Almost-Euclidean Subspaces of l1N\ell_1^N via Tensor Products: A Simple Approach to Randomness Reduction

It has been known since 1970's that the N-dimensional l1- space contains almost Euclidean subspaces whose dimension is Ω(N). However, proofs of existence of such subspaces were probabilistic, hence non-constructive, which made the results not-quite-suitable for subsequently discovered applications to high-dimensional nearest neighbor search, error-correcting codes over the reals, compressive sensing and other computational problems. In this paper we present a "low-tech" scheme which, for any γ > 0, allows us to exhibit almost Euclidean Ω(N)- dimensional subspaces of l1N while using only Nγ random bits. Our results extend and complement (particularly) recent work by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1) simplicity (we use only tensor products) and (2) yielding almost Euclidean subspaces with arbitrarily small distortions.

[1]  W. Beckner.  Inequalities in Fourier analysis , 1975 .

[2]  J. Marcinkiewicz,et al.  Quelques inégalités pour les opérations linéaires , 1939 .

[3]  T. Figiel,et al.  The dimension of almost spherical sections of convex bodies , 1976 .

[4]  Inequalities for scalar-valued linear operators that extend to their vector-valued analogues , 1980 .

[5]  Shachar Lovett,et al.  Almost Euclidean sections of the N-dimensional cross-polytope using O(N) random bits , 2007, Electron. Colloquium Comput. Complex..

[6]  Piotr Indyk,et al.  Uncertainty principles, extractors, and explicit embeddings of l2 into l1 , 2007, STOC '07.

[7]  Roman Vershynin,et al.  Uncertainty Principles and Vector Quantization , 2006, IEEE Transactions on Information Theory.

[8]  V. Milman,et al.  Logarithmic reduction of the level of randomness in some probabilistic geometric constructions , 2006 .

[9]  Sorin C. Popescu,et al.  Lidar Remote Sensing , 2011 .

[10]  K. Ball.  An elementary introduction to modern convex geometry, in flavors of geometry , 1997 .

[11]  V. Milman.  Topics in Asymptotic Geometric Analysis , 2000 .

[12]  J. Lindenstrauss,et al.  Handbook of geometry of Banach spaces , 2001 .

[13]  James R. Lee,et al.  Almost Euclidean subspaces of e N 1 via expander codes , 2008, SODA 2008.

[14]  Venkatesan Guruswami,et al.  Euclidean Sections of with Sublinear Randomness and Error-Correction over the Reals , 2008, APPROX-RANDOM.

[15]  N. Tomczak-Jaegermann.  Banach-Mazur distances and finite-dimensional operator ideals , 1989 .

[16]  M. Rudelson,et al.  Euclidean embeddings in spaces of finite volume ratio via random matrices , 2005 .

[17]  Борис Сергеевич Кашин,et al.  Замечание о задаче сжатого измерения@@@A Remark on Compressed Sensing , 2007 .

[18]  V. Milman.  New proof of the theorem of A. Dvoretzky on intersections of convex bodies , 1971 .

[19]  Andrej Yu. Garnaev,et al.  On widths of the Euclidean Ball , 1984 .

[20]  G. Pisier.  The volume of convex bodies and Banach space geometry , 1989 .

[21]  S. Szarek.  Convexity, complexity, and high dimensions , 2006 .

[22]  Guillaume Aubrun,et al.  Hastings’s Additivity Counterexample via Dvoretzky’s Theorem , 2010, 1003.4925.

[23]  Silvio Levy,et al.  Flavors of Geometry , 1997 .

[24]  Piotr Indyk.  Dimensionality reduction techniques for proximity problems , 2000, SODA '00.

[25]  B. Bollobás.  THE VOLUME OF CONVEX BODIES AND BANACH SPACE GEOMETRY (Cambridge Tracts in Mathematics 94) , 1991 .

[26]  W. Rudin.  Trigonometric Series with Gaps , 1960 .

[27]  G. Schechtman.  A remark concerning the dependence on ɛ in dvoretzky's theorem , 1989 .

[28]  K. Ball.  An Elementary Introduction to Modern Convex Geometry , 1997 .

[29]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[30]  Yuval Rabani,et al.  Explicit Dimension Reduction and Its Applications , 2011, 2011 IEEE 26th Annual Conference on Computational Complexity.

[31]  Large subspaces ofl∞n and estimates of the gordon-lewis constantand estimates of the gordon-lewis constant , 1980 .

[32]  S. Szarek,et al.  Chapter 8 - Local Operator Theory, Random Matrices and Banach Spaces , 2001 .

[33]  Y. Gordon.  Some inequalities for Gaussian processes and applications , 1985 .

[34]  Noga Alon,et al.  Construction of asymptotically good low-rate error-correcting codes through pseudo-random graphs , 1992, IEEE Trans. Inf. Theory.