972 research outputs found
Non-negative Principal Component Analysis: Message Passing Algorithms and Sharp Asymptotics
Principal component analysis (PCA) aims at estimating the direction of
maximal variability of a high-dimensional dataset. A natural question is: does
this task become easier, and estimation more accurate, when we exploit
additional knowledge on the principal vector? We study the case in which the
principal vector is known to lie in the positive orthant. Similar constraints
arise in a number of applications, ranging from analysis of gene expression
data to spike sorting in neural signal processing.
In the unconstrained case, the estimation performances of PCA has been
precisely characterized using random matrix theory, under a statistical model
known as the `spiked model.' It is known that the estimation error undergoes a
phase transition as the signal-to-noise ratio crosses a certain threshold.
Unfortunately, tools from random matrix theory have no bearing on the
constrained problem. Despite this challenge, we develop an analogous
characterization in the constrained case, within a one-spike model.
In particular: ~We prove that the estimation error undergoes a similar
phase transition, albeit at a different threshold in signal-to-noise ratio that
we determine exactly; ~We prove that --unlike in the unconstrained case--
estimation error depends on the spike vector, and characterize the least
favorable vectors; ~We show that a non-negative principal component can
be approximately computed --under the spiked model-- in nearly linear time.
This despite the fact that the problem is non-convex and, in general, NP-hard
to solve exactly.Comment: 51 pages, 7 pdf figure
Link Prediction in Graphs with Autoregressive Features
In the paper, we consider the problem of link prediction in time-evolving
graphs. We assume that certain graph features, such as the node degree, follow
a vector autoregressive (VAR) model and we propose to use this information to
improve the accuracy of prediction. Our strategy involves a joint optimization
procedure over the space of adjacency matrices and VAR matrices which takes
into account both sparsity and low rank properties of the matrices. Oracle
inequalities are derived and illustrate the trade-offs in the choice of
smoothing parameters when modeling the joint effect of sparsity and low rank
property. The estimate is computed efficiently using proximal methods through a
generalized forward-backward agorithm.Comment: NIPS 201
Alien Registration- Richard, Emile (Rumford, Oxford County)
https://digitalmaine.com/alien_docs/12415/thumbnail.jp
A statistical model for tensor PCA
We consider the Principal Component Analysis problem for large tensors of
arbitrary order under a single-spike (or rank-one plus noise) model. On the
one hand, we use information theory, and recent results in probability theory,
to establish necessary and sufficient conditions under which the principal
component can be estimated using unbounded computational resources. It turns
out that this is possible as soon as the signal-to-noise ratio becomes
larger than (and in particular can remain bounded as
the problem dimensions increase).
On the other hand, we analyze several polynomial-time estimation algorithms,
based on tensor unfolding, power iteration and message passing ideas from
graphical models. We show that, unless the signal-to-noise ratio diverges in
the system dimensions, none of these approaches succeeds. This is possibly
related to a fundamental limitation of computationally tractable estimators for
this problem.
We discuss various initializations for tensor power iteration, and show that
a tractable initialization based on the spectrum of the matricized tensor
outperforms significantly baseline methods, statistically and computationally.
Finally, we consider the case in which additional side information is available
about the unknown signal. We characterize the amount of side information that
allows the iterative algorithms to converge to a good estimate.Comment: Neural Information Processing Systems (NIPS) 2014 (slightly expanded:
30 pages, 6 figures
Strongly Refuting Random CSPs Below the Spectral Threshold
Random constraint satisfaction problems (CSPs) are known to exhibit threshold
phenomena: given a uniformly random instance of a CSP with variables and
clauses, there is a value of beyond which the CSP will be
unsatisfiable with high probability. Strong refutation is the problem of
certifying that no variable assignment satisfies more than a constant fraction
of clauses; this is the natural algorithmic problem in the unsatisfiable regime
(when ).
Intuitively, strong refutation should become easier as the clause density
grows, because the contradictions introduced by the random clauses become
more locally apparent. For CSPs such as -SAT and -XOR, there is a
long-standing gap between the clause density at which efficient strong
refutation algorithms are known, , and the
clause density at which instances become unsatisfiable with high probability,
.
In this paper, we give spectral and sum-of-squares algorithms for strongly
refuting random -XOR instances with clause density in time or in
rounds of the sum-of-squares hierarchy, for any
and any integer . Our algorithms provide a smooth
transition between the clause density at which polynomial-time algorithms are
known at , and brute-force refutation at the satisfiability
threshold when . We also leverage our -XOR results to obtain
strong refutation algorithms for SAT (or any other Boolean CSP) at similar
clause densities. Our algorithms match the known sum-of-squares lower bounds
due to Grigoriev and Schonebeck, up to logarithmic factors.
Additionally, we extend our techniques to give new results for certifying
upper bounds on the injective tensor norm of random tensors
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