New algorithm demixes sparse signals from nonlinear noisy observations with 35x accuracy gain

Researchers led by Raziyeh Takbiri tackle the challenge of recovering two sparse vectors from a small number of nonlinear observations of their superposition—a problem common in signal processing, compressed sensing, and machine learning. Their proposed framework replaces the conventional squared-loss with a Huberized data fidelity term, which is robust to heavy-tailed noise and outliers, while using generalized folded-concave penalties (SCAD and MCP) to enforce sparsity. A two-block proximal alternating algorithm with backtracking (NLD-PALM) guarantees convergence to critical points under the Kurdyka–Łojasiewicz property with local linear rates.

On the statistical side, the paper establishes restricted strong convexity of the nonlinear loss via an exact sign-definite decomposition, yielding estimation error bounds of order σ√(s log n/m) at every localized stationary point. Under a beta-min condition, an oracle rate σ√(s/m) free of log n and shrinkage bias is achieved. Experiments at n=512 show an earlier phase transition than convex ℓ₁ demixing and greedy hard-thresholding baselines, and a 35× accuracy boost under 5% gross outliers. The method also successfully demixes spike-plus-background signals observed through a saturating amplifier, making it highly practical for real-world sensor and communication systems.