36 citations · 61 across the 23 of their papers we have counts for
27 papers
A stable rank-adaptive step-and-truncate finite volume method for Vlasov transport on domains with piecewise linear boundaries
André Uschmajew, Andreas Zeiser
We consider the numerical solution of the linear Vlasov transport equation on bounded spatial domains with inflow boundary conditions based on low-rank approximation. We combine a…
Stochastic Optimization of Tree Tensor Networks
Marius Willner, Maximilian Scharf, André Uschmajew +2
Tensor networks, originally developed for quantum many-body physics, are promising models for machine learning. We derive stochastic Riemannian optimizers for tree tensor networks…
On low-rank tensor train approximability for linear nearest neighbor systems
Patrick Gelß, Sebastian Matera, Reinhold Schneider +1
Low-rank tensor methods are an important tool in the numerical treatment of equations with a high-dimensional state space. Nearest neighbor interaction systems like the Ising model…
Natural Riemannian gradient for learning functional tensor networks
Nikolas Klug, Michael Ulbrich, André Uschmajew +1
We consider machine learning tasks with low-rank functional tree tensor networks (TTN) as the learning model. While in the case of least-squares regression, low-rank functional TTN…
Numerically stable variants of overrelaxation for operator Sinkhorn iteration
Henrik Eisenmann, Tasuku Soma, Xun Tang +1
We consider accelerated versions of the operator Sinkhorn iteration (OSI) for solving scaling problems for completely positive maps. Based on the interpretation of OSI as alternati…
On the randomized SVD in infinite dimensions
Daniel Kressner, David Persson, André Uschmajew
Randomized methods, such as the randomized SVD (singular value decomposition) and Nyström approximation, are an effective way to compute low-rank approximations of large matrices.…
Discontinuous Galerkin discretization of conservative dynamical low-rank approximation schemes for the Vlasov-Poisson equation
André Uschmajew, Andreas Zeiser
A numerical dynamical low-rank approximation (DLRA) scheme for the solution of the Vlasov-Poisson equation is presented. Based on the formulation of the DLRA equations as Friedrich…
Dynamical low-rank tensor approximations to high-dimensional parabolic problems: existence and convergence of spatial discretizations
Markus Bachmayr, Henrik Eisenmann, André Uschmajew
We consider dynamical low-rank approximations to parabolic problems on higher-order tensor manifolds in Hilbert spaces. In addition to existence of solutions and their stability wi…
Gauss-Southwell type descent methods for low-rank matrix optimization
Guillaume Olikier, André Uschmajew, Bart Vandereycken
We consider gradient-related methods for low-rank matrix optimization with a smooth cost function. The methods operate on single factors of the low-rank factorization and share asp…
On the approximation of vector-valued functions by volume sampling
Daniel Kressner, Tingting Ni, André Uschmajew
Given a Hilbert space and a finite measure space , the approximation of a vector-valued function by a -dimensional subspace $\mathcal U \sub…
Dynamical low-rank approximation of the Vlasov-Poisson equation with piecewise linear spatial boundary
André Uschmajew, Andreas Zeiser
We consider dynamical low-rank approximation (DLRA) for the numerical simulation of Vlasov--Poisson equations based on separation of space and velocity variables, as proposed in se…
Time-Varying Semidefinite Programming: Path Following a Burer-Monteiro Factorization
Antonio Bellon, Mareike Dressler, Vyacheslav Kungurtsev +2
We present an online algorithm for time-varying semidefinite programs (TV-SDPs), based on the tracking of the solution trajectory of a low-rank matrix factorization, also known as…
Kronecker Product Approximation of Operators in Spectral Norm via Alternating SDP
Mareike Dressler, André Uschmajew, Venkat Chandrasekaran
The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The appr…
Local convergence of alternating low-rank optimization methods with overrelaxation
Ivan V. Oseledets, Maxim V. Rakhuba, André Uschmajew
The local convergence of alternating optimization methods with overrelaxation for low-rank matrix and tensor problems is established. The analysis is based on the linearization of…
Maximum relative distance between real rank-two and rank-one tensors
Henrik Eisenmann, André Uschmajew
It is shown that the relative distance in Frobenius norm of a real symmetric order- tensor of rank two to its best rank-one approximation is upper bounded by $\sqrt{1-(1-1/d)^{d…
A note on the optimal convergence rate of descent methods with fixed step sizes for smooth strongly convex functions
André Uschmajew, Bart Vandereycken
Based on a result by Taylor, Hendrickx, and Glineur (J. Optim. Theory Appl., 178(2):455--476, 2018) on the attainable convergence rate of gradient descent for smooth and strongly c…
Riemannian thresholding methods for row-sparse and low-rank matrix recovery
Henrik Eisenmann, Felix Krahmer, Max Pfeffer +1
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…
A note on overrelaxation in the Sinkhorn algorithm
Tobias Lehmann, Max-K. von Renesse, Alexander Sambale +1
We derive an a priori parameter range for overrelaxation of the Sinkhorn algorithm, which guarantees global convergence and a strictly faster asymptotic local convergence. Guided b…
Existence of dynamical low-rank approximations to parabolic problems
Markus Bachmayr, Henrik Eisenmann, Emil Kieri +1
The existence and uniqueness of weak solutions to dynamical low-rank evolution problems for parabolic partial differential equations in two spatial dimensions is shown, covering al…
Chebyshev polynomials and best rank-one approximation ratio
Andrei Agrachev, Khazhgali Kozhasov, André Uschmajew
We establish a new extremal property of the classical Chebyshev polynomials in the context of best rank-one approximation of tensors. We also give some necessary conditions for a t…
Alternating least squares as moving subspace correction
Ivan Oseledets, Maxim Rakhuba, André Uschmajew
In this note we take a new look at the local convergence of alternating optimization methods for low-rank matrices and tensors. Our abstract interpretation as sequential optimizati…
On orthogonal tensors and best rank-one approximation ratio
Zhening Li, Yuji Nakatsukasa, Tasuku Soma +1
As is well known, the smallest possible ratio between the spectral norm and the Frobenius norm of an matrix with is and is (up to scalar scaling…
Tensor Networks for Latent Variable Analysis. Part I: Algorithms for Tensor Train Decomposition
Anh-Huy Phan, Andrzej Cichocki, Andre Uschmajew +3
Decompositions of tensors into factor matrices, which interact through a core tensor, have found numerous applications in signal processing and machine learning. A more general ten…
Finding a low-rank basis in a matrix subspace
Yuji Nakatsukasa, Tasuku Soma, André Uschmajew
For a given matrix subspace, how can we find a basis that consists of low-rank matrices? This is a generalization of the sparse vector problem. It turns out that when the subspace…