176 research outputs found
Linear Asymptotic Convergence of Anderson Acceleration: Fixed-Point Analysis
We study the asymptotic convergence of AA(), i.e., Anderson acceleration
with window size for accelerating fixed-point methods ,
. Convergence acceleration by AA() has been widely observed but
is not well understood. We consider the case where the fixed-point iteration
function is differentiable and the convergence of the fixed-point method
itself is root-linear. We identify numerically several conspicuous properties
of AA() convergence: First, AA() sequences converge
root-linearly but the root-linear convergence factor depends strongly on the
initial condition. Second, the AA() acceleration coefficients
do not converge but oscillate as converges to . To shed light on
these observations, we write the AA() iteration as an augmented fixed-point
iteration , and analyze the continuity
and differentiability properties of and . We find that the
vector of acceleration coefficients is not continuous at the fixed
point . However, we show that, despite the discontinuity of ,
the iteration function is Lipschitz continuous and directionally
differentiable at for AA(1), and we generalize this to AA() with
for most cases. Furthermore, we find that is not differentiable at
. We then discuss how these theoretical findings relate to the observed
convergence behaviour of AA(). The discontinuity of at
allows to oscillate as converges to , and the
non-differentiability of allows AA() sequences to converge with
root-linear convergence factors that strongly depend on the initial condition.
Additional numerical results illustrate our findings
Self-supervised Spatio-temporal Representation Learning for Videos by Predicting Motion and Appearance Statistics
We address the problem of video representation learning without
human-annotated labels. While previous efforts address the problem by designing
novel self-supervised tasks using video data, the learned features are merely
on a frame-by-frame basis, which are not applicable to many video analytic
tasks where spatio-temporal features are prevailing. In this paper we propose a
novel self-supervised approach to learn spatio-temporal features for video
representation. Inspired by the success of two-stream approaches in video
classification, we propose to learn visual features by regressing both motion
and appearance statistics along spatial and temporal dimensions, given only the
input video data. Specifically, we extract statistical concepts (fast-motion
region and the corresponding dominant direction, spatio-temporal color
diversity, dominant color, etc.) from simple patterns in both spatial and
temporal domains. Unlike prior puzzles that are even hard for humans to solve,
the proposed approach is consistent with human inherent visual habits and
therefore easy to answer. We conduct extensive experiments with C3D to validate
the effectiveness of our proposed approach. The experiments show that our
approach can significantly improve the performance of C3D when applied to video
classification tasks. Code is available at
https://github.com/laura-wang/video_repres_mas.Comment: CVPR 201
Local Fourier analysis for saddle-point problems
The numerical solution of saddle-point problems has attracted considerable interest in
recent years, due to their indefiniteness and often poor spectral properties that make
efficient solution difficult. While much research already exists, developing efficient
algorithms remains challenging. Researchers have applied finite-difference, finite element,
and finite-volume approaches successfully to discretize saddle-point problems,
and block preconditioners and monolithic multigrid methods have been proposed for
the resulting systems. However, there is still much to understand.
Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in
the presence of electromagnetic fields. Often, the discretization and linearization of
MHD leads to a saddle-point system. We present vector-potential formulations of
MHD and a theoretical analysis of the existence and uniqueness of solutions of both
the continuum two-dimensional resistive MHD model and its discretization.
Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid
and other multilevel algorithms. We first adapt LFA to analyse the properties of
multigrid methods for both finite-difference and finite-element discretizations of the
Stokes equations, leading to saddle-point systems. Monolithic multigrid methods,
based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From
this LFA, optimal parameters are proposed for these multigrid solvers. Numerical
experiments are presented to validate our theoretical results. A modified two-level
LFA is proposed for high-order finite-element methods for the Lapalce problem, curing
the failure of classical LFA smoothing analysis in this setting and providing a reliable
way to estimate actual multigrid performance. Finally, we extend LFA to analyze the
balancing domain decomposition by constraints (BDDC) algorithm, using a new choice
of basis for the space of Fourier harmonics that greatly simplifies the application of
LFA. Improved performance is obtained for some two- and three-level variants
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