arXiv subjects

Yuehaw Khoo

Publications and source records attributed to Yuehaw Khoo.

At least 37 records · Page 2Linked to original sources

arXiv2025

Method of Moments for Estimation of Noisy Curves

In this paper, we study the problem of recovering a ground truth high dimensional piecewise linear curve $C^*(t):[0, 1]\to\mathbb{R}^d$ from a high noise Gaussian point cloud with covariance $σ^2I$ centered around the curve. We establish that the sample complexity of recovering $C^*$ from data scales with order at least $σ^6$. We then show that recovery of a piecewise linear curve from the third moment is locally well-posed, and hence $O(σ^6)$ samples is also sufficient for recovery. We propose methods to recover a curve from data based on a fitting to the third moment tensor with a careful initialization strategy and conduct some numerical experiments verifying the ability of our methods to recover curves. All code for our numerical experiments is publicly available on GitHub.

math.ST↗

arXiv2025

Subspace method of moments for ab initio 3-D single-particle cryo-EM reconstruction

Cryo-electron microscopy (cryo-EM) is a widely used technique for recovering the 3-D structure of biological molecules from a large number of experimentally generated noisy 2-D tomographic projection images of the 3-D structure, taken from unknown viewing angles. Through computationally intensive algorithms, these observed images are processed to reconstruct the 3-D structures. Many popular computational methods rely on estimating the unknown angles as part of the reconstruction process, which becomes particularly challenging at low signal-to-noise ratios. The method of moments (MoM) offers an alternative approach that circumvents the estimation of viewing orientations of individual projection images by instead estimating the underlying distribution of the viewing angles, and is robust to noise given sufficiently many images. However, the method of moments typically entails computing higher-order moments of the projection images, incurring significant computational and memory costs. To mitigate this, we propose a new approach called the subspace method of moments (SubspaceMoM), which compresses the first three moments using data-driven low-rank tensor techniques as well as expansion into a suitable function basis. The compressed moments can be efficiently computed from the set of projection images using numerical quadrature and can be employed to jointly reconstruct the 3-D structure and the distribution of viewing orientations. We illustrate the practical applicability of SubspaceMoM through numerical experiments using up to the third-order moment on synthetic datasets with a simplified cryo-EM image formation model, which significantly improves the reconstruction resolution compared to previous MoM approaches.

math.NA↗

arXiv2025

Tensor Density Estimator by Convolution-Deconvolution

We propose a linear algebraic framework for performing density estimation. It consists of three simple steps: convolving the empirical distribution with certain smoothing kernels to remove the exponentially large variance; compressing the empirical distribution after convolution as a tensor train, with efficient tensor decomposition algorithms; and finally, applying a deconvolution step to recover the estimated density from such tensor-train representation. Numerical results demonstrate the high accuracy and efficiency of the proposed methods.

math.NA↗

arXiv2025

Solving cluster moment relaxation with hierarchical matrix

Convex relaxation methods are powerful tools for studying the lowest energy of many-body problems. By relaxing the representability conditions for marginals to a set of local constraints, along with a global semidefinite constraint, a polynomial-time solvable semidefinite program (SDP) that provides a lower bound for the energy can be derived. In this paper, we propose accelerating the solution of such an SDP relaxation by imposing a hierarchical structure on the positive semidefinite (PSD) primal and dual variables. Furthermore, these matrices can be updated efficiently using the algebra of the compressed representations within an augmented Lagrangian method. We achieve quadratic and even near-linear time per-iteration complexity. Through experimentation on the quantum transverse field Ising model, we showcase the capability of our approach to provide a sufficiently accurate lower bound for the exact ground-state energy.

math.OC↗

arXiv2025

Optimization-Free Diffusion Model -- A Perturbation Theory Approach

Diffusion models have emerged as a powerful framework in generative modeling, typically relying on optimizing neural networks to estimate the score function via forward SDE simulations. In this work, we propose an alternative method that is both optimization-free and forward SDE-free. By expanding the score function in a sparse set of eigenbasis of the backward Kolmogorov operator associated with the diffusion process, we reformulate score estimation as the solution to a linear system, avoiding iterative optimization and time-dependent sample generation. We analyze the approximation error using perturbation theory and demonstrate the effectiveness of our method on high-dimensional Boltzmann distributions and real-world datasets.

math.NA↗

arXiv2025

On a spherically lifted spin model at finite temperature

We investigate an $n$-vector model over $k$ sites with generic pairwise interactions and spherical constraints. The model is a lifting of the Ising model whereby the support of the spin is lifted to a hypersphere. We show that the $n$-vector model converges to a limiting distribution at a rate of $n^{-1/2 + o(1)}$. We show that the limiting distribution for $n \to \infty$ is determined by the solution of an equality-constrained maximization task over positive definite matrices. We prove that the obtained maximal value and maximizer, respectively, give rise to the free energy and correlation function of the limiting distribution. In the finite temperature regime, the maximization task is a log-determinant regularization of the semidefinite program (SDP) in the Goemans-Williamson algorithm. Moreover, the inverse temperature determines the regularization strength, with the zero temperature limit converging to the SDP in Goemans-Williamson. Our derivation draws a curious connection between the semidefinite relaxation of integer programming and the spherical lifting of sampling on a hypercube. To the authors' best knowledge, this work is the first to solve the setting of fixed $k$ and infinite $n$ under unstructured pairwise interactions.

math.PR↗

arXiv2025

Initialization and training of matrix product state probabilistic models

Modeling probability distributions via the wave function of a quantum state is central to quantum-inspired generative modeling and quantum state tomography (QST). We investigate a common failure mode in training randomly initialized matrix product states (MPS) using gradient descent. The results show that the trained MPS models do not accurately predict the strong interactions between boundary sites in periodic spin chain models. In the case of the Born machine algorithm, we further identify a causality trap, where the trained MPS models resemble causal models that ignore the non-local correlations in the true distribution. We propose two complementary strategies to overcome the training failure -- one through optimization and one through initialization. First, we develop a natural gradient descent (NGD) method, which approximately simulates the gradient flow on tensor manifolds and significantly enhances training efficiency. Numerical experiments show that NGD avoids local minima in both Born machines and in general MPS tomography. Remarkably, we show that NGD with line search can converge to the global minimum in only a few iterations. Second, for the BM algorithm, we introduce a warm-start initialization based on the TTNS-Sketch algorithm. We show that gradient descent under a warm initialization does not encounter the causality trap and admits rapid convergence to the ground truth.

math.NA↗

arXiv2025

Multivariate Density Estimation via Variance-Reduced Sketching

Multivariate density estimation is of great interest in various scientific and engineering disciplines. In this work, we introduce a new framework called Variance-Reduced Sketching (VRS), specifically designed to estimate multivariate density functions with a reduced curse of dimensionality. Our VRS framework conceptualizes multivariate functions as infinite-size matrices/tensors, and facilitates a new sketching technique motivated by the numerical linear algebra literature to reduce the variance in density estimation problems. We demonstrate the robust numerical performance of VRS through a series of simulated experiments and real-world data applications. Notably, VRS shows remarkable improvement over existing neural network density estimators and classical kernel methods in numerous distribution models. Additionally, we offer theoretical justifications for VRS to support its ability to deliver density estimation with a reduced curse of dimensionality.

stat.ML↗

arXiv2025

Multi-Frequency Progressive Refinement for Learned Inverse Scattering

Interpreting scattered acoustic and electromagnetic wave patterns is a computational task that enables remote imaging in a number of important applications, including medical imaging, geophysical exploration, sonar and radar detection, and nondestructive testing of materials. However, accurately and stably recovering an inhomogeneous medium from far-field scattered wave measurements is a computationally difficult problem, due to the nonlinear and non-local nature of the forward scattering process. We design a neural network, called Multi-Frequency Inverse Scattering Network (MFISNet), and a training method to approximate the inverse map from far-field scattered wave measurements at multiple frequencies. We consider three variants of MFISNet, with the strongest performing variant inspired by the recursive linearization method--a commonly used technique for stably inverting scattered wavefield data--that progressively refines the estimate with higher frequency content. MFISNet outperforms past methods in regimes with high-contrast, heterogeneous large objects, and inhomogeneous unknown backgrounds.

physics.comp-ph↗

arXiv2024

S-SOS: Stochastic Sum-Of-Squares for Parametric Polynomial Optimization

Global polynomial optimization is an important tool across applied mathematics, with many applications in operations research, engineering, and physical sciences. In various settings, the polynomials depend on external parameters that may be random. We discuss a stochastic sum-of-squares (S-SOS) algorithm based on the sum-of squares hierarchy that constructs a series of semidefinite programs to jointly find strict lower bounds on the global minimum and extract candidates for parameterized global minimizers. We prove quantitative convergence of the hierarchy as the degree increases and use it to solve unconstrained and constrained polynomial optimization problems parameterized by random variables. By employing $n$-body priors from condensed matter physics to induce sparsity, we can use S-SOS to produce solutions and uncertainty intervals for sensor network localization problems containing up to 40 variables and semidefinite matrix sizes surpassing $800 \times 800$.

math.OC↗

arXiv2024

Augmented Lagrangian method for coupled-cluster

We propose to improve the convergence properties of the single-reference coupled cluster (CC) method through an augmented Lagrangian formalism. The conventional CC method changes a linear high-dimensional eigenvalue problem with exponential size into a problem of determining the roots of a nonlinear system of equations that has a manageable size. However, current numerical procedures for solving this system of equations to get the lowest eigenvalue suffer from two practical issues: First, solving the CC equations may not converge, and second, when converging, they may converge to other -- potentially unphysical -- states, which are stationary points of the CC energy expression. We show that both issues can be dealt with when a suitably defined energy is minimized in addition to solving the original CC equations. We further propose an augmented Lagrangian method for coupled cluster (alm-CC) to solve the resulting constrained optimization problem. We numerically investigate the proposed augmented Lagrangian formulation showing that the convergence towards the ground state is significantly more stable and that the optimization procedure is less susceptible to local minima. Furthermore, the computational cost of alm-CC is comparable to the conventional CC method.

physics.comp-ph↗

arXiv2024

Deep Neural-network Prior for Orbit Recovery from Method of Moments

Orbit recovery problems are a class of problems that often arise in practice and various forms. In these problems, we aim to estimate an unknown function after being distorted by a group action and observed via a known operator. Typically, the observations are contaminated with a non-trivial level of noise. Two particular orbit recovery problems of interest in this paper are multireference alignment and single-particle cryo-EM modelling. In order to suppress the noise, we suggest using the method of moments approach for both problems while introducing deep neural network priors. In particular, our neural networks should output the signals and the distribution of group elements, with moments being the input. In the multireference alignment case, we demonstrate the advantage of using the NN to accelerate the convergence for the reconstruction of signals from the moments. Finally, we use our method to reconstruct simulated and biological volumes in the cryo-EM setting.

stat.ME↗

arXiv2023

Convex Relaxation for Fokker-Planck

We propose an approach to directly estimate the moments or marginals for a high-dimensional equilibrium distribution in statistical mechanics, via solving the high-dimensional Fokker-Planck equation in terms of low-order cluster moments or marginals. With this approach, we bypass the exponential complexity of estimating the full high-dimensional distribution and directly solve the simplified partial differential equations for low-order moments/marginals. Moreover, the proposed moment/marginal relaxation is fully convex and can be solved via off-the-shelf solvers. We further propose a time-dependent version of the convex programs to study non-equilibrium dynamics. We show the proposed method can recover the meanfield approximation of an equilibrium density. Numerical results are provided to demonstrate the performance of the proposed algorithm for high-dimensional systems.

math.NA↗

arXiv2023

Joint Community Detection and Rotational Synchronization via Semidefinite Programming

In the presence of heterogeneous data, where randomly rotated objects fall into multiple underlying categories, it is challenging to simultaneously classify them into clusters and synchronize them based on pairwise relations. This gives rise to the joint problem of community detection and synchronization. We propose a series of semidefinite relaxations, and prove their exact recovery when extending the celebrated stochastic block model to this new setting where both rotations and cluster identities are to be determined. Numerical experiments demonstrate the efficacy of our proposed algorithms and confirm our theoretical result which indicates a sharp phase transition for exact recovery.

stat.ML↗

arXiv2023

Tensorizing flows: a tool for variational inference

Fueled by the expressive power of deep neural networks, normalizing flows have achieved spectacular success in generative modeling, or learning to draw new samples from a distribution given a finite dataset of training samples. Normalizing flows have also been applied successfully to variational inference, wherein one attempts to learn a sampler based on an expression for the log-likelihood or energy function of the distribution, rather than on data. In variational inference, the unimodality of the reference Gaussian distribution used within the normalizing flow can cause difficulties in learning multimodal distributions. We introduce an extension of normalizing flows in which the Gaussian reference is replaced with a reference distribution that is constructed via a tensor network, specifically a matrix product state or tensor train. We show that by combining flows with tensor networks on difficult variational inference tasks, we can improve on the results obtained by using either tool without the other.

cs.LG↗

arXiv2023

Autocorrelation analysis for cryo-EM with sparsity constraints: Improved sample complexity and projection-based algorithms

The number of noisy images required for molecular reconstruction in single-particle cryo-electron microscopy (cryo-EM) is governed by the autocorrelations of the observed, randomly-oriented, noisy projection images. In this work, we consider the effect of imposing sparsity priors on the molecule. We use techniques from signal processing, optimization, and applied algebraic geometry to obtain new theoretical and computational contributions for this challenging non-linear inverse problem with sparsity constraints. We prove that molecular structures modeled as sums of Gaussians are uniquely determined by the second-order autocorrelation of their projection images, implying that the sample complexity is proportional to the square of the variance of the noise. This theory improves upon the non-sparse case, where the third-order autocorrelation is required for uniformly-oriented particle images and the sample complexity scales with the cube of the noise variance. Furthermore, we build a computational framework to reconstruct molecular structures which are sparse in the wavelet basis. This method combines the sparse representation for the molecule with projection-based techniques used for phase retrieval in X-ray crystallography.

math.NA↗

arXiv2022

High-dimensional density estimation with tensorizing flow

We propose the tensorizing flow method for estimating high-dimensional probability density functions from the observed data. The method is based on tensor-train and flow-based generative modeling. Our method first efficiently constructs an approximate density in the tensor-train form via solving the tensor cores from a linear system based on the kernel density estimators of low-dimensional marginals. We then train a continuous-time flow model from this tensor-train density to the observed empirical distribution by performing a maximum likelihood estimation. The proposed method combines the optimization-less feature of the tensor-train with the flexibility of the flow-based generative models. Numerical results are included to demonstrate the performance of the proposed method.

cs.LG↗

arXiv2022

A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization

Community detection and orthogonal group synchronization are both fundamental problems with a variety of important applications in science and engineering. In this work, we consider the joint problem of community detection and orthogonal group synchronization which aims to recover the communities and perform synchronization simultaneously. To this end, we propose a simple algorithm that consists of a spectral decomposition step followed by a blockwise column pivoted QR factorization (CPQR). The proposed algorithm is efficient and scales linearly with the number of edges in the graph. We also leverage the recently developed `leave-one-out' technique to establish a near-optimal guarantee for exact recovery of the cluster memberships and stable recovery of the orthogonal transforms. Numerical experiments demonstrate the efficiency and efficacy of our algorithm and confirm our theoretical characterization of it.

stat.ML↗