Hierarchical alternating nonlinear least squares for nonnegative matrix factorization using rational functions

Abstract

Nonnegative Matrix Factorization (NMF) models are widely used to recover linearly mixed nonnegative data. When the data is made of samplings of continuous signals, the factors in NMF can be constrained to be samples of nonnegative rational functions, which allow fairly general models; this is referred to as NMF using rational functions (R-NMF). We first show that, under mild assumptions, R-NMF has an essentially unique factorization unlike NMF, which is crucial in applications where ground-truth factors need to be recovered such as blind source separation problems. Then we present different approaches to solve R-NMF: the R-HANLS, R-ANLS and R-NLS methods. From our tests, no method significantly outperforms the others, and a trade-off should be done between time and accuracy. Indeed, R-HANLS is fast and accurate for large problems, while R-ANLS is more accurate, but also more resources demanding, both in time and memory. R-NLS is very accurate but only for small problems. Moreover, we show that R-NMF outperforms NMF in various tasks including the recovery of semi-synthetic continuous signals, and a classification problem of real hyperspectral signals.

An ocean exploration model based on geometric problems and least squares multibeam detection techniques

“…This measurement begins as the ship sets off from the center of the sea area, and continues as the ship travels in a direction diverging from the angle between lines. This methodology thus enables the estimation of coverage width in given situations [1][2][3]. At the same time, the paper discusses how to design the coverage width and overlap rate of the transmission line reasonably by using the sea depth data measured by a single beam in certain sea area several years ago, so as to improve the data quality and survey efficiency by using the least squares [4][5][6][7] data, to provide help for the measurement wiring of multi-beam survey ship.…”

Functional Decomposition of Retinal Ganglion Cell Receptive Fields

“…Despite its old age and the seemingly daunting growth of deep learning, the field of non-negative matrix factorization (NMF) (D. D. is still under active investigation with numerous publications in recent years (for instance Ang and Gillis 2019;Hautecoeur et al 2022; reporting faster or more accurate implementations. Because certain aspects, like its non-convex nature or the convergence to a stable solution , have not been fully elucidated, NMF remains interesting to mathematicians and computer scientists from a theoretical perspective.…”