Related papers: Geometry-Aware Maximum Likelihood Estimation of In…

We propose a framework of principal manifolds to model high-dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve…

Accurate statistical inference in logistic regression models remains a critical challenge when the ratio between the number of parameters and sample size is not negligible. This is because approximations based on either classical asymptotic…

The size of datasets has been increasing rapidly both in terms of number of variables and number of events. As a result, the empty space phenomenon and the curse of dimensionality complicate the extraction of useful information. But, in…

We introduce numerical algebraic geometry methods for computing lower bounds on the reach, local feature size, and the weak feature size of the real part of an equidimensional and smooth algebraic variety using the variety's defining…

Probabilistic Latent Variable Models (LVMs) excel at modeling complex, high-dimensional data through lower-dimensional representations. Recent advances show that equipping these latent representations with a Riemannian metric unlocks…

Monte Carlo maximum likelihood (MCML) provides an elegant approach to find maximum likelihood estimators (MLEs) for latent variable models. However, MCML algorithms are computationally expensive when the latent variables are…

We propose POLAR, a novel radar-guided depth estimation method that introduces polynomial fitting to efficiently transform scaleless depth predictions from pretrained monocular depth estimation (MDE) models into metric depth maps. Unlike…

For data living in a manifold $M\subseteq \mathbb{R}^m$ and a point $p\in M$ we consider a statistic $U_{k,n}$ which estimates the variance of the angle between pairs of vectors $X_i-p$ and $X_j-p$, for data points $X_i$, $X_j$, near $p$,…

We study the properties of nonparametric least squares regression using deep neural networks. We derive non-asymptotic upper bounds for the prediction error of the empirical risk minimizer of feedforward deep neural regression. Our error…

The need for efficiently comparing and representing datasets with unknown alignment spans various fields, from model analysis and comparison in machine learning to trend discovery in collections of medical datasets. We use manifold learning…

This paper considers the problem of finding a meaningful template function that represents the common pattern of a sample of curves. To address this issue, a novel algorithm based on a robust version of the isometric featuring mapping…

We develop a new method to estimate the area, and more generally the intrinsic volumes, of a compact subset $X$ of $\mathbb{R}^d$ from a set $Y$ that is close in the Hausdorff distance. This estimator enjoys a linear rate of convergence as…

The question of representation of 3D geometry is of vital importance when it comes to leveraging the recent advances in the field of machine learning for geometry processing tasks. For common unstructured surface meshes state-of-the-art…

Information about intrinsic dimension is crucial to perform dimensionality reduction, compress information, design efficient algorithms, and do statistical adaptation. In this paper we propose an estimator for the intrinsic dimension of a…

Manifold learning techniques have become increasingly valuable as data continues to grow in size. By discovering a lower-dimensional representation (embedding) of the structure of a dataset, manifold learning algorithms can substantially…

Traditional manifold learning algorithms assumed that the embedded manifold is globally or locally isometric to Euclidean space. Under this assumption, they divided manifold into a set of overlapping local patches which are locally…

This paper introduces the Intrinsic Dimension Estimating Autoencoder (IDEA), which identifies the underlying intrinsic dimension of a wide range of datasets whose samples lie on either linear or nonlinear manifolds. Beyond estimating the…

Autoencoders have emerged as powerful models for visualization and dimensionality reduction based on the fundamental assumption that high-dimensional data is generated from a low-dimensional manifold. A critical challenge in autoencoder…

We consider a problem of great practical interest: the repairing and recovery of a low-dimensional manifold embedded in high-dimensional space from noisy scattered data. Suppose that we observe a point cloud sampled from the low-dimensional…

Imbalanced classification presents a formidable challenge in machine learning, particularly when tabular datasets are plagued by noise and overlapping class boundaries. From a geometric perspective, the core difficulty lies in the…