7,727 research outputs found
Towards Graph-Aware Diffusion Modeling for Collaborative Filtering
Recovering masked feedback with neural models is a popular paradigm in
recommender systems. Seeing the success of diffusion models in solving
ill-posed inverse problems, we introduce a conditional diffusion framework for
collaborative filtering that iteratively reconstructs a user's hidden
preferences guided by its historical interactions. To better align with the
intrinsic characteristics of implicit feedback data, we implement forward
diffusion by applying synthetic smoothing filters to interaction signals on an
item-item graph. The resulting reverse diffusion can be interpreted as a
personalized process that gradually refines preference scores. Through graph
Fourier transform, we equivalently characterize this model as an anisotropic
Gaussian diffusion in the graph spectral domain, establishing both forward and
reverse formulations. Our model outperforms state-of-the-art methods by a large
margin on one dataset and yields competitive results on the others.Comment: 13 pages, 6 figure
Behavioral analysis of anisotropic diffusion in image processing
©1996 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or distribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.DOI: 10.1109/83.541424In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik (1990). The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator
An asymptotic preserving scheme for strongly anisotropic elliptic problems
In this article we introduce an asymptotic preserving scheme designed to
compute the solution of a two dimensional elliptic equation presenting large
anisotropies. We focus on an anisotropy aligned with one direction, the
dominant part of the elliptic operator being supplemented with Neumann boundary
conditions. A new scheme is introduced which allows an accurate resolution of
this elliptic equation for an arbitrary anisotropy ratio.Comment: 21 page
Asymptotic-Preserving scheme for a bi-fluid Euler-Lorentz model
The present work is devoted to the simulation of a strongly magnetized plasma
considered as a mixture of an ion fluid and an electron fluid. For the sake of
simplicity, we assume that the model is isothermal and described by Euler
equations coupled with a term representing the Lorentz force. Moreover we
assume that both Euler systems are coupled through a quasi-neutrality
constraint. The numerical method which is described in the present document is
based on an Asymptotic-Preserving semi-discretization in time of a variant of
this two-fluid Euler-Lorentz model with a small perturbation of the
quasi-neutrality constraint. Firstly, we present the two-fluid model and the
motivations for introducing a small perturbation into the quasi-neutrality
equation, then we describe the time semi-discretization of the perturbed model
and a fully-discrete finite volume scheme based on it. Finally, we present some
numerical results which have been obtained with this method
Dendrites with corners
A phase-field model for diffusion-limited crystal growth is formulated that
is capable of handling highly anisotropic interfaces. It uses a Willmore
regularization that yields corners of finite size. An asymptotic analysis
reveals that Herring's law is recovered for the advancing surfaces. The model
is validated by conducting simulations of dendritic growth for low anistorpies
and comparing the results to the data from the literature. The model makes it
possible to simulate high anisotropy dendrites for which the standard
phase-field models are ill-posed. In this regime, the interplay between a
Herring instability on the dendrite flanks and the corner regularization
creates zig-zag shaped corrugations and leads to a non-monotonic trend of tip
velocity as a function of anisotropy strength.Comment: 7 pages, 2 figures, proceedings of the International Conference on
Solidification Science and Processing (ICSSP), Jodhpur, India, 202
Duality-based Asymptotic-Preserving method for highly anisotropic diffusion equations
The present paper introduces an efficient and accurate numerical scheme for
the solution of a highly anisotropic elliptic equation, the anisotropy
direction being given by a variable vector field. This scheme is based on an
asymptotic preserving reformulation of the original system, permitting an
accurate resolution independently of the anisotropy strength and without the
need of a mesh adapted to this anisotropy. The counterpart of this original
procedure is the larger system size, enlarged by adding auxiliary variables and
Lagrange multipliers. This Asymptotic-Preserving method generalizes the method
investigated in a previous paper [arXiv:0903.4984v2] to the case of an
arbitrary anisotropy direction field
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