The aim of this study is to provide a foundation to understand the
relationship between non-negative matrix factorization (NMF) and non-negative
autoencoders enabling proper interpretation and understanding of
autoencoder-based alternatives to NMF. Since its introduction, NMF has been a
popular tool for extracting interpretable, low-dimensional representations of
high-dimensional data. However, recently, several studies have proposed to
replace NMF with autoencoders. This increasing popularity of autoencoders
warrants an investigation on whether this replacement is in general valid and
reasonable. Moreover, the exact relationship between non-negative autoencoders
and NMF has not been thoroughly explored. Thus, a main aim of this study is to
investigate in detail the relationship between non-negative autoencoders and
NMF. We find that the connection between the two models can be established
through convex NMF, which is a restricted case of NMF. In particular, convex
NMF is a special case of an autoencoder. The performance of NMF and
autoencoders is compared within the context of extraction of mutational
signatures from cancer genomics data. We find that the reconstructions based on
NMF are more accurate compared to autoencoders, while the signatures extracted
using both methods show comparable consistencies and values when externally
validated. These findings suggest that the non-negative autoencoders
investigated in this article do not provide an improvement of NMF in the field
of mutational signature extraction.

通过对非负矩阵因式分解 (NMF) 和非负自动编码器之间的关系进行详细探究，发现这两种模型之间的连接可以通过凸 NMF 建立，该模型是自动编码器的特例。在从癌症基因组数据中提取突变特征的背景下，将 NMF 和自动编码器的性能进行比较，结果显示 NMF 基于重构的准确性优于自动编码器，而使用两种方法提取的特征在外部验证时表现出可比性和一致性，这表明本文中研究的非负自动编码器在突变特征提取领域并没有改进 NMF 的效果。

自编码器与非负矩阵分解的关系及其在突变特征提取中的应用

On the Relation Between Autoencoders and Non-negative Matrix  Factorization, and Their Application for Mutational Signature Extraction

Non-negative matrix factorization (NMF) is a fundamental matrix decomposition
technique that is used primarily for dimensionality reduction and is increasing
in popularity in the biological domain. Although finding a unique NMF is
generally not possible, there are various iterative algorithms for NMF
optimization that converge to locally optimal solutions. Such techniques can
also serve as a starting point for deep learning methods that unroll the
algorithmic iterations into layers of a deep network. Here we develop unfolded
deep networks for NMF and several regularized variants in both a supervised and
an unsupervised setting. We apply our method to various mutation data sets to
reconstruct their underlying mutational signatures and their exposures. We
demonstrate the increased accuracy of our approach over standard formulations
in analyzing simulated and real mutation data.

通过发展 NMF 及其正则化变体的展开深度网络，我们在监督和无监督的情况下将其应用于不同的突变数据集以重建其基础突变特征以及暴露情况，实验证明与标准公式相比，我们的方法在分析模拟和真实突变数据方面具有更高的准确性。