[Paper Review] The PhaseLift for Non-quadratic Gaussian Measurements

Christos Thrampoulidis, Ankit Singh Rawat|arXiv (Cornell University)|Dec 11, 2017

Advanced X-ray Imaging Techniques31 references3 citations

TL;DR

This paper proposes a semidefinite optimization method, PhaseLift for non-quadratic Gaussian measurements, to recover structured signals from nonlinear measurements when traditional least-squares and Lasso methods fail due to zero linear sensitivity ($\mu_\ell = 0$). By lifting the problem into a higher-dimensional space, it effectively treats nonlinear link functions as linear in the lifted domain, achieving consistent recovery with error bounds that depend on a few interpretable parameters capturing nonlinearity.

ABSTRACT

We study the problem of recovering a structured signal $\mathbf{x}_0$ from high-dimensional measurements of the form $y=f(\mathbf{a}^T\mathbf{x}_0)$ for some nonlinear function $f$. When the measurement vector $\mathbf a$ is iid Gaussian, Brillinger observed in his 1982 paper that $\mu_\ell\cdot\mathbf{x}_0 = \min_{\mathbf{x}}\mathbb{E}(y - \mathbf{a}^T\mathbf{x})^2$, where $\mu_\ell=\mathbb{E}_{\gamma}[\gamma f(\gamma)]$ with $\gamma$ being a standard Gaussian random variable. Based on this simple observation, he showed that, in the classical statistical setting, the least-squares method is consistent. More recently, Plan \& Vershynin extended this result to the high-dimensional setting and derived error bounds for the generalized Lasso. Unfortunately, both least-squares and the Lasso fail to recover $\mathbf{x}_0$ when $\mu_\ell=0$. For example, this includes all even link functions. We resolve this issue by proposing and analyzing an appropriate generic semidefinite-optimization based method. In a nutshell, our idea is to treat such link functions as if they were linear in a lifted space of higher-dimension. An appealing feature of our error analysis is that it captures the effect of the nonlinearity in a few simple summary parameters, which can be particularly useful in system design.

Motivation & Objective

• To address the failure of least-squares and Lasso methods in high-dimensional signal recovery when the link function is even, leading to zero linear sensitivity ($\mu_\ell = 0$).
• To develop a robust, generic method for recovering structured signals from nonlinear measurements of the form $y = f(\mathbf{a}^T\mathbf{x}_0)$ with i.i.d. Gaussian $\mathbf{a}$.
• To provide a theoretical error analysis that captures the impact of nonlinearity through a small set of summary parameters, enabling practical system design.

Proposed method

• Lift the original signal recovery problem into a higher-dimensional space where nonlinear measurements are treated as linear, enabling semidefinite programming (SDP) formulation.
• Formulate the recovery problem as a convex optimization via PhaseLift, leveraging the structure of the measurement model and the second-order moment of the nonlinear function.
• Use the key insight from Brillinger (1982) that $\mu_\ell = \mathbb{E}[\gamma f(\gamma)]$ determines the effective linear sensitivity of the link function $f$.
• Design a semidefinite program that minimizes a convex surrogate of the original non-convex problem, ensuring consistency under high-dimensional asymptotics.
• Introduce summary parameters derived from $f$ and the Gaussian distribution to quantify the effect of nonlinearity on recovery error.
• Establish error bounds that depend on these parameters, enabling performance prediction and system design.

Experimental results

Research questions

• RQ1 Can a convex optimization framework recover structured signals from nonlinear Gaussian measurements when $\mu_\ell = 0$?
• RQ2 How can the nonlinearity in the measurement function $f$ be captured in a way that enables theoretical error analysis?
• RQ3 What is the performance of a lifted semidefinite program in recovering signals when standard methods fail due to zero linear sensitivity?
• RQ4 How do the parameters $\mu_\ell$ and higher-order moments of $f$ influence the recovery error in high-dimensional settings?
• RQ5 Can the error bounds be expressed in terms of interpretable, system-level parameters that guide measurement design?

Key findings

• The proposed semidefinite optimization method successfully recovers structured signals even when $\mu_\ell = 0$, overcoming the failure of least-squares and Lasso in such cases.
• The method achieves consistent recovery by lifting the problem into a higher-dimensional space where the nonlinear measurement model becomes effectively linear.
• Error bounds are derived in terms of a few summary parameters capturing the nonlinearity, such as $\mu_\ell = \mathbb{E}[\gamma f(\gamma)]$, enabling interpretable performance analysis.
• The error analysis explicitly accounts for the structure of the nonlinear function $f$, providing a framework for system design and performance prediction.
• The method generalizes prior results by Plan & Vershynin to the case of even link functions and other nonlinearities where linear methods fail.