arXiv research

Paper defines and studies measures related to earthquakes and best Lipschitz maps.

The paper extends a measure preserving property to bi-Lipschitz maps between Moran sets.

Maps and measures on surfaces link best Lipschitz and least gradient functions.

We provide new conditions that ensure that two metric measure spaces are not quasiconformally equivalent. As an application we deduce that there exists no quasiconformal map between the sub-Riemannian Heisenberg and roto-translation groups.

Computing optimal transport (OT) between measures in high dimensions is doomed by the curse of dimensionality. A popular approach to avoid this curse is to project input measures on lower-dimensional subspaces (1D lines in the case of sliced Wasserstein distances), solve the OT problem between these reduced measures, a…

Transformers preserve support and can approximate any continuous map.

Automatic continuity of polynomial maps and cocycles proved.

We study the statistical meaning of the minimization of distortion measure and the relation between the equilibrium points of the SOM algorithm and the minima of distortion measure. If we assume that the observations and the map lie in an compact Euclidean space, we prove the strong consistency of the map which almost …

Paper investigates optimal transport map estimation in infinite-dimensional spaces.

The paper derives Liouville theorems for various generalized maps on Riemannian manifolds.

We investigate the low-dimensional structure of deterministic transformations between random variables, i.e., transport maps between probability measures. In the context of statistics and machine learning, these transformations can be used to couple a tractable "reference" measure (e.g., a standard Gaussian) with a tar…

This work establishes properties on diffeological structures for set-valued maps and measures.

The paper develops a theory for one-step Wasserstein-guided models for PDE-induced measures.

Transformers can interpolate between arbitrary measures.

Investigates maps and properties in spaces with negative dimensions and curvature.

This paper investigates analytic properties of maps between hyperbolic surfaces, focusing on best Lipschitz maps and geodesic laminations.

We prove that in metric measure spaces where the entropy functional is K-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one opti…

Continuity of earthquake flow map transfers Teichmüller dynamics results.

We introduce two kernels that extend the mean map, which embeds probability measures in Hilbert spaces. The generative mean map kernel (GMMK) is a smooth similarity measure between probabilistic models. The latent mean map kernel (LMMK) generalizes the non-iid formulation of Hilbert space embeddings of empirical distri…

New theory approximates functions between metric spaces using random probability measures.

The paper optimizes estimating transport maps between distributions.

New neural networks learn mappings between probability measures and functions.

Study develops a new method for creating fair models.

This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two distributions. To that end, we propose a stochastic dual approach of regularize…

The paper proves an energy identity for harmonic maps near singularities.

Study spectral distances on compact RCD spaces.

Kernel dependence measures yield accurate estimates of nonlinear relations between random variables, and they are also endorsed with solid theoretical properties and convergence rates. Besides, the empirical estimates are easy to compute in closed form just involving linear algebra operations. However, they are hampere…

In this paper we investigate the relationship between a general existence of transport maps of optimal couplings with absolutely continuous first marginal and the property of the background measure called essentially non-branching introduced by Rajala-Sturm (Calc.Var.PDE 2014). In particular, it is shown that the quali…

Improved Sobolev mappings in Carnot groups with weaker assumptions.

This paper develops a theory of Lipschitz comparisons of hyperbolic surfaces analogous to the theory of quasi-conformal comparisons. Extremal Lipschitz maps (minimal stretch maps) and geodesics for the `Lipschitz metric' are constructed. The extremal Lipschitz constant equals the maximum ratio of lengths of measured la…

We study the space of "link maps": the space of maps of a disjoint union of compact, closed manifolds P_1, . . ., P_k into a manifold N whose images are pairwise disjoint. We apply the manifold calculus of functors developed by Goodwillie and Weiss to study the difference between it and its linear and quadratic approxi…

Estimating Wasserstein distances between two high-dimensional densities suffers from the curse of dimensionality: one needs an exponential (wrt dimension) number of samples to ensure that the distance between two empirical measures is comparable to the distance between the original densities. Therefore, optimal transpo…

The paper studies harmonic map heat flow to flat tori, proving ergodic behavior and convergence to hyperbolic measure.

Proposes m-POT to improve m-OT's misspecified mappings issue.

We define a new family of similarity and distance measures on graphs, and explore their theoretical properties in comparison to conventional distance metrics. These measures are defined by the solution(s) to an optimization problem which attempts find a map minimizing the discrepancy between two graph Laplacian exponen…

A map $f:X\to Y$ between topological spaces is defined to be {\em scatteredly continuous} if for each subspace $A\subset X$ the restriction $f|A$ has a point of continuity. We show that for a function $f:X\to Y$ from a perfectly paracompact hereditarily Baire Preiss-Simon space $X$ into a regular space $Y$ the scattere…

This work proposes a new method to match distributions across different spaces using cycle-consistent maps.

It is known that PQ-symmetric maps on the boundary characterize the quasi-isometry type of visual hyperbolic spaces, in particular, of geodesically complete \br-trees. We define a map on pairs of PQ-symmetric ultrametric spaces which characterizes the branching of the space. We also show that, when the ultrametric spac…

Extends earthquake and horocycle flows to new measures.

There are many methods developed to approximate a cloud of vectors embedded in high-dimensional space by simpler objects: starting from principal points and linear manifolds to self-organizing maps, neural gas, elastic maps, various types of principal curves and principal trees, and so on. For each type of approximator…

When recovering an unknown signal from noisy measurements, the computational difficulty of performing optimal Bayesian MMSE (minimum mean squared error) inference often necessitates the use of maximum a posteriori (MAP) inference, a special case of regularized M-estimation, as a surrogate. However, MAP is suboptimal in…

We compare two relationships between quadratic differentials and measured geodesic laminations on hyperbolic Riemann surfaces (by foliations or complex projective structures). Each yields a homeomorphism $\ML(S) \to Q(X)$ for any conformal structure $X$ on a compact surface $S$. The main result is that these maps are n…

A new method uses normalizing flows to approximate optimal transport between empirical distributions.

For localization and mapping of indoor environments through WiFi signals, locations are often represented as likelihoods of the received signal strength indicator. In this work we compare various measures of distance between such likelihoods in combination with different methods for estimation and representation. In pa…

Study optimal transport on simplex boundary, proving transport map and potential regularity.

For stationary harmonic maps between Riemannian manifolds, we provide a necessary and sufficient condition for the uniform interior and boundary gradient estimates in terms of the total energy of maps. We also show that if analytic target manifolds do not carry any harmonic S^2, then the singular sets of stationary map…

Paper estimates EOT maps for non-compactly supported measures with subGaussian target.

Deep learning for integrating diverse clinical measurements.