{"ID":6536150,"CreatedAt":"2026-07-14T01:21:01.169441415Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.10618","arxiv_id":"2607.10618","title":"Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization","abstract":"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 \\emph{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 \\emph{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.","short_abstract":"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 c...","url_abs":"https://arxiv.org/abs/2607.10618","url_pdf":"https://arxiv.org/pdf/2607.10618v1","authors":"[\"Raziyeh Takbiri\"]","published":"2026-07-12T07:29:39Z","proceeding":"stat.ML","tasks":"[\"stat.ML\",\"cs.LG\",\"eess.SP\"]","methods":"[]","has_code":false}