Fast Algorithms for Demixing Sparse Signals from Nonlinear Observations

Mohammadreza Soltani, Chinmay Hegde

    Research output: Contribution to journalArticle

    Abstract

    We study the problem of demixing a pair of sparse signals from noisy, nonlinear observations of their superposition. Mathematically, we consider a nonlinear signal observation model, $y-i = g(a-i^Tx) + e-i, \ i=1,\ldots,m$, where $x = \Phi w+\Psi z$ denotes the superposition signal, $\Phi$ and $\Psi$ are orthonormal bases in $\mathbb {R}^n$, and $w, z\in \mathbb {R}^n$ are sparse coefficient vectors of the constituent signals, and $e-i$ represents the noise. Moreover, $g$ represents a nonlinear link function, and $a-i\in \mathbb {R}^n$ is the $i$th row of the measurement matrix $A\in \mathbb {R}^{m\times n}$. Problems of this nature arise in several applications ranging from astronomy, computer vision, and machine learning. In this paper, we make some concrete algorithmic progress for the above demixing problem. Specifically, we consider two scenarios: first, the case when the demixing procedure has no knowledge of the link function, and second is the case when the demixing algorithm has perfect knowledge of the link function. In both cases, we provide fast algorithms for recovery of the constituents $w$ and $z$ from the observations. Moreover, we support these algorithms with a rigorous theoretical analysis and derive (nearly) tight upper bounds on the sample complexity of the proposed algorithms for achieving stable recovery of the component signals. We also provide a range of numerical simulations to illustrate the performance of the proposed algorithms on both real and synthetic signals and images.

    Original languageEnglish (US)
    Article number7931568
    Pages (from-to)4209-4222
    Number of pages14
    JournalIEEE Transactions on Signal Processing
    Volume65
    Issue number16
    DOIs
    StatePublished - Aug 15 2017

    Keywords

    • Demixing
    • incoherence
    • nonlinear measurements
    • sparse recovery

    ASJC Scopus subject areas

    • Signal Processing
    • Electrical and Electronic Engineering

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