Information-Geometric Decomposition of Generalization Error in Unsupervised Learning

• URL: http://arxiv.org/abs/2604.12340v1
• Date: Tue, 14 Apr 2026 06:23:18 GMT
• Title: Information-Geometric Decomposition of Generalization Error in Unsupervised Learning
• Abstract summary: We decompose the Kullback--Leibler generalization error (GE) of unsupervised learning into three non-negative components: model error, data bias, and variance.<n>The decomposition is exact for any e-flat model class.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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 $ε$-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 $ε$. Although rank-constrained $ε$-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 $λ_{\mathrm{cut}}^{*} = ε$ -- 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 $ε_{*}(α)$, where $α$ is the dimension-to-sample-size ratio. All claims are verified numerically.

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