16 research outputs found
Tensor-Based Algorithms for Image Classification
Interest in machine learning with tensor networks has been growing rapidly in recent years. We show that tensor-based methods developed for learning the governing equations of dynamical systems from data can, in the same way, be used for supervised learning problems and propose two novel approaches for image classification. One is a kernel-based reformulation of the previously introduced multidimensional approximation of nonlinear dynamics (MANDy), the other an alternating ridge regression in the tensor train format. We apply both methods to the MNIST and fashion MNIST data set and show that the approaches are competitive with state-of-the-art neural network-based classifiers
Tensor-based dynamic mode decomposition
Dynamic mode decomposition (DMD) is a recently developed tool for the
analysis of the behavior of complex dynamical systems. In this paper, we will
propose an extension of DMD that exploits low-rank tensor decompositions of
potentially high-dimensional data sets to compute the corresponding DMD modes
and eigenvalues. The goal is to reduce the computational complexity and also
the amount of memory required to store the data in order to mitigate the curse
of dimensionality. The efficiency of these tensor-based methods will be
illustrated with the aid of several different fluid dynamics problems such as
the von K\'arm\'an vortex street and the simulation of two merging vortices
Multidimensional approximation of nonlinear dynamical systems
A key task in the field of modeling and analyzing nonlinear dynamical systems is the recovery of unknown governing equations from measurement data only. There is a wide range of application areas for this important instance of system identification, ranging from industrial engineering and acoustic signal processing to stock market models. In order to find appropriate representations of underlying dynamical systems, various data-driven methods have been proposed by different communities. However, if the given data sets are high-dimensional, then these methods typically suffer from the curse of dimensionality. To significantly reduce the computational costs and storage consumption, we propose the method multidimensional approximation of nonlinear dynamical systems (MANDy) which combines data-driven methods with tensor network decompositions. The efficiency of the introduced approach will be illustrated with the aid of several high-dimensional nonlinear dynamical systems
Nearest-Neighbor Interaction Systems in the Tensor-Train Format
Low-rank tensor approximation approaches have become an important tool in the
scientific computing community. The aim is to enable the simulation and
analysis of high-dimensional problems which cannot be solved using conventional
methods anymore due to the so-called curse of dimensionality. This requires
techniques to handle linear operators defined on extremely large state spaces
and to solve the resulting systems of linear equations or eigenvalue problems.
In this paper, we present a systematic tensor-train decomposition for
nearest-neighbor interaction systems which is applicable to a host of different
problems. With the aid of this decomposition, it is possible to reduce the
memory consumption as well as the computational costs significantly.
Furthermore, it can be shown that in some cases the rank of the tensor
decomposition does not depend on the network size. The format is thus feasible
even for high-dimensional systems. We will illustrate the results with several
guiding examples such as the Ising model, a system of coupled oscillators, and
a CO oxidation model
Continuous optimization methods for the graph isomorphism problem
The graph isomorphism problem looks deceptively simple, but although
polynomial-time algorithms exist for certain types of graphs such as planar
graphs and graphs with bounded degree or eigenvalue multiplicity, its
complexity class is still unknown. Information about potential isomorphisms
between two graphs is contained in the eigenvalues and eigenvectors of their
adjacency matrices. However, symmetries of graphs often lead to repeated
eigenvalues so that associated eigenvectors are determined only up to basis
rotations, which complicates graph isomorphism testing. We consider orthogonal
and doubly stochastic relaxations of the graph isomorphism problem, analyze the
geometric properties of the resulting solution spaces, and show that their
complexity increases significantly if repeated eigenvalues exist. By
restricting the search space to suitable subspaces, we derive an efficient
Frank-Wolfe based continuous optimization approach for detecting isomorphisms.
We illustrate the efficacy of the algorithm with the aid of various highly
symmetric graphs
Existence and Uniqueness of Solutions of the Koopman--von Neumann Equation on Bounded Domains
The Koopman--von Neumann equation describes the evolution of a complex-valued
wavefunction corresponding to the probability distribution given by an
associated classical Liouville equation. Typically, it is defined on the whole
Euclidean space. The investigation of bounded domains, particularly in
practical scenarios involving quantum-based simulations of dynamical systems,
has received little attention so far. We consider the Koopman--von Neumann
equation associated with an ordinary differential equation on a bounded domain
whose trajectories are contained in the set's closure. Our main results are the
construction of a strongly continuous semigroup together with the existence and
uniqueness of solutions of the associated initial value problem. To this end, a
functional-analytic framework connected to Sobolev spaces is proposed and
analyzed. Moreover, the connection of the Koopman--von Neumann framework to
transport equations is highlighted
Tensor-based computation of metastable and coherent sets
Recent years have seen rapid advances in the data-driven analysis of
dynamical systems based on Koopman operator theory -- with extended dynamic
mode decomposition (EDMD) being a cornerstone of the field. On the other hand,
low-rank tensor product approximations -- in particular the tensor train (TT)
format -- have become a valuable tool for the solution of large-scale problems
in a number of fields. In this work, we combine EDMD and the TT format,
enabling the application of EDMD to high-dimensional problems in conjunction
with a large set of features. We derive efficient algorithms to solve the EDMD
eigenvalue problem based on tensor representations of the data, and to project
the data into a low-dimensional representation defined by the eigenvectors. We
extend this method to perform canonical correlation analysis (CCA) of
non-reversible or time-dependent systems. We prove that there is a physical
interpretation of the procedure and demonstrate its capabilities by applying
the method to several benchmark data sets
Feature space approximation for kernel-based supervised learning
We propose a method for the approximation of high- or even
infinite-dimensional feature vectors, which play an important role in
supervised learning. The goal is to reduce the size of the training data,
resulting in lower storage consumption and computational complexity.
Furthermore, the method can be regarded as a regularization technique, which
improves the generalizability of learned target functions. We demonstrate
significant improvements in comparison to the computation of data-driven
predictions involving the full training data set. The method is applied to
classification and regression problems from different application areas such as
image recognition, system identification, and oceanographic time series
analysis
Improved local models and new Bell inequalities via Frank-Wolfe algorithms
In Bell scenarios with two outcomes per party, we algorithmically consider
the two sides of the membership problem for the local polytope: constructing
local models and deriving separating hyperplanes, that is, Bell inequalities.
We take advantage of the recent developments in so-called Frank-Wolfe
algorithms to significantly increase the convergence rate of existing methods.
As an application, we study the threshold value for the nonlocality of
two-qubit Werner states under projective measurements. Here, we improve on both
the upper and lower bounds present in the literature. Importantly, our bounds
are entirely analytical; moreover, they yield refined bounds on the value of
the Grothendieck constant of order three: . We also demonstrate the efficiency of our approach in
multipartite Bell scenarios, and present the first local models for all
projective measurements with visibilities noticeably higher than the
entanglement threshold. We make our entire code accessible as a Julia library
called BellPolytopes.jl.Comment: 16 pages, 3 figure