Non-negative matrix factorization (NMF) is a problem of decomposing multivariate data into a set of features and their corresponding activations. When applied to experimental data, NMF has to cope with noise, which is often highly correlated. We show that correlated noise can break the Donoho and Stodden separability conditions of a dataset and a regular NMF algorithm will fail to decompose it, even when given freedom to be able to represent the noise as a separate feature. To cope with this issue, we present an algorithm for NMF with a generalized least squares objective function (glsNMF) and derive multiplicative updates for the method together with proving their convergence. The new algorithm successfully recovers the true representation from the noisy data. Robust performance can make glsNMF a valuable tool for analyzing empirical data.

中文翻译：

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相关噪声：它如何破坏 NMF，以及如何处理它。

非负矩阵分解 (NMF) 是将多元数据分解为一组特征及其相应激活的问题。当应用于实验数据时，NMF 必须处理噪声，这通常是高度相关的。我们表明相关噪声可以打破数据集的 Donoho 和 Stodden 可分离性条件，并且常规 NMF 算法将无法分解它，即使给予能够将噪声表示为单独特征的自由。为了解决这个问题，我们提出了一种具有广义最小二乘目标函数 (glsNMF) 的 NMF 算法，并推导出该方法的乘法更新并证明它们的收敛性。新算法成功地从嘈杂的数据中恢复了真实的表示。强大的性能可以使 glsNMF 成为分析经验数据的宝贵工具。