Rank Suggestion in Non-negative Matrix Factorization: Residual Sensitivity to Initial Conditions (RSIC)
Abstract: Determining the appropriate rank in Non-negative Matrix Factorization (NMF) is a critical challenge that often requires extensive parameter tuning and domain-specific knowledge. Traditional methods for rank determination focus on identifying a single optimal rank, which may not capture the complex structure inherent in real-world datasets. In this study, we introduce a novel approach called Residual Sensitivity to Intial Conditions (RSIC) that suggests potentially multiple ranks of interest by analyzing the sensitivity of the relative residuals (e.g., relative reconstruction error) to different initializations. By computing the Mean Coordinatewise Interquartile Range (MCI) of the residuals across multiple random initializations, our method identifies regions where the NMF solutions are less sensitive to initial conditions and potentially more meaningful. We evaluate RSIC on a diverse set of datasets, including single-cell gene expression data, image data, and text data, and compare it against current state-of-the-art rank determination methods. Our experiments demonstrate that RSIC effectively identifies relevant ranks consistent with the underlying structure of the data, outperforming traditional methods in scenarios where they are computationally infeasible or less accurate. This approach provides a more scalable and generalizable solution for rank determination in NMF that does not rely on domain-specific knowledge or assumptions.
Submission Length: Long submission (more than 12 pages of main content)
Changes Since Last Submission: We added in the author information and removed the coloring which had been used to indicate changes made.
We also made a few trivial typographic changes: ensuring consistent punctuation around "e.g.", removing a space before a period, adding a missing period at the end of a paragraph, adding "is" in a missing spot, and addressing minor figure/table/footnote placement issues due to adding author information.
Code: https://doi.org/10.5281/zenodo.15161345
Assigned Action Editor: ~Patrick_Flaherty1
Submission Number: 3518
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