3 research outputs found
Combinatorial Compressive Sampling with Applications.
We simplify and improve the deterministic Compressed Sensing (CS) results of Cormode and Muthukrishnan (CM). A simple relaxation of our deterministic CS technique then generates a new randomized CS result similar to those derived by CM. Finally, our CS techniques are applied to two computational problems of wide interest: The calculation of a periodic function's Fourier transform, and matrix multiplication. Short descriptions of our results follow.
(i) Sublinear-Time Sparse Fourier Transforms: Suppose is -sparse in frequency (e.g., is an exact superposition of sinusoids with frequencies in ). Then we may recover in time by deterministically sampling it at points. If succeeding with high probability is sufficient, we may sample at points and then reconstruct it in time via a randomized version of our deterministic Fourier algorithm. If is much larger than , both algorithms run in sublinear-time in the sense that they will outrun any procedure which samples at least times (e.g., both algorithms are faster than a fast Fourier transform).
In addition to developing new sublinear-time Fourier methods we have implemented previously existing sublinear-time Fourier algorithms. The resulting implementations, called AAFFT 0.5/0.9, are empirically evaluated. The results are promising: AAFFT 0.9 outperforms standard FFTs (e.g., FFTW 3.1) on signals containing about energetic frequencies spread over a bandwidth of or more. Furthermore, AAFFT utilizes significantly less memory than a standard FFT on large signals since it only needs to sample a fraction of the input signal's entries.
(ii) Fast matrix multiplication: Suppose both and are dense matrices. It is conjectured that can be computed in -time. If is known to be -sparse/compressible in each column (e.g., each column of contains only a few non-zero entries) we show that may be calculated in -time. Thus, we generalize previous rapid rectangular matrix multiplication results due to D. Coppersmith.Ph.D.Applied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/61558/1/markiwen_1.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/61558/2/markiwen_2.pd