Imperial College London
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Dr Mahdi Cheraghchi

Faculty of Engineering, Department of Computing

Honorary Senior Lecturer
 
 
 
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Contact

 

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Location

 

353ACE ExtensionSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@inproceedings{Cheraghchi:2016:10.1137/1.9781611974331.ch23,
author = {Cheraghchi, M and Indyk, P},
doi = {10.1137/1.9781611974331.ch23},
pages = {298--317},
publisher = {SIAM},
title = {Nearly optimal deterministic algorithm for sparse Walsh-Hadamard transform},
url = {http://dx.doi.org/10.1137/1.9781611974331.ch23},
year = {2016}
}
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RIS format (EndNote, RefMan)

TY  - CPAPER
AB  - 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 iV-dimensional vector x ∈ N in time k1+α(log N)°(1) Specifically, the algorithm is given query access to x and computes a k-sparse x ∈ .N satisfying ||x-||1 ≤ c|| - Hk()\\1, for an absolute constant c > 0, where x is the transform of x and Hk(x) 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 l1/l1 compressed sensing scheme based on general lossless condensers that is equipped with a fast reconstruction algorithm running in time k1+α(logN)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)(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 (k log3 N).
AU  - Cheraghchi,M
AU  - Indyk,P
DO  - 10.1137/1.9781611974331.ch23
EP  - 317
PB  - SIAM
PY  - 2016///
SP  - 298
TI  - Nearly optimal deterministic algorithm for sparse Walsh-Hadamard transform
UR  - http://dx.doi.org/10.1137/1.9781611974331.ch23
UR  - http://hdl.handle.net/10044/1/32447
ER  -
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