相关论文: Persistent Cohomology and Circular Coordinates

Contrastive Learning (CL) has shown promising performance in collaborative filtering. The key idea is to generate augmentation-invariant embeddings by maximizing the Mutual Information between different augmented views of the same instance.…

This paper studies first the differential inequalities that make it possible to build a global theory of pseudo-holomorphic functions in the case of one or several complex variables. In the case of one complex dimension, we prove that the…

Hamiltonian systems of ordinary and partial differential equations are fundamental mathematical models spanning virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is…

We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…

We propose Path Signatures Logistic Regression (PSLR), a semi-parametric framework for classifying vector-valued functional data with scalar covariates. Classical functional logistic regression models rely on linear assumptions and fixed…

Similarity-driven multi-view linear reconstruction (SiMLR) is an algorithm that exploits inter-modality relationships to transform large scientific datasets into smaller, more well-powered and interpretable low-dimensional spaces. SiMLR…

Nonsingular estimation of high dimensional covariance matrices is an important step in many statistical procedures like classification, clustering, variable selection an future extraction. After a review of the essential background…

Modulo imaging enables high dynamic range (HDR) acquisition by cyclically wrapping saturated intensities, but accurate reconstruction remains challenging due to ambiguities between natural image edges and artificial wrap discontinuities.…

We propose a method for efficiently computing orientation-preserving and approximately continuous correspondences between non-rigid shapes, using the functional maps framework. We first show how orientation preservation can be formulated…

Persistence diagrams, which summarize the birth and death of homological features extracted from data, are employed as stable signatures for applications in image analysis and other areas. Besides simply considering the multiset of…

A number of recent studies have proposed that linear representations are appropriate for solving nonlinear dynamical systems with quantum computers, which fundamentally act linearly on a wave function in a Hilbert space. Linear…

Bilevel programming has recently received a great deal of attention due to its abundant applications in many areas. The optimal value function approach provides a useful reformulation of the bilevel problem, but its utility is often limited…

Embedding high-dimensional data onto a low-dimensional manifold is of both theoretical and practical value. In this paper, we propose to combine deep neural networks (DNN) with mathematics-guided embedding rules for high-dimensional data…

There has been a lot of interest in sufficient dimension reduction (SDR) methodologies as well as nonlinear extensions in the statistics literature. In this note, we use classical results regarding metric spaces and positive definite…

Many problems in machine learning can be expressed by means of a graph with nodes representing training samples and edges representing the relationship between samples in terms of similarity, temporal proximity, or label information. Graphs…

Homography estimation is an important task in computer vision applications, such as image stitching, video stabilization, and camera calibration. Traditional homography estimation methods heavily depend on the quantity and distribution of…

0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the…

Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…

Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMAP have demonstrated impressive visualization performance on many real world datasets. One tension that has always faced these methods is the trade-off between preservation of…

We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms…