Nearly optimal deterministic algorithm for sparse walsh-hadamard transform

For every fixed constant α > 0, we design an algorithm for computing the k-sparse Walsh-Hadamard transform (i.e., Discrete Fourier Transform over the Boolean cube) of an N-dimensional vector x ∈ RN in time k1+α(log N)O(1). Specifically, the algorithm is given query access to x and computes a k-sparse x ∈ RN satisfying ||[EQUATION]||1 ≤c||[EQUATION] -- Hk([EQUATION])||1, for an absolute constant c > 0, where [EQUATION] is the transform of x and Hk([EQUATION]) is its best k-sparse approximation. Our algorithm is fully deterministic and only uses non-adaptive queries to x (i.e., all queries are determined and performed in parallel when the algorithm starts). An important technical tool that we use is a construction of nearly optimal and linear lossless condensers which is a careful instantiation of the GUV condenser (Guruswami, Umans, Vadhan, JACM 2009). Moreover, we design a deterministic and non-adaptive e1/e1 compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time k1+α(log N)O(1) (for the GUV-based condenser) and is of independent interest. Our scheme significantly simplifies and improves an earlier expander-based construction due to Berinde, Gilbert, Indyk, Karloff, Strauss (Allerton 2008). Our methods use linear lossless condensers in a black box fashion; therefore, any future improvement on explicit constructions of such condensers would immediately translate to improved parameters in our framework (potentially leading to k(log N)O(1) reconstruction time with a reduced exponent in the poly-logarithmic factor, and eliminating the extra parameter α). By allowing the algorithm to use randomness, while still using non-adaptive queries, the running time of the algorithm can be improved to O(k log3 N).

[1]  Mahdi Cheraghchi,et al.  Applications of Derandomization Theory in Coding , 2011, ArXiv.

[2]  Martin Vetterli,et al.  A fast Hadamard transform for signals with sub-linear sparsity , 2013, 2013 51st Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[3]  Adi Akavia,et al.  Deterministic Sparse Fourier Approximation Via Approximating Arithmetic Progressions , 2014, IEEE Transactions on Information Theory.

[4]  Piotr Indyk,et al.  Nearly optimal sparse fourier transform , 2012, STOC '12.

[5]  V. Shoup New algorithms for finding irreducible polynomials over finite fields , 1990 .

[6]  Ely Porat,et al.  Polynomials: a new tool for length reduction in binary discrete convolutions , 2014, ArXiv.

[7]  Amnon Ta-Shma,et al.  Loss-less condensers, unbalanced expanders, and extractors , 2001, STOC '01.

[8]  Leonid A. Levin,et al.  A hard-core predicate for all one-way functions , 1989, STOC '89.

[9]  Venkatesan Guruswami,et al.  Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes , 2009, JACM.

[10]  Piotr Indyk,et al.  Combining geometry and combinatorics: A unified approach to sparse signal recovery , 2008, 2008 46th Annual Allerton Conference on Communication, Control, and Computing.

[11]  Piotr Indyk,et al.  Recent Developments in the Sparse Fourier Transform: A compressed Fourier transform for big data , 2014, IEEE Signal Processing Magazine.

[12]  Holger Rauhut,et al.  A Mathematical Introduction to Compressive Sensing , 2013, Applied and Numerical Harmonic Analysis.

[13]  Martin Vetterli,et al.  A Fast Hadamard Transform for Signals With Sublinear Sparsity in the Transform Domain , 2015, IEEE Trans. Inf. Theory.

[14]  Mark A. Iwen,et al.  Combinatorial Sublinear-Time Fourier Algorithms , 2010, Found. Comput. Math..

[15]  A.C. Gilbert,et al.  A Tutorial on Fast Fourier Sampling , 2008, IEEE Signal Processing Magazine.

[16]  M. A. Iwen,et al.  Improved Approximation Guarantees for Sublinear-Time Fourier Algorithms , 2010, ArXiv.

[17]  Eyal Kushilevitz,et al.  Learning Decision Trees Using the Fourier Spectrum , 1993, SIAM J. Comput..

[18]  Leonid A. Levin,et al.  Pseudo-random generation from one-way functions , 1989, STOC '89.