Stochastic complexity of vectors containing cluster structure

The Normalized Maximum Likelihood (NML) codelength, or stochastic complexity, represents a principled criterion for universal coding. While recent coarea-based formulations provided a calculation method for smooth models, this framework collapses for the non-smooth estimators ubiquitous in modern machine learning (e.g., Lasso, Sparse SVMs). In this work, we provide a rigorous framework for computing the NML for regular path-differentiable Lipschitz (PDL) estimators. By applying classical geometric measure theory and bridging the coarea formula with conservative Jacobians, we prove that the stochastic complexity for non-smooth models is well-posed and theoretically consistent with the outputs of modern Automatic Differentiation. To compute this quantity exactly, we introduce the Propose-and-Project Metropolis-Hastings (PDL-PPMH) sampler, a geometric MCMC algorithm capable of traversing the non-differentiable level sets of the maximum likelihood estimator. We theoretically justify its components, including a stochastic tangent space proposal and a provably convergent non-smooth projection solver. We demonstrate the method's robustness by sampling from a high-dimensional Lasso posterior ($P=2000$), while simultaneously quantifying the computational scaling that governs the trade-off between exactness and mixing time. Crucially, we empirically demonstrate that our exact NML criterion provides a highly data-efficient alternative to cross-validation, achieving statistically indistinguishable predictive optima without requiring data splitting. Altogether, our work paves the way for the theoretical analysis of the NML codelength for regular non-smooth models.

We study inference in stochastic block models (SBMs) through the lens of optimal transport (OT). We first establish that maximum likelihood variational inference (MLVI) can be interpreted as a semi-relaxed Gromov-Wasserstein (srGW) projection with entropic regularization. While this formulation yields accurate clustering, the entropic regularization prevents transport plans to be sparse, hindering intrinsic model selection. Consequently, we investigate unregularized srGW estimators, and prove that they consistently recover both the SBM connectivity matrix and latent cluster assignments in the asymptotic regime. However, this asymptotic property does not translate into reliable model selection in finite samples, and calls for additional mechanisms to promote sparsity in the inferred cluster proportions. We empirically show that such a regularized formulation yields estimators that simultaneously recover model parameters and select the number of clusters in a single optimization problem, thereby avoiding costly grid search or heuristic model selection procedures.

The exact computation of the Normalized Maximum Likelihood (NML) codelength for regular non-smooth estimators (e.g., Lasso) has been historically limited by the cubic scaling walls of manifold-constrained projection and volume integration. At each step of the geometric Propose-and-Project Metropolis--Hastings (PPMH) sampler, evaluating the projection operator requires inverting an $(N+k) \times (N+k)$ generalized KKT matrix, while calculating the volume factor requires the determinant of an $(N-k) \times (N-k)$ Gram matrix. This paper presents an exact, mathematically equivalent formulation that bypasses both bottlenecks by utilizing the block Schur complement and Sylvester's determinant identity. We prove that the computational complexity of both operations collapses from $\mathcal{O}(N^3)$ to $\mathcal{O}(k^3 + N^2 k)$ per step. We generalize this reduction to Sparse Support Vector Machines (SVMs), Elastic Net, and Group Lasso. Finally, we provide a rigorous numerical stability analysis and evaluate the sampler's efficiency using the Effective Sample Size (ESS) per second. Our empirical benchmarks on high-dimensional datasets confirm a constant speedup exceeding $14{,}100\times$ while maintaining double-precision numerical equivalence, rendering exact non-smooth NML estimation highly tractable for large-scale statistical inference.