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The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.
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Analysis of Infrared and Raman Imaging Data Using Alternating and Modified Alternating Least Squares
Modified alternating least squares (MALS) outperforms alternating least squares (ALS) in the analysis of infrared and Raman image spectral data. MALS offers superior stability thanks to ridge regression and a substantial speed advantage due to the kernel nature of the algorithm, reducing computational overhead. MALS excels in resolving basis vectors even in low signal-to-noise, nearly collinear data, whereas ALS often falls short. For spectroscopic imaging, both MALS and other ALS methods rely on spatial resolution between sample components, as low spatial resolution leads to increased mixing of components. Spectroscopic imaging combines spectroscopy and digital imaging to extract chemical composition. Multivariate curve resolution (MCR)’s foundation in ALS regression makes it a vital tool for this analysis, enabling a comprehensive examination of complex spectroscopic images. This tutorial delves into the mathematical techniques necessary for extracting chemical insights from infrared and Raman spectroscopic images. While this discussion focuses on two-dimensional spatial data, the methodology can be extended to three-dimensional data.
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- Award ID(s):
- 2003867
- PAR ID:
- 10589738
- Editor(s):
- Walsh, David
- Publisher / Repository:
- MJH Associates.
- Date Published:
- Journal Name:
- Spectroscopy
- Volume:
- 38
- Issue:
- s11
- ISSN:
- 0887-6703
- Page Range / eLocation ID:
- 22 to 25
- Subject(s) / Keyword(s):
- multivariate curve resolution, image analysis, modified alternating least squares, alternating least squares
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.56530/spectroscopy.ai1281l4
