Recovering linear images of sparse signals from indirect observations

We consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition: $y_i=g(\inner{\ba_i}{\bPhi\bw^\ast+\bPsi\bz^\ast})+e_i$, $i=1,\dots,m$, with $m\ll n$, incoherent orthonormal bases $\bPhi,\bPsi$, a scalar link $g$, and noise $e_i$ that may be heavy-tailed or contaminated. We propose a regularization-based framework combining a Huberized data fidelity with generalized folded-concave penalties (SCAD, MCP), and a two-block proximal alternating algorithm with backtracking (NLD-PALM) whose whole iterate sequence provably converges to critical points under the Kurdyka--Łojasiewicz property, with local linear rates. On the statistical side we establish restricted strong convexity of the Huberized nonlinear loss through an exact sign-definite decomposition, and derive estimation error bounds of order $σ\sqrt{s\log(n)/m}$ that hold at every localized stationary point, an oracle rate $σ\sqrt{s/m}$ free of $\log n$ and shrinkage bias under a beta-min condition, and a co-equal recovery theorem for unknown monotone links via a linear surrogate and a clipped Plan--Vershynin decoupling. The estimator requires no knowledge of the sparsity levels, and its guarantees hold under symmetric noise with only finite variance. Experiments at $n=512$ under a frozen data-driven regularization rule show an earlier phase transition than convex $\ell_1$ demixing and greedy hard-thresholding baselines, a $35\times$ accuracy advantage over squared-loss estimation under $5\%$ gross outliers, and successful demixing of spike-plus-background signals observed through a saturating amplifier.

We consider the problem of support recovery for sparse binary signals from noisy linear measurements. For sparse Gaussian measurement matrices we identify sufficient conditions on the minimal sample size for maximum-likelihood recovery in the high-SNR regime $ds/p \to \infty$, where $p$ denotes the signal dimension, $s$ the number of non-zero components of the signal, and $d$ the expected number of non-zero components per row of measurement. Combined with known lower bounds, this yields an information-theoretic threshold of order $s\log(p/s) / \log(ds/p)$, making explicit the price of measurement sparsity. In particular, we highlight a regime where the sample-complexity loss from measurement sparsity is logarithmic while the computational gain is nearly linear. Second, we study recovery after sparsifying an originally dense Gaussian design: the observations are generated from the dense design, while estimation uses an independently sparsified design and a rescaled response. In the proportional regime $s=αp$, $d=ψp$, we prove that, for every fixed target error level $δ$ and every slack $\varepsilon>0$, a sample size of order $p/ψ^2$ is sufficient for support recovery for arbitrarily small $ψ$.

Recovery from linear measurements under sparse adversarial corruption is typically formulated as an exact-recovery problem: one seeks structural conditions on $\mathbf{A}$ (e.g., restricted isometry property) guaranteeing unique recovery of $\mathbf{x}^\star$ from $\mathbf{y} = \mathbf{A}\mathbf{x}^\star + \mathbf{e}$ with $\|\mathbf{e}\|_0 \leq q$. However, these guarantees provide no guidance once exact recovery fails. This limitation obscures simple robustness phenomena -- for instance, repeated rows in $\mathbf{A}$ can preserve nontrivial information about $\mathbf{x}^\star$ under sparse corruption. In this paper, we study what information about $\mathbf{x}^\star$ can be uniformly recovered from $\mathbf{y} = \mathbf{A}\mathbf{x}^\star + \mathbf{e}$ for arbitrary $\mathbf{A}\in\mathbb{R}^{m\times n}$ and any $q$-sparse $\mathbf{e}$. We show that the robust information is precisely $\mathbf{x}^\star + \ker(\mathbf{U})$, where $\mathbf{U}$ is the orthogonal projection onto the intersection of rowspaces of all submatrices of $\mathbf{A}$ obtained by deleting $2q$ rows. This clarifies how the row structure of $\mathbf{A}$ governs whether a $q$-sparse corruption allows exact, partial, or only trivial recovery. We further prove every $\mathbf{x}$ minimizing $\|\mathbf{y} - \mathbf{A} \mathbf{x}\|_0$ belongs to $\mathbf{x}^\star + \ker(\mathbf{U})$, yielding a constructive approach to recover this set. For i.i.d. Gaussian matrices, we establish a sharp phase transition between exact and trivial recovery. We sketch two applications: robust network tomography and signal reconstruction from oversampled DCT.