Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.

Optimal transport is widely used to learn distributions, enforce distributional constraints, and model uncertainty. In applications, transport losses are often computed from samples through tractable representations, such as one-dimensional sorting formulas or sliced Wasserstein costs, making them practical components in training pipelines. We study parameterized objectives defined by sampled transport costs and prove graphical convergence of their subdifferentials to the subdifferential of the population objective. In particular, this ensures that standard subgradient methods consistently approach stationary points of the population-level problem. We illustrate the results in several settings, including risk-averse optimization, fairness-constrained learning, and sliced Wasserstein problems. Our analysis highlights that smooth parameterizations provide a favorable interface between statistical consistency and optimization. By contrast, transport objectives with nonsmooth costs and models may exhibit unstable derivatives in the large-sample limit.

Neural posterior estimation has emerged as a powerful tool for amortized inference, with growing adoption across scientific and applied domains. In many of these applications, the conditioning variable is a set of observations whose elements depend not only on the target but also on unknown factors shared across the set. Optimal inference therefore requires treating the set jointly, which in turn requires training the estimator at the deployment set size -- a regime where memory and compute quickly become prohibitive. We introduce a simple, theoretically grounded strategy that decouples representation learning from posterior modeling. Our method trains a mean-pool Deep Set on sets of size at most two, producing an encoder that generalizes to arbitrary set sizes. The inference head is then finetuned on pre-aggregated embeddings, making training cost essentially independent of the deployment set size N. Across scalar, image, multi-view 3D, molecular, and high-dimensional conditional generation benchmarks with N in the thousands, our approach matches or outperforms standard baselines at a fraction of the compute.

A nonparametric model using a sequence of Bernstein polynomials is constructed to approximate arbitrary isotropic covariance functions valid in $\mathbb{R}^\infty$ and related approximation properties are investigated using the popular $L_{\infty}$ norm and $L_2$ norms. A computationally efficient sieve maximum likelihood (sML) estimation is then developed to nonparametrically estimate the unknown isotropic covaraince function valid in $\mathbb{R}^\infty$. Consistency of the proposed sieve ML estimator is established under increasing domain regime. The proposed methodology is compared numerically with couple of existing nonparametric as well as with commonly used parametric methods. Numerical results based on simulated data show that our approach outperforms the parametric methods in reducing bias due to model misspecification and also the nonparametric methods in terms of having significantly lower values of expected $L_{\infty}$ and $L_2$ norms. Application to precipitation data is illustrated to showcase a real case study. Additional technical details and numerical illustrations are also made available.

We derive the first finite-time logarithmic Bayes regret upper bounds for Bayesian bandits. In a multi-armed bandit, we obtain O(c logn)and O(ch log2 n)upper bounds for an upper confidence bound algorithm, where ch and c are constants depending on the prior distribution and the gaps of bandit instances sampled from it, respectively. The latter bound asymptotically matches the lower bound of Lai (1987). Our proofs are a major technical departure from prior works, while being simple and general. To show the generality of our techniques, we apply them to linear bandits. Our results provide insights on the value of prior in the Bayesian setting, both in the objective and as a side information given to the learner. They significantly improve upon existing O( n)bounds, which have become standard in the literature despite the logarithmic lower bound of Lai (1987).

We consider general Gaussian latent tree models in which the observed variables are not restricted to be leaves of the tree. Extending related recent work, we give a full semi-algebraic description of the set of covariance matrices of any such model. In other words, we find polynomial constraints that characterize when a matrix is the covariance matrix of a distribution in a given latent tree model. However, leveraging these constraints to test a given such model is often complicated by the number of constraints being large and by singularities of individual polynomials, which may invalidate standard approximations to relevant probability distributions. Illustrating with the star tree, we propose a new testing methodology that circumvents singularity issues by trading off some statistical estimation efficiency and handles cases with many constraints through recent advances on Gaussian approximation for maxima of sums of high-dimensional random vectors. Our test avoids the need to maximize the possibly multimodal likelihood function of such models and is applicable to models with larger number of variables. These points are illustrated in numerical experiments.