Assessing the similarity of real matrices with arbitrary shape (2403.17687v3)
Abstract: Assessing the similarity of matrices is valuable for analyzing the extent to which data sets exhibit common features in tasks such as data clustering, dimensionality reduction, pattern recognition, group comparison, and graph analysis. Methods proposed for comparing vectors, such as cosine similarity, can be readily generalized to matrices. However, this approach usually neglects the inherent two-dimensional structure of matrices. Here, we propose singular angle similarity (SAS), a measure for evaluating the structural similarity between two arbitrary, real matrices of the same shape based on singular value decomposition. After introducing the measure, we compare SAS with standard measures for matrix comparison and show that only SAS captures the two-dimensional structure of matrices. Further, we characterize the behavior of SAS in the presence of noise and as a function of matrix dimensionality. Finally, we apply SAS to two use cases: square non-symmetric matrices of probabilistic network connectivity, and non-square matrices representing neural brain activity. For synthetic data of network connectivity, SAS matches intuitive expectations and allows for a robust assessment of similarities and differences. For experimental data of brain activity, SAS captures differences in the structure of high-dimensional responses to different stimuli. We conclude that SAS is a suitable measure for quantifying the shared structure of matrices with arbitrary shape.
- Mark EJ Newman “The structure and function of complex networks” In SIAM Review 45.2, 2003, pp. 167–256 DOI: 10.1137/S003614450342480
- Brittny Calsbeek and Charles J Goodnight “Empirical comparison of G matrix test statistics: finding biologically relevant change” In Evolution 63.10 Blackwell Publishing Inc Malden, USA, 2009, pp. 2627–2635
- Ricard V. Sol and M. Montoya “Complexity and Fragility in Ecological Networks” In Proceedings of the Royal Society of London. Series B: Biological Sciences 268.1480 The Royal Society, 2001, pp. 2039–2045 DOI: 10.1098/rspb.2001.1767
- Carlo Piccardi, Lisa Calatroni and Fabio Bertoni “Clustering Financial Time Series by Network Community Analysis” In International Journal of Modern Physics C 22.01 World Scientific Publishing Co., 2011, pp. 35–50 DOI: 10.1142/S012918311101604X
- Peter H Schönemann “A generalized solution of the orthogonal procrustes problem” In Psychometrika 31.1 Springer, 1966, pp. 1–10
- “A unifying tool for linear multivariate statistical methods: the RV-coefficient” In Journal of the Royal Statistical Society Series C: Applied Statistics 25.3 Oxford University Press, 1976, pp. 257–265
- Harold Hotelling “Relations Between Two Sets Of Variates” In Biometrika 28.3-4, 1936, pp. 321–377 DOI: 10.1093/biomet/28.3-4.321
- Gene H Golub and Hongyuan Zha “The canonical correlations of matrix pairs and their numerical computation” In Linear Algebra for Signal Processing 69, The IMA Volumes in Mathematics and its Applications Springer, 1995 DOI: https://doi.org/10.1007/978-1-4612-4228-4˙3
- Robin Gutzen, Sonja Grün and Michael Denker “Evaluating the statistical similarity of neural network activity and connectivity via eigenvector angles” In BioSystems 223 Elsevier, 2023, pp. 104813 DOI: https://doi.org/10.1016/j.biosystems.2022.104813
- Lloyd N Trefethen and David Bau “Numerical linear algebra” Siam, 2022
- “On random graphs” In Publicationes Mathematicae 6, 1959, pp. 290–297
- Juyong Park and Mark EJ Newman “Statistical mechanics of networks” In Physical Review E 70.6 APS, 2004, pp. 066117
- “The Size of the Largest Strongly Connected Component of a Random Digraph with a Given Degree Sequence” In Combinatorics, Probability and Computing 13.3 Cambridge University Press, 2004, pp. 319–337 DOI: 10.1017/S096354830400611X
- D. J. Watts and S. H. Strogatz “Collective dynamics of small-world networks” In Nature 393, 1998, pp. 440–442 DOI: 10.1038/30918
- “Statistical mechanics of complex networks” In Reviews of Modern Physics 74, 2002, pp. 47–97
- “1024-Channel Electrophysiological Recordings in Macaque V1 and V4 during Resting State” In Scientific Data 9.1, 2022, pp. 77 DOI: 10.1038/s41597-022-01180-1
- Hans Supèr and Pieter R. Roelfsema “Chronic multiunit recordings in behaving animals: Advantages and limitations” In Progress in Brain Research 147.SPEC. ISS. Elsevier, 2005, pp. 263–282 DOI: 10.1016/S0079-6123(04)47020-4
- “Neural manifolds in V1 change with top-down signals from V4 targeting the foveal region” In BioRxiv Cold Spring Harbor Laboratory, 2023, pp. 2023–06 DOI: https://doi.org/10.1101/2023.06.14.544966
- Vladimir Alexandrovich Marchenko and Leonid Andreevich Pastur “Distribution of eigenvalues for some sets of random matrices” In Matematicheskii Sbornik 114.4 Russian Academy of Sciences, Steklov Mathematical Institute of Russian …, 1967, pp. 507–536
- Jacob Cohen “Statistical Power Analysis for the The Behavioral Sciences” L. Erlbaum Associates, 1988, pp. 567
- Student “The probable error of a mean” In Biometrika JSTOR, 1908, pp. 1–25
- Bernard L Welch “The generalization of Student’s problem when several different population varlances are involved” In Biometrika 34.1-2 Oxford University Press, 1947, pp. 28–35 DOI: https://doi.org/10.1093/biomet/34.1-2.28
- T Tony Cai, Jianqing Fan and Tiefeng Jiang “Distributions of angles in random packing on spheres” In Journal of Machine Learning Research 14, 2013, pp. 1837–1864 DOI: 10.1098/rspb.2001.1767
- Lev Davidovich Landau and Evgenii Mikhailovich Lifshitz “Quantum mechanics: non-relativistic theory” Elsevier, 2013
- “Numerische Mathematik 1: Eine algorithmisch orientierte Einführung” Walter de Gruyter, 2008 DOI: 10.1515/9783110203554
- “Deoxyglucose analysis of retinotopic organization in primate striate cortex” In Science 218.4575 American Association for the Advancement of Science, 1982, pp. 902–904 DOI: 10.1126/science.713498
- D. H. Hubel and T. N. Wiesel “Receptive fields of single neurones in the cat’s striate cortex” In Journal of Physiology 148, 1959, pp. 574–591
- Russell L De Valois, Duane G Albrecht and Lisa G Thorell “Spatial frequency selectivity of cells in macaque visual cortex” In Vision Research 22.5 Elsevier, 1982, pp. 545–559
- Klaus Scharnhorst “Angles in complex vector spaces” In Acta Applicandae Mathematicae 69 Springer, 2001, pp. 95–103 DOI: 10.1023/A:1012692601098
- Clive WJ Granger “Investigating causal relations by econometric models and cross-spectral methods” In Econometrica: journal of the Econometric Society JSTOR, 1969, pp. 424–438 DOI: https://doi.org/10.2307/1912791
- Thomas Schreiber “Measuring information transfer” In Physical Review Letters 85.2 APS, 2000, pp. 461
- “Reproducible Neural Network Simulations: Statistical Methods for Model Validation on the Level of Network Activity Data” In Frontiers in Neuroinformatics 12, 2018, pp. 90 DOI: 10.3389/fninf.2018.00090