Related papers: The flag manifold as a tool for analyzing and comp…

Pairwise comparison of graphs is key to many applications in Machine learning ranging from clustering, kernel-based classification/regression and more recently supervised graph prediction. Distances between graphs usually rely on…

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint…

Parametrizations of data manifolds in shape spaces can be computed using the rich toolbox of Riemannian geometry. This, however, often comes with high computational costs, which raises the question if one can learn an efficient neural…

Topological data analysis provides a set of tools to uncover low-dimensional structure in noisy point clouds. Prominent amongst the tools is persistence homology, which summarizes birth-death times of homological features using data objects…

Given dense image feature correspondences of a non-rigidly moving object across multiple frames, this paper proposes an algorithm to estimate its 3D shape for each frame. To solve this problem accurately, the recent state-of-the-art…

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however…

The Grassmann angle improves upon similar angles between subspaces that measure volume contraction in orthogonal projections. It works in real or complex spaces, with important differences, and is asymmetric, what makes it more efficient…

Various non-trivial spaces are becoming popular for embedding structured data such as graphs, texts, or images. Following spherical and hyperbolic spaces, more general product spaces have been proposed. However, searching for the best…

Recent advances in self-supervised learning and neural network scaling have enabled the creation of large models, known as foundation models, which can be easily adapted to a wide range of downstream tasks. The current paradigm for…

Fusion and inference from multiple and massive disparate data sources - the requirement for our most challenging data analysis problems and the goal of our most ambitious statistical pattern recognition methodologies - -has many and varied…

A standard problem in applied topology is how to discover topological invariants of data from a noisy point cloud that approximates it. We consider the case where a sample is drawn from a properly embedded C1-submanifold without boundary in…

Due to advancements in digital cameras, it is easy to gather multiple images (or videos) from an object under different conditions. Therefore, image-set classification has attracted more attention, and different solutions were proposed to…

We derive three families of orthogonally-equivariant matrix submanifold models for the Grassmann, flag, and Stiefel manifolds respectively. These families are exhaustive -- every orthogonally-equivariant submanifold model of the lowest…

Measuring the similarity between data points often requires domain knowledge, which can in parts be compensated by relying on unsupervised methods such as latent-variable models, where similarity/distance is estimated in a more compact…

Many statistical and machine learning approaches rely on pairwise distances between data points. The choice of distance metric has a fundamental impact on performance of these procedures, raising questions about how to appropriately…

In image set classification, a considerable progress has been made by representing original image sets on Grassmann manifolds. In order to extend the advantages of the Euclidean based dimensionality reduction methods to the Grassmann…

Drawing motivation from the manifold hypothesis, which posits that most high-dimensional data lies on or near low-dimensional manifolds, we apply manifold learning to the space of neural networks. We learn manifolds where datapoints are…

Learning efficient graph representation is the key to favorably addressing downstream tasks on graphs, such as node or graph property prediction. Given the non-Euclidean structural property of graphs, preserving the original graph data's…

This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold $M$ into clusters lying around low-dimensional submanifolds of $M$. Important examples of $M$, for which the proposed clustering algorithm is…

We propose a novel technique for training deep networks with the objective of obtaining feature representations that exist in a Euclidean space and exhibit strong clustering behavior. Our desired features representations have three traits:…