Related papers: Learning Manifolds with K-Means and K-Flats

K-means is a classical clustering algorithm with wide applications. However, soft K-means, or fuzzy c-means at m=1, remains unsolved since 1981. To address this challenging open problem, we propose a novel clustering model, i.e.…

Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches…

We introduce a method for constructing skills capable of solving tasks drawn from a distribution of parameterized reinforcement learning problems. The method draws example tasks from a distribution of interest and uses the corresponding…

Consider the set of solutions to a system of polynomial equations in many variables. An algebraic manifold is an open submanifold of such a set. We introduce a new method for computing integrals and sampling from distributions on algebraic…

Clustering is a popular form of unsupervised learning for geometric data. Unfortunately, many clustering algorithms lead to cluster assignments that are hard to explain, partially because they depend on all the features of the data in a…

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a…

The manifold hypothesis suggests that high-dimensional data often lie on or near a low-dimensional manifold. Estimating the dimension of this manifold is essential for leveraging its structure, yet existing work on dimension estimation is…

K-fold cross-validation is a widely used tool for assessing classifier performance. The reproducibility crisis faced by artificial intelligence partly results from the irreproducibility of reported k-fold cross-validation-based performance…

We present a framework for learning probability distributions on topologically non-trivial manifolds, utilizing normalizing flows. Current methods focus on manifolds that are homeomorphic to Euclidean space, enforce strong structural priors…

Choosing the right representation for geometry is crucial for making 3D models compatible with existing applications. Focusing on piecewise-smooth man-made shapes, we propose a new representation that is usable in conventional CAD modeling…

Analyzing large volumes of high-dimensional data requires dimensionality reduction: finding meaningful low-dimensional structures hidden in their high-dimensional observations. Such practice is needed in atomistic simulations of complex…

Manifold Learning is a class of algorithms seeking a low-dimensional non-linear representation of high-dimensional data. Thus manifold learning algorithms are, at least in theory, most applicable to high-dimensional data and sample sizes to…

A common belief in high-dimensional data analysis is that data are concentrated on a low-dimensional manifold. This motivates simultaneous dimension reduction and regression on manifolds. We provide an algorithm for learning gradients on…

K-means clustering is a workhorse of unsupervised learning, but it is notoriously brittle to outliers, distribution shifts, and limited sample sizes. Viewing k-means as Lloyd--Max quantization of the empirical distribution, we develop a…

We investigate compact complex manifolds endowed with SKT or balanced metrics. In each case we define a new functional whose critical points are proved to be precisely the K\"ahler metrics, if any, on the manifold. As general manifolds of…

We are interested in derivative-free optimization of high-dimensional functions. The sample complexity of existing methods is high and depends on problem dimensionality, unlike the dimensionality-independent rates of first-order methods.…

In this study, we develop a method for multi-task manifold learning. The method aims to improve the performance of manifold learning for multiple tasks, particularly when each task has a small number of samples. Furthermore, the method also…

Solving geometric optimization problems over uncertain data have become increasingly important in many applications and have attracted a lot of attentions in recent years. In this paper, we study two important geometric optimization…

This paper has two purposes. First it partially extends the result in the author's previous work concerning the asymptotic expansion of the Tian-Yau metrics, by considering a slightly larger class of quasi-projective manifolds. This text is…

We survey some of the recent advances in mean estimation and regression function estimation. In particular, we describe sub-Gaussian mean estimators for possibly heavy-tailed data both in the univariate and multivariate settings. We focus…