Nonnegative matrix factorization (NMF) approximates a nonnegative matrix, $X$, by the product of two nonnegative factors, $WH$, where $W$ has $r$ columns and $H$ has $r$ rows. In this paper, we consider NMF using the component-wise L1 norm as the error measure (L1-NMF), which is suited for data corrupted by heavy-tailed noise, such as Laplace noise or salt and pepper noise, or in the presence of outliers. Our first contribution is an NP-hardness proof for L1-NMF, even when $r=1$, in contrast to the standard NMF that uses least squares. Our second contribution is to show that L1-NMF strongly enforces sparsity in the factors for sparse input matrices, thereby favoring interpretability. However, if the data is affected by false zeros, too sparse solutions might degrade the model. Our third contribution is a new, more general, L1-NMF model for sparse data, dubbed weighted L1-NMF (wL1-NMF), where the sparsity of the factorization is controlled by adding a penalization parameter to the entries of $WH$ associated with zeros in the data. The fourth contribution is a new coordinate descent (CD) approach for wL1-NMF, denoted as sparse CD (sCD), where each subproblem is solved by a weighted median algorithm. To the best of our knowledge, sCD is the first algorithm for L1-NMF whose complexity scales with the number of nonzero entries in the data, making it efficient in handling large-scale, sparse data. We perform extensive numerical experiments on synthetic and real-world data to show the effectiveness of our new proposed model (wL1-NMF) and algorithm (sCD).

翻译：非负矩阵分解（NMF）将非负矩阵$X$近似表示为两个非负因子$WH$的乘积，其中$W$有$r$列、$H$有$r$行。本文研究采用分量L1范数作为误差度量的NMF（L1-NMF），该方法适用于受重尾噪声（如拉普拉斯噪声、椒盐噪声）干扰或存在异常值的数据。我们的第一个贡献是证明L1-NMF的NP难性，即便在$r=1$时也成立——这与采用最小二乘法的标准NMF形成对比。第二个贡献是表明对于稀疏输入矩阵，L1-NMF会强力促使因子稀疏化，从而提升可解释性。然而，若数据受到零值误判影响，过于稀疏的解可能会降低模型性能。第三个贡献是针对稀疏数据提出一种更通用的新型L1-NMF模型，称为加权L1-NMF（wL1-NMF），其通过对$WH$中与数据零值对应的元素添加惩罚参数来控制分解的稀疏度。第四个贡献是为wL1-NMF设计了一种新的坐标下降（CD）方法——稀疏坐标下降（sCD），其中每个子问题通过加权中值算法求解。据我们所知，sCD是首个时间复杂度与数据非零元素数量成比例的L1-NMF算法，使其能高效处理大规模稀疏数据。我们在合成和真实数据上进行了大量数值实验，验证了所提新模型（wL1-NMF）与算法（sCD）的有效性。