Online Sparse System Identification and Signal Reconstruction using Projections onto Weighted $\ell_1$ Balls

TL;DR

This paper introduces a new projection-based adaptive algorithm for sparse system identification and signal reconstruction that efficiently incorporates sparsity constraints using weighted $\\ell_1$ balls and hyperslabs, demonstrating competitive performance.

Contribution

The paper proposes a novel projection-based adaptive algorithm utilizing hyperslabs and weighted \(\ell_1\) balls for efficient sparse system identification, with improved computational complexity.

Findings

Algorithm exhibits linear complexity with respect to system order.

Performance validated against LASSO and recent adaptive sparse algorithms.

Numerical results confirm effectiveness in sparse signal reconstruction.

Abstract

This paper presents a novel projection-based adaptive algorithm for sparse signal and system identification. The sequentially observed data are used to generate an equivalent sequence of closed convex sets, namely hyperslabs. Each hyperslab is the geometric equivalent of a cost criterion, that quantifies "data mismatch". Sparsity is imposed by the introduction of appropriately designed weighted $\ell_1$ balls. The algorithm develops around projections onto the sequence of the generated hyperslabs as well as the weighted $\ell_1$ balls. The resulting scheme exhibits linear dependence, with respect to the unknown system's order, on the number of multiplications/additions and an $\mathcal{O}(L\log_2L)$ dependence on sorting operations, where $L$ is the length of the system/signal to be estimated. Numerical results are also given to validate the performance of the proposed method against the LASSO algorithm and two very recently developed adaptive sparse LMS and LS-type of adaptive algorithms, which are considered to belong to the same algorithmic family.