Sampling from constrained statistical distributions is a fundamental task in various fields including Bayesian statistics, computational chemistry, and statistical physics. This article considers sampling from a constrained distribution that is described by an unconstrained density, as well as additional equality and/or inequality constraints, which often make the constraint set nonconvex. Existing methods struggle in the presence of such nonconvex constraints, as they rely on projections, which are computationally expensive or intractable, are specialized to either inequality or equality constraints, and often lack rigorous quantitative convergence guarantees. In this paper, we introduce Overdamped Langevin with LAnding (OLLA), a new framework that can design overdamped Langevin dynamics accommodating both nonlinear equality and inequality constraints. The proposed dynamics also deterministically corrects trajectories along the normal direction of the constraint surface, thus obviating the need for explicit projections. We show that, under suitable regularity conditions on the target density and the feasible set Σ Rd, OLLA converges exponentially fast in 2-Wasserstein distance to the constrained target density ρΣ(x) exp( f(x))dσΣ. Lastly, through experiments, we demonstrate the efficiency of OLLA compared to known constrained Langevin algorithms and their slack variable variants, highlighting its favorable computational cost and fast empirical mixing.1

Expanding on neural operators, we propose a novel framework for stochastic process learning across arbitrary domains. In particular, we develop operator flow matching (OFM) for learning stochastic process priors on function spaces. OFM provides the probability density of the values of any collection of points and enables mathematically tractable functional regression at new points with mean and density estimation. Our method outperforms state-of-the-art models in stochastic process learning, functional regression, and prior learning.

Regularization is one of the most fundamental topics in optimization, statistics and machine learning. To get sparsity in estimating a parameter u Rd, an ℓq penalty term, u q, is usually added to the objective function. What is the probabilistic distribution corresponding to such ℓq penalty? What is the correct stochastic process corresponding to u q when we model functions u Lq? This is important for statistically modeling high-dimensional objects such as images, with penalty to preserve certain properties, e.g.

Spectral clustering and co-clustering are well-known techniques in data analysis, and recent work has extended spectral clustering to square, symmetric tensors and hypermatrices derived from a network. We develop a new tensor spectral co-clustering method that simultaneously clusters the rows, columns, and slices of a nonnegative three-mode tensor and generalizes to tensors with any number of modes. The algorithm is based on a new random walk model which we call the super-spacey random surfer. We show that our method out-performs state-of-the-art co-clustering methods on several synthetic datasets with ground truth clusters and then use the algorithm to analyze several real-world datasets.