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

1 CERN Cognitive Festival in Georgia GTU, October 22 – 26, 2018 Some Properties of Hadamard Matrices V. Kvaratskhelia, M. Menteshashvili, G. Giorgobiani Hadamard matrix - 𝐼 = (ℎ 𝑗𝑘 ) 𝑗, 𝑘 = 1,2, … , 𝑜; 1 < 𝑜 < ∞ ℎ 𝑗𝑘 = ±1 〈ℎ 𝑗 ⊥ ℎ 𝑙 〉 = 0, 𝑗 ≠ 𝑙 ℎ 𝑗 ≡ (ℎ 𝑗1 , ℎ 𝑗2 , ⋯ , ℎ 𝑗𝑜 ) ∈ ℝ 𝑜 𝓘 𝒐 - 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

2 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 other 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.

3 In 2002 V. Kvaratskhelia (in “Some inequalities related to Hadamard matrices”. Functional Analysis and Its Applications ) considered the following characteristic: ℝ 𝑜 𝑗𝑡 𝑓𝑟𝑣𝑗𝑞𝑞𝑓𝑒 𝑥𝑗𝑢ℎ 𝑚 𝑞 − 𝑜𝑝𝑠𝑛, 1 ≤ 𝑞 ≤ ∞ ‖𝑦‖ 𝑞 = √|𝑦 1 | 𝑞 + ⋯ |𝑦 𝑜 | 𝑞 𝑞 ‖𝑦‖ ∞ = 𝑛𝑏𝑦{|𝑦 1 |, … , |𝑦 𝑜 |} 𝑦 = (𝑦 1 , … , 𝑦 𝑜 ) ∈ ℝ 𝑜 . 𝜛 𝑞,𝐼 ≡ 𝑛𝑏𝑦{‖ℎ 1 ‖ 𝑞 , ‖ℎ 1 + ℎ 2 ‖ 𝑞 , ⋯ , ‖ℎ 1 + ℎ 2 + ⋯ + ℎ 𝑜 ‖ 𝑞 }, 𝜷 𝒒,𝓘 𝒐 ≡ 𝒏𝒃𝒚 𝑰∈𝓘 𝒐 𝝕 𝒒,𝑰 ( 1 𝑞 + 1 ( 1 𝑞 + 1 2 ) ≤ 𝜷 𝒒,𝓘 𝒐 ≤ 𝑜 1 2 ) , 1 ≤ 𝑞 ≤ 2 , (1) √2 ∙ 𝑜 𝜷 𝒒,𝓘 𝒐 = 𝑜 , 2 ≤ 𝑞 ≤ ∞ . (2) Naturally arises the question to estimate the minimum 𝝏 𝒒,𝓘 𝒐 ≡ 𝒏𝒋𝒐 𝑰∈𝓘 𝒐 𝝕 𝒒,𝑰 Submitted paper (2018): G. Giorgobiani, V. Kvaratskhelia. Maximum inequalities and their applications to Orthogonal and Hadamard matrices.

4 The following estimations are valid: a) when 1 ≤ p < ∞ (1 𝑞 +1 2) √ 7 ln 𝑜 ; 𝜕 𝑞,ℋ 𝑜 ≤ 𝑜 b) when p = 2 𝜕 2,ℋ 𝑜 ≤ 𝑜; c) when 𝑞 = ∞ , for some absolute constant 𝐿 𝜕 ∞,ℋ 𝑜 ≤ 𝐿 √𝑜 . ( 1 𝑞 + 1 2 ) √ 7 ln 𝑜 is asymptoticly Case 𝟑 < 𝒒 < ∞ : the bound 𝑜 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 × 10 11 ) . Case 𝒒 = ∞ : 𝝏 ∞,𝓘 𝒐 ≪ 𝜷 ∞,𝓘 𝒐 . Algorithms Sign-Algorithms – Spencer; Lovett & Meka: Partial Coloring Lemma (Herding algorithms of the Machin Learning) Permutation-Algorithm – S. Chobanyan.

5 Generalization. Complex Hadamard matrices ℎ 11 ⋯ ℎ 1𝑜 𝐼 = [ ⋯ ⋯ ⋯ ] ℎ 𝑜1 ⋯ ℎ 𝑜𝑜 ℎ 𝑗𝑘 ∈ ℂ |ℎ 𝑗𝑘 | = 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 𝑜 ] 𝑙,𝑘 = 1 √𝑜 𝑓 2𝜌𝒋(𝑙−1)(𝑘−1)/𝑜 , 𝑙, 𝑘 = 1, … , 𝑜, [𝐺

6 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.

7 Thank you for your attention