Information-Geometric Decomposition of Generalization Error in Unsupervised Learning

arXiv:2604.12340v1 Announce Type: cross Abstract: We decompose the Kullback--Leibler generalization error (GE) -- the expected KL divergence from the data distribution to the trained model -- of unsupervised learning into three non-negative components: model error, data bias, and variance. The decomposition is exact for any e-flat model class and follows from two identities of information geometry: the generalized Pythagorean theorem and a dual e-mixture variance identity. As an analytically tractable demonstration, we apply the framework to epsilon-PCA, a regularized principal component analysis in which the empirical covariance is truncated at rank N_K and discarded directions are pinned at a fixed noise floor epsilon. Although rank-constrained epsilon-PCA is not itself e-flat, it admits a technical reformulation with the same total GE on isotropic Gaussian data, under which each component of the decomposition takes closed form. The optimal rank emerges as the cutoff lambda_{cut}^{} = epsilon -- the model retains exactly those empirical eigenvalues exceeding the noise floor -- with the cutoff reflecting a marginal-rate balance between model-error gain and data-bias cost. A boundary comparison further yields a three-regime phase diagram -- retain-all, interior, and collapse -- separated by the lower Marchenko--Pastur edge and an analytically computable collapse threshold epsilon_{}(alpha), where alpha is the dimension-to-sample-size ratio. All claims are verified numerically.

Source: arXiv cs.LG | 2026-04-15

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