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Tuning-Free Structured Sparse Recovery of Multiple Measurement Vectors using Implicit Regularization

Lakshmi Jayalal, Sheetal Kalyani

TL;DR

The paper tackles the MMV recovery problem with joint row sparsity by introducing a tuning-free approach based on implicit regularization through overparameterization. It reparameterizes the estimate as $\mathbf{X} = (\mathbf{g}^{\odot 2} \mathbf{1}_L) \odot \mathbf{V}$ and optimizes a standard least-squares loss via gradient descent on the factors, promoting row-sparse solutions without explicit regularizers or prior knowledge of sparsity or noise variance. Theoretical results reveal invariant global and row-wise balancedness and a momentum-like dynamics that favor learning rows with larger norms, along with a convergence guarantee to a rank-$K$ trajectory when initialized near the origin. Simulations show the method achieving performance comparable to traditional MMV algorithms while remaining tuning-free, underscoring its practical relevance and bridging implicit-regularization theory with MMV structured recovery.

Abstract

Recovering jointly sparse signals in the multiple measurement vectors (MMV) setting is a fundamental problem in machine learning, but traditional methods like multiple measurement vectors orthogonal matching pursuit (M-OMP) and multiple measurement vectors FOCal Underdetermined System Solver (M-FOCUSS) often require careful parameter tuning or prior knowledge of the sparsity of the signal and/or noise variance. We introduce a novel tuning-free framework that leverages Implicit Regularization (IR) from overparameterization to overcome this limitation. Our approach reparameterizes the estimation matrix into factors that decouple the shared row-support from individual vector entries. We show that the optimization dynamics inherently promote the desired row-sparse structure by applying gradient descent to a standard least-squares objective on these factors. We prove that with a sufficiently small and balanced initialization, the optimization dynamics exhibit a "momentum-like" effect, causing the norms of rows in the true support to grow significantly faster than others. This formally guarantees that the solution trajectory converges towards an idealized row-sparse solution. Additionally, empirical results demonstrate that our approach achieves performance comparable to established methods without requiring any prior information or tuning.

Tuning-Free Structured Sparse Recovery of Multiple Measurement Vectors using Implicit Regularization

TL;DR

The paper tackles the MMV recovery problem with joint row sparsity by introducing a tuning-free approach based on implicit regularization through overparameterization. It reparameterizes the estimate as and optimizes a standard least-squares loss via gradient descent on the factors, promoting row-sparse solutions without explicit regularizers or prior knowledge of sparsity or noise variance. Theoretical results reveal invariant global and row-wise balancedness and a momentum-like dynamics that favor learning rows with larger norms, along with a convergence guarantee to a rank- trajectory when initialized near the origin. Simulations show the method achieving performance comparable to traditional MMV algorithms while remaining tuning-free, underscoring its practical relevance and bridging implicit-regularization theory with MMV structured recovery.

Abstract

Recovering jointly sparse signals in the multiple measurement vectors (MMV) setting is a fundamental problem in machine learning, but traditional methods like multiple measurement vectors orthogonal matching pursuit (M-OMP) and multiple measurement vectors FOCal Underdetermined System Solver (M-FOCUSS) often require careful parameter tuning or prior knowledge of the sparsity of the signal and/or noise variance. We introduce a novel tuning-free framework that leverages Implicit Regularization (IR) from overparameterization to overcome this limitation. Our approach reparameterizes the estimation matrix into factors that decouple the shared row-support from individual vector entries. We show that the optimization dynamics inherently promote the desired row-sparse structure by applying gradient descent to a standard least-squares objective on these factors. We prove that with a sufficiently small and balanced initialization, the optimization dynamics exhibit a "momentum-like" effect, causing the norms of rows in the true support to grow significantly faster than others. This formally guarantees that the solution trajectory converges towards an idealized row-sparse solution. Additionally, empirical results demonstrate that our approach achieves performance comparable to established methods without requiring any prior information or tuning.

Paper Structure

This paper contains 17 sections, 16 theorems, 106 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminaries
  4. IR-MMV
  5. Theoretical Analysis
  6. Simulation
  7. Balancedness and Incremental Learning
  8. Effect of Initialization
  9. Performance with Varying Noise
  10. Conclusion
  11. Gradient Updates
  12. Characterization of the gradient flow
  13. Supporting Lemmas and the Proof of the Main Theorem
  14. Proof of Theorem V.1
  15. Proof of Corollary 1
  16. ...and 2 more sections

Key Result

Lemma 5.1

For the system evolving under continuous gradient flow for the set of update equations eqn:IRMMV_gUpdate, eqn:IRMMV_vUpdate and eqn:IRMMV_xUpdate, derived from the loss function $\mathcal{L}(\mathbf{g}, \mathbf{V})=\Vert \mathbf{Y}-\mathbf{A}(\mathbf{g}^{\odot2}_{}\mathbf{1}_L\mathbf{V})\Vert _{F}^{

Figures (4)

  • Figure 1: (a) Evolution of the norms of the components for rows in the support and non-support sets. (b) Loss versus iterations for different initialization values of $\alpha$.
  • Figure 2: The evolution of coordinate norms during training for different initialization parameters $\alpha_g^{}$.
  • Figure 3: Performance of *irmmv averaged over $20$ trials.
  • Figure 4: Comparison of MNIST digit reconstruction using different methods. The first row shows the original images.

Theorems & Definitions (39)

  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Definition 3.4
  • Lemma 5.1: Balancedness
  • proof : Proof sketch (for detailed proof see \ref{['lemma:MMV_BalancednessApp']})
  • Remark 5.1
  • Lemma 5.2: Dynamics of row norm
  • proof : Proof sketch (for detailed proof see Lemma \ref{['lemma:MMV_gradientFlowCompleteInequalityRowNorm']})
  • Lemma 5.3: Dynamics of row norms under perfect balancedness
  • ...and 29 more