3,708 research outputs found
Low-Rank Matrices on Graphs: Generalized Recovery & Applications
Many real world datasets subsume a linear or non-linear low-rank structure in
a very low-dimensional space. Unfortunately, one often has very little or no
information about the geometry of the space, resulting in a highly
under-determined recovery problem. Under certain circumstances,
state-of-the-art algorithms provide an exact recovery for linear low-rank
structures but at the expense of highly inscalable algorithms which use nuclear
norm. However, the case of non-linear structures remains unresolved. We revisit
the problem of low-rank recovery from a totally different perspective,
involving graphs which encode pairwise similarity between the data samples and
features. Surprisingly, our analysis confirms that it is possible to recover
many approximate linear and non-linear low-rank structures with recovery
guarantees with a set of highly scalable and efficient algorithms. We call such
data matrices as \textit{Low-Rank matrices on graphs} and show that many real
world datasets satisfy this assumption approximately due to underlying
stationarity. Our detailed theoretical and experimental analysis unveils the
power of the simple, yet very novel recovery framework \textit{Fast Robust PCA
on Graphs
Regularization of Toda lattices by Hamiltonian reduction
The Toda lattice defined by the Hamiltonian with , which
exhibits singular (blowing up) solutions if some of the , can be
viewed as the reduced system following from a symmetry reduction of a subsystem
of the free particle moving on the group G=SL(n,\Real ). The subsystem is
, where consists of the determinant one matrices with
positive principal minors, and the reduction is based on the maximal nilpotent
group . Using the Bruhat decomposition we show that the full
reduced system obtained from , which is perfectly regular, contains
Toda lattices. More precisely, if is odd the reduced system
contains all the possible Toda lattices having different signs for the .
If is even, there exist two non-isomorphic reduced systems with different
constituent Toda lattices. The Toda lattices occupy non-intersecting open
submanifolds in the reduced phase space, wherein they are regularized by being
glued together. We find a model of the reduced phase space as a hypersurface in
{\Real}^{2n-1}. If for all , we prove for that the
Toda phase space associated with is a connected component of this
hypersurface. The generalization of the construction for the other simple Lie
groups is also presented.Comment: 42 pages, plain TeX, one reference added, to appear in J. Geom. Phy
Chiron: A Robust Recommendation System with Graph Regularizer
Recommendation systems have been widely used by commercial service providers
for giving suggestions to users. Collaborative filtering (CF) systems, one of
the most popular recommendation systems, utilize the history of behaviors of
the aggregate user-base to provide individual recommendations and are effective
when almost all users faithfully express their opinions. However, they are
vulnerable to malicious users biasing their inputs in order to change the
overall ratings of a specific group of items. CF systems largely fall into two
categories - neighborhood-based and (matrix) factorization-based - and the
presence of adversarial input can influence recommendations in both categories,
leading to instabilities in estimation and prediction. Although the robustness
of different collaborative filtering algorithms has been extensively studied,
designing an efficient system that is immune to manipulation remains a
significant challenge. In this work we propose a novel "hybrid" recommendation
system with an adaptive graph-based user/item similarity-regularization -
"Chiron". Chiron ties the performance benefits of dimensionality reduction
(through factorization) with the advantage of neighborhood clustering (through
regularization). We demonstrate, using extensive comparative experiments, that
Chiron is resistant to manipulation by large and lethal attacks
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