Non-square matrix sensing without spurious local minima via the Burer-Monteiro approach (1609.03240v2)

Abstract: We consider the non-square matrix sensing problem, under restricted isometry property (RIP) assumptions. We focus on the non-convex formulation, where any rank-$r$ matrix $X \in \mathbb{R}^{m \times n}$ is represented as $UV^\top$, where $U \in \mathbb{R}^{m \times r}$ and $V \in \mathbb{R}^{n \times r}$. In this paper, we complement recent findings on the non-convex geometry of the analogous PSD setting [5], and show that matrix factorization does not introduce any spurious local minima, under RIP.

• The paper utilizes the Burer-Monteiro factorization for non-square matrix sensing and proves the absence of spurious local minima in the resulting non-convex problem under standard assumptions.
• It demonstrates that any local minimum is quantitatively near a global minimum and confirms strict saddle properties for the regularized objective function.
• This research extends the applicability of non-convex optimization techniques to non-square matrix recovery, relevant for areas like sensor networks and signal processing.

Non-square Matrix Sensing Without Spurious Local Minima via the Burer-Monteiro Approach

The paper addresses the problem of non-square matrix sensing, which involves inferring a low-rank matrix from linear measurements under the restricted isometry property (RIP). The paper leverages the Burer-Monteiro approach, representing the rank-$r$ matrix as $UV^\top$, where $U \in R^{m \times r}$ and $V \in R^{n \times r}$. The authors significantly advance the understanding of non-convex formulations by demonstrating the absence of spurious local minima in this framework, a result that complements existing findings in square, positive semidefinite (PSD) settings.

Problem Formulation

The task is formulated as a minimization problem: $$\underset{X \in R^{m \times n}}{\text{minimize}} \quad f(X) := \|\mathcal{A}(X) - b\|_2^2 \quad \text{subject to} \quad \text{rank}(X) \leq r,$$ where $b$ denotes observations and $\mathcal{A}$ is a sensing linear map. Despite multiple approaches to solve the convex and non-convex versions, including spectral decomposition integer computations, large-scale scenarios necessitate alternative solutions that embed low-rank constraints directly into the objective.

Methodology: Non-convex Reformulation

By employing a factorization trick, the problem reduces to optimizing over $$W = \begin{bmatrix} U & V \end{bmatrix}$$: $$\underset{U \in R^{m \times r}, V \in R^{n \times r}}{\text{minimize}} \quad f(UV^\top) := \|\mathcal{A}(UV^\top) - b\|_2^2.$$ The inherent non-convexity due to bilinearity raises concerns about potential spurious local minima. The authors introduce a regularization term $g(W) = \lambda \|W^\top W\|^2$ to their formulation, ensuring a focus on balanced factorizations, eliminating suboptimal factorizations.

Key Findings and Implications

The paper establishes that the re-parametrized non-convex problem retains a geometry without erroneous local minima under standard RIP assumptions:

• Demonstrated via thorough analysis and bounded fault terms, any local minima are proximal to the global minimum with a quantifiable bound.
• The paper generalizes results from matrix completion and sensing, transcending to non-square cases, affirming Burer-Monteiro's application scope without spurious local formations.

Specific results indicate, in noiseless conditions, any local minimum achieves global optimality assuming $\delta_{4r} < 0.0363$. For noisy cases or high-rank scenarios, the distance to true solution demonstrates proportional constraints to residual noise levels and best rank-$r$ approximations, respectively.

Saddle Points and Practical Considerations

It further attunes the discussed non-square matrix sensing problem to practical settings where saddle points could trap conventional first-order methods. The paper leverages recent advancements indicating these can be overcome using stochastic or small-step gradient descent methodologies, confirming $f + g$ exhibits the required strict saddle properties.

Theoretical and Practical Contributions

The implications reach both theoretical and application terrains:

• Extending the usability and understanding of non-convex methodologies in matrix retrieval and sensing applications.
• Practical tools enhancing sensor network design, quantum state tomography, and computational metrics in signal processing.

Future Outlook

In closing, this paper propels further exploration into robust sensing methodologies across diverse applications while fueling debate on non-convex optimization robustness. The insights encourage continued innovation in parameter enhancement and application-specific tweaks, allowing for productive advances in AI-driven sensing mechanisms.