6,707 research outputs found
Symmetric Tensor Decomposition by an Iterative Eigendecomposition Algorithm
We present an iterative algorithm, called the symmetric tensor eigen-rank-one
iterative decomposition (STEROID), for decomposing a symmetric tensor into a
real linear combination of symmetric rank-1 unit-norm outer factors using only
eigendecompositions and least-squares fitting. Originally designed for a
symmetric tensor with an order being a power of two, STEROID is shown to be
applicable to any order through an innovative tensor embedding technique.
Numerical examples demonstrate the high efficiency and accuracy of the proposed
scheme even for large scale problems. Furthermore, we show how STEROID readily
solves a problem in nonlinear block-structured system identification and
nonlinear state-space identification
Sparse Volterra and Polynomial Regression Models: Recoverability and Estimation
Volterra and polynomial regression models play a major role in nonlinear
system identification and inference tasks. Exciting applications ranging from
neuroscience to genome-wide association analysis build on these models with the
additional requirement of parsimony. This requirement has high interpretative
value, but unfortunately cannot be met by least-squares based or kernel
regression methods. To this end, compressed sampling (CS) approaches, already
successful in linear regression settings, can offer a viable alternative. The
viability of CS for sparse Volterra and polynomial models is the core theme of
this work. A common sparse regression task is initially posed for the two
models. Building on (weighted) Lasso-based schemes, an adaptive RLS-type
algorithm is developed for sparse polynomial regressions. The identifiability
of polynomial models is critically challenged by dimensionality. However,
following the CS principle, when these models are sparse, they could be
recovered by far fewer measurements. To quantify the sufficient number of
measurements for a given level of sparsity, restricted isometry properties
(RIP) are investigated in commonly met polynomial regression settings,
generalizing known results for their linear counterparts. The merits of the
novel (weighted) adaptive CS algorithms to sparse polynomial modeling are
verified through synthetic as well as real data tests for genotype-phenotype
analysis.Comment: 20 pages, to appear in IEEE Trans. on Signal Processin
A unified framework for solving a general class of conditional and robust set-membership estimation problems
In this paper we present a unified framework for solving a general class of
problems arising in the context of set-membership estimation/identification
theory. More precisely, the paper aims at providing an original approach for
the computation of optimal conditional and robust projection estimates in a
nonlinear estimation setting where the operator relating the data and the
parameter to be estimated is assumed to be a generic multivariate polynomial
function and the uncertainties affecting the data are assumed to belong to
semialgebraic sets. By noticing that the computation of both the conditional
and the robust projection optimal estimators requires the solution to min-max
optimization problems that share the same structure, we propose a unified
two-stage approach based on semidefinite-relaxation techniques for solving such
estimation problems. The key idea of the proposed procedure is to recognize
that the optimal functional of the inner optimization problems can be
approximated to any desired precision by a multivariate polynomial function by
suitably exploiting recently proposed results in the field of parametric
optimization. Two simulation examples are reported to show the effectiveness of
the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic
Control (2014
The linearization problem of a binary quadratic problem and its applications
We provide several applications of the linearization problem of a binary
quadratic problem. We propose a new lower bounding strategy, called the
linearization-based scheme, that is based on a simple certificate for a
quadratic function to be non-negative on the feasible set. Each
linearization-based bound requires a set of linearizable matrices as an input.
We prove that the Generalized Gilmore-Lawler bounding scheme for binary
quadratic problems provides linearization-based bounds. Moreover, we show that
the bound obtained from the first level reformulation linearization technique
is also a type of linearization-based bound, which enables us to provide a
comparison among mentioned bounds. However, the strongest linearization-based
bound is the one that uses the full characterization of the set of linearizable
matrices. Finally, we present a polynomial-time algorithm for the linearization
problem of the quadratic shortest path problem on directed acyclic graphs. Our
algorithm gives a complete characterization of the set of linearizable matrices
for the quadratic shortest path problem
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