selected publications

1. Conference

   Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis

   Kaheon Kim, Bohan Zhou, Changbo Zhu, and Xiaohui Chen

   In International Conference on Learning Representations (ICLR) 2026

   Abs Bib PDF

   This paper introduces a new constraint-free concave dual formulation for the Wasserstein barycenter. Tailoring the vanilla dual gradient ascent algorithm to the Sobolev geometry, we derive a scalable Sobolev gradient ascent (SGA) algorithm to compute the barycenter for input distributions supported on a regular grid. Despite the algorithmic simplicity, we provide a global convergence analysis that achieves the same rate as the classical subgradient descent methods for minimizing nonsmooth convex functions in the Euclidean space. A central feature of our SGA algorithm is that the computationally expensive c-concavity projection operator enforced on the Kantorovich dual potentials is unnecessary to guarantee convergence, leading to significant algorithmic and theoretical simplifications over all existing primal and dual methods for computing the exact barycenter. Our numerical experiments demonstrate the superior empirical performance of SGA over the existing optimal transport barycenter solvers.

   @inproceedings{kim2025sobolevgradientascentoptimal,
     author = {Kim, Kaheon and Zhou, Bohan and Zhu, Changbo and Chen, Xiaohui},
     title = {{Sobolev Gradient Ascent for Optimal Transport: Barycenter Optimization and Convergence Analysis}},
     year = {2026},
     booktitle = {International Conference on Learning Representations (ICLR)},
   }

2. Conference

   Scalable Second-order Riemannian Optimization for K-means Clustering

   Peng Xu, Chun-Ying Hou, Xiaohui Chen, and Richard Y. Zhang

   In International Conference on Learning Representations (ICLR) 2026

   Abs Bib PDF Code

   Clustering is a hard discrete optimization problem. Nonconvex approaches such as low-rank semidefinite programming (SDP) have recently demonstrated promising statistical and local algorithmic guarantees for cluster recovery. Due to the combinatorial structure of the K-means clustering problem, current relaxation algorithms struggle to balance their constraint feasibility and objective optimality, presenting tremendous challenges in computing the second-order critical points with rigorous guarantees. In this paper, we provide a new formulation of the K-means problem as a smooth unconstrained optimization over a submanifold and characterize its Riemannian structures to allow it to be solved using a second-order cubic-regularized Riemannian Newton algorithm. By factorizing the K-means manifold into a product manifold, we show how each Newton subproblem can be solved in linear time. Our numerical experiments show that the proposed method converges significantly faster than the state-of-the-art first-order nonnegative low-rank factorization method, while achieving similarly optimal statistical accuracy.

   @inproceedings{xu2025scalablesecondorderriemannianoptimization,
     author = {Xu, Peng and Hou, Chun-Ying and Chen, Xiaohui and Zhang, Richard Y.},
     title = {{Scalable Second-order Riemannian Optimization for $K$-means Clustering}},
     year = {2026},
     booktitle = {International Conference on Learning Representations (ICLR)},
   }

3. Journal

   Holographic deep thermalization for secure and efficient quantum random state generation

   Bingzhi Zhang, Peng Xu, Xiaohui Chen, and Quntao Zhuang

   Nature Communications 2025

   Abs Bib PDF Code

   Quantum randomness, especially random pure states, underpins fundamental questions like black hole physics and quantum complexity, as well as in practical applications such as quantum device benchmarking and quantum advantage certification. The conventional approach for generating genuine random states, known as ‘deep thermalization’, faces significant challenges, including scalability issues due to the need for a large ancilla system and susceptibility to attacks, as demonstrated in this work. We introduce holographic deep thermalization, a secure and hardware-efficient quantum random state generator. Via a sequence of scrambling-measure-reset processes, it continuously trades space with time, and substantially reduces the required ancilla size to as small as a system-size-independent constant; at the same time, it guarantees security by eliminating quantum correlation between the data system and potential attackers. Thanks to the resource reduction, our circuit-based implementation on IBM Quantum devices achieves genuine 5-qubit random state generation utilizing only a total of 8 qubits.

   @article{zhang2024holographicdeepthermalization,
     author = {Zhang, Bingzhi and Xu, Peng and Chen, Xiaohui and Zhuang, Quntao},
     date = {2025/07/09},
     date-added = {2025-07-09 22:04:02 -0700},
     date-modified = {2025-07-09 22:04:02 -0700},
     doi = {10.1038/s41467-025-61546-y},
     id = {Zhang2025},
     isbn = {2041-1723},
     journal = {Nature Communications},
     number = {1},
     pages = {6341},
     title = {Holographic deep thermalization for secure and efficient quantum random state generation},
     url = {https://doi.org/10.1038/s41467-025-61546-y},
     volume = {16},
     year = {2025},
     bdsk-url-1 = {https://doi.org/10.1038/s41467-025-61546-y}
   }

4. arXiv

   Learning Density Evolution from Snapshot Data

   Rentian Yao, Atsushi Nitanda, Xiaohui Chen, and Yun Yang

   Feb 2025

   Abs Bib PDF

   Motivated by learning dynamical structures from static snapshot data, this paper presents a distribution-on-scalar regression approach for estimating the density evolution of a stochastic process from its noisy temporal point clouds. We propose an entropy-regularized nonparametric maximum likelihood estimator (E-NPMLE), which leverages the entropic optimal transport as a smoothing regularizer for the density flow. We show that the E-NPMLE has almost dimension-free statistical rates of convergence to the ground truth distributions, which exhibit a striking phase transition phenomenon in terms of the number of snapshots and per-snapshot sample size. To efficiently compute the E-NPMLE, we design a novel particle-based and grid-free coordinate KL divergence gradient descent (CKLGD) algorithm and prove its polynomial iteration complexity. Moreover, we provide numerical evidence on synthetic data to support our theoretical findings. This work contributes to the theoretical understanding and practical computation of estimating density evolution from noisy observations in arbitrary dimensions.

   @misc{YNCY2025_snapshot-learning,
     author = {Yao, Rentian and Nitanda, Atsushi and Chen, Xiaohui and Yang, Yun},
     month = feb,
     title = {{Learning Density Evolution from Snapshot Data}},
     year = {2025},
   }

5. Journal

   Wasserstein proximal coordinate gradient algorithms

   Rentian Yao, Xiaohui Chen, and Yun Yang

   Journal of Machine Learning Research May 2024

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   Motivated by approximation Bayesian computation using mean-field variational approximation and the computation of equilibrium in multi-species systems with cross-interaction, this paper investigates the composite geodesically convex optimization problem over multiple distributions. The objective functional under consideration is composed of a convex potential energy on a product of Wasserstein spaces and a sum of convex self-interaction and internal energies associated with each distribution. To efficiently solve this problem, we introduce the Wasserstein Proximal Coordinate Gradient (WPCG) algorithms with parallel, sequential and random update schemes. Under a quadratic growth (QC) condition that is weaker than the usual strong convexity requirement on the objective functional, we show that WPCG converges exponentially fast to the unique global optimum. In the absence of the QG condition, WPCG is still demonstrated to converge to the global optimal solution, albeit at a slower polynomial rate. Numerical results for both motivating examples are consistent with our theoretical findings.

   @article{YaoChenYang2023_WPCG,
     author = {Yao, Rentian and Chen, Xiaohui and Yang, Yun},
     date-added = {2023-10-09 22:22:12 -0700},
     date-modified = {2024-10-15 07:29:34 -0700},
     journal = {Journal of Machine Learning Research},
     month = may,
     number = {269},
     pages = {1-66},
     title = {Wasserstein proximal coordinate gradient algorithms},
     volume = {25},
     year = {2024},
   }

6. Journal

   Generative quantum machine learning via denoising diffusion probabilistic models

   Bingzhi Zhang, Peng Xu, Xiaohui Chen, and Quntao Zhuang

   Physical Review Letters May 2024

   Abs Bib PDF Code

   Deep generative models are key-enabling technology to computer vision, text generation, and large language models. Denoising diffusion probabilistic models (DDPMs) have recently gained much attention due to their ability to generate diverse and high-quality samples in many computer vision tasks, as well as to incorporate flexible model architectures and a relatively simple training scheme. Quantum generative models, empowered by entanglement and superposition, have brought new insight to learning classical and quantum data. Inspired by the classical counterpart, we propose the quantum denoising diffusion probabilistic model (QuDDPM) to enable efficiently trainable generative learning of quantum data. QuDDPM adopts sufficient layers of circuits to guarantee expressivity, while it introduces multiple intermediate training tasks as interpolation between the target distribution and noise to avoid barren plateau and guarantee efficient training. We provide bounds on the learning error and demonstrate QuDDPM’s capability in learning correlated quantum noise model, quantum many-body phases, and topological structure of quantum data. The results provide a paradigm for versatile and efficient quantum generative learning.

   @article{zhang2023generative,
     author = {Zhang, Bingzhi and Xu, Peng and Chen, Xiaohui and Zhuang, Quntao},
     date-added = {2023-10-09 22:08:13 -0700},
     date-modified = {2024-01-31 18:03:04 -0800},
     journal = {Physical Review Letters},
     primaryclass = {quant-ph},
     title = {Generative quantum machine learning via denoising diffusion probabilistic models},
     year = {2024},
   }

7. ConferenceOral

   Statistically Optimal K-means Clustering via Nonnegative Low-rank Semidefinite Programming

   Yubo Zhuang, Xiaohui Chen, Yun Yang, and Richard Y. Zhang

   In International Conference on Learning Representations (ICLR) May 2024

   Oral Presentation [One of 85/7262 submissions. Top 1.2%]

   Abs Bib PDF Code Poster

   K-means clustering is a widely used machine learning method for identifying patterns in large datasets. Semidefinite programming (SDP) relaxations have recently been proposed for solving the K-means optimization problem that enjoy strong statistical optimality guarantees, but the prohibitive cost of implementing an SDP solver renders these guarantees inaccessible to practical datasets. By contrast, nonnegative matrix factorization (NMF) is a simple clustering algorithm that is widely used by machine learning practitioners, but without a solid statistical underpinning nor rigorous guarantees. In this paper, we describe an NMF-like algorithm that works by solving a nonnegative low-rank restriction of the SDP relaxed K-means formulation using a nonconvex Burer–Monteiro factorization approach. The resulting algorithm is just as simple and scalable as state-of-the-art NMF algorithms, while also enjoying the same strong statistical optimality guarantees as the SDP. In our experiments, we observe that our algorithm achieves substantially smaller mis-clustering errors compared to the existing state-of-the-art.

   @inproceedings{ZhuangChenYangZhang2023_NMF-BM,
     author = {Zhuang, Yubo and Chen, Xiaohui and Yang, Yun and Zhang, Richard Y.},
     booktitle = {International Conference on Learning Representations (ICLR)},
     date-added = {2023-06-04 11:47:33 -0700},
     date-modified = {2024-01-16 21:35:09 -0800},
     month = may,
     title = {Statistically Optimal K-means Clustering via Nonnegative Low-rank Semidefinite Programming},
     year = {2024},
   }

8. Journal

   Cutoff for Exact Recovery of Gaussian Mixture Models

   Xiaohui Chen, and Yun Yang

   IEEE Transactions on Information Theory Jun 2021

   Abs Bib PDF

   We determine the information-theoretic cutoff value on separation of cluster centers for exact recovery of cluster labels in a K-component Gaussian mixture model with equal cluster sizes. Moreover, we show that a semidefinite programming (SDP) relaxation of the K-means clustering method achieves such sharp threshold for exact recovery without assuming the symmetry of cluster centers.

   @article{9366690,
     author = {Chen, Xiaohui and Yang, Yun},
     date-added = {2022-08-29 10:07:29 -0500},
     date-modified = {2022-09-30 08:53:58 -0500},
     doi = {10.1109/TIT.2021.3063155},
     issn = {1557-9654},
     journal = {IEEE Transactions on Information Theory},
     month = jun,
     number = {6},
     pages = {4223-4238},
     title = {Cutoff for Exact Recovery of Gaussian Mixture Models},
     volume = {67},
     year = {2021},
     bdsk-url-1 = {https://doi.org/10.1109/TIT.2021.3063155}
   }

9. Journal

   Diffusion K-means clustering on manifolds: Provable exact recovery via semidefinite relaxations

   Xiaohui Chen, and Yun Yang

   Applied and Computational Harmonic Analysis May 2021

   Abs Bib PDF Code

   We introduce the diffusion K-means clustering method on Riemannian submanifolds, which maximizes the within-cluster connectedness based on the diffusion distance. The diffusion K-means constructs a random walk on the similarity graph with vertices as data points randomly sampled on the manifolds and edges as similarities given by a kernel that captures the local geometry of manifolds. The diffusion K-means is a multi-scale clustering tool that is suitable for data with non-linear and non-Euclidean geometric features in mixed dimensions. Given the number of clusters, we propose a polynomial-time convex relaxation algorithm via the semidefinite programming (SDP) to solve the diffusion K-means. In addition, we also propose a nuclear norm regularized SDP that is adaptive to the number of clusters. In both cases, we show that exact recovery of the SDPs for diffusion K-means can be achieved under suitable between-cluster separability and within-cluster connectedness of the submanifolds, which together quantify the hardness of the manifold clustering problem. We further propose the localized diffusion K-means by using the local adaptive bandwidth estimated from the nearest neighbors. We show that exact recovery of the localized diffusion K-means is fully adaptive to the local probability density and geometric structures of the underlying submanifolds.

   @article{CHEN2021303,
     author = {Chen, Xiaohui and Yang, Yun},
     date-added = {2022-08-29 09:04:40 -0500},
     date-modified = {2022-09-30 08:54:12 -0500},
     doi = {https://doi.org/10.1016/j.acha.2020.03.002},
     issn = {1063-5203},
     journal = {Applied and Computational Harmonic Analysis},
     keywords = {Manifold clustering, K-means, Riemannian submanifolds, Diffusion distance, Semidefinite programming, Random walk on random graphs, Laplace-Beltrami operator, Mixing times, Adaptivity},
     month = may,
     pages = {303-347},
     title = {Diffusion K-means clustering on manifolds: Provable exact recovery via semidefinite relaxations},
     url = {https://www.sciencedirect.com/science/article/pii/S106352032030021X},
     volume = {52},
     year = {2021},
     bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S106352032030021X},
     bdsk-url-2 = {https://doi.org/10.1016/j.acha.2020.03.002}
   }

10. Journal

    Finite sample change point inference and identification for high-dimensional mean vectors

    Mengjia Yu, and Xiaohui Chen

    Journal of the Royal Statistical Society: Series B (Statistical Methodology) Apr 2021

    Abs Bib PDF

    Abstract Cumulative sum (CUSUM) statistics are widely used in the change point inference and identification. For the problem of testing for existence of a change point in an independent sample generated from the mean-shift model, we introduce a Gaussian multiplier bootstrap to calibrate critical values of the CUSUM test statistics in high dimensions. The proposed bootstrap CUSUM test is fully data dependent and it has strong theoretical guarantees under arbitrary dependence structures and mild moment conditions. Specifically, we show that with a boundary removal parameter the bootstrap CUSUM test enjoys the uniform validity in size under the null and it achieves the minimax separation rate under the sparse alternatives when the dimension p can be larger than the sample size n. Once a change point is detected, we estimate the change point location by maximising the ℓ∞-norm of the generalised CUSUM statistics at two different weighting scales corresponding to covariance stationary and non-stationary CUSUM statistics. For both estimators, we derive their rates of convergence and show that dimension impacts the rates only through logarithmic factors, which implies that consistency of the CUSUM estimators is possible when p is much larger than n. In the presence of multiple change points, we propose a principled bootstrap-assisted binary segmentation (BABS) algorithm to dynamically adjust the change point detection rule and recursively estimate their locations. We derive its rate of convergence under suitable signal separation and strength conditions. The results derived in this paper are non-asymptotic and we provide extensive simulation studies to assess the finite sample performance. The empirical evidence shows an encouraging agreement with our theoretical results.

    @article{https://doi.org/10.1111/rssb.12406,
      author = {Yu, Mengjia and Chen, Xiaohui},
      date-added = {2022-08-29 10:16:32 -0500},
      date-modified = {2022-09-30 08:54:35 -0500},
      doi = {https://doi.org/10.1111/rssb.12406},
      journal = {Journal of the Royal Statistical Society: Series B (Statistical Methodology)},
      keywords = {binary segmentation, bootstrap, CUSUM, change point analysis, Gaussian approximation, high-dimensional data},
      month = apr,
      number = {2},
      pages = {247-270},
      title = {Finite sample change point inference and identification for high-dimensional mean vectors},
      url = {https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12406},
      volume = {83},
      year = {2021},
      bdsk-url-1 = {https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12406},
      bdsk-url-2 = {https://doi.org/10.1111/rssb.12406}
    }

11. Journal

    Jackknife multiplier bootstrap: finite sample approximations to the U-process supremum with applications

    Xiaohui Chen, and Kengo Kato

    Probability Theory and Related Fields Jul 2020

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    This paper is concerned with finite sample approximations to the supremum of a non-degenerate U-process of a general order indexed by a function class. We are primarily interested in situations where the function class as well as the underlying distribution change with the sample size, and the U-process itself is not weakly convergent as a process. Such situations arise in a variety of modern statistical problems. We first consider Gaussian approximations, namely, approximate the U-process supremum by the supremum of a Gaussian process, and derive coupling and Kolmogorov distance bounds. Such Gaussian approximations are, however, not often directly applicable in statistical problems since the covariance function of the approximating Gaussian process is unknown. This motivates us to study bootstrap-type approximations to the U-process supremum. We propose a novel jackknife multiplier bootstrap (JMB) tailored to the U-process, and derive coupling and Kolmogorov distance bounds for the proposed JMB method. All these results are non-asymptotic, and established under fairly general conditions on function classes and underlying distributions. Key technical tools in the proofs are new local maximal inequalities for U-processes, which may be useful in other problems. We also discuss applications of the general approximation results to testing for qualitative features of nonparametric functions based on generalized local U-processes.

    @article{ChenKato2020_PTRF,
      author = {Chen, Xiaohui and Kato, Kengo},
      date-added = {2022-08-29 09:04:40 -0500},
      date-modified = {2022-09-30 08:55:34 -0500},
      day = {31},
      doi = {10.1007/s00440-019-00936-y},
      issn = {1432-2064},
      journal = {Probability Theory and Related Fields},
      month = jul,
      pages = {1097--1163},
      title = {Jackknife multiplier bootstrap: finite sample approximations to the U-process supremum with applications},
      url = {https://doi.org/10.1007/s00440-019-00936-y},
      volume = {176},
      year = {2020},
      bdsk-url-1 = {https://doi.org/10.1007/s00440-019-00936-y}
    }

12. Journal

    Randomized incomplete U-statistics in high dimensions

    Xiaohui Chen, and Kengo Kato

    The Annals of Statistics Jul 2019

    Abs Bib PDF

    This paper studies inference for the mean vector of a high-dimensional U-statistic. In the era of big data, the dimension d of the U-statistic and the sample size n of the observations tend to be both large, and the computation of the U-statistic is prohibitively demanding. Data-dependent inferential procedures such as the empirical bootstrap for U-statistics is even more computationally expensive. To overcome such a computational bottleneck, incomplete U-statistics obtained by sampling fewer terms of the U-statistic are attractive alternatives. In this paper, we introduce randomized incomplete U-statistics with sparse weights whose computational cost can be made independent of the order of the U-statistic. We derive nonasymptotic Gaussian approximation error bounds for the randomized incomplete U-statistics in high dimensions, namely in cases where the dimension d is possibly much larger than the sample size n, for both nondegenerate and degenerate kernels. In addition, we propose generic bootstrap methods for the incomplete U-statistics that are computationally much less demanding than existing bootstrap methods, and establish finite sample validity of the proposed bootstrap methods. Our methods are illustrated on the application to nonparametric testing for the pairwise independence of a high-dimensional random vector under weaker assumptions than those appearing in the literature.

    @article{10.1214/18-AOS1773,
      author = {Chen, Xiaohui and Kato, Kengo},
      date-added = {2022-08-29 15:34:05 -0500},
      date-modified = {2022-09-30 08:57:52 -0500},
      doi = {10.1214/18-AOS1773},
      journal = {The Annals of Statistics},
      keywords = {Bernoulli sampling, bootstrap, Divide and conquer, Gaussian approximation, incomplete $U$-statistics, randomized inference, sampling with replacement},
      number = {6},
      pages = {3127 -- 3156},
      publisher = {Institute of Mathematical Statistics},
      title = {{Randomized incomplete $U$-statistics in high dimensions}},
      url = {https://doi.org/10.1214/18-AOS1773},
      volume = {47},
      year = {2019},
      bdsk-url-1 = {https://doi.org/10.1214/18-AOS1773}
    }

13. Journal

    Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications

    Xiaohui Chen

    The Annals of Statistics Jul 2018

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    This paper studies the Gaussian and bootstrap approximations for the probabilities of a nondegenerate U-statistic belonging to the hyperrectangles in \mathbbR^d when the dimension d is large. A two-step Gaussian approximation procedure that does not impose structural assumptions on the data distribution is proposed. Subject to mild moment conditions on the kernel, we establish the explicit rate of convergence uniformly in the class of all hyperrectangles in \mathbbR^d that decays polynomially in sample size for a high-dimensional scaling limit, where the dimension can be much larger than the sample size. We also provide computable approximation methods for the quantiles of the maxima of centered U-statistics. Specifically, we provide a unified perspective for the empirical bootstrap, the randomly reweighted bootstrap and the Gaussian multiplier bootstrap with the jackknife estimator of covariance matrix as randomly reweighted quadratic forms and we establish their validity. We show that all three methods are inferentially first-order equivalent for high-dimensional U-statistics in the sense that they achieve the same uniform rate of convergence over all d-dimensional hyperrectangles. In particular, they are asymptotically valid when the dimension d can be as large as O(e^n^c) for some constant c∈(0,1/7). The bootstrap methods are applied to statistical applications for high-dimensional non-Gaussian data including: (i) principled and data-dependent tuning parameter selection for regularized estimation of the covariance matrix and its related functionals; (ii) simultaneous inference for the covariance and rank correlation matrices. In particular, for the thresholded covariance matrix estimator with the bootstrap selected tuning parameter, we show that for a class of sub-Gaussian data, error bounds of the bootstrapped thresholded covariance matrix estimator can be much tighter than those of the minimax estimator with a universal threshold. In addition, we also show that the Gaussian-like convergence rates can be achieved for heavy-tailed data, which are less conservative than those obtained by the Bonferroni technique that ignores the dependency in the underlying data distribution.

    @article{10.1214/17-AOS1563,
      author = {Chen, Xiaohui},
      date-added = {2022-08-29 10:35:00 -0500},
      date-modified = {2022-09-30 08:54:48 -0500},
      doi = {10.1214/17-AOS1563},
      journal = {The Annals of Statistics},
      keywords = {bootstrap, Gaussian approximation, high-dimensional inference, U-statistics},
      number = {2},
      pages = {642 -- 678},
      publisher = {Institute of Mathematical Statistics},
      title = {{Gaussian and bootstrap approximations for high-dimensional U-statistics and their applications}},
      url = {https://doi.org/10.1214/17-AOS1563},
      volume = {46},
      year = {2018},
      bdsk-url-1 = {https://doi.org/10.1214/17-AOS1563}
    }