A universal, sample-optimal convergence theorem for the original BIHT algorithm is proved and a scalar lower bound is proved 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.
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 $\widetilde O(s/\epsilon)$ measurements, a deterministic finite-time iterate has directional error at most $\epsilon$, 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 $\tau$ 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 $\widetilde O(s/\epsilon)$ 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.
This paper provides the first near-optimal lower bounds for one-bit compressed sensing of approximately sparse signals lying in a scaled $\ell_1$ ball, which is a commonly adopted relaxation of the exactly $k$-sparse assumption. In prior works, the best known upper bounds on uniform Euclidean error are of order $\widetilde{O}((k/m)^{1/3})$, where $m$ is the number of measurements. Under sub-Gaussian matrices, we establish nearly matching lower bounds for both the canonical one-bit compressed sensing model and the uniformly dithered model. Our argument is to first embed a small Euclidean ball into the signal set, which is straightforward for the dithered model but relies on a lifting map for the canonical model, and then construct two signals in this small ball that are separated in Euclidean distance by at least $(k/m)^{1/3}$ (up to logarithmic factor) but are indistinguishable from the binary measurements. Moreover, our argument extends to approximately sparse signals that live in a properly scaled $\ell_q$ ball $(q\in [0,1])$, yielding a lower bound $\widetilde{\Omega}((k/m)^{\frac{2-q}{2+q}})$ that smoothly bridges the cases of exact sparsity ($q=0$) and $\ell_1$ sparsity ($q=1$). Finally, we discuss the extensions of our lower bounds to sub-Weibull matrices, adversarial bit flipping, matrix recovery, and characterize the transition to the non-sparse case.
Junren Chen, Arya Mazumdar, Ming Yuan· 0 citations
A dimension-free version of the retained-energy form of the Mallat--Zeitouni conjecture is established, showing that the KL basis is within this factor of the optimal basis, and shows that the possible advantage of optimizing over all orthonormal bases vanishes as $d$ grows.
Minbo Gao, Zheng-Feng Ji, Cheng-Hua Liu· 0 citations
This work eliminates the residual-stage $O(d)$-bit payload and reduces the leading upper-bound constant by a factor of approximately $5.93$ compared with the two-stage construction of Feng et al.
Honghao Lin, V. Mirrokni, David P. Woodruff· 0 citations
Noise-shaped one-bit coefficients in normalized discrete polynomial Fourier extension are studied, and exact orthogonality identities, fourth-moment formulas, local kernel estimates, and oscillatory transfer bounds are established.
A randomized fully non-adaptive protocol is constructed that fixes all queries before observing the data and matches the optimal adaptive sample complexity, giving a negative answer to the COLT 2026 open problem asking whether interaction is necessary for order-optimal one-bit mean estimation.
Among $n+1$ equiprobable equal-energy signals in $\R^n$ under additive white Gaussian noise with maximum-likelihood decoding, which arrangement maximizes the probability of correct decoding? The question is Shannon's, recorded by Rice in 1950. Mulgund proved in 2026 that the regular-simplex value bounds the correct-decoding probability of every signal set at every signal-to-noise ratio, leaving open whether the simplex is the only maximizer. This paper determines the equality cases in a form stronger than uniqueness. A signal set other than a regular simplex falls strictly below the bound at every positive signal-to-noise ratio. Hence a code meeting the bound at one positive operating point is already a regular simplex, up to vertex relabeling and an orthogonal map. In probabilistic form, among the correlation matrices that signal sets induce, any matrix other than the identity gives a lower-orthant probability strictly above its independent counterpart at every finite threshold, leaving no room for a nontrivial equality. No code of ambient dimension below $n$ attains the bound. Under an energy budget $E$ with unrestricted blocklength the optimal codebook is uniquely the regular simplex of circumradius $\sqrt{E}$. Every optimal codeword therefore exhausts its allowance. Equality in the Simplex Mean Width Conjecture likewise occurs only at the regular simplex. The proof strengthens the first self-convolution step of Mulgund's argument with Royen's correlation theorem. The single-parameter rigidity is machine-checked in Lean 4.
Mengwei Su, Kaiwen Yang, Hao Xu et al.· 1 citation
