Generative models have been studied experimentally and theoretically as priors for inverse problems such as compressed sensing. Recent work by Gunn et al. studied the use of generative priors with tunable complexity, where a family of generative priors with varying complexity is maintained and a specific complexity can be selected at inversion time. They demonstrated that lower reconstruction errors can be experimentally attained for a variety of inverse problems by appropriately tuning the complexity of the generative prior. In the present paper, we establish theory for compressed sensing in the setting of a tunable family of linear generative priors naturally related through their singular value decompositions. We prove that in noiseless Gaussian compressed sensing, the full-dimensional linear prior attains the minimum expected reconstruction error over the entire family of linear priors. Thus, in this idealized linear noiseless setting, tuning to a lower-complexity prior does not improve the expected reconstruction error. This result is in contract to the behavior of denoising, where lower complexity priors attain lower reconstruction errors due to a standard bias-variance tradeoff. This result indicates that the experimental benefits of tunability in compressed sensing with neural network priors arises due to nonlinearities in the generative models.
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.
Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family, \[ (κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}. \] of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to $α, κ$ and thus allows a universal approximation scheme for the precision and covariance matrices of the entire $(α, κ)$-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.