AI 中文总结
研究1比特压缩感知中二进制迭代硬阈值处理（BIHT）算法，证明无噪声时原始BIHT算法有通用样本最优收敛定理，符号损坏时未归一化的BIHT能达鲁棒误差下限但不稳定，刻画了每次迭代归一化在算法上的必要性。
AI 中文摘要
二进制迭代硬阈值处理（BIHT）是一种从1比特符号测量中恢复稀疏向量的简单而有效的贪心方法。原始形式的BIHT执行一个“梯度下降”步骤，然后进行硬阈值处理。[Jac+11]的介绍性工作中未解决该算法的收敛分析，十多年来一直未解决，后续的严格分析研究了归一化变体，该变体还将每次迭代投影到单位球面上。本文解决了这一差距，并刻画了每次迭代归一化在算法上何时必要。在无噪声设置中，证明了原始BIHT算法的通用样本最优收敛定理。在符号损坏情况下，证明了明显的分离。未进行每次迭代归一化的BIHT在早期迭代中仍能达到鲁棒误差下限，但其恢复不稳定。证明了一个标量下限，表明任何非平凡的损坏模式都会使迭代无限振荡。因此，在符号损坏下，BIHT不存在一般的最后迭代收敛定理，而归一化的替代方法可避免这种情况。
英文摘要
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.
Comments28 pages, 2 figures
