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.
翻译:多个右侧( MNNNLS) 的非负最小方问题在依赖累进线性组合的模型中出现。 特别是, 这些模型是大多数非负矩阵因子化算法的核心, 并且有许多应用程序。 非惯性限制是自然偏向偏斜性, 也就是说, 解决方法只有很少的非零条目。 但是, 通常可以进一步加强这种宽度, 因为它可以改善结果的可解释性, 并有助于减少噪音, 这导致了稀疏的 MNNNLS 问题。 与大多数以前执行松散性柱或行的工程相比, 我们首先对稀散的 MNNNNLS 进行新颖的配方, 并带有母体性限制。 然后, 我们提出一个两步法的配方, 以最小化的模型形式, 以构建更精确或大致的每列一个列。 我们两个步骤的解算法为每个子题的PAretofront, 也就是, 代表我们两个步骤的折叠性解决方案, 既在重建的轨道上, 也以最小化的路径来显示更细化的路径, 在重建中, 的路径上, 重建中, 显示, 我们的路径上的, 重建中, 显示, 在重建中, 和神经的, 的, 的, 我们的, 的, 我们的, 将这些矩阵的, 在重建的, 进行两个步骤的, 进行两个步骤, 在重建中, 的, 的, 在重建, 的, 的, 的, 的, 我们的, 的, 进行更精确的, 的, 的, 进行更精确的, 进行更精确的, 和 进行更精确的, 进行。