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What is Learnable in Valiant's Theory of the Learnable? 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Deviance-style normalization for jointly overdispersed counts
[Submitted on 24 Jun 2026] · 2026-06-25 · via math.ST updates on arXiv.org

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Abstract:We introduce a Dirichlet--multinomial (DM) deviance residualization for sparse, jointly overdispersed count matrices, the regime that dominates sequencing-based biochemical assays. The DM null treats each sample's count vector as a fixed-total composition with a single scalar concentration $\alpha_0$ governing overdispersion, and arises exactly by conditioning independent negative-binomial feature counts on the observed sample total -- making the DM the joint conditional analogue of standard feature-wise overdispersed count models. The resulting transform preserves exact sparsity, evaluates in constant time per nonzero entry, agrees with multinomial residuals on singleton counts, shrinks repeated-count residuals according to the overdispersion the null tolerates, and recovers the multinomial residual as $\alpha_0\to\infty$. The same fixed-dispersion comparison principle extends to ordered and tree-structured features via the generalized DM and the Dirichlet-tree multinomial, giving a single residual family that subsumes joint and feature-wise count nulls under a common compositional logic and is computationally lightweight enough to drop into existing sparse pipelines.

Submission history

From: Akshay Balsubramani [view email]
[v1] Wed, 24 Jun 2026 17:36:07 UTC (3,593 KB)

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