Sparse regularization plays a central role in solving inverse problems arising from incomplete or corrupted measurements. Different regularizers correspond to different prior assumptions about the structure of the unknown signal, and reconstruction performance depends on how well these priors match the intrinsic sparsity of the data. This work investigates the effect of sparsity priors in inverse problems by comparing conventional L1 regularization with the Variational Garrote (VG), a probabilistic method that approximates L0 sparsity through variational binary gating variables. A unified experimental framework is constructed across multiple reconstruction tasks including signal resampling, signal denoising, and sparse-view computed tomography. To enable consistent comparison across models with different parameterizations, regularization strength is swept across wide ranges and reconstruction behavior is analyzed through train-generalization error curves. Experiments reveal characteristic bias-variance tradeoff patterns across tasks and demonstrate that VG frequently achieves lower minimum generalization error and improved stability in strongly underdetermined regimes where accurate support recovery is critical. These results suggest that sparsity priors closer to spike-and-slab structure can provide advantages when the underlying coefficient distribution is strongly sparse. The study highlights the importance of prior-data alignment in sparse inverse problems and provides empirical insights into the behavior of variational L0-type methods across different information bottlenecks.

翻译：稀疏正则化在解决由不完整或损坏测量产生的逆问题中起着核心作用。不同的正则化器对应于对未知信号结构的不同先验假设，而重建性能取决于这些先验与数据内在稀疏性的匹配程度。本研究通过比较传统的L1正则化与变分门控方法，探究了稀疏先验在逆问题中的效果。变分门控是一种概率方法，通过变分二元门控变量来逼近L0稀疏性。我们在多个重建任务上构建了统一的实验框架，包括信号重采样、信号去噪和稀疏视图计算机断层扫描。为了使不同参数化模型之间能够进行一致比较，我们在宽范围内扫描正则化强度，并通过训练-泛化误差曲线分析重建行为。实验揭示了跨任务的特征性偏差-方差权衡模式，并证明在准确支撑集恢复至关重要的强欠定区域中，变分门控方法经常能够实现更低的最小泛化误差和更高的稳定性。这些结果表明，当底层系数分布具有强稀疏性时，更接近尖峰-平板结构的稀疏先验能够提供优势。本研究强调了先验-数据对齐在稀疏逆问题中的重要性，并对变分L0型方法在不同信息瓶颈下的行为提供了实证见解。