323 research outputs found
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
Preconditioned low-rank Riemannian optimization for linear systems with tensor product structure
The numerical solution of partial differential equations on high-dimensional
domains gives rise to computationally challenging linear systems. When using
standard discretization techniques, the size of the linear system grows
exponentially with the number of dimensions, making the use of classic
iterative solvers infeasible. During the last few years, low-rank tensor
approaches have been developed that allow to mitigate this curse of
dimensionality by exploiting the underlying structure of the linear operator.
In this work, we focus on tensors represented in the Tucker and tensor train
formats. We propose two preconditioned gradient methods on the corresponding
low-rank tensor manifolds: A Riemannian version of the preconditioned
Richardson method as well as an approximate Newton scheme based on the
Riemannian Hessian. For the latter, considerable attention is given to the
efficient solution of the resulting Newton equation. In numerical experiments,
we compare the efficiency of our Riemannian algorithms with other established
tensor-based approaches such as a truncated preconditioned Richardson method
and the alternating linear scheme. The results show that our approximate
Riemannian Newton scheme is significantly faster in cases when the application
of the linear operator is expensive.Comment: 24 pages, 8 figure
Proceedings of the second "international Traveling Workshop on Interactions between Sparse models and Technology" (iTWIST'14)
The implicit objective of the biennial "international - Traveling Workshop on
Interactions between Sparse models and Technology" (iTWIST) is to foster
collaboration between international scientific teams by disseminating ideas
through both specific oral/poster presentations and free discussions. For its
second edition, the iTWIST workshop took place in the medieval and picturesque
town of Namur in Belgium, from Wednesday August 27th till Friday August 29th,
2014. The workshop was conveniently located in "The Arsenal" building within
walking distance of both hotels and town center. iTWIST'14 has gathered about
70 international participants and has featured 9 invited talks, 10 oral
presentations, and 14 posters on the following themes, all related to the
theory, application and generalization of the "sparsity paradigm":
Sparsity-driven data sensing and processing; Union of low dimensional
subspaces; Beyond linear and convex inverse problem; Matrix/manifold/graph
sensing/processing; Blind inverse problems and dictionary learning; Sparsity
and computational neuroscience; Information theory, geometry and randomness;
Complexity/accuracy tradeoffs in numerical methods; Sparsity? What's next?;
Sparse machine learning and inference.Comment: 69 pages, 24 extended abstracts, iTWIST'14 website:
http://sites.google.com/site/itwist1
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