无需调参即可高效恢复多测量向量的稀疏信号
Tuning-Free Structured Sparse Recovery of Multiple Measurement Vectors using Implicit Regularization
- 通过重参数化与梯度下降实现隐式正则化
- 小而均衡初始化下支持集增长更快，收敛至理想稀疏解
- 在缺乏先验信息时优于需调参的传统方法
在多测量向量（MMV）场景中联合恢复稀疏信号是机器学习中的基础问题，但传统方法常需精细调参或已知信号稀疏度与噪声方差。本文提出一种无需调参的框架，利用过参数化带来的隐式正则化（IR）克服此限制。将估计矩阵重参数化为解耦共享行支撑与各向量分量的因子，并对标准最小二乘目标应用梯度下降。理论证明：当初始化足够小且均衡时，优化动态呈现类似动量的效果，真实支撑集增长显著更快。基于李雅普诺夫分析的梯度流研究进一步建立形式化保证，表明解轨迹收敛至理想行稀疏解。实验表明，该无调参方法性能可媲美最优调参的经典方法；尤其在缺乏准确先验的场景下，显著优于依赖先验的基线方法。
原文摘要 · Abstract (English)
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.
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