矩阵式$\ell_0$约束稀疏非负最小二乘
Matrix-wise $\ell_0$-constrained Sparse Nonnegative Least Squares
摘要：
多右端非负最小二乘问题（MNNLS）出现在依赖于加法线性组合的模型中。特别是，它们是大多数非负矩阵分解算法的核心，并有许多应用。已知非负性约束自然有利于稀疏性，即具有很少非零项的解。然而，进一步增强这种稀疏性通常是有用的，因为它提高了结果的可解释性，并有助于减少噪声，从而导致稀疏MNNLS问题。在本文中，相对于大多数以前的作品，执行稀疏列或行明智的，我们首先介绍了一种新的制定稀疏MNNLS，矩阵明智的稀疏约束。然后，我们提出了一个两步算法来解决这个问题。第一步将稀疏MNNLS划分为子问题，原始问题的每列一个。然后，它使用不同的算法来产生，精确或近似，帕累托前每个子问题，也就是说，产生一组代表重建误差和稀疏性之间的不同权衡的解决方案。第二步在这些Pareto前沿中选择解决方案，以构建使重构误差最小化的稀疏约束矩阵。我们在面部和高光谱图像上进行了实验，结果表明，我们提出的两步方法比列式和全局应用的最先进的稀疏编码算法提供了更准确的结果。
英文摘要：
Nonnegative least squares problems with multiple right-hand sides (MNNLS) arise in models that rely on additive linear combinations. In particular, they are at the core of most nonnegative matrix factorization algorithms and have many applications. The nonnegativity constraint is known to naturally favor sparsity, that is, solutions with few non-zero entries. However, it is often useful to further enhance this sparsity, as it improves the interpretability of the results and helps reducing noise, which leads to the sparse MNNLS problem. In this paper, as opposed to most previous works that enforce sparsity column- or row-wise, we first introduce a novel formulation for sparse MNNLS, with a matrix-wise sparsity constraint. Then, we present a two-step algorithm to tackle this problem. The first step divides sparse MNNLS in subproblems, one per column of the original problem. It then uses different algorithms to produce, either exactly or approximately, a Pareto front for each subproblem, that is, to produce a set of solutions representing different tradeoffs between reconstruction error and sparsity. The second step selects solutions among these Pareto fronts in order to build a sparsity-constrained matrix that minimizes the reconstruction error. We perform experiments on facial and hyperspectral images, and we show that our proposed two-step approach provides more accurate results than state-of-the-art sparse coding heuristics applied both column-wise and globally.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
