22,345 research outputs found
Manifold Elastic Net: A Unified Framework for Sparse Dimension Reduction
It is difficult to find the optimal sparse solution of a manifold learning
based dimensionality reduction algorithm. The lasso or the elastic net
penalized manifold learning based dimensionality reduction is not directly a
lasso penalized least square problem and thus the least angle regression (LARS)
(Efron et al. \cite{LARS}), one of the most popular algorithms in sparse
learning, cannot be applied. Therefore, most current approaches take indirect
ways or have strict settings, which can be inconvenient for applications. In
this paper, we proposed the manifold elastic net or MEN for short. MEN
incorporates the merits of both the manifold learning based dimensionality
reduction and the sparse learning based dimensionality reduction. By using a
series of equivalent transformations, we show MEN is equivalent to the lasso
penalized least square problem and thus LARS is adopted to obtain the optimal
sparse solution of MEN. In particular, MEN has the following advantages for
subsequent classification: 1) the local geometry of samples is well preserved
for low dimensional data representation, 2) both the margin maximization and
the classification error minimization are considered for sparse projection
calculation, 3) the projection matrix of MEN improves the parsimony in
computation, 4) the elastic net penalty reduces the over-fitting problem, and
5) the projection matrix of MEN can be interpreted psychologically and
physiologically. Experimental evidence on face recognition over various popular
datasets suggests that MEN is superior to top level dimensionality reduction
algorithms.Comment: 33 pages, 12 figure
Geometric Structures on Spaces of Weighted Submanifolds
In this paper we use a diffeo-geometric framework based on manifolds that are
locally modeled on "convenient" vector spaces to study the geometry of some
infinite dimensional spaces. Given a finite dimensional symplectic manifold
, we construct a weak symplectic structure on each leaf of a foliation of the space of compact oriented isotropic submanifolds
in equipped with top degree forms of total measure 1. These forms are
called weightings and such manifolds are said to be weighted. We show that this
symplectic structure on the particular leaves consisting of weighted Lagrangian
submanifolds is equivalent to a heuristic weak symplectic structure of
Weinstein [Adv. Math. 82 (1990), 133-159]. When the weightings are positive,
these symplectic spaces are symplectomorphic to reductions of a weak symplectic
structure of Donaldson [Asian J. Math. 3 (1999), 1-15] on the space of
embeddings of a fixed compact oriented manifold into . When is compact,
by generalizing a moment map of Weinstein we construct a symplectomorphism of
each leaf consisting of positive weighted isotropic
submanifolds onto a coadjoint orbit of the group of Hamiltonian
symplectomorphisms of equipped with the Kirillov-Kostant-Souriau symplectic
structure. After defining notions of Poisson algebras and Poisson manifolds, we
prove that each space can also be identified with a
symplectic leaf of a Poisson structure. Finally, we discuss a kinematic
description of spaces of weighted submanifolds
Dimensionality Reduction Mappings
A wealth of powerful dimensionality reduction methods has been established which can be used for data visualization and preprocessing. These are accompanied by formal evaluation schemes, which allow a quantitative evaluation along general principles and which even lead to further visualization schemes based on these objectives. Most methods, however, provide a mapping of a priorly given finite set of points only, requiring additional steps for out-of-sample extensions. We propose a general view on dimensionality reduction based on the concept of cost functions, and, based on this general principle, extend dimensionality reduction to explicit mappings of the data manifold. This offers simple out-of-sample extensions. Further, it opens a way towards a theory of data visualization taking the perspective of its generalization ability to new data points. We demonstrate the approach based on a simple global linear mapping as well as prototype-based local linear mappings.
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