3,316 research outputs found
Multivariate GARCH estimation via a Bregman-proximal trust-region method
The estimation of multivariate GARCH time series models is a difficult task
mainly due to the significant overparameterization exhibited by the problem and
usually referred to as the "curse of dimensionality". For example, in the case
of the VEC family, the number of parameters involved in the model grows as a
polynomial of order four on the dimensionality of the problem. Moreover, these
parameters are subjected to convoluted nonlinear constraints necessary to
ensure, for instance, the existence of stationary solutions and the positive
semidefinite character of the conditional covariance matrices used in the model
design. So far, this problem has been addressed in the literature only in low
dimensional cases with strong parsimony constraints. In this paper we propose a
general formulation of the estimation problem in any dimension and develop a
Bregman-proximal trust-region method for its solution. The Bregman-proximal
approach allows us to handle the constraints in a very efficient and natural
way by staying in the primal space and the Trust-Region mechanism stabilizes
and speeds up the scheme. Preliminary computational experiments are presented
and confirm the very good performances of the proposed approach.Comment: 35 pages, 5 figure
A Riemannian Trust Region Method for the Canonical Tensor Rank Approximation Problem
The canonical tensor rank approximation problem (TAP) consists of
approximating a real-valued tensor by one of low canonical rank, which is a
challenging non-linear, non-convex, constrained optimization problem, where the
constraint set forms a non-smooth semi-algebraic set. We introduce a Riemannian
Gauss-Newton method with trust region for solving small-scale, dense TAPs. The
novelty of our approach is threefold. First, we parametrize the constraint set
as the Cartesian product of Segre manifolds, hereby formulating the TAP as a
Riemannian optimization problem, and we argue why this parametrization is among
the theoretically best possible. Second, an original ST-HOSVD-based retraction
operator is proposed. Third, we introduce a hot restart mechanism that
efficiently detects when the optimization process is tending to an
ill-conditioned tensor rank decomposition and which often yields a quick escape
path from such spurious decompositions. Numerical experiments show improvements
of up to three orders of magnitude in terms of the expected time to compute a
successful solution over existing state-of-the-art methods
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