For every fixed constant α>0, we design an algorithm for computing
the k-sparse Walsh-Hadamard transform of an N-dimensional vector x∈RN in time k1+α(logN)O(1). Specifically, the
algorithm is given query access to x and computes a k-sparse x~∈RN satisfying ∥x~−x^∥1≤c∥x^−Hk(x^)∥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(logN)O(1) reconstruction time with a reduced exponent in
the poly-logarithmic factor, and eliminating the extra parameter α).
Finally, by allowing the algorithm to use randomness, while still using
non-adaptive queries, the running time of the algorithm can be improved to
O~(klog3N)