Related papers: Comparing Morse Complexes Using Optimal Transport:…
Measuring similarity between complex objects is a fundamental task in many scientific fields. When objects are represented as graphs, graph similarity/distance measures offer a powerful framework for quantifying structural resemblance.…
In this work, we address the problem of cross-modal comparison of aerial data streams. A variety of simulated automobile trajectories are sensed using two different modalities: full-motion video, and radio-frequency (RF) signals received by…
We lay out the phenomenological behavior of event-shape observables evaluated by solving optimal transport problems between collider events and reference geometries -- which we name 'manifold distances' -- to provide guidance regarding…
We consider the Monge problem of optimal transport between a compactly supported source measure and a target probability measure with unbounded support. We consider the convergence of optimal maps and potential functions when the target…
Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical…
Complex networks are a powerful modeling tool, allowing the study of countless real-world systems. They have been used in very different domains such as computer science, biology, sociology, management, etc. Authors have been trying to…
Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from…
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal…
The Morse-Smale complex of a function $f$ decomposes the sample space into cells where $f$ is increasing or decreasing. When applied to nonparametric density estimation and regression, it provides a way to represent, visualize, and compare…
Given a compact smooth manifold $M$ with non-empty boundary and a Morse function, a pseudo-gradient Morse-Smale vector field adapted to the boundary allows one to build a Morse complex whose homology is isomorphic to the (absolute or…
Distances on merge trees facilitate visual comparison of collections of scalar fields. Two desirable properties for these distances to exhibit are 1) the ability to discern between scalar fields which other, less complex topological…
The Morse-Smale complex is an important tool for global topological analysis in various problems of computational geometry and topology. Algorithms for Morse-Smale complexes have been presented in case of piecewise linear manifolds.…
Coupling probability measures lies at the core of many problems in statistics and machine learning, from domain adaptation to transfer learning and causal inference. Yet, even when restricted to deterministic transports, such couplings are…
Topological structures such as the merge tree provide an abstract and succinct representation of scalar fields. They facilitate effective visualization and interactive exploration of feature-rich data. A merge tree captures the topology of…
Brain pathologies often manifest as partial or complete loss of tissue. The goal of many neuroimaging studies is to capture the location and amount of tissue changes with respect to a clinical variable of interest, such as disease…
Optimal transport maps define a one-to-one correspondence between probability distributions, and as such have grown popular for machine learning applications. However, these maps are generally defined on empirical observations and cannot be…
Graphs are used in almost every scientific discipline to express relations among a set of objects. Algorithms that compare graphs, and output a closeness score, or a correspondence among their nodes, are thus extremely important. Despite…
We propose a simple subsampling scheme for fast randomized approximate computation of optimal transport distances. This scheme operates on a random subset of the full data and can use any exact algorithm as a black-box back-end, including…
Graph kernel is a powerful tool measuring the similarity between graphs. Most of the existing graph kernels focused on node labels or attributes and ignored graph hierarchical structure information. In order to effectively utilize graph…
We study Morse theory on noncompact manifolds equipped with exhaustions by compact pieces, defining the Morse homology of a pair which consists of the manifold and related geometric/homotopy data. We construct a collection of Morse data…