔
  • Graphs with pseudo-arboricity 𝑠: each edge can be mapped

to a vertex such that at most 𝑠 edges are mapped to any vertex

6/17/2016 30

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Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

  • 𝜷 = 𝛁

𝐦𝐩𝐑 𝒍

  • For 𝑠-sparse graph: 𝜷 = 𝑷 𝒔
  • Graphs with pseudo-arboricity 𝑠: each edge can be mapped

to a vertex such that at most 𝑠 edges are mapped to any vertex

  • Would mean constant approximation factor for trees, grids,

planar graphs, …, and in particular 𝑃(𝑠)-approximation for 𝑠- degree graphs

6/17/2016 31

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SLIDE 32

Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

  • 𝜷 = 𝛁

𝐦𝐩𝐑 𝒍

  • For 𝑠-sparse graph: 𝜷 = 𝑷 𝒔
  • Graphs with pseudo-arboricity 𝑠: each edge can be mapped

to a vertex such that at most 𝑠 edges are mapped to any vertex

  • Would mean constant approximation factor for trees, grids,

planar graphs, …, and in particular 𝑃(𝑠)-approximation for 𝑠- degree graphs

  • 𝜷 is very close to one in experiments

6/17/2016 32

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SLIDE 33

Results

  • Prove bounds for the pruning gap
  • 𝜷 = 𝑷

𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 π₯

  • 𝜷 = 𝛁

𝐦𝐩𝐑 𝒍

  • For 𝑠-sparse graph: 𝜷 = 𝑷 𝒔
  • Graphs with pseudo-arboricity 𝑠: each edge can be mapped

to a vertex such that at most 𝑠 edges are mapped to any vertex

  • Would mean constant approximation factor for trees, grids,

planar graphs, …, and in particular 𝑃(𝑠)-approximation for 𝑠- degree graphs

  • 𝜷 is very close to one in experiments

6/17/2016 33

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SLIDE 34

Overview of the proof for

𝜷 = 𝑷 𝐦𝐩𝐑 𝒍 𝐦𝐩𝐑 𝐦𝐩𝐑 𝒍

6/17/2016 Sepideh Mahabadi 34

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SLIDE 35

0-Extension Problem [Kar98]

6/17/2016 35

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0-Extension Problem [Kar98]

  • The input:
  • a graph 𝐼 π‘Š, 𝐹

6/17/2016 36

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SLIDE 37

0-Extension Problem [Kar98]

  • The input:
  • a graph 𝐼 π‘Š, 𝐹
  • a weight function π‘₯ 𝑓

6/17/2016 37

2 1 2 1 2

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0-Extension Problem [Kar98]

  • The input:
  • a graph 𝐼 π‘Š, 𝐹
  • a weight function π‘₯ 𝑓
  • a set of terminals π‘ˆ βŠ‚ π‘Š

6/17/2016 38

2 1 2 1 2

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0-Extension Problem [Kar98]

  • The input:
  • a graph 𝐼 π‘Š, 𝐹
  • a weight function π‘₯ 𝑓
  • a set of terminals π‘ˆ βŠ‚ π‘Š
  • The goal: find a mapping 𝑔: π‘Š β†’ π‘ˆ s.t.
  • Each terminal is mapped to itself
  • Minimize 𝑣,𝑀 ∈𝐹 π‘₯ 𝑣, 𝑀 𝑒(𝑔 𝑣 , 𝑔 𝑀 )

6/17/2016 39

2 1 2 1 2

Cost = 1 β‹… 𝑒(𝑒1, 𝑒2)

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SLIDE 40

0-Extension Problem

Prior work: [CKR05, FHRT03, AFHKTT04, LN04]

6/17/2016 40

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0-Extension Problem

Prior work: [CKR05, FHRT03, AFHKTT04, LN04]

  • Upper bounds:
  • E.g. 𝑃(

log |π‘ˆ| log log |π‘ˆ|) approximation algorithm [CKR05]

6/17/2016 41

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SLIDE 42

0-Extension Problem

Prior work: [CKR05, FHRT03, AFHKTT04, LN04]

  • Upper bounds:
  • E.g. 𝑃(

log |π‘ˆ| log log |π‘ˆ|) approximation algorithm [CKR05]

  • consider the metric relaxation of the LP for the

problem

  • Solve LP
  • Round the solution

6/17/2016 42

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SLIDE 43

0-Extension Problem

Prior work: [CKR05, FHRT03, AFHKTT04, LN04]

  • Upper bounds:
  • E.g. 𝑃(

log |π‘ˆ| log log |π‘ˆ|) approximation algorithm [CKR05]

  • consider the metric relaxation of the LP for the

problem

  • Solve LP
  • Round the solution
  • Ξ©( log |π‘ˆ|) integrality gap [FHRT03]

6/17/2016 43

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SLIDE 44

0-Extension Problem

Prior work: [CKR05, FHRT03, AFHKTT04, LN04]

  • Upper bounds:
  • E.g. 𝑃(

log |π‘ˆ| log log |π‘ˆ|) approximation algorithm [CKR05]

  • consider the metric relaxation of the LP for the

problem

  • Solve LP
  • Round the solution
  • Ξ©( log |π‘ˆ|) integrality gap [FHRT03]
  • Efficient if the number of terminals is low

6/17/2016 44

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SLIDE 45

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

6/17/2016 45

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SLIDE 46

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

6/17/2016 46

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SLIDE 47

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

2. SNN can be solved using 0-extension in a black-box manner

  • Set: π‘ˆ = 𝑄 , π‘Š = 𝑅 βˆͺ 𝑄 , π‘₯ = 1 , 𝐼 = 𝐻 βˆͺ

π‘Ÿπ‘—, π‘žπ‘— 𝑗}

6/17/2016 47

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SLIDE 48

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

2. SNN can be solved using 0-extension in a black-box manner

  • Set: π‘ˆ = 𝑄 , π‘Š = 𝑅 βˆͺ 𝑄 , π‘₯ = 1 , 𝐼 = 𝐻 βˆͺ

π‘Ÿπ‘—, π‘žπ‘— 𝑗}

  • giving 𝑷(

π’Žπ’‘π’‰ 𝒐 π’Žπ’‘π’‰ π’Žπ’‘π’‰ 𝒐) approximation algorithm

6/17/2016 48

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SLIDE 49

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

2. SNN can be solved using 0-extension in a black-box manner

  • Set: π‘ˆ = 𝑄 , π‘Š = 𝑅 βˆͺ 𝑄 , π‘₯ = 1 , 𝐼 = 𝐻 βˆͺ

π‘Ÿπ‘—, π‘žπ‘— 𝑗}

  • giving 𝑷(

π’Žπ’‘π’‰ 𝒐 π’Žπ’‘π’‰ π’Žπ’‘π’‰ 𝒐) approximation algorithm

3. Improve to depend only on 𝒍 not n

6/17/2016 49

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SLIDE 50

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

2. SNN can be solved using 0-extension in a black-box manner

  • Set: π‘ˆ = 𝑄 , π‘Š = 𝑅 βˆͺ 𝑄 , π‘₯ = 1 , 𝐼 = 𝐻 βˆͺ

π‘Ÿπ‘—, π‘žπ‘— 𝑗}

  • giving 𝑷(

π’Žπ’‘π’‰ 𝒐 π’Žπ’‘π’‰ π’Žπ’‘π’‰ 𝒐) approximation algorithm

3. Improve to depend only on 𝒍 not n

  • Analyzing INN using 0-extension in a β€œgrey-box” manner

6/17/2016 50

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SLIDE 51

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

2. SNN can be solved using 0-extension in a black-box manner

  • Set: π‘ˆ = 𝑄 , π‘Š = 𝑅 βˆͺ 𝑄 , π‘₯ = 1 , 𝐼 = 𝐻 βˆͺ

π‘Ÿπ‘—, π‘žπ‘— 𝑗}

  • giving 𝑷(

π’Žπ’‘π’‰ 𝒐 π’Žπ’‘π’‰ π’Žπ’‘π’‰ 𝒐) approximation algorithm

3. Improve to depend only on 𝒍 not n

  • Analyzing INN using 0-extension in a β€œgrey-box” manner
  • Using subtle properties of existing algorithms for 0-extension

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SLIDE 52

Connection to SNN

  • SNN: (𝑸, 𝑹, 𝑯)
  • 0-Extension: (𝑾, 𝑰, 𝒙, 𝑼)

1. 0-extension can be solved using generalized SNN

  • 𝑅 = π‘Š , 𝑄 = π‘ˆ , πœ‡π‘—,π‘˜ = π‘₯(𝑗, π‘˜) , πœ†π‘— = ∞ 𝑗𝑔 π‘Ÿπ‘— ∈ π‘ˆ π‘π‘œπ‘’ 0 𝑃. π‘₯.

2. SNN can be solved using 0-extension in a black-box manner

  • Set: π‘ˆ = 𝑄 , π‘Š = 𝑅 βˆͺ 𝑄 , π‘₯ = 1 , 𝐼 = 𝐻 βˆͺ

π‘Ÿπ‘—, π‘žπ‘— 𝑗}

  • giving 𝑷(

π’Žπ’‘π’‰ 𝒐 π’Žπ’‘π’‰ π’Žπ’‘π’‰ 𝒐) approximation algorithm

3. Improve to depend only on 𝒍 not n

  • Analyzing INN using 0-extension in a β€œgrey-box” manner
  • Using subtle properties of existing algorithms for 0-extension
  • Leads to an 𝑷(

π’Žπ’‘π’‰ 𝒍 π’Žπ’‘π’‰ π’Žπ’‘π’‰ 𝒍) approximation

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SLIDE 53

Experiments

6/17/2016 Sepideh Mahabadi 53

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SLIDE 54

Experimental Results

  • De-noising problem
  • Each pixel is a query point
  • Data set 𝑄 : all 256 3 possible colors
  • Graph: the grid

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SLIDE 55

Experimental Results

  • De-noising problem
  • Each pixel is a query point
  • Data set 𝑄 : all 256 3 possible colors
  • Graph: the grid
  • Algorithm
  • Only consider the colors that appear in the noisy image

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SLIDE 56

Experimental Results

  • De-noising problem
  • Each pixel is a query point
  • Data set 𝑄 : all 256 3 possible colors
  • Graph: the grid
  • Algorithm
  • Only consider the colors that appear in the noisy image
  • Result: empirical pruning gap 𝛽 is very close to 1,

at most 1.024

6/17/2016 56

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SLIDE 57

Experimental Results

6/17/2016 57

Image Noisy Image De-noised using all colors De-noised using noisy image colors

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SLIDE 58

Experimental Results

6/17/2016 59

Image Half-Noisy De-noised

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SLIDE 59

Conclusion

  • Summary of Results

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SLIDE 60

Conclusion

  • Summary of Results
  • Presented Independent NN pruning

6/17/2016 61

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SLIDE 61

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽

6/17/2016 62

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SLIDE 62

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙) 6/17/2016 63

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SLIDE 63

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications

6/17/2016 64

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SLIDE 64

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications
  • 𝛽 β‰ˆ 1 in the denoising experiments

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SLIDE 65

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications
  • 𝛽 β‰ˆ 1 in the denoising experiments
  • Open Problems

6/17/2016 66

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SLIDE 66

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications
  • 𝛽 β‰ˆ 1 in the denoising experiments
  • Open Problems
  • Prove tighter bounds for 𝛽

6/17/2016 67

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SLIDE 67

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications
  • 𝛽 β‰ˆ 1 in the denoising experiments
  • Open Problems
  • Prove tighter bounds for 𝛽
  • Get better guarantees using different algorithm, i.e., instead of picking the

closest point pick a few points.

6/17/2016 68

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SLIDE 68

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications
  • 𝛽 β‰ˆ 1 in the denoising experiments
  • Open Problems
  • Prove tighter bounds for 𝛽
  • Get better guarantees using different algorithm, i.e., instead of picking the

closest point pick a few points.

  • Solve the general case of the problem, i.e.,
  • where the metrics 𝑒𝑍(π‘Ÿπ‘—, π‘žπ‘—) and π‘’π‘Œ(π‘žπ‘—, π‘žπ‘˜) are different
  • There are weights

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SLIDE 69

Conclusion

  • Summary of Results
  • Presented Independent NN pruning
  • Induces an extra factor 𝛽
  • 𝛽 = 𝑃(

log 𝑙 log log 𝑙) , 𝛽 = Ξ©( log 𝑙)

  • 𝛽 = 𝑃(1) for sparse graphs that are mostly used in applications
  • 𝛽 β‰ˆ 1 in the denoising experiments
  • Open Problems
  • Prove tighter bounds for 𝛽
  • Get better guarantees using different algorithm, i.e., instead of picking the

closest point pick a few points.

  • Solve the general case of the problem, i.e.,
  • where the metrics π‘’π‘Œ(π‘Ÿπ‘—, π‘žπ‘—) and 𝑒𝑍(π‘žπ‘—, π‘žπ‘˜) are different
  • There are weights

6/17/2016 70

Thank You!