NDLI: Nearly optimal deterministic algorithm for sparse walsh-hadamard transform

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Nearly optimal deterministic algorithm for sparse walsh-hadamard transform

Content Provider	ACM Digital Library
Author	Indyk, Piotr Cheraghchi, Mahdi
Abstract	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 ∈ $R^{N}$ in time $k^{1+α}(log$ $N)^{O(1)}.$ Specifically, the algorithm is given query access to x and computes a k-sparse x ∈ $R^{N}$ satisfying $\|\|[EQUATION]\|\|_{1}$ ≤c\|\|[EQUATION] -- $H_{k}([EQUATION])\|\|_{1},$ for an absolute constant c > 0, where [EQUATION] is the transform of x and $H_{k}([EQUATION])$ 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 $ℓ_{1}/ℓ_{1}$ compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time $k^{1+α}(log$ $N)^{O(1)}$ (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 Õ(k $log^{3}$ N).
Ending Page	317
Page Count	20
Starting Page	298
File Format	PDF
ISBN	9781611974331
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2016-01-10
Access Restriction	Subscribed
Content Type	Text
Resource Type	Article

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