Related papers: Efficient Manifold and Subspace Approximations wit…
Manifold hypothesis states that data points in high-dimensional space actually lie in close vicinity of a manifold of much lower dimension. In many cases this hypothesis was empirically verified and used to enhance unsupervised and…
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…
Sparse principal component analysis (PCA) and sparse canonical correlation analysis (CCA) are two essential techniques from high-dimensional statistics and machine learning for analyzing large-scale data. Both problems can be formulated as…
Single-cell RNA sequencing (scRNA-seq) has revolutionized our ability to analyze gene expression at the cellular level. By providing data on gene expression for each individual cell, scRNA-seq generates large datasets with thousands of…
Anisotropic decompositions using representation systems such as curvelets, contourlet, or shearlets have recently attracted significantly increased attention due to the fact that they were shown to provide optimally sparse approximations of…
This paper describes the systematic application of local topological methods for detecting interfaces and related anomalies in complicated high-dimensional data. By examining the topology of small regions around each point, one can…
The manifold hypothesis, which assumes that data lies on or close to an unknown manifold of low intrinsic dimension, is a staple of modern machine learning research. However, recent work has shown that real-world data exhibits distinct…
Fitting an unknown number of hyperplanes to data is a fundamental yet challenging problem in machine learning, characterized by its non-convexity, non-differentiability, and unknown model order. Existing approaches often struggle with local…
Consider a set of $n$ data points in the Euclidean space $\mathbb{R}^d$. This set is called dataset in machine learning and data science. Manifold hypothesis states that the dataset lies on a low-dimensional submanifold with high…
We introduce an adaptive scattered data fitting scheme as extension of local least squares approximations to hierarchical spline spaces. To efficiently deal with non-trivial data configurations, the local solutions are described in terms of…
When analyzing parametric statistical models, a useful approach consists in modeling geometrically the parameter space. However, even for very simple and commonly used hierarchical models like statistical mixtures or stochastic deep neural…
Many representation systems on the sphere have been proposed in the past, such as spherical harmonics, wavelets, or curvelets. Each of these data representations is designed to extract a specific set of features, and choosing the best fixed…
We propose a new sparse principal component analysis (SPCA) method in which the solutions are obtained by projecting the full cardinality principal components onto subsets of variables. The resulting components are guaranteed to explain a…
We propose a method to reconstruct and cluster incomplete high-dimensional data lying in a union of low-dimensional subspaces. Exploring the sparse representation model, we jointly estimate the missing data while imposing the intrinsic…
In this paper, we propose a novel lower dimensional representation of a shape sequence. The proposed dimension reduction is invertible and computationally more efficient in comparison to other related works. Theoretically, the differential…
Riemannian diffusion models draw inspiration from standard Euclidean space diffusion models to learn distributions on general manifolds. Unfortunately, the additional geometric complexity renders the diffusion transition term inexpressible…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
Even with the rise in popularity of over-parameterized models, simple dimensionality reduction and clustering methods, such as PCA and k-means, are still routinely used in an amazing variety of settings. A primary reason is the combination…
In this paper, we study the estimation of partially linear models for spatial data distributed over complex domains. We use bivariate splines over triangulations to represent the nonparametric component on an irregular two-dimensional…
We introduce a generalized attention mechanism for spherical domains, enabling Transformer architectures to natively process data defined on the two-dimensional sphere - a critical need in fields such as atmospheric physics, cosmology, and…