arXiv research

A new clustering method for intersecting surfaces using curvature constraints.

Study on metrics with positive scalar curvature and convex boundary.

CoMPNetX uses neural networks to efficiently solve constrained motion planning problems.

Paper tackles efficient navigation in constrained environments using supervised and reinforcement learning.

We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as mos…

The present work extends the randomized shortest-paths framework (RSP), interpolating between shortest-path and random-walk routing in a network, in three directions. First, it shows how to deal with equality constraints on a subset of transition probabilities and develops a generic algorithm for solving this constrain…

Optimal transport with path constraints for distributions of different masses.

Develops methods for estimating constrained function-valued parameters in infinite-dimensional models.

A bounded curvature path is a continuously differentiable piecewise $C^2$ path with a bounded absolute curvature that connects two points in the tangent bundle of a surface. In this work, we analyze the homotopy classes of bounded curvature paths for points in the tangent bundle of the Euclidean plane. We show the exis…

Bayesian method detects Markov order in network paths more reliably.

Unified framework for measuring causality-based fairness in machine learning.

Study shows infinitely many path components for positive Ricci curvature metrics on certain spin manifolds.

We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical…

This article addresses the problem of approximating the price of options on discrete and continuous arithmetic average of the underlying, i.e. discretely and continuously monitored Asian options, in local volatility models. A path-integral-type expression for option prices is obtained using a Brownian bridge representa…

The study examines mean curvature flow and Heegaard surfaces in lens spaces.

The paper develops fair machine learning models using causal path-specific effects.

Study of most probable paths for anisotropic Brownian motions on manifolds.

The paper studies surfaces with spherical curvature lines and their generation by constrained elastic curves.

Choose two points in the tangent bundle of the Euclidean plane $(x,X),(y,Y)\in T{ \mathbb R}^2$. In this work we characterise the immersed length minimising paths with a prescribed bound on the curvature starting at $x$, tangent to $X$; finishing at $y$, tangent to $Y$, in each connected component of the space of paths…

Classifies 3D manifolds with specific structures and automorphisms.

Energy quantization for surfaces with area, volume, and mean curvature constraints.

This paper studies a class of non$-$Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a $Z-$constrained BSDE, with dynamics associated to a non singular underlying forward process. Du…

In this paper we prove the path connectedness of the moduli spaces of metrics with positive isotropic curvature on certain compact four-dimensional manifolds.

Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.

The aim of this paper is to associate a measure for certain sets of paths in the Euclidean plane $\mathbb{R}^2$ with fixed starting and ending points. Then, working on parameterized surfaces with a specific Riemannian metric, we define and calculate the integral of the length over the set of paths obtained as the image…

The study examines moduli spaces of metrics with positive Ricci or non-negative sectional curvature on sphere bundles.

The study explores metrics on real projective spaces with nonnegative sectional or positive Ricci curvature.

Develops a new solver for path-dependent PDEs using signature kernels.

The paper classifies path structures on 3D Lie groups and reduces non-flat ones to Z/2Z-structures.

The paper studies curvatures and austere properties of orbits in symmetric spaces.

Study of large area-constrained Willmore surfaces in Schwarzschild-like manifolds.

A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.

Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.

Many introductory courses in quantum mechanics include Feynman's time-slicing definition of the path integral, with a complete derivation of the propagator in the simplest of cases. However, attempts to generalize this, for instance to non-quadratic potentials, encounter formidable analytic issues in showing the succes…

Deep learning has become a powerful and popular tool for a variety of machine learning tasks. However, it is challenging to understand the mechanism of deep learning from a theoretical perspective. In this work, we propose a random active path model to study collective properties of deep neural networks with binary syn…

We derive a curvature-variation formula for a path of left-invariant metrics on a compact Lie group, beginning at a bi-invariant metric. We prove rigidity theorems for paths which remain nonnegatively curved, and we make progress towards a classification of the left-invariant metrics with nonnegative curvature on SO(4)…

Study shows infinitely many metrics with nonnegative sectional or positive Ricci curvature on specific 5D quotients.

Controller-Augmented Hidden Markov Models (CHMMs) are a framework for constrained sequential inference.

We compare alternative computing strategies for solving the constrained lasso problem. As its name suggests, the constrained lasso extends the widely-used lasso to handle linear constraints, which allow the user to incorporate prior information into the model. In addition to quadratic programming, we employ the alterna…

The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. PDE-constrained LDDMM methods are formulated as constrained variational problems, wher…

We establish existence of compact minimizers of the prescribed mean curvature problem with volume constraint in periodic media. As a consequence, we construct compact approximate solutions to the prescribed mean curvature equation. We also show convergence after rescaling of the volume-constrained minimizers towards a …

In this paper, we report a "new" continuity path which links the constant scalar curvature equation to a second order elliptic equation. This is largely an expository article where we describes various aspects of geometry and analysis associated with path.

Proves path connectedness of asymptotically flat metrics with boundary.

The paper proves geometric inequalities in sphere using locally constrained flows.

Paper explores rough path theory for frictionless markets, linking NCFL to unbiased rough integrators.

We provide explicit examples which show that mean convexity (i.e. positivity of the mean curvature) and positivity of the scalar curvature are non-preserved curvature conditions for hypersurfaces of the Euclidean space evolving under either the volume- or the area preserving mean curvature flow. The relevance of our ex…

Proves CLT for Brownian paths on pinched negative curvature manifolds.

Geometrically represents path integral reduction Jacobian for interacting systems.