On the Role of Normalization in Binary Iterative Hard Thresholding for 1-bit Compressed Sensing

2026-07-16

AI Summary

This work addresses the long-standing absence of convergence theory for the original Binary Iterative Hard Thresholding (BIHT) algorithm in 1-bit compressed sensing by establishing, for the first time, a sample-optimal finite-step convergence guarantee. Through deterministic finite-time analysis, directional error bound estimation, and an adversarial sign-flip noise model, the study proves that both the original and normalized BIHT achieve a sample complexity of Õ(s/ε) in the noiseless setting. However, under sign-flip noise, the original BIHT fails to ensure convergence of its final iterate, whereas per-iteration normalization plays an essential role in stabilizing the algorithm and effectively preventing oscillations. This result fills a critical theoretical gap that has persisted in the field for over a decade.

Abstract

Binary Iterative Hard Thresholding (BIHT) is a simple, yet effective, greedy method for recovering a sparse vector from one-bit sign measurements. In its original form, BIHT performs a ``gradient-descent'' step, followed by hard thresholding. A convergence analysis of this algorithm was left open in the introductory work of [Jac+11] and has remained unresolved for over a decade, with subsequent sharp analyses studying a normalized variant instead, that additionally projects every iterate onto the unit sphere. This paper resolves that gap and characterizes when per-iteration normalization is algorithmically necessary. In the noiseless setting, we prove a universal, sample-optimal convergence theorem for the original BIHT algorithm. Specifically, with $\widetilde O(s/ε)$ measurements, a deterministic finite-time iterate has directional error at most $ε$, simultaneously for every $s$-sparse unit vector. This matches the optimal sample dependence achieved by normalized BIHT in prior work. Thus, in the noiseless regime, per-iterate normalization is unnecessary for optimal recovery. Under sign corruptions, we prove a sharp separation. If at most a $τ$ fraction of signs are flipped adversarially, then BIHT, without per-iterate normalization, still reaches the robust error floor at an early iterate with a matching $\widetilde O(s/ε)$ sample complexity rate as its normalized variant. This recovery, however, is not stable. We prove a scalar lower bound showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely. Consequently, no general last-iterate convergence theorem can hold for BIHT under sign corruptions, while its normalized surrogate provably escapes this instance.

Problem

• Binary Iterative Hard Thresholding
• 1-bit Compressed Sensing
• Normalization
• Convergence Analysis
• Sign Corruptions

Innovation

• Binary Iterative Hard Thresholding
• 1-bit Compressed Sensing
• Normalization
• Convergence Analysis
• Robust Recovery