Lakshmi Jayalal

Lakshmi Jayalal, Sheetal Kalyani

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.

Lakshmi Jayalal, Gokularam Muthukrishnan, Sheetal Kalyani

The performance of the standard Online Robust Principal Component Analysis (OR-PCA) technique depends on the optimum tuning of the explicit regularizers and this tuning is dataset sensitive. We aim to remove the dependency on these tuning parameters by using implicit regularization. We propose to use the implicit regularization effect of various modified gradient descents to make OR-PCA tuning free. Our method incorporates three different versions of modified gradient descent that separately but naturally encourage sparsity and low-rank structures in the data. The proposed method performs comparable or better than the tuned OR-PCA for both simulated and real-world datasets. Tuning-free ORPCA makes it more scalable for large datasets since we do not require dataset-dependent parameter tuning.