649 research outputs found
Barycentric Subspace Analysis on Manifolds
This paper investigates the generalization of Principal Component Analysis
(PCA) to Riemannian manifolds. We first propose a new and general type of
family of subspaces in manifolds that we call barycentric subspaces. They are
implicitly defined as the locus of points which are weighted means of
reference points. As this definition relies on points and not on tangent
vectors, it can also be extended to geodesic spaces which are not Riemannian.
For instance, in stratified spaces, it naturally allows principal subspaces
that span several strata, which is impossible in previous generalizations of
PCA. We show that barycentric subspaces locally define a submanifold of
dimension k which generalizes geodesic subspaces.Second, we rephrase PCA in
Euclidean spaces as an optimization on flags of linear subspaces (a hierarchy
of properly embedded linear subspaces of increasing dimension). We show that
the Euclidean PCA minimizes the Accumulated Unexplained Variances by all the
subspaces of the flag (AUV). Barycentric subspaces are naturally nested,
allowing the construction of hierarchically nested subspaces. Optimizing the
AUV criterion to optimally approximate data points with flags of affine spans
in Riemannian manifolds lead to a particularly appealing generalization of PCA
on manifolds called Barycentric Subspaces Analysis (BSA).Comment: Annals of Statistics, Institute of Mathematical Statistics, A
Para\^itr
On Quasi-Newton Forward--Backward Splitting: Proximal Calculus and Convergence
We introduce a framework for quasi-Newton forward--backward splitting
algorithms (proximal quasi-Newton methods) with a metric induced by diagonal
rank- symmetric positive definite matrices. This special type of
metric allows for a highly efficient evaluation of the proximal mapping. The
key to this efficiency is a general proximal calculus in the new metric. By
using duality, formulas are derived that relate the proximal mapping in a
rank- modified metric to the original metric. We also describe efficient
implementations of the proximity calculation for a large class of functions;
the implementations exploit the piece-wise linear nature of the dual problem.
Then, we apply these results to acceleration of composite convex minimization
problems, which leads to elegant quasi-Newton methods for which we prove
convergence. The algorithm is tested on several numerical examples and compared
to a comprehensive list of alternatives in the literature. Our quasi-Newton
splitting algorithm with the prescribed metric compares favorably against
state-of-the-art. The algorithm has extensive applications including signal
processing, sparse recovery, machine learning and classification to name a few.Comment: arXiv admin note: text overlap with arXiv:1206.115
A family of quadrilateral finite elements
We present a novel family of quadrilateral finite elements, which
define global spaces over a general quadrilateral mesh with vertices of
arbitrary valency. The elements extend the construction by (Brenner and Sung,
J. Sci. Comput., 2005), which is based on polynomial elements of tensor-product
degree , to all degrees . Thus, we call the family of
finite elements Brenner-Sung quadrilaterals. The proposed quadrilateral
can be seen as a special case of the Argyris isogeometric element of (Kapl,
Sangalli and Takacs, CAGD, 2019). The quadrilateral elements possess similar
degrees of freedom as the classical Argyris triangles. Just as for the Argyris
triangle, we additionally impose continuity at the vertices. In this
paper we focus on the lower degree cases, that may be desirable for their lower
computational cost and better conditioning of the basis: We consider indeed the
polynomial quadrilateral of (bi-)degree~, and the polynomial degrees
and by employing a splitting into or polynomial
pieces, respectively.
The proposed elements reproduce polynomials of total degree . We show that
the space provides optimal approximation order. Due to the interpolation
properties, the error bounds are local on each element. In addition, we
describe the construction of a simple, local basis and give for
explicit formulas for the B\'{e}zier or B-spline coefficients of the basis
functions. Numerical experiments by solving the biharmonic equation demonstrate
the potential of the proposed quadrilateral finite element for the
numerical analysis of fourth order problems, also indicating that (for )
the proposed element performs comparable or in general even better than the
Argyris triangle with respect to the number of degrees of freedom
Adaptive, Anisotropic and Hierarchical cones of Discrete Convex functions
We address the discretization of optimization problems posed on the cone of
convex functions, motivated in particular by the principal agent problem in
economics, which models the impact of monopoly on product quality. Consider a
two dimensional domain, sampled on a grid of N points. We show that the cone of
restrictions to the grid of convex functions is in general characterized by N^2
linear inequalities; a direct computational use of this description therefore
has a prohibitive complexity. We thus introduce a hierarchy of sub-cones of
discrete convex functions, associated to stencils which can be adaptively,
locally, and anisotropically refined. Numerical experiments optimize the
accuracy/complexity tradeoff through the use of a-posteriori stencil refinement
strategies.Comment: 35 pages, 11 figures. (Second version fixes a small bug in Lemma 3.2.
Modifications are anecdotic.
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