# Demixing Sparse Signals from Nonlinear Observations using Generalized Non-convex Regularization

arXiv: https://arxiv.org/abs/2607.10618 · HTML: https://mlsanity.com/paper/2607.10618/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 12 Jul 2026.

## Builds on (most cited first, 7 of 7)

- [Regularized M-estimators with nonconvexity: Statistical and algorithmic theory for local optima](https://mlsanity.com/paper/1305.2436/) (2013)
- [Calculus of the exponent of Kurdyka-Lojasiewicz inequality and its applications to linear convergence of first-order methods](https://mlsanity.com/paper/1602.02915/) (2016)
- [Statistical consistency and asymptotic normality for high-dimensional robust M-estimators](https://mlsanity.com/paper/1501.00312/) (2015)
- [Support recovery without incoherence: A case for nonconvex regularization](https://mlsanity.com/paper/1412.5632/) (2014)
- [A Survey on Nonconvex Regularization Based Sparse and Low-Rank Recovery in Signal Processing, Statistics, and Machine Learning](https://mlsanity.com/paper/1808.05403/) (2018)
- [Fast Algorithms for Demixing Sparse Signals from Nonlinear Observations](https://mlsanity.com/paper/1608.01234/) (2016)
- [Demixing Structured Superposition Signals from Periodic and Aperiodic Nonlinear Observations](https://mlsanity.com/paper/1708.02999/) (2017)

## Abstract (arXiv)

We consider the recovery of a pair of sparse vectors from a limited number of nonlinear observations of their superposition: yi=g(innerbaibPhibw^ast+bPsibz^ast)+ei, i=1,dots,m, with mll n, incoherent orthonormal bases bPhi,bPsi, a scalar link g, and noise ei 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--Lojasiewicz 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√slog(n)/m that hold at every localized stationary point, an oracle rate σ√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 ℓ1 demixing and greedy hard-thresholding baselines, a 35× accuracy advantage over squared-loss estimation under 5% gross outliers, and successful demixing of spike-plus-background signals observed through a saturating amplifier.