16 research outputs found
A new convergence proof for the higher-order power method and generalizations
A proof for the point-wise convergence of the factors in the higher-order
power method for tensors towards a critical point is given. It is obtained by
applying established results from the theory of \L{}ojasiewicz inequalities to
the equivalent, unconstrained alternating least squares algorithm for best
rank-one tensor approximation
Riemannian thresholding methods for row-sparse and low-rank matrix recovery
In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular, a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show near-optimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity
An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank
An approximate projection onto the tangent cone to the variety of third-order
tensors of bounded tensor-train rank is proposed and proven to satisfy a better
angle condition than the one proposed by Kutschan (2019). Such an approximate
projection enables, e.g., to compute gradient-related directions in the tangent
cone, as required by algorithms aiming at minimizing a continuously
differentiable function on the variety, a problem appearing notably in tensor
completion. A numerical experiment is presented which indicates that, in
practice, the angle condition satisfied by the proposed approximate projection
is better than both the one satisfied by the approximate projection introduced
by Kutschan and the proven theoretical bound
A Riemannian rank-adaptive method for low-rank matrix completion
The low-rank matrix completion problem can be solved by Riemannian
optimization on a fixed-rank manifold. However, a drawback of the known
approaches is that the rank parameter has to be fixed a priori. In this paper,
we consider the optimization problem on the set of bounded-rank matrices. We
propose a Riemannian rank-adaptive method, which consists of fixed-rank
optimization, rank increase step and rank reduction step. We explore its
performance applied to the low-rank matrix completion problem. Numerical
experiments on synthetic and real-world datasets illustrate that the proposed
rank-adaptive method compares favorably with state-of-the-art algorithms. In
addition, it shows that one can incorporate each aspect of this rank-adaptive
framework separately into existing algorithms for the purpose of improving
performance.Comment: 22 pages, 12 figures, 1 tabl
Bayesian inversion with a hierarchical tensor representation
The statistical Bayesian approach is a natural setting to resolve the ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametric deterministic representation of the forward model, a sampling-free approach to Bayesian inversion with an explicit representation of the parameter densities is developed. The approximation of the involved randomness inevitably leads to several high dimensional expressions, which are often tackled with classical sampling methods such as MCMC. To speed up these methods, the use of a surrogate model is beneficial since it allows for faster evaluation with respect to calibration parameters. However, the inherently slow convergence can not be remedied by this. As an alternative, a complete functional treatment of the inverse problem is feasible as demonstrated in this work, with functional representations of the parametric forward solution as well as the probability densities of the calibration parameters, determined by Bayesian inversion. The proposed sampling-free approach is discussed in the context of hierarchical tensor representations, which are employed for the adaptive evaluation of a random PDE (the forward problem) in generalized chaos polynomials and the subsequent high-dimensional quadrature of the log-likelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the involved low rank (solution) manifolds. All required computations can be carried out efficiently in the low-rank format. A priori convergence is examined, considering all approximations that occur in the method. Numerical experiments demonstrate the performance and verify the theoretical results
Adaptive stochastic Galerkin FEM with hierarchical tensor representations
The solution of PDE with stochastic data commonly leads to very high-dimensional algebraic problems, e.g. when multiplicative noise is present. The Stochastic Galerkin FEM considered in this paper then suffers from the curse of dimensionality. This is directly related to the number of random variables required for an adequate representation of the random fields included in the PDE. With the presented new approach, we circumvent this major complexity obstacle by combining two highly efficient model reduction strategies, namely a modern low-rank tensor representation in the tensor train format of the problem and a refinement algorithm on the basis of a posteriori error estimates to adaptively adjust the different employed discretizations. The adaptive adjustment includes the refinement of the FE mesh based on a residual estimator, the problem-adapted stochastic discretization in anisotropic Legendre Wiener chaos and the successive increase of the tensor rank. Computable a posteriori error estimators are derived for all error terms emanating from the discretizations and the iterative solution with a preconditioned ALS scheme of the problem. Strikingly, it is possible to exploit the tensor structure of the problem to evaluate all error terms very efficiently. A set of benchmark problems illustrates the performance of the adaptive algorithm with higher-order FE. Moreover, the influence of the tensor rank on the approximation quality is investigated
First-order optimization on stratified sets
We consider the problem of minimizing a differentiable function with locally
Lipschitz continuous gradient on a stratified set and present a first-order
algorithm designed to find a stationary point of that problem. Our assumptions
on the stratified set are satisfied notably by the determinantal variety (i.e.,
matrices of bounded rank), its intersection with the cone of
positive-semidefinite matrices, and the set of nonnegative sparse vectors. The
iteration map of the proposed algorithm applies a step of projected-projected
gradient descent with backtracking line search, as proposed by Schneider and
Uschmajew (2015), to its input but also to a projection of the input onto each
of the lower strata to which it is considered close, and outputs a point among
those thereby produced that maximally reduces the cost function. Under our
assumptions on the stratified set, we prove that this algorithm produces a
sequence whose accumulation points are stationary, and therefore does not
follow the so-called apocalypses described by Levin, Kileel, and Boumal (2022).
We illustrate the apocalypse-free property of our method through a numerical
experiment on the determinantal variety