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insight - Dimensionality reduction - # Non-negative matrix factorization on noisy data with negative values

Core Concepts

This paper presents two novel algorithms, Shift-NMF and Nearly-NMF, that can handle noisy data with negative values in non-negative matrix factorization (NMF) while maintaining the non-negativity constraints on the templates and coefficients.

Abstract

The paper starts by introducing the problem of handling negative values in NMF, which commonly arise due to noise in the observed data even when the true underlying signal is strictly positive. Prior NMF methods have not treated negative data in a statistically consistent manner, which becomes problematic for low signal-to-noise data with many negative values.
The paper then presents two new NMF algorithms, Shift-NMF and Nearly-NMF, that can handle both the noisiness of the input data and any introduced negativity. Both algorithms use the negative data space without clipping or masking and recover non-negative signals without any introduced positive offset.
The paper demonstrates the effectiveness of these algorithms numerically on both simple and more realistic examples, and proves that both algorithms have monotonically decreasing update rules. Nearly-NMF is shown to be superior to Shift-NMF in terms of convergence speed while achieving similar final objective values.
The paper concludes by discussing the broader applicability of these methods beyond the astronomical context in which they were developed, such as in NMR spectroscopy, neuroscience, and bioinformatics, where negative data can arise from noise in the measurement process.

Stats

The paper presents the following key metrics and figures:
Simulated quasar spectra dataset with 130,000 training and 70,000 validation samples
11,050 wavelength pixels per spectrum, with 61.1% missing data and 11.4% negative values
Computational scaling analysis showing Nearly-NMF scales linearly with number of quasars, templates, and data pixels

Quotes

"Even if all data values are positive, the noise values intrinsic in data collection mean that standard NMF as presented in (1) will perform suboptimally and will attempt to fit noise values when generating template and coefficient matrices."
"While the standard NMF update rules cannot correctly handle negative values of the data, it is possible to consider the negative components of the data in the fit while still maintaining the non-negativity constraint on the coefficients and templates."

Key Insights Distilled From

by Dylan Green,... at **arxiv.org** 04-01-2024

Deeper Inquiries

To extend the Shift-NMF and Nearly-NMF algorithms to handle constraints beyond non-negativity, such as sparsity or smoothness, in the template or coefficient matrices, additional terms can be introduced into the objective function. For example, to enforce sparsity, a regularization term like the L1 norm penalty can be added to the objective function. This penalty encourages the coefficients or templates to have many zero entries, promoting sparsity. Similarly, to enforce smoothness, a penalty term that penalizes large changes between neighboring elements in the matrices can be included. By incorporating these additional constraints into the objective function, the algorithms can be adapted to learn sparse or smooth representations while handling noisy data with negative values.

One potential limitation of the Shift-NMF and Nearly-NMF approaches compared to other techniques for handling noisy data, such as robust PCA or Bayesian matrix factorization methods, is their reliance on the Euclidean norm as the objective function. While the Euclidean norm is commonly used and effective in many cases, it may not always capture the underlying structure of the data optimally, especially in scenarios where the noise is non-Gaussian or the data distribution is non-linear. Robust PCA, for example, explicitly models the noise in the data and separates it from the underlying signal, making it more robust to outliers and non-Gaussian noise. Bayesian matrix factorization methods, on the other hand, provide a probabilistic framework that can capture uncertainty in the data and model complex relationships more flexibly. Additionally, these methods may offer better interpretability and generalization to unseen data compared to the Shift-NMF and Nearly-NMF approaches.

The algorithms could be adapted to work with other matrix factorization objectives beyond the Euclidean norm, such as the Kullback-Leibler (KL) divergence commonly used for count data, by modifying the update rules to minimize the KL divergence between the reconstructed matrix and the original data matrix. The KL divergence measures the difference between two probability distributions, making it suitable for count data where the values are non-negative integers. By incorporating the KL divergence into the objective function and deriving update rules that minimize this divergence, the algorithms can be tailored to handle count data and provide a more appropriate fit for such datasets.

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