arXiv subjects

Lexing Ying

Publications and source records attributed to Lexing Ying.

At least 55 records · Page 3Linked to original sources

arXiv2024

A note on continuous-time online learning

In online learning, the data is provided in a sequential order, and the goal of the learner is to make online decisions to minimize overall regrets. This note is concerned with continuous-time models and algorithms for several online learning problems: online linear optimization, adversarial bandit, and adversarial linear bandit. For each problem, we extend the discrete-time algorithm to the continuous-time setting and provide a concise proof of the optimal regret bound.

stat.ML↗

arXiv2024

Quantum wave packet transforms with compact frequency support

Different kinds of wave packet transforms are widely used for extracting multi-scale structures in signal processing tasks. This paper introduces the quantum circuit implementation of a broad class of wave packets, including Gabor atoms and wavelets, with compact frequency support. Our approach operates in the frequency space, involving reallocation and reshuffling of signals tailored for manipulation on quantum computers. The resulting implementation is different from the existing quantum algorithms for spatially compactly supported wavelets and can be readily extended to quantum transforms of other wave packets with compact frequency support.

quant-ph↗

arXiv2024

Quantum Hamiltonian Learning for the Fermi-Hubbard Model

This work proposes a protocol for Fermionic Hamiltonian learning. For the Hubbard model defined on a bounded-degree graph, the Heisenberg-limited scaling is achieved while allowing for state preparation and measurement errors. To achieve $ε$-accurate estimation for all parameters, only $\tilde{\mathcal{O}}(ε^{-1})$ total evolution time is needed, and the constant factor is independent of the system size. Moreover, our method only involves simple one or two-site Fermionic manipulations, which is desirable for experiment implementation.

quant-ph↗

arXiv2024

Orthogonal Bootstrap: Efficient Simulation of Input Uncertainty

Bootstrap is a popular methodology for simulating input uncertainty. However, it can be computationally expensive when the number of samples is large. We propose a new approach called \textbf{Orthogonal Bootstrap} that reduces the number of required Monte Carlo replications. We decomposes the target being simulated into two parts: the \textit{non-orthogonal part} which has a closed-form result known as Infinitesimal Jackknife and the \textit{orthogonal part} which is easier to be simulated. We theoretically and numerically show that Orthogonal Bootstrap significantly reduces the computational cost of Bootstrap while improving empirical accuracy and maintaining the same width of the constructed interval.

stat.ME↗

arXiv2024

A perturbative analysis for noisy spectral estimation

Spectral estimation is a fundamental task in signal processing. Recent algorithms in quantum phase estimation are concerned with the large noise, large frequency regime of the spectral estimation problem. The recent work in Ding-Epperly-Lin-Zhang shows that the ESPRIT algorithm exhibits superconvergence behavior for the spike locations in terms of the maximum frequency. This note provides a perturbative analysis to explain this behavior. It also extends the discussion to the case where the noise grows with the sampling frequency.

math.NA↗

arXiv2024

Solving high-dimensional Kolmogorov backward equations with functional hierarchical tensor operators

Solving high-dimensional partial differential equations necessitates methods free of exponential scaling in the dimension of the problem. This work introduces a tensor network approach for the Kolmogorov backward equation via approximating directly the Markov operator. We show that the high-dimensional Markov operator can be obtained under a functional hierarchical tensor (FHT) ansatz with a hierarchical sketching algorithm. When the terminal condition admits an FHT ansatz, the proposed operator outputs an FHT ansatz for the PDE solution through an efficient functional tensor network contraction procedure. In addition, the proposed operator-based approach also provides an efficient way to solve the Kolmogorov forward equation when the initial distribution is in an FHT ansatz. We apply the proposed approach successfully to two challenging time-dependent Ginzburg-Landau models with hundreds of variables.

math.NA↗

arXiv2024

Eigenmatrix for unstructured sparse recovery

This note considers the unstructured sparse recovery problems in a general form. Examples include rational approximation, spectral function estimation, Fourier inversion, Laplace inversion, and sparse deconvolution. The main challenges are the noise in the sample values and the unstructured nature of the sample locations. This note proposes the eigenmatrix, a data-driven construction with desired approximate eigenvalues and eigenvectors. The eigenmatrix offers a new way for these sparse recovery problems. Numerical results are provided to demonstrate the efficiency of the proposed method.

math.NA↗

arXiv2024

A Sinkhorn-type Algorithm for Constrained Optimal Transport

Entropic optimal transport (OT) and the Sinkhorn algorithm have made it practical for machine learning practitioners to perform the fundamental task of calculating transport distance between statistical distributions. In this work, we focus on a general class of OT problems under a combination of equality and inequality constraints. We derive the corresponding entropy regularization formulation and introduce a Sinkhorn-type algorithm for such constrained OT problems supported by theoretical guarantees. We first bound the approximation error when solving the problem through entropic regularization, which reduces exponentially with the increase of the regularization parameter. Furthermore, we prove a sublinear first-order convergence rate of the proposed Sinkhorn-type algorithm in the dual space by characterizing the optimization procedure with a Lyapunov function. To achieve fast and higher-order convergence under weak entropy regularization, we augment the Sinkhorn-type algorithm with dynamic regularization scheduling and second-order acceleration. Overall, this work systematically combines recent theoretical and numerical advances in entropic optimal transport with the constrained case, allowing practitioners to derive approximate transport plans in complex scenarios.

math.OC↗

arXiv2024

Multidimensional unstructured sparse recovery via eigenmatrix

This note considers the multidimensional unstructured sparse recovery problems. Examples include Fourier inversion and sparse deconvolution. The eigenmatrix is a data-driven construction with desired approximate eigenvalues and eigenvectors proposed for the one-dimensional problems. This note extends the eigenmatrix approach to multidimensional problems. Numerical results are provided to demonstrate the performance of the proposed method.

math.NA↗

arXiv2024

A sublinear-time randomized algorithm for column and row subset selection based on strong rank-revealing QR factorizations

In this work, we analyze a sublinear-time algorithm for selecting a few rows and columns of a matrix for low-rank approximation purposes. The algorithm is based on an initial uniformly random selection of rows and columns, followed by a refinement of this choice using a strong rank-revealing QR factorization. We prove bounds on the error of the corresponding low-rank approximation (more precisely, the CUR approximation error) when the matrix is a perturbation of a low-rank matrix that can be factorized into the product of matrices with suitable incoherence and/or sparsity assumptions.

math.NA↗

arXiv2024

Convergence Analysis of Discrete Diffusion Model: Exact Implementation through Uniformization

Diffusion models have achieved huge empirical success in data generation tasks. Recently, some efforts have been made to adapt the framework of diffusion models to discrete state space, providing a more natural approach for modeling intrinsically discrete data, such as language and graphs. This is achieved by formulating both the forward noising process and the corresponding reversed process as Continuous Time Markov Chains (CTMCs). In this paper, we investigate the theoretical properties of the discrete diffusion model. Specifically, we introduce an algorithm leveraging the uniformization of continuous Markov chains, implementing transitions on random time points. Under reasonable assumptions on the learning of the discrete score function, we derive Total Variation distance and KL divergence guarantees for sampling from any distribution on a hypercube. Our results align with state-of-the-art achievements for diffusion models in $\mathbb{R}^d$ and further underscore the advantages of discrete diffusion models in comparison to the $\mathbb{R}^d$ setting.

stat.ML↗

arXiv2024

Understanding the Generalization Benefits of Late Learning Rate Decay

Why do neural networks trained with large learning rates for a longer time often lead to better generalization? In this paper, we delve into this question by examining the relation between training and testing loss in neural networks. Through visualization of these losses, we note that the training trajectory with a large learning rate navigates through the minima manifold of the training loss, finally nearing the neighborhood of the testing loss minimum. Motivated by these findings, we introduce a nonlinear model whose loss landscapes mirror those observed for real neural networks. Upon investigating the training process using SGD on our model, we demonstrate that an extended phase with a large learning rate steers our model towards the minimum norm solution of the training loss, which may achieve near-optimal generalization, thereby affirming the empirically observed benefits of late learning rate decay.

cs.LG↗

arXiv2024

Accelerating Sinkhorn Algorithm with Sparse Newton Iterations

Computing the optimal transport distance between statistical distributions is a fundamental task in machine learning. One remarkable recent advancement is entropic regularization and the Sinkhorn algorithm, which utilizes only matrix scaling and guarantees an approximated solution with near-linear runtime. Despite the success of the Sinkhorn algorithm, its runtime may still be slow due to the potentially large number of iterations needed for convergence. To achieve possibly super-exponential convergence, we present Sinkhorn-Newton-Sparse (SNS), an extension to the Sinkhorn algorithm, by introducing early stopping for the matrix scaling steps and a second stage featuring a Newton-type subroutine. Adopting the variational viewpoint that the Sinkhorn algorithm maximizes a concave Lyapunov potential, we offer the insight that the Hessian matrix of the potential function is approximately sparse. Sparsification of the Hessian results in a fast $O(n^2)$ per-iteration complexity, the same as the Sinkhorn algorithm. In terms of total iteration count, we observe that the SNS algorithm converges orders of magnitude faster across a wide range of practical cases, including optimal transportation between empirical distributions and calculating the Wasserstein $W_1, W_2$ distance of discretized densities. The empirical performance is corroborated by a rigorous bound on the approximate sparsity of the Hessian matrix.

math.OC↗

arXiv2024

Multimodal Sampling via Approximate Symmetries

Sampling from multimodal distributions is a challenging task in scientific computing. When a distribution has an exact symmetry between the modes, direct jumps among them can accelerate the samplings significantly. However, the distributions from most applications do not have exact symmetries. This paper considers the distributions with approximate symmetries. We first construct an exactly symmetric reference distribution from the target one by averaging over the group orbit associated with the approximate symmetry. Next, we can apply the multilevel Monte Carlo methods by constructing a continuation path between the reference and target distributions. We discuss how to implement these steps with annealed importance sampling and tempered transitions. Compared with traditional multilevel methods, the proposed approach can be more effective since the reference and target distributions are much closer. Numerical results of the Ising models are presented to illustrate the efficiency of the proposed method.

math.NA↗

arXiv2023

Solving high-dimensional Fokker-Planck equation with functional hierarchical tensor

This work is concerned with solving high-dimensional Fokker-Planck equations with the novel perspective that solving the PDE can be reduced to independent instances of density estimation tasks based on the trajectories sampled from its associated particle dynamics. With this approach, one sidesteps error accumulation occurring from integrating the PDE dynamics on a parameterized function class. This approach significantly simplifies deployment, as one is free of the challenges of implementing loss terms based on the differential equation. In particular, we introduce a novel class of high-dimensional functions called the functional hierarchical tensor (FHT). The FHT ansatz leverages a hierarchical low-rank structure, offering the advantage of linearly scalable runtime and memory complexity relative to the dimension count. We introduce a sketching-based technique that performs density estimation over particles simulated from the particle dynamics associated with the equation, thereby obtaining a representation of the Fokker-Planck solution in terms of our ansatz. We apply the proposed approach successfully to three challenging time-dependent Ginzburg-Landau models with hundreds of variables.

math.NA↗

arXiv2023

Statistical Spatially Inhomogeneous Diffusion Inference

Inferring a diffusion equation from discretely-observed measurements is a statistical challenge of significant importance in a variety of fields, from single-molecule tracking in biophysical systems to modeling financial instruments. Assuming that the underlying dynamical process obeys a $d$-dimensional stochastic differential equation of the form $$\mathrm{d}\boldsymbol{x}_t=\boldsymbol{b}(\boldsymbol{x}_t)\mathrm{d} t+Σ(\boldsymbol{x}_t)\mathrm{d}\boldsymbol{w}_t,$$ we propose neural network-based estimators of both the drift $\boldsymbol{b}$ and the spatially-inhomogeneous diffusion tensor $D = ΣΣ^{T}$ and provide statistical convergence guarantees when $\boldsymbol{b}$ and $D$ are $s$-Hölder continuous. Notably, our bound aligns with the minimax optimal rate $N^{-\frac{2s}{2s+d}}$ for nonparametric function estimation even in the presence of correlation within observational data, which necessitates careful handling when establishing fast-rate generalization bounds. Our theoretical results are bolstered by numerical experiments demonstrating accurate inference of spatially-inhomogeneous diffusion tensors.

stat.ML↗

arXiv2023

On low-depth algorithms for quantum phase estimation

Quantum phase estimation is one of the critical building blocks of quantum computing. For early fault-tolerant quantum devices, it is desirable for a quantum phase estimation algorithm to (1) use a minimal number of ancilla qubits, (2) allow for inexact initial states with a significant mismatch, (3) achieve the Heisenberg limit for the total resource used, and (4) have a diminishing prefactor for the maximum circuit length when the overlap between the initial state and the target state approaches one. In this paper, we prove that an existing algorithm from quantum metrology can achieve the first three requirements. As a second contribution, we propose a modified version of the algorithm that also meets the fourth requirement, which makes it particularly attractive for early fault-tolerant quantum devices.

quant-ph↗

arXiv2023

On adaptive low-depth quantum algorithms for robust multiple-phase estimation

This paper is an algorithmic study of quantum phase estimation with multiple eigenvalues. We present robust multiple-phase estimation (RMPE) algorithms with Heisenberg-limited scaling. The proposed algorithms improve significantly from the idea of single-phase estimation methods by combining carefully designed signal processing routines and an adaptive determination of runtime amplifying factors. They address both the {\em integer-power} model, where the unitary $U$ is given as a black box with only integer runtime accessible, and the {\em real-power} model, where $U$ is defined through a Hamiltonian $H$ by $U = \exp(-2π\mathrm{i} H)$ with any real runtime allowed. These algorithms are particularly suitable for early fault-tolerant quantum computers in the following senses: (1) a minimal number of ancilla qubits are used, (2) an imperfect initial state with a significant residual is allowed, (3) the prefactor in the maximum runtime can be arbitrarily small given that the residual is sufficiently small and a gap among the dominant eigenvalues is known in advance. Even if the eigenvalue gap does not exist, the proposed RMPE algorithms can achieve the Heisenberg limit while maintaining (1) and (2).

quant-ph↗