Sampling Requirements and Accelerated Schemes for Sparse Linear Regression with Orthogonal Least-Squares

TL;DR

This paper introduces the Accelerated Orthogonal Least-Squares (AOLS) algorithm, which efficiently recovers sparse vectors from linear measurements with improved performance and lower computational costs compared to existing methods.

Contribution

The paper proposes AOLS, a novel accelerated greedy algorithm for sparse recovery that reduces computational complexity and improves recovery guarantees over traditional OLS.

Findings

AOLS recovers k-sparse vectors with high probability using O(k log(m/(k+L-1))) measurements.

AOLS outperforms existing algorithms in accuracy and speed in simulations.

AOLS is effective in high-dimensional clustering tasks on union of subspaces.

Abstract

We study the problem of inferring a sparse vector from random linear combinations of its components. We propose the Accelerated Orthogonal Least-Squares (AOLS) algorithm that improves performance of the well-known Orthogonal Least-Squares (OLS) algorithm while requiring significantly lower computational costs. While OLS greedily selects columns of the coefficient matrix that correspond to non-zero components of the sparse vector, AOLS employs a novel computationally efficient procedure that speeds up the search by anticipating future selections via choosing $L$ columns in each step, where $L$ is an adjustable hyper-parameter. We analyze the performance of AOLS and establish lower bounds on the probability of exact recovery for both noiseless and noisy random linear measurements. In the noiseless scenario, it is shown that when the coefficients are samples from a Gaussian distribution, AOLS with high probability recovers a $k$-sparse $m$-dimensional sparse vector using ${\cal O}(k\log \frac{m}{k+L-1})$ measurements. Similar result is established for the bounded-noise scenario where an additional condition on the smallest nonzero element of the unknown vector is required. The asymptotic sampling complexity of AOLS is lower than the asymptotic sampling complexity of the existing sparse reconstruction algorithms. In simulations, AOLS is compared to state-of-the-art sparse recovery techniques and shown to provide better performance in terms of accuracy, running time, or both. Finally, we consider an application of AOLS to clustering high-dimensional data lying on the union of low-dimensional subspaces and demonstrate its superiority over existing methods.