36 research outputs found
Tensors in modelling multi-particle interactions
In this work we present recent results on application of low-rank tensor
decompositions to modelling of aggregation kinetics taking into account
multi-particle collisions (for three and more particles). Such kinetics can be
described by system of nonlinear differential equations with right-hand side
requiring operations for its straight-forward evaluation, where is
number of particles size classes and is number of particles colliding
simultaneously. Such a complexity can be significantly reduced by application
low rank tensor decompositions (either Tensor Train or Canonical Polyadic) to
acceleration of evaluation of sums and convolutions from right-hand side.
Basing on this drastic reduction of complexity for evaluation of right-hand
side we further utilize standard second order Runge-Kutta time integration
scheme and demonstrate that our approach allows to obtain numerical solutions
of studied equations with very high accuracy in modest times. We also show
preliminary results on parallel scalability of novel approach and conclude that
it can be efficiently utilized with use of supercomputers.Comment: LaTEX, 8 pages, 3 figures, submitted to proceedings of LSSC'19
conference, Sozopol, Bulgari
Neural Networks Compression for Language Modeling
In this paper, we consider several compression techniques for the language
modeling problem based on recurrent neural networks (RNNs). It is known that
conventional RNNs, e.g, LSTM-based networks in language modeling, are
characterized with either high space complexity or substantial inference time.
This problem is especially crucial for mobile applications, in which the
constant interaction with the remote server is inappropriate. By using the Penn
Treebank (PTB) dataset we compare pruning, quantization, low-rank
factorization, tensor train decomposition for LSTM networks in terms of model
size and suitability for fast inference.Comment: Keywords: LSTM, RNN, language modeling, low-rank factorization,
pruning, quantization. Published by Springer in the LNCS series, 7th
International Conference on Pattern Recognition and Machine Intelligence,
201
Low rank approximation of multidimensional data
In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics
community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in
the literature but either on separate papers or into a pure applied
mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.IV Research and Transfer Plan of the University of SevillaInstitut CarnotJunta de Andaluc铆aIDEX program of the University of Bordeau
Tensor Product Approximation (DMRG) and Coupled Cluster method in Quantum Chemistry
We present the Copupled Cluster (CC) method and the Density matrix
Renormalization Grooup (DMRG) method in a unified way, from the perspective of
recent developments in tensor product approximation. We present an introduction
into recently developed hierarchical tensor representations, in particular
tensor trains which are matrix product states in physics language. The discrete
equations of full CI approximation applied to the electronic Schr\"odinger
equation is casted into a tensorial framework in form of the second
quantization. A further approximation is performed afterwards by tensor
approximation within a hierarchical format or equivalently a tree tensor
network. We establish the (differential) geometry of low rank hierarchical
tensors and apply the Driac Frenkel principle to reduce the original
high-dimensional problem to low dimensions. The DMRG algorithm is established
as an optimization method in this format with alternating directional search.
We briefly introduce the CC method and refer to our theoretical results. We
compare this approach in the present discrete formulation with the CC method
and its underlying exponential parametrization.Comment: 15 pages, 3 figure
Approximating turbulent and non-turbulent events with the Tensor Train decomposition method
Low-rank multilevel approximation methods are often suited to attack high-dimensional problems successfully and they allow very compact representation of large data sets. Specifically, hierarchical tensor product decomposition methods, e.g., the Tree-Tucker format and the Tensor Train format emerge as a promising approach for application to data that are concerned with cascade-of-scales problems as, e.g., in turbulent fluid dynamics. Beyond multilinear mathematics, those tensor formats are also successfully applied in e.g., physics or chemistry, where they are used in many body problems and quantum states. Here, we focus on two particular objectives, that is, we aim at capturing self-similar structures that might be hidden in the data and we present the reconstruction capabilities of the Tensor Train decomposition method tested with 3D channel turbulence flow data
Geometric methods on low-rank matrix and tensor manifolds
In this chapter we present numerical methods for low-rank matrix and tensor problems that explicitly make use of the geometry of rank constrained matrix and tensor spaces. We focus on two types of problems: The first are optimization problems, like matrix and tensor completion, solving linear systems and eigenvalue problems. Such problems can be solved by numerical optimization for manifolds, called Riemannian optimization methods. We will explain the basic elements of differential geometry in order to apply such methods efficiently to rank constrained matrix and tensor spaces. The second type of problem is ordinary differential equations, defined on matrix and tensor spaces. We show how their solution can be approximated by the dynamical low-rank principle, and discuss several numerical integrators that rely in an essential way on geometric properties that are characteristic to sets of low rank matrices and tensors