Ivan Oseledets homepage

Google mycitations service looks great. Here is mine

I decided to by on a twitter also (now posts only in Russian, but who knows )

oseledetsivan

Finally, after more than 2 years of reviewing process, the basic paper on the TT-format is published in SISC!

You can get the journal version here or directly at the SIAM website

The first text was based on the paper “Compact matrix form of the d-dimensional tensor decomposition”. However, it has improved (I believe) a lot, and can be considered as as separate paper.

Also, this paper is the first one, where my 1-year old daugther Nastya has written a small part: look at the Algorithm 2, step 2 at the end. These slashes were typed by her, but I only noticed it after publishing

The new paper,

S.V. Dolgov, I.V. Oseledets, Solution of linear systems and matrix inversion in the TT-format
describes a DMRG-type method for the solution of linear systems with both the matrix and the tensor in the TT-format. The method is able to solve certain structured linear systems of order 2^d, where d can be of order several hundreds, and TT-ranks can be of order tensor or hundreds. The solver is available as a part of the TT-Toolbox 2.1. Moreover, some test data of the article can be downloaded (the new Toolbox is required to be installed). You can download it from the page Benchmarks and data or directly download test data as gzipped tar archive

This archive contains .mat files with A,x,rhs, where A is a TT-matrix, rhs is a TT-vector (TT-tensor), x is an approximate solution of A*x = rhs. You can check this directly by computing in MATLAB norm(A*x – rhs)/norm(rhs)

Be aware, that computing full(A) is prohibitive for all examples, but full(x) or full(rhs) sometimes is not!

I plan to add more benchmarking data to this page, with more TT-matrices and TT-vectors.

TT-Toolbox 2.1 is released. It contains several bug fixes. My great thanks to Sergey Dolgov for writing many important routines and to Vladimir Kazeev for providing his code for the construction of the QTT-representation of discrete Laplace operator (it can be done for extremely high dimensions in a second!).

TT-Toolbox 2.1 can be downloaded from here or from the page

The short introduction can be also downloaded.

All comments and suggestions are welcome. In the next release we plan to add the conversion procedures from the T. Kolda/B. Bader Tensor Toolbox (conversion from ktensor and sptensor classes), and also to provide more benchmarking examples.

New version of TT-Toolbox is released.

TT Toolbox version 2.0 introduces several major innovations compared to version 1.0.

• New classes tt_tensor and tt_matrix, which now represent TT-tensors and TT-matrices.
• Object-oriented approach allowed to overload many standard MATLAB functions, including addition, subtraction,
  multiplication by number, scalar product, norm, Kronecker products, matrix-by-vector product etc
• Complex arithmetics is supported
• Several subroutines to generate basic TT-tensors and TT-matrices: matrices of all ones, identity matrix,
  random TT-tensors with fixed core sizes
• Advances operations, like computation of F(TT) using cross method, matrix-by-vector product using Krylov methods, dmrg solver for linear systems…

TT-Toolbox 2.0 can be downloaded from this page, or directly.

Brief introduction to functionality and changes

DMRG (Density Matrix Renormalization Group) is a very efficient algorithm for computation of low-lying eigenstates of quantum spin systems. QTT (Quantics Tensor Train) consists in tensorization of one-dimensional objects (i.e. vectors of values of function on a grid with 2^d points) into a d-dimensional tensor, and application of Tensor Train (TT) format to such tensor. What do they have in common? Check our new paper, DMRG+QTT approach to high-dimensional quantum molecular dynamics

I have written a one-page article that describes vital operations in TT-format. Hope you will find it convincing, that TT-format is indeed a very elegant representation for tensors.

In our paper Algebraic wavelet transform via quantics tensor train decomposition we show, how quantics tensor train approach, which consists in tensorization of vectors and matrices, can be interpreted as algebraic wavelet transform . Advantages are especially clear for two dimensional functions. Besides new results, this approach, called WTT (wavelet tensor trains) describes several open and interesting problems

The aim of Preprint 2010-04 is to construct explicit tensor-train representations for certain function-related tensors and vectors, which are constructed on the basis of introduced functional tensor train decomposition. These results are then used to construct explicit quantics tensor train decomposition for polynomial and sine functions.