Recovering jointly sparse signals in the multiple measurement vectors (MMV) setting is a fundamental problem in machine learning, but traditional methods often require careful parameter tuning or prior knowledge of the sparsity of the signal and/or noise variance. We propose a 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 and applies gradient descent to a standard least-squares objective. We prove that with a sufficiently small and balanced initialization, the optimization dynamics exhibit a "momentum-like" effect where the true support grows significantly faster. Leveraging a Lyapunov-based analysis of the gradient flow, we further establish formal guarantees that the solution trajectory converges towards an idealized row-sparse solution. Empirical results demonstrate that our tuning-free approach achieves performance comparable to optimally tuned established methods. Furthermore, our framework significantly outperforms these baselines in scenarios where accurate priors are unavailable to the baselines.

翻译：在多测量向量（MMV）场景下恢复联合稀疏信号是机器学习中的一个基础性问题，但传统方法通常需要精细的参数调整或对信号稀疏度及噪声方差的先验知识。本文提出一种无需调参的框架，利用过参数化产生的隐式正则化（IR）来克服这一局限。该方法将估计矩阵重新参数化为若干因子，从而将共享的行支撑与各向量分量解耦，并对标准最小二乘目标函数应用梯度下降。我们证明，在足够小且平衡的初始化条件下，优化动态会呈现一种"类动量"效应，使得真实支撑集的增长速度显著加快。基于对梯度流的李雅普诺夫分析，我们进一步建立了形式化保证，证明解轨迹会收敛于理想化的行稀疏解。实验结果表明，本方法在无需调参的情况下取得了与经过最优调参的经典方法相当的性能。此外，在先验信息无法准确获取的场景中，本框架显著优于这些基线方法。