On the Computational and Statistical Efficiency of the Empirical Maximum Entropy on the Mean Method

The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the empirical MEM method. We establish a parametric convergence rate of $O(n^{-1/2})$ in expectation for empirical MEM, improving upon the previously established $O(n^{-1/4})$ guarantee by King-Roskamp et al. (2026). Our proof is based on a novel stability analysis of the primal and dual optimization problems under perturbations of the underlying probability measure, relying only on foundational tools from convex analysis and probability. We further show that the MEM dual problem admits a reformulation as an expected risk minimization problem, thereby placing MEM within the modern framework of stochastic optimization and enabling scalable stochastic gradient algorithms for large-scale inverse problems. Together, these results place empirical MEM as a statistically and computationally efficient methodology for data-driven inverse problems.

Soft-Constrained Optimization of Latent Space in Variational Autoencoders

The usefulness of a variational autoencoder (VAE) depends on two properties of its latent space that are hard to obtain together: high encoding capacity in the individual latent variables, and a low-dimensional, disentangled organization of those variables. Weakening the Kullback-Leibler regularization raises capacity but degrades disentanglement, while strengthening it prunes latent variables away entirely. We formulate VAE training as a soft-constrained optimization problem that addresses both. First, we impose an entropy-based constraint (EC) on individual latent variables, showing that the entropy of a latent code upper-bounds the mutual information it carries about the generative factors of the data. Second, we propose a weight-filter method that exploits the slack of the soft constraint to prune low-entropy dimensions during downstream training. On dSprites, the EC raises the aggregate latent-variable activation score by 43-62% over a vanilla VAE, attains the highest FactorVAE score among the \b{eta} \b{eta}-VAE variants (0.891 vs 0.847), and lowers reconstruction error by up to 38%. On MNIST, the weight filter reduces the latent dimensionality supplied to a downstream classifier from ten to two while holding accuracy above 90%, converging in 37% fewer epochs than the same procedure without the EC. We also find that low-entropy discrete factors tend to merge into a single latent variable, whereas high-entropy continuous factors are distributed across several.

Self-Balancing Sequential Sampling: Fast Convergence with Controlled Predictability

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

Information Gap and Feasibility-Aware Inference in Binomial Logistic Mixtures

This paper studies the information gap between mixture detection and label recovery in binomial logistic mixtures. Standard likelihood-based criteria such as the Bayesian information criterion (BIC) can detect the presence of two components, but this does not guarantee that the corresponding labels are recoverable. We show that this gap is intrinsic to binomial logistic mixtures with a fixed number of trials: observed-data evidence for mixture structure and per-observation information for label recovery have different local orders in the component separation, and only the former accumulates with the sample size. As a result, there exists a detectable-but-unrecoverable regime in which BIC selects two components while the posterior labels remain essentially uninformative. To address this issue, we propose two feasibility-aware inference procedures: a recoverability-aware BIC with a posterior-entropy penalty and an entropy-regularized estimator that mitigates the tendency of the maximum likelihood estimator to produce overly separated components and overly concentrated posterior responsibilities. Numerical experiments confirm the predicted gap and demonstrate that the proposed methods avoid misleading component selections and improve the calibration of posterior label probabilities.