Related papers: Measure differential equations
Stochastic differential equations (SDEs) offer powerful and accessible mathematical models for capturing both deterministic and probabilistic aspects of dynamic behavior across a wide range of physical, financial, and social systems.…
A flat membrane with given shape is displayed; two points in the membrane are randomly selected; the probability that the separation between the points have a specified value is sought. A simple method to evaluate the probability density is…
Distances between probability distributions are a key component of many statistical machine learning tasks, from two-sample testing to generative modeling, among others. We introduce a novel distance between measures that compares them…
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, with solutions over the…
Mesh deformation plays a pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent…
We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact…
A common feature of methods for analyzing samples of probability density functions is that they respect the geometry inherent to the space of densities. Once a metric is specified for this space, the Fr\'echet mean is typically used to…
Nowadays stochastic computer simulations with both numeral and distribution inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This paper studies the design and analysis issues of such computer…
The machine learning methods for data-driven identification of partial differential equations (PDEs) are typically defined for a given number of spatial dimensions and a choice of coordinates the data have been collected in. This dependence…
We present a method for approximating solutions of Stochastic Differential Equations (SDEs) with arbitrary rates. This approximation is derived for bounded and measurable test functions. Specifically, we demonstrate that, leveraging the…
Low-dimensional structure in real-world data plays an important role in the success of generative models, which motivates diffusion models defined on intrinsic data manifolds. Such models are driven by stochastic differential equations…
Towards understanding the fundamental limits of estimation from data of varied quality, we study the problem of estimating a mean parameter from heteroskedastic Gaussian observations where the variances are unknown and may vary arbitrarily…
The Wasserstein metric or earth mover's distance (EMD) is a useful tool in statistics, machine learning and computer science with many applications to biological or medical imaging, among others. Especially in the light of increasingly…
The problem of measurement in quantum mechanics is reanalyzed within a general, strictly probabilistic framework (without reduction postulate). Based on a novel comprehensive definition of measurement the natural emergence of objective…
The Sliced Wasserstein (SW) distance has become a popular alternative to the Wasserstein distance for comparing probability measures. Widespread applications include image processing, domain adaptation and generative modelling, where it is…
We study the dynamics of measure-valued solutions of what we call the EPDiff equations, standing for the {\it Euler-Poincar\'e equations associated with the diffeomorphism group (of $\mathbb{R}^n$ or an $n$-dimensional manifold $M$)}. Our…
The Maximum Mean Discrepancy (MMD) has found numerous applications in statistics and machine learning, most recently as a penalty in the Wasserstein Auto-Encoder (WAE). In this paper we compute closed-form expressions for estimating the…
Partial differential equations (PDEs) with multiple scales or those defined over sufficiently large domains arise in various areas of science and engineering and often present problems when approximating the solutions numerically. Machine…
We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these…
A novel probabilistic numerical method for quantifying the uncertainty induced by the time integration of ordinary differential equations (ODEs) is introduced. Departing from the classical strategy to randomize ODE solvers by adding a…