Inexact Gradient Projection and Fast Data Driven Compressed Sensing

TL;DR

This paper analyzes the convergence of inexact projected gradient algorithms for nonconvex sets, demonstrating that approximate oracles can maintain convergence rates and applying these results to accelerate data-driven compressed sensing with efficient nearest neighbor searches.

Contribution

The paper introduces convergence guarantees for inexact IPG algorithms with approximate oracles and applies these to improve compressed sensing efficiency using cover trees.

Findings

Approximate oracles can achieve similar accuracy as exact IPG.

Linear convergence rate maintained under certain conditions.

Logarithmic complexity achieved for low-dimensional datasets.

Abstract

We study convergence of the iterative projected gradient (IPG) algorithm for arbitrary (possibly nonconvex) sets and when both the gradient and projection oracles are computed approximately. We consider different notions of approximation of which we show that the Progressive Fixed Precision (PFP) and the $(1+\epsilon)$-optimal oracles can achieve the same accuracy as for the exact IPG algorithm. We show that the former scheme is also able to maintain the (linear) rate of convergence of the exact algorithm, under the same embedding assumption. In contrast, the $(1+\epsilon)$-approximate oracle requires a stronger embedding condition, moderate compression ratios and it typically slows down the convergence. We apply our results to accelerate solving a class of data driven compressed sensing problems, where we replace iterative exhaustive searches over large datasets by fast approximate nearest neighbour search strategies based on the cover tree data structure. For datasets with low intrinsic dimensions our proposed algorithm achieves a complexity logarithmic in terms of the dataset population as opposed to the linear complexity of a brute force search. By running several numerical experiments we conclude similar observations as predicted by our theoretical analysis.