250 research outputs found
Monotone discretizations of levelset convex geometric PDEs
We introduce a novel algorithm that converges to level-set convex viscosity
solutions of high-dimensional Hamilton-Jacobi equations. The algorithm is
applicable to a broad class of curvature motion PDEs, as well as a recently
developed Hamilton-Jacobi equation for the Tukey depth, which is a statistical
depth measure of data points. A main contribution of our work is a new monotone
scheme for approximating the direction of the gradient, which allows for
monotone discretizations of pure partial derivatives in the direction of, and
orthogonal to, the gradient. We provide a convergence analysis of the algorithm
on both regular Cartesian grids and unstructured point clouds in any dimension
and present numerical experiments that demonstrate the effectiveness of the
algorithm in approximating solutions of the affine flow in two dimensions and
the Tukey depth measure of high-dimensional datasets such as MNIST and
FashionMNIST.Comment: 42 pages including reference
The back-and-forth method for Wasserstein gradient flows
We present a method to efficiently compute Wasserstein gradient flows. Our
approach is based on a generalization of the back-and-forth method (BFM)
introduced by Jacobs and L\'eger to solve optimal transport problems. We evolve
the gradient flow by solving the dual problem to the JKO scheme. In general,
the dual problem is much better behaved than the primal problem. This allows us
to efficiently run large-scale simulations for a large class of internal
energies including singular and non-convex energies
Deep JKO: time-implicit particle methods for general nonlinear gradient flows
We develop novel neural network-based implicit particle methods to compute
high-dimensional Wasserstein-type gradient flows with linear and nonlinear
mobility functions. The main idea is to use the Lagrangian formulation in the
Jordan--Kinderlehrer--Otto (JKO) framework, where the velocity field is
approximated using a neural network. We leverage the formulations from the
neural ordinary differential equation (neural ODE) in the context of continuous
normalizing flow for efficient density computation. Additionally, we make use
of an explicit recurrence relation for computing derivatives, which greatly
streamlines the backpropagation process. Our methodology demonstrates
versatility in handling a wide range of gradient flows, accommodating various
potential functions and nonlinear mobility scenarios. Extensive experiments
demonstrate the efficacy of our approach, including an illustrative example
from Bayesian inverse problems. This underscores that our scheme provides a
viable alternative solver for the Kalman-Wasserstein gradient flow.Comment: 23 page
Many-Body Quadrupolar Sum Rule for Higher-Order Topological Insulator
The modern theory of polarization establishes the bulk-boundary
correspondence for the bulk polarization. In this paper, we attempt to extend
it to a sum rule of the bulk quadrupole moment by employing a many-body
operator introduced in [Phys. Rev. B 100, 245134 (2019)] and [Phys. Rev. B 100,
245135 (2019)]. The sum rule that we propose consists of the alternating sum of
four observables, which are the phase factors of the many-body operator in
different boundary conditions. We demonstrate its validity through extensive
numerical computations for various non-interacting tight-binding models. We
also observe that individual terms in the sum rule correspond to the bulk
quadrupole moment, the edge-localized polarizations, and the corner charge in
the thermodynamic limit on some models.Comment: 13 pages (3 figures
Monotone Generative Modeling via a Gromov-Monge Embedding
Generative Adversarial Networks (GANs) are powerful tools for creating new
content, but they face challenges such as sensitivity to starting conditions
and mode collapse. To address these issues, we propose a deep generative model
that utilizes the Gromov-Monge embedding (GME). It helps identify the
low-dimensional structure of the underlying measure of the data and then maps
it, while preserving its geometry, into a measure in a low-dimensional latent
space, which is then optimally transported to the reference measure. We
guarantee the preservation of the underlying geometry by the GME and
-cyclical monotonicity of the generative map, where is an intrinsic
embedding cost employed by the GME. The latter property is a first step in
guaranteeing better robustness to initialization of parameters and mode
collapse. Numerical experiments demonstrate the effectiveness of our approach
in generating high-quality images, avoiding mode collapse, and exhibiting
robustness to different starting conditions.Comment: 29 pages including main text and appendi
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