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.
Recent years have seen rapid advances in the data-driven analysis of dynamical systems based on Koopman operator theory and related approaches. 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 Koopman-based models and the TT format, enabling their application to high-dimensional problems in conjunction with a rich set of basis functions or features. We derive efficient algorithms to obtain a reduced matrix representation of the system’s evolution operator starting from an appropriate low-rank representation of the data. These algorithms can be applied to both stationary and non-stationary systems. We establish the infinite-data limit of these matrix representations, and demonstrate our methods’ capabilities using several benchmark data sets.
•We present AMUSEt, a method to approximate evolution operators on tensor spaces.•We analyze the convergence properties of the method in the limit of infinite data.•We present an alternative formulation of the method using HOCUR decompositions.•We apply our methods to benchmark molecular dynamics and fluid dynamics data sets.
In multiscale modeling of heterogeneous catalytic processes, one crucial point is the solution of a Markovian master equation describing the stochastic reaction kinetics. Usually, this is too ...high-dimensional to be solved with standard numerical techniques and one has to rely on sampling approaches based on the kinetic Monte Carlo method. In this study we break the curse of dimensionality for the direct solution of the Markovian master equation by exploiting the Tensor Train Format for this purpose. The performance of the approach is demonstrated on a first principles based, reduced model for the CO oxidation on the RuO2(110) surface. We investigate the complexity for increasing system size and for various reaction conditions. The advantage over the stochastic simulation approach is illustrated by a problem with increased stiffness.
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.
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.
Clonal evolution is believed to be a main driver for progression of various types of cancer and implicated in facilitating resistance to drugs. However, the hierarchical organization of malignant ...clones in the hematopoiesis of myelodysplastic syndromes (MDS) and its impact on response to drug therapy remain poorly understood. Using high-throughput sequencing of patient and xenografted cells, we evaluated the intratumoral heterogeneity (n= 54) and reconstructed mutational trajectories (n = 39) in patients suffering from MDS (n = 52) and chronic myelomonocytic leukemia-1 (n = 2). We identified linear and also branching evolution paths and confirmed on a patient-specific level that somatic mutations in epigenetic regulators and RNA splicing genes frequently constitute isolated disease-initiating events. Using high-throughput exome- and/or deep-sequencing, we analyzed 103 chronologically acquired samples from 22 patients covering a cumulative observation time of 75 years MDS disease progression. Our data revealed highly dynamic shaping of complex oligoclonal architectures, specifically upon treatment with lenalidomide and other drugs. Despite initial clinical response to treatment, patients' marrow persistently remained clonal with rapid outgrowth of founder-, sub-, or even fully independent clones, indicating an increased dynamic rate of clonal turnover. The emergence and disappearance of specific clones frequently correlated with changes of clinical parameters, highlighting their distinct and far-reaching functional properties. Intriguingly, increasingly complex mutational trajectories are frequently accompanied by clinical progression during the course of disease. These data substantiate a need for regular broad molecular monitoring to guide clinical treatment decisions in MDS.
•Mutational trajectories are defined by complex patterns of molecular heterogeneity in MDS, including lower-risk cases.•Therapeutic intervention dynamically reshapes mutational patterns often resulting in branched or independent evolution of MDS clones.
Abstract
We derive symmetric and antisymmetric kernels by symmetrizing and antisymmetrizing conventional kernels and analyze their properties. In particular, we compute the feature space dimensions ...of the resulting polynomial kernels, prove that the reproducing kernel Hilbert spaces induced by symmetric and antisymmetric Gaussian kernels are dense in the space of symmetric and antisymmetric functions, and propose a Slater determinant representation of the antisymmetric Gaussian kernel, which allows for an efficient evaluation even if the state space is high-dimensional. Furthermore, we show that by exploiting symmetries or antisymmetries the size of the training data set can be significantly reduced. The results are illustrated with guiding examples and simple quantum physics and chemistry applications.
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