This paper considers the minimization of a continuously differentiable function over a cardinality constraint. We focus on smooth and relatively smooth functions. These smoothness criteria result in new descent lemmas. Based on the new descent lemmas, novel optimality conditions and algorithms are developed, which extend the previously proposed hard-thresholding algorithms. We give a theoretical analysis of these algorithms and extend previous results on properties of iterative hard thresholding-like algorithms. In particular, we focus on the weighted $\ell_2$ norm, which requires efficient solution of convex subproblems. We apply our algorithms to compressed sensing problems to demonstrate the theoretical findings and the enhancements achieved through the proposed framework.

翻译：本文研究在基数约束下连续可微函数的最小化问题。我们聚焦于光滑函数与相对光滑函数。这些光滑性准则导出了新的下降引理。基于这些新的下降引理，我们提出了新颖的最优性条件与算法，扩展了先前提出的硬阈值算法。我们对这些算法进行了理论分析，并扩展了关于类迭代硬阈值算法性质的已有结论。特别地，我们重点关注加权$\ell_2$范数，这需要高效求解凸子问题。我们将所提算法应用于压缩感知问题，以验证理论发现并展示通过所提框架实现的性能提升。