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ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES.

ABSTRACT: This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix M⋆∈ℝn×n , yet only a randomly perturbed version M is observed. The noise matrix M - M ^⋆ is composed of independent (but not necessarily homoscedastic) entries and is, therefore, not symmetric in general. This might arise if, for example, we have two independent samples for each entry of M ^⋆ and arrange them in an asymmetric fashion. The aim is to estimate the leading eigenvalue and the leading eigenvector of M ^⋆. We demonstrate that the leading eigenvalue of the data matrix M can be O(n) times more accurate (up to some log factor) than its (unadjusted) leading singular value of M in eigenvalue estimation. Moreover, the eigen-decomposition approach is fully adaptive to heteroscedasticity of noise, without the need of any prior knowledge about the noise distributions. In a nutshell, this curious phenomenon arises since the statistical asymmetry automatically mitigates the bias of the eigenvalue approach, thus eliminating the need of careful bias correction. Additionally, we develop appealing non-asymptotic eigenvector perturbation bounds; in particular, we are able to bound the perterbation of any linear function of the leading eigenvector of M (e.g. entrywise eigenvector perturbation). We also provide partial theory for the more general rank-r case. The takeaway message is this: arranging the data samples in an asymmetric manner and performing eigen-decomposition could sometimes be quite beneficial.

SUBMITTER: Chen Y

PROVIDER: S-EPMC8300484 | biostudies-literature |

REPOSITORIES: biostudies-literature

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Project description:This article investigates a form of rank deficiency in phenotypic covariance matrices derived from geometric morphometric data, and its impact on measures of phenotypic integration. We first define a type of rank deficiency based on information theory then demonstrate that this deficiency impairs the performance of phenotypic integration metrics in a model system. Lastly, we propose methods to treat for this information rank deficiency. Our first goal is to establish how the rank of a typical geometric morphometric covariance matrix relates to the information entropy of its eigenvalue spectrum. This requires clear definitions of matrix rank, of which we define three: the full matrix rank (equal to the number of input variables), the mathematical rank (the number of nonzero eigenvalues), and the information rank or "effective rank" (equal to the number of nonredundant eigenvalues). We demonstrate that effective rank deficiency arises from a combination of methodological factors-Generalized Procrustes analysis, use of the correlation matrix, and insufficient sample size-as well as phenotypic covariance. Secondly, we use dire wolf jaws to document how differences in effective rank deficiency bias two metrics used to measure phenotypic integration. The eigenvalue variance characterizes the integration change incorrectly, and the standardized generalized variance lacks the sensitivity needed to detect subtle changes in integration. Both metrics are impacted by the inclusion of many small, but nonzero, eigenvalues arising from a lack of information in the covariance matrix, a problem that usually becomes more pronounced as the number of landmarks increases. We propose a new metric for phenotypic integration that combines the standardized generalized variance with information entropy. This metric is equivalent to the standardized generalized variance but calculated only from those eigenvalues that carry nonredundant information. It is the standardized generalized variance scaled to the effective rank of the eigenvalue spectrum. We demonstrate that this metric successfully detects the shift of integration in our dire wolf sample. Our third goal is to generalize the new metric to compare data sets with different sample sizes and numbers of variables. We develop a standardization for matrix information based on data permutation then demonstrate that Smilodon jaws are more integrated than dire wolf jaws. Finally, we describe how our information entropy-based measure allows phenotypic integration to be compared in dense semilandmark data sets without bias, allowing characterization of the information content of any given shape, a quantity we term "latent dispersion". [Canis dirus; Dire wolf; effective dispersion; effective rank; geometric morphometrics; information entropy; latent dispersion; modularity and integration; phenotypic integration; relative dispersion.].

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ASYMMETRY HELPS: EIGENVALUE AND EIGENVECTOR ANALYSES OF ASYMMETRICALLY PERTURBED LOW-RANK MATRICES.

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