Related papers: G-invariant diffusion maps
Diffusion models have demonstrated remarkable capabilities in synthesizing realistic images, spurring interest in using their representations for various downstream tasks. To better understand the robustness of these representations, we…
We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs $(\mathbf{y}, g \cdot \mathbf{y})$, where $g$ is drawn from a finite group acting on the data. Our method jointly learns a latent space and a…
Diffusion-based manifold learning methods have proven useful in representation learning and dimensionality reduction of modern high dimensional, high throughput, noisy datasets. Such datasets are especially present in fields like biology…
Recently, diffusion models have been used to solve various inverse problems in an unsupervised manner with appropriate modifications to the sampling process. However, the current solvers, which recursively apply a reverse diffusion step…
Diffusion maps are a nonlinear manifold learning technique based on harmonic analysis of a diffusion process over the data. Out-of-sample extensions with computational complexity $\mathcal{O}(N)$, where $N$ is the number of points…
This work studies the classical spectral clustering algorithm which embeds the vertices of some graph $G=(V_G, E_G)$ into $\mathbb{R}^k$ using $k$ eigenvectors of some matrix of $G$, and applies $k$-means to partition $V_G$ into $k$…
Spectral techniques are popular and robust approaches to data analysis. A prominent example is the use of eigenvectors of a Laplacian, constructed from data affinities, to identify natural data groupings or clusters, or to produce a…
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man…
We present a novel nonparametric algorithm for symmetry-based disentangling of data manifolds, the Geometric Manifold Component Estimator (GEOMANCER). GEOMANCER provides a partial answer to the question posed by Higgins et al. (2018): is it…
We introduce Hodge Diffusion Maps, a novel manifold learning algorithm designed to analyze and extract topological information from high-dimensional data-sets. This method approximates the exterior derivative acting on differential forms,…
Our goal is to efficiently compute low-dimensional latent coordinates for nodes in an input graph -- known as graph embedding -- for subsequent data processing such as clustering. Focusing on finite graphs that are interpreted as uniform…
In this paper, we develop a theory about the relationship between invariant and equivariant maps with regard to a group $G$. We then leverage this theory in the context of deep neural networks with group symmetries in order to obtain novel…
Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space…
In this paper, we consider a closed Riemannian manifold $M^{n+1}$ with dimension $3\leq n+1\leq 7$, and a compact Lie group $G$ acting as isometries on $M$ with cohomogeneity at least $3$. Suppose the union of non-principal orbits…
We introduce {\em vector diffusion maps} (VDM), a new mathematical framework for organizing and analyzing massive high dimensional data sets, images and shapes. VDM is a mathematical and algorithmic generalization of diffusion maps and…
We analyze the spectral clustering procedure for identifying coarse structure in a data set $x_1, \dots, x_n$, and in particular study the geometry of graph Laplacian embeddings which form the basis for spectral clustering algorithms. More…
Treating neural network inputs and outputs as random variables, we characterize the structure of neural networks that can be used to model data that are invariant or equivariant under the action of a compact group. Much recent research has…
Generative modeling seeks to uncover the underlying factors that give rise to observed data that can often be modeled as the natural symmetries that manifest themselves through invariances and equivariances to certain transformation laws.…
The analysis of data sets arising from multiple sensors has drawn significant research attention over the years. Traditional methods, including kernel-based methods, are typically incapable of capturing nonlinear geometric structures. We…
We introduce and study equivariant Seiberg-Witten invariants for $4$-manifolds equipped with a smooth action of a finite group $G$. Our invariants come in two types: cohomological, valued in the group cohomology of $G$ and $K$-theoretic,…