arXiv subjects

Yuehaw Khoo

Publications and source records attributed to Yuehaw Khoo.

At least 19 recordsLinked to original sources

arXiv2026

Recursive Sketched Interpolation: Efficient Hadamard Products of Tensor Trains

The Hadamard product of two tensors in the tensor-train (TT) format is a fundamental operation across various applications, such as TT-based function multiplication for nonlinear differential equations or convolutions. However, conventional methods for computing this product typically scale as at least $\mathcal{O}(χ^4)$ with respect to the TT bond dimension (TT-rank) $χ$, creating a severe computational bottleneck in practice. By combining randomized tensor-train sketching with slice selection via interpolative decomposition, we introduce Recursive Sketched Interpolation (RSI), a ``scale product'' algorithm that computes the Hadamard product of TTs at a computational cost of $\mathcal{O}(χ^3)$. Benchmarks across various TT scenarios demonstrate that RSI offers superior scalability compared to traditional methods while maintaining comparable accuracy. We generalize RSI to compute more complex operations, including Hadamard products of multiple TTs and other element-wise nonlinear mappings, without increasing the complexity beyond $\mathcal{O}(χ^3)$.

quant-ph↗

arXiv2026

Accelerated Learning of High Dimensional Functions with a Tensor-Featured Training Network

In this work we present a method to accelerate the optimization of learning high dimensional functions using deep neural network (DNN). This optimization procedure introduces contextual features into the first layer of a DNN. The parameters of DNN are optimized via standard gradient descent while keeping the input-feature basis fixed. After optimization of the DNN parameters, the feature layer is provided a chance to update and change before DNN optimization resumes. The feature layer has two types of functions: those that can be evaluated quickly in a matrix-free way on the domain (i.e. rank-1 features) and more complex features that must first be decomposed using tensor network (TN) decomposition strategies (tensor features). In particular, we study the effect of adding features which distill pretrained DNN into TNs using a discretize and decompose strategy. To efficiently decompose high-dimensional functions constructed from discretized DNN, we leverage a randomized tensor decomposition strategy. Using randomization, we are able to reduce the storage cost of decomposing high dimensional functions by at least 8 orders of magnitude. Using this approach, we are able to efficiently train models between 5 and 40 dimensions.

cs.LG↗

arXiv2026

Cluster-Aware Matching via Laplacian Optimal Transport

In many applications of matching, the point clouds to be matched are not merely unstructured sets of points but rather samples from distributions with an intrinsic cluster structure. In such cases, as individual points are often interchangeable within a coherent region, finding a robust region-to-region alignment is more desirable than establishing a precise point-to-point correspondence. To this end, we propose a novel approach for cluster-aware matching based on Laplacian Optimal Transport (LapOT). The key idea is to regularize the optimal transport problem with quadratic Laplacian terms constructed from similarity graphs of the point clouds, which encourages the optimal coupling to respect the cluster structure of both point sets. We also introduce Refined Simultaneous Clustering (RSC), a method that leverages the cluster-aware coupling obtained from LapOT to produce consistent partitions across the point sets, which can overcome the limitations of independent clustering and yield more stable and interpretable results. We demonstrate the effectiveness of our approach through theoretical analysis and empirical experiments, showing that LapOT indeed produces cluster-aware matching that leads to more consistent and meaningful alignments between point clouds.

stat.ML↗

arXiv2026

Convex Relaxations for the Optimization of Markov Processes

In this paper, we study the problem of optimizing Markov processes that interpolate between two prescribed probability distributions while minimizing a given cost. The main computational challenge is the curse of dimensionality: in high-dimensional state spaces, representing the full distribution is intractable. To address this, we reformulate the problem in terms of sequential couplings and develop convex relaxations based on local marginals and cluster moments. These relaxations exploit locality and sparse interaction structure, provide computable lower bounds, and recover low-order statistics of the intermediate laws. We identify dynamic optimal transport as a special case of our Markov process optimization problem and develop a procedure for recovering the underlying Benamou--Brenier dynamics from the relaxed solution. We also show that the procedure extends to more general Markov processes and illustrate it with a constrained process between Ising models.

math.OC↗

arXiv2026

Convex relaxation approaches for high-dimensional optimal transport

Optimal transport (OT) is a powerful tool in mathematics and data science but faces severe computational and statistical challenges in high dimensions. We propose convex relaxation approaches based on marginal and cluster moment relaxations that exploit locality in the distributions. These methods approximate high-dimensional couplings using low-order marginals and sparse moment statistics, yielding semidefinite programs that provide lower bounds on the OT cost with greatly reduced complexity. For Gaussian measures with sparse correlations, we prove an exponential convergence rate for the cluster moment relaxation and an improved statistical error bound. We also establish approximation error bounds for the marginal relaxation when the reference measures are local perturbations of mean-field measures. In addition, we demonstrate how to extract transport maps from our relaxations, offering a simpler and interpretable alternative to neural networks in generative modeling. Extensive numerical experiments demonstrate strong empirical performance across a range of distributions. Our results suggest that convex relaxations can provide a promising path for dimension reduction in high-dimensional OT.

math.OC↗

arXiv2026

Permutation Recovery on Manifold Data via Spectral Seriation

Data points in many scientific experiments originate from an ordered structure, yet this ordering is often unavailable.We consider noisy data points with the correct ordering to be recovered. The underlying structure naturally places the data on a 1-dimensional manifold. Because eigenfunctions of 1-dimensional manifold Laplacian are trigonometric functions, and the manifold Laplacian can be approximated by the graph data Laplacian, the data ordering can be recovered by inverting the data Laplacian eigenvectors.We propose two spectral algorithms, one for the periodic structure (closed loop) and one for the non-periodic structure (open curve). We have derived the uniform error bound for the algorithms, which is composed of two parts: the discretization error between the manifold eigenfunctions and the noiseless graph Laplacian eigenvectors, and the eigenvectors error caused by data noise. In numerical studies, our spectral seriation algorithms outperform other manifold learning methods. The superior performance of our algorithms is demonstrated further on a biomolecule data example.

stat.ME↗

arXiv2026

Adaptive tensor train metadynamics for high-dimensional free energy exploration

A key challenge for molecular dynamics simulations is efficient exploration of free energy landscapes over relevant collective variables (CV). Common methods for enhancing sampling become prohibitively inefficient beyond only a few CVs; in the case of the widely-used metadynamics method, the computational cost of evaluating and storing the bias potential grows exponentially with the number of dimensions. Here, we introduce TT-Metadynamics, in which the accumulated sum of Gaussian functions in the original metadynamics method is periodically compressed into a low-rank tensor train (TT) representation. The TT enables efficient memory use and prevents the computational cost of evaluating the bias potential from increasing with simulation time. We present a "sketching" algorithm that allows us to construct the TT with linear scaling in the number of CVs. Applied to benchmark systems with up to 14 CVs, the accuracy of TT-Metadynamics matches or exceeds that of standard metadynamics in long simulations, particularly in systems with high barriers. These results establish TT-Metadynamics as a scalable and effective method for computing free energies that are functions of several CVs.

physics.chem-ph↗

arXiv2026

Sketch Tomography: Hybridizing Classical Shadow and Matrix Product State

We introduce Sketch Tomography, an efficient procedure for quantum state tomography based on the classical shadow protocol used for quantum observable estimations. The procedure applies to the case where the ground truth quantum state is a matrix product state (MPS). The density matrix of the ground truth state admits a tensor train ansatz as a result of the MPS assumption, and we estimate the tensor components of the ansatz through a series of observable estimations, thus outputting an approximation of the density matrix. The procedure is provably convergent with a sample complexity that scales quadratically in the system size. We conduct extensive numerical experiments to show that the procedure outputs an accurate approximation to the quantum state. For observable estimation tasks involving moderately large subsystems, we show that our procedure gives rise to a more accurate estimation than the classical shadow protocol. We also show that sketch tomography is more accurate in observable estimation than quantum states trained from the maximum likelihood estimation formulation.

quant-ph↗

arXiv2026

Inlier Recovery for Robust Registration via Gram-Matrix Overlap

Robust point-set registration in the presence of noise and outliers is challenging because the matched points (inliers) must be identified before reliable alignment can be performed. Existing robust registration methods typically optimize over the transformation space and are often designed for regimes with a nonvanishing fraction of inliers. In this paper, we study the inlier recovery problem arising in robust registration by comparing two datasets through the Hadamard product of their Gram matrices. This formulation converts the inlier identification into a structured recovery problem and avoids direct optimization over the rotation group. Based on this idea, we develop two methods: an eigenvector matching method based on the leading eigenvector of the Gram-matrix overlap, and a row-sum matching method based on aggregated entrywise comparison. We show that the eigenvector method achieves weak recovery when the dimension and sample size are of the same order, while the row-sum method achieves exact recovery under a broader range of dimensional scalings. In particular, when the dimension is comparable to the sample size, exact recovery is possible even when the inlier fraction vanishes, with the number of inliers as small as order $\sqrt{n}$, up to logarithmic factors. We also discuss a parallel implementation for large-scale settings. Numerical experiments on brain imaging data and image examples demonstrate that the proposed methods effectively identify matched structure under substantial corruption.

stat.ME↗

arXiv2026

Non-Uniform Quantum Fourier Transform

The Discrete Fourier Transform (DFT) is central to the analysis of uniformly sampled signals, yet many practical applications involve non-uniform sampling, requiring the Non-Uniform Discrete Fourier Transform (NUDFT). While quantum algorithms for the standard DFT are well established, a corresponding framework for the non-uniform case remains underdeveloped. This work introduces a quantum algorithm for the Non-Uniform Quantum Fourier Transform (NUQFT) based on a low-rank factorization of the NUDFT matrix. The factorization is translated into an explicit quantum construction using block encodings, Quantum Signal Processing, and the Linear Combination of Unitaries framework, yielding an $ε$-accurate block encoding of the NUDFT matrix with controlled approximation error from both classical truncation and quantum implementation. Under standard oracle access assumptions for non-uniform sampling points, we derive explicit, non-asymptotic gate-level resource estimates. The resulting complexity scales polylogarithmically with target precision, quadratically with the number of qubits through the quantum Fourier transform, and logarithmically with a geometry-dependent conditioning parameter induced by the non-uniform grid. This establishes a concrete and resource-efficient quantum analogue of the NUDFT and provides a foundation for quantum algorithms on irregularly sampled data.

quant-ph↗

arXiv2026

Re-anchoring Quantum Monte Carlo with Tensor-Train Sketching

We propose a novel algorithm for calculating the ground-state energy of quantum many-body systems by combining auxiliary-field quantum Monte Carlo (AFQMC) with tensor-train sketching. In AFQMC, a good trial wavefunction to guide the random walk is crucial for improving the sampling efficiency and controlling the sign problem. Our proposed method iterates between determining a new trial wavefunction in the form of a tensor train, derived from the current walkers, and using this updated trial wavefunction to anchor the next phase of AFQMC. Numerical results demonstrate that the algorithm is highly accurate for large spin systems. The overlap between the estimated trial wavefunction and the ground-state wavefunction also achieves high fidelity. We additionally provide a convergence analysis, highlighting how an effective trial wavefunction can reduce the variance in the AFQMC energy estimation. From a complementary perspective, our algorithm also extends the reach of tensor-train methods for studying quantum many-body systems.

math.NA↗

arXiv2026

Reinforced Inverse Scattering

Inverse wave scattering aims at determining the properties of an object using data on how the object scatters incoming waves. In order to collect information, sensors are put in different locations to send and receive waves from each other. The choice of sensor positions and incident wave frequencies determines the reconstruction quality of scatterer properties. This paper introduces reinforcement learning to develop precision imaging that decides sensor positions and wave frequencies adaptive to different scatterers in an intelligent way, thus obtaining a significant improvement in reconstruction quality with limited imaging resources. Extensive numerical results will be provided to demonstrate the superiority of the proposed method over existing methods.

cs.LG↗

arXiv2026

Generative Modeling via Hierarchical Tensor Sketching

We propose a hierarchical tensor-network approach for approximating high-dimensional probability density via empirical distribution. This leverages randomized singular value decomposition (SVD) techniques and involves solving linear equations for tensor cores in this tensor network. The complexity of the resulting algorithm scales linearly in the dimension of the high-dimensional density. An analysis of estimation error demonstrates the effectiveness of this method through several numerical experiments.

math.NA↗

arXiv2025

Robust Point Matching with Distance Profiles

Computational difficulty of quadratic matching and the Gromov-Wasserstein distance has led to various approximation and relaxation schemes. One of such methods, relying on the notion of distance profiles, has been widely used in practice, but its theoretical understanding is limited. By delving into the statistical complexity of the previously proposed method based on distance profiles, we show that it suffers from the curse of dimensionality unless we make certain assumptions on the underlying metric measure spaces. Building on this insight, we propose and analyze a modified matching procedure that can be used to robustly match points under a certain probabilistic setting. We demonstrate the performance of the proposed methods using simulations and real data applications to complement the theoretical findings. As a result, we contribute to the literature by providing theoretical underpinnings of the matching procedures based on distance invariants like distance profiles, which have been widely used in practice but rarely analyzed theoretically.

stat.ME↗

arXiv2025

Fast operator learning for mapping correlations

We propose a fast, optimization-free method for learning the transition operators of high-dimensional Markov processes. The central idea is to perform a Galerkin projection of the transition operator to a suitable set of low-order bases that capture the correlations between the dimensions. Such a discretized operator can be obtained from moments corresponding to our choice of basis without curse of dimensionality. Furthermore, by exploiting its low-rank structure and the spatial decay of correlations, we can obtain a compressed representation with computational complexity of order $\mathcal{O}(dN)$, where $d$ is the dimensionality and $N$ is the sample size. We further theoretically analyze the approximation error of the proposed compressed representation. We numerically demonstrate that the learned operator allows efficient prediction of future events and solving high-dimensional boundary value problems. This gives rise to a simple linear algebraic method for high-dimensional rare-events simulations.

math.NA↗

arXiv2025

Combining Monte Carlo and Tensor-network Methods for Partial Differential Equations via Sketching

In this paper, we propose a general framework for solving high-dimensional partial differential equations with tensor networks. Our approach uses Monte-Carlo simulations to update the solution and re-estimates the new solution from samples as a tensor-network using a recently proposed tensor train sketching technique. We showcase the versatility and flexibility of our approach by applying it to two specific scenarios: simulating the Fokker-Planck equation through Langevin dynamics and quantum imaginary time evolution via auxiliary-field quantum Monte Carlo. We also provide convergence guarantees and numerical experiments to demonstrate the efficacy of the proposed method.

math.NA↗

arXiv2025

A Model-Guided Neural Network Method for the Inverse Scattering Problem

Inverse medium scattering is an ill-posed, nonlinear wave-based imaging problem arising in medical imaging, remote sensing, and non-destructive testing. Machine learning (ML) methods offer increased inference speed and flexibility in capturing prior knowledge of imaging targets relative to classical optimization-based approaches; however, they perform poorly in regimes where the scattering behavior is highly nonlinear. A key limitation is that ML methods struggle to incorporate the physics governing the scattering process, which are typically inferred implicitly from the training data or loosely enforced via architectural design. In this paper, we present a method that endows a machine learning framework with explicit knowledge of problem physics, in the form of a differentiable solver representing the forward model. The proposed method progressively refines reconstructions of the scattering potential using measurements at increasing wave frequencies, following a classical strategy to stabilize recovery. Empirically, we find that our method provides high-quality reconstructions at a fraction of the computational or sampling costs of competing approaches.

physics.comp-ph↗

arXiv2025

A Bregman ADMM for Bethe variational problem

In this work, we propose a novel Bregman ADMM with nonlinear dual update to solve the Bethe variational problem (BVP), a key optimization formulation in graphical models and statistical physics. Our algorithm provides rigorous convergence guarantees, even if the objective function of BVP is non-convex and non-Lipschitz continuous on the boundary. A central result of our analysis is proving that the entries in local minima of BVP are strictly positive, effectively resolving non-smoothness issues caused by zero entries. Beyond theoretical guarantees, the algorithm possesses high level of separability and parallelizability to achieve highly efficient subproblem computation. Our Bregman ADMM can be easily extended to solve the quantum Bethe variational problem. Numerical experiments are conducted to validate the effectiveness and robustness of the proposed method. Based on this research, we have released an open-source package of the proposed method at https://github.com/TTYmath/BADMM-BVP.

math.OC↗