YoonHaeng Hur

Xun Tang, Yoonhaeng Hur, Yuehaw Khoo et al.

In this paper, we present a density estimation framework based on tree tensor-network states. The proposed method consists of determining the tree topology with Chow-Liu algorithm, and obtaining a linear system of equations that defines the tensor-network components via sketching techniques. Novel choices of sketch functions are developed in order to consider graphical models that contain loops. Sample complexity guarantees are provided and further corroborated by numerical experiments.

Kulunu Dharmakeerthi, YoonHaeng Hur, Tengyuan Liang

Practitioners often deploy a learned prediction model in a new environment where the joint distribution of covariate and response has shifted. In observational data, the distribution shift is often driven by unobserved confounding factors lurking in the environment, with the underlying mechanism unknown. Confounding can obfuscate the definition of the best prediction model (concept shift) and shift covariates to domains yet unseen (covariate shift). Therefore, a model maximizing prediction accuracy in the source environment could suffer a significant accuracy drop in the target environment. This motivates us to study the domain adaptation problem with observational data: given labeled covariate and response pairs from a source environment, and unlabeled covariates from a target environment, how can one predict the missing target response reliably? We root the adaptation problem in a linear structural causal model to address endogeneity and unobserved confounding. We study the necessity and benefit of leveraging exogenous, invariant covariate representations to cure concept shifts and improve target prediction. This further motivates a new representation learning method for adaptation that optimizes for a lower-dimensional linear subspace and, subsequently, a prediction model confined to that subspace. The procedure operates on a non-convex objective-that naturally interpolates between predictability and stability/invariance-constrained on the Stiefel manifold. We study the optimization landscape and prove that, when the regularization is sufficient, nearly all local optima align with an invariant linear subspace resilient to both concept and covariate shift. In terms of predictability, we show a model that uses the learned lower-dimensional subspace can incur a nearly ideal gap between target and source risk. Three real-world data sets are investigated to validate our method and theory.

YoonHaeng Hur, Yuehaw Khoo

We show the outlier robustness and noise stability of practical matching procedures based on distance profiles. Although the idea of matching points based on invariants like distance profiles has a long history in the literature, there has been little understanding of the theoretical properties of such procedures, especially in the presence of outliers and noise. We provide a theoretical analysis showing that under certain probabilistic settings, the proposed matching procedure is successful with high probability even in the presence of outliers and noise. We demonstrate the performance of the proposed method using a real data example and provide simulation studies to complement the theoretical findings. Lastly, we extend the concept of distance profiles to the abstract setting and connect the proposed matching procedure to the Gromov-Wasserstein distance and its lower bound, with a new sample complexity result derived based on the properties of distance profiles. This paper contributes to the literature by providing theoretical underpinnings of the matching procedures based on invariants like distance profiles, which have been widely used in practice but have rarely been analyzed theoretically.

Wenxuan Guo, YoonHaeng Hur, Tengyuan Liang et al.

Motivated by robust dynamic resource allocation in operations research, we study the \textit{Online Learning to Transport} (OLT) problem where the decision variable is a probability measure, an infinite-dimensional object. We draw connections between online learning, optimal transport, and partial differential equations through an insight called the minimal selection principle, originally studied in the Wasserstein gradient flow setting by \citet{Ambrosio_2005}. This allows us to extend the standard online learning framework to the infinite-dimensional setting seamlessly. Based on our framework, we derive a novel method called the \textit{minimal selection or exploration (MSoE) algorithm} to solve OLT problems using mean-field approximation and discretization techniques. In the displacement convex setting, the main theoretical message underpinning our approach is that minimizing transport cost over time (via the minimal selection principle) ensures optimal cumulative regret upper bounds. On the algorithmic side, our MSoE algorithm applies beyond the displacement convex setting, making the mathematical theory of optimal transport practically relevant to non-convex settings common in dynamic resource allocation.

YoonHaeng Hur, Wenxuan Guo, Tengyuan Liang

This paper introduces a new simulation-based inference procedure to model and sample from multi-dimensional probability distributions given access to i.i.d.\ samples, circumventing the usual approaches of explicitly modeling the density function or designing Markov chain Monte Carlo. Motivated by the seminal work on distance and isomorphism between metric measure spaces, we propose a new notion called the Reversible Gromov-Monge (RGM) distance and study how RGM can be used to design new transform samplers to perform simulation-based inference. Our RGM sampler can also estimate optimal alignments between two heterogeneous metric measure spaces $(\cX, μ, c_{\cX})$ and $(\cY, ν, c_{\cY})$ from empirical data sets, with estimated maps that approximately push forward one measure $μ$ to the other $ν$, and vice versa. We study the analytic properties of the RGM distance and derive that under mild conditions, RGM equals the classic Gromov-Wasserstein distance. Curiously, drawing a connection to Brenier's polar factorization, we show that the RGM sampler induces bias towards strong isomorphism with proper choices of $c_{\cX}$ and $c_{\cY}$. Statistical rate of convergence, representation, and optimization questions regarding the induced sampler are studied. Synthetic and real-world examples showcasing the effectiveness of the RGM sampler are also demonstrated.