Sublinear approximation of signals

Abstract

It has recently been observed that sparse and compressible signals can be sketched using very few nonadaptive linear measurements in comparison with the length of the signal. This sketch can be viewed as an embedding of an entire class of compressible signals into a low-dimensional space. In particular, <i>d</i>-dimensional signals with <i>m</i> nonzero entries (<i>m</i>-sparse signals) can be embedded in <i>O</i>(<i>m</i> log <i>d</i>) dimensions. To date, most algorithms for approximating or reconstructing the signal from the sketch, such as the linear programming approach proposed by Candes-Tao and Donoho, require time polynomial in the signal length. This paper develops a new method, called Chaining Pursuit, for sketching both <i>m</i>-sparse and compressible signals with <i>O</i>(<i>m</i> polylog <i>d</i>) nonadaptive linear measurements. The algorithm can reconstruct the original signal in time <i>O</i>(<i>m</i> polylog <i>d</i>) with an error proportional to the optimal <i>m</i>-term approximation error. In particular, <i>m</i>-sparse signals are recovered perfectly and compressible signals are recovered with polylogarithmic distortion. Moreover, the algorithm can operate in small space <i>O</i>(<i>m</i> polylog <i>d</i>), so it is appropriate for streaming data.