Related papers: Manifold learning with bi-stochastic kernels
Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…
Diffusion models excel in content generation by implicitly learning the data manifold, yet they lack a practical method to leverage this manifold - unlike other deep generative models equipped with latent spaces. This paper introduces a…
In supervised learning with distributional inputs in the two-stage sampling setup, relevant to applications like learning-based medical screening or causal learning, the inputs (which are probability distributions) are not accessible in the…
We present a novel kernel-based machine learning algorithm for identifying the low-dimensional geometry of the effective dynamics of high-dimensional multiscale stochastic systems. Recently, the authors developed a mathematical framework…
Data sites selected from modeling high-dimensional problems often appear scattered in non-paternalistic ways. Except for sporadic clustering at some spots, they become relatively far apart as the dimension of the ambient space grows. These…
This work incorporates the multi-modality of the data distribution into a Gaussian Process regression model. We approach the problem from a discriminative perspective by learning, jointly over the training data, the target space variance in…
The logarithmic representation of infinitesimal generators is generalized to the cases when the evolution operator is unbounded. The generalized result is applicable to the representation of infinitesimal generators of unbounded evolution…
Recently, there has been much interest in spectral approaches to learning manifolds---so-called kernel eigenmap methods. These methods have had some successes, but their applicability is limited because they are not robust to noise. To…
Infinitesimal gradient boosting (Dombry and Duchamps, 2021) is defined as the vanishing-learning-rate limit of the popular tree-based gradient boosting algorithm from machine learning. It is characterized as the solution of a nonlinear…
Recent advances in diffusion models have demonstrated their remarkable ability to capture complex image distributions, but the geometric properties of the learned data manifold remain poorly understood. We address this gap by introducing a…
Geometric representation learning in preserving the intrinsic geometric and topological properties for discrete non-Euclidean data is crucial in scientific applications. Previous research generally mapped non-Euclidean discrete data into…
We establish the asymptotic normality of the regression estimator in a fixed-design setting when the errors are given by a field of dependent random variables. The result applies to martingale-difference or strongly mixing random fields. On…
We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in…
For the analysis of molecular processes, the estimation of time-scales, i.e., transition rates, is very important. Estimating the transition rates between molecular conformations is -- from a mathematical point of view -- an invariant…
Kernel density estimation is a widely used nonparametric approach to estimate an unknown distribution. Recent work in Bayesian predictive inference has considered stochastic processes formed by specifying the predictive distribution for the…
Statistical inference from high-dimensional data with low-dimensional structures has recently attracted lots of attention. In machine learning, deep generative modeling approaches implicitly estimate distributions of complex objects by…
We consider determinantal point processes on a compact complex manifold X in the limit of many particles. The correlation kernels of the processes are the Bergman kernels associated to a a high power of a given Hermitian holomorphic line…
Since its foundations, more than one hundred years ago, the field of structural biology has strived to understand and analyze the properties of molecules and their interactions by studying the structure that they take in 3D space. However,…
Random Fourier features (RFFs) provide a promising way for kernel learning in a spectral case. Current RFFs-based kernel learning methods usually work in a two-stage way. In the first-stage process, learning the optimal feature map is often…
We study the theoretical behavior of denoising score matching--the learning task associated to diffusion models--when the data distribution is supported on a low-dimensional manifold and the score is parameterized using a random feature…