Some Properties of Hadamard Matrices V. Kvaratskhelia, M. - - PDF document

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CERN Cognitive Festival in Georgia GTU, October 22 – 26, 2018

Some Properties of Hadamard Matrices

• V. Kvaratskhelia, M. Menteshashvili, G. Giorgobiani

Hadamard matrix - 𝐼 = (ℎ𝑗𝑘) 𝑗, 𝑘 = 1,2, … , 𝑜; 1 < 𝑜 < ∞

ℎ𝑗𝑘 = ±1 〈ℎ𝑗 ⊥ ℎ𝑙〉 = 0, 𝑗 ≠ 𝑙 ℎ𝑗 ≡ (ℎ𝑗1, ℎ𝑗2, ⋯ , ℎ𝑗𝑜) ∈ ℝ𝑜

𝓘𝒐 - the class of all Hadamard matrices of order 𝑜 = 4𝑙.

Example: 8 × 8 Hadamard matrix [ 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 −𝟐 −𝟐 – 𝟐 𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 −𝟐 −𝟐 – 𝟐 𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 −𝟐 −𝟐 −𝟐 𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 𝟐 𝟐 𝟐 −𝟐 −𝟐 −𝟐 −𝟐 𝟐]

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Practical use:

• Error-correcting codes - in early satellite transmissions.

For example: 1971 – 72, Mariner 9’s mission to Mars, 54 billion bits of data had been transmitted; Flybys of the outer planets in the solar system.

• Modern CDMA cellphones - minimize interference with
• ther transmissions to the base station.
• New applications are everywhere about us such as in:

pattern recognition, neuroscience, optical communication and information hiding.

• Compressive Sensing (Signal Reconstruction).
• D. J. Lum et al. Fast Hadamard transforms for compressive sensing. 2015
• In Chemical Physics - Construction of the orthogonal

set of molecular orbitals.

• K. Balasubramanian. Molecular orbitals and Hadamard matrices 1993.

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In 2002 V. Kvaratskhelia (in “Some inequalities related to Hadamard matrices”. Functional Analysis and Its Applications) considered the following characteristic: ℝ𝑜 𝑗𝑡 𝑓𝑟𝑣𝑗𝑞𝑞𝑓𝑒 𝑥𝑗𝑢ℎ 𝑚𝑞 − 𝑜𝑝𝑠𝑛, 1 ≤ 𝑞 ≤ ∞ ‖𝑦‖𝑞 = √|𝑦1|𝑞 + ⋯ |𝑦𝑜|𝑞

𝑞

‖𝑦‖∞ = 𝑛𝑏𝑦{|𝑦1|, … , |𝑦𝑜|} 𝑦 = (𝑦1, … , 𝑦𝑜) ∈ ℝ𝑜. 𝜛 𝑞,𝐼 ≡ 𝑛𝑏𝑦{‖ℎ1‖𝑞, ‖ℎ1 + ℎ2‖𝑞, ⋯ , ‖ℎ1 + ℎ2 + ⋯ + ℎ𝑜‖𝑞}, 𝜷 𝒒,𝓘𝒐 ≡ 𝒏𝒃𝒚

𝑰∈𝓘𝒐 𝝕𝒒,𝑰 1 √2 ∙ 𝑜 (1

𝑞 +1 2) ≤ 𝜷𝒒,𝓘𝒐 ≤ 𝑜

( 1

𝑞 +1 2), 1 ≤ 𝑞 ≤ 2, (1)

𝜷𝒒,𝓘𝒐 = 𝑜, 2 ≤ 𝑞 ≤ ∞ . (2) Naturally arises the question to estimate the minimum 𝝏𝒒,𝓘𝒐 ≡ 𝒏𝒋𝒐

𝑰∈𝓘𝒐 𝝕 𝒒,𝑰

Submitted paper (2018):

• G. Giorgobiani, V. Kvaratskhelia. Maximum inequalities and their applications to

Orthogonal and Hadamard matrices.

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The following estimations are valid: a) when 1 ≤ p < ∞ 𝜕𝑞,ℋ𝑜 ≤ 𝑜

(1 𝑞 +1 2) √ 7 ln 𝑜 ;

b) when p = 2 𝜕2,ℋ𝑜 ≤ 𝑜; c) when 𝑞 = ∞, for some absolute constant 𝐿 𝜕∞,ℋ𝑜 ≤ 𝐿 √𝑜 . Case 𝟑 < 𝒒 < ∞: the bound 𝑜

(1

𝑞 +1 2) √ 7 ln 𝑜 is asymptoticly

smaller then 𝑜 of (2) and this is achieved for extremely large 𝑜-s (𝑗𝑔 𝑞 = 25, 𝑜 ≥ 33; 𝑗𝑔 𝑞 = 2.5, 𝑜 > 2 × 1011). Case 𝒒 = ∞: 𝝏∞,𝓘𝒐 ≪ 𝜷∞,𝓘𝒐.

Algorithms

Sign-Algorithms – Spencer; Lovett & Meka: Partial Coloring Lemma (Herding algorithms of the Machin Learning) Permutation-Algorithm – S. Chobanyan.

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Generalization. Complex Hadamard matrices

𝐼 = [ ℎ11 ⋯ ℎ1𝑜 ⋯ ⋯ ⋯ ℎ𝑜1 ⋯ ℎ𝑜𝑜 ] ℎ𝑗𝑘 ∈ ℂ |ℎ𝑗𝑘| = 1 𝐼𝐼∗ = 𝑜𝐽 𝐼∗ − 𝑑𝑝𝑜𝑘𝑣𝑕𝑏𝑢𝑓 𝑢𝑠𝑏𝑜𝑡𝑞𝑝𝑡𝑓, 𝐽 − 𝑗𝑒𝑓𝑜𝑢𝑗𝑢𝑧 They are Unitary matrices after rescaling. Example: rescaled Fourier Matrix, 𝑜 ≥ 1 𝐼 = √𝑜[𝐺

𝑜]𝑙,𝑘

[𝐺

𝑜]𝑙,𝑘 = 1

√𝑜 𝑓2𝜌𝒋(𝑙−1)(𝑘−1)/𝑜, 𝑙, 𝑘 = 1, … , 𝑜,

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Unitary (complex) matrices are important in Particle Physics:

• CKM (Cabibbo-Kobayashi-Maskawa) matrix,

appears in the coupling of quarks to 𝑋± bosons;

• Reconstruction Problem of a unitary matrix see e.g.

Auberson, G., Martin A., Mennessier G. “On the reconstruction of a unitary matrix from its moduli”. The CERN Theory Department:1990 - Report # CERN-TH-5809-90.

Applications of Complex Hadamard matrices (in 90-ies)

• various branches of mathematics,
• quantum optics,
• high-energy physics,
• quantum teleportation.

We plan to transfer our results for Real Hadamard matrices to the Complex case.

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Thank you for your attention