Peng Xu

Peng Xu

Publications

You can also find my articles on my Google Scholar profile.

Preprints

Scalable Second-order Riemannian Optimization for \(K\)-means Clustering

Published in Preprint, 2025

Preprint

Abstract 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.

Journal Articles

Holographic Deep Thermalization for Secure and Efficient Quantum Random State Generation

Published in Nature Communications, 2025

Link Preprint

Abstract Randomness is a cornerstone of science, underpinning fields such as statistics, information theory, dynamical systems, and thermodynamics. In quantum science, quantum randomness, especially random pure states, plays a pivotal role in 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, called "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. By adopting a sequential application of a scrambling-measure-reset process, 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 removing quantum correlation between the data system and 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.

Generative Quantum Machine Learning via Denoising Diffusion Probabilistic Models

Published in Physical Review Letters, 2024

Link Preprint Code

Abstract 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.

orthoDr: Semiparametric Dimension Reduction via Orthogonality Constrained Optimization

Published in The R Journal, 2019

Link Preprint R Package

Abstract orthoDr is a package in R that solves dimension reduction problems using orthogonality constrained optimization approach. The package serves as a unified framework for many regression and survival analysis dimension reduction models that utilize semiparametric estimating equations. The main computational machinery of `orthoDr` is a first-order algorithm developed by Wen and Yin (2012) for optimization within the Stiefel manifold. We implement the algorithm through Rcpp and OpenMP for fast computation. In addition, we developed a general-purpose solver for such constrained problems with user-specified objective functions, which works as a drop-in version of optim(). The package also serves as a platform for future methodology developments along this line of work.

Conference Papers

Embedding Empirical Distributions for Computing Optimal Transport Maps

Published in 2025 IEEE International Symposium on Information Theory, 2025

Preprint Code

Abstract Distributional data have become increasingly prominent in modern signal processing, highlighting the necessity of computing optimal transport (OT) maps across multiple probability distributions. Nevertheless, recent studies on neural OT methods predominantly focused on the efficient computation of a single map between two distributions. To address this challenge, we introduce a novel approach to learning transport maps for new empirical distributions. Specifically, we employ the transformer architecture to produce embeddings from distributional data of varying length; these embeddings are then fed into a hypernetwork to generate neural OT maps. Various numerical experiments were conducted to validate the embeddings and the generated OT maps. The model implementation and the code are provided on this https URL.

Approximate Leave-One-Out for Fast Parameter Tuning in High Dimensions

Published in INFORMS Data Mining and Decision Analysis Workshop, 2019

Link R Package

Abstract Consider the class of learning schemes which is composed of a sum of losses over every data point and a regularizer that imposes special structures on the model parameter and controls the model complexity. A tuning parameter, typically adjusting the amount of regularization, is necessary for this framework to work well. Finding the optimal tuning is a challenging problem in high-dimensional regimes where both the sample size and the dimension of the parameter space are large. We propose two frameworks to obtain a computationally efficient approximation of the leave-one-out cross validation (LOOCV) risk for nonsmooth losses and regularizers. Our two frameworks are based on the primal and dual formulations of the aforementioned learning scheme. We then prove the equivalence of the two approaches under smoothness conditions. This equivalence enables us to justify the accuracy of both methods under such conditions. Finally we apply our approaches to several standard problems, including generalized LASSO and support vector machines, and empirically demonstrate the effectiveness of our results.

Best Theoretical Paper Finalists, INFORMS 2019 Data Mining and Decision Analytics Workshop Best Paper Competition Awards.