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极小化问题。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。基于新的下降引理，我们提出了扩展先前硬阈值算法的最优性条件与算法框架。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，我们重点研究了加权$\ell_2$范数情形，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，以验证理论结果并展示该框架带来的性能提升。