parallel-tensor-train

This repository discusses parallel and scalable tensor train decomposition techiniques for high dimensional tensors. Presently parallel algorithms are implemented in Matlab. Distributed memory implementations of the parallel algorithms in C++ are currently underway.

We proposed mainly 4 approaches to obtain tensor train approximations of high dimensional tensors. These are

1. STTA
2. LSR
3. SLSB
4. LSB

STTA is a sequential approach, while others are parallel. We refer the user to our paper for more details about these approaches.

How to run code of this repository?

Requirements (Matlab code)

• Matlab/Octave

Enter into the structured folder (all matlab files are in this directory)

>> cd structured

Rank based examples

>> A = rand(4,4,4,4);
>> r=2

>> Gsvd = tensorTrainCompressionFixedRankSVD(A, r);
>> approx_error = computeError(A, Gsvd)
Core tensor size = 1 X 4 X 2
Core tensor size = 2 X 4 X 2
Core tensor size = 2 X 4 X 2
Core tensor size = 2 X 4 X 1
ne =

    8   16   16    8

total_elements =  48
compression_ratio_percentage =  81.250
approx_error =  4.2098

>> Gqrcpsvd = tensorTrainCompressionFixedRankQRCPSVD(A, r);
>> computeError(A, Gqrcpsvd)
Core tensor size = 1 X 4 X 2
Core tensor size = 2 X 4 X 2
Core tensor size = 2 X 4 X 2
Core tensor size = 2 X 4 X 1
ne =

    8   16   16    8

total_elements =  48
compression_ratio_percentage =  81.250
ans =  4.4020

>> G3qrsvd = parallelTensorTrainCompressionLSBFixedRankQRCPSVD(A, r);
>> approx_error = computeError(A, G3qrsvd)
Core tensor size = 1 X 4 X 2
Core tensor size = 2 X 4 X 2
Core tensor size = 2 X 4 X 2
Core tensor size = 2 X 4 X 1
ne =

    8   16   16    8

total_elements =  48
compression_ratio_percentage =  81.250
approx_error =  4.6781

Long run example(takes a long time):

 >> caller()
Enter the tensor choice:
1) A tensor with 4^12 elements
2) A tensor with 16^6 elements
1
Enter the low rank function
1) Log function
2) Sin function
3) Inverse square function
4) Inverse cube function
5) Inverse penta function
1
Enter the prescribed accuracy:0.001
Results for sequential tensor train approximation
Core tensor size = 1 X 4 X 3
Core tensor size = 3 X 4 X 3
Core tensor size = 3 X 4 X 3
Core tensor size = 3 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 1
ne =

   12   36   36   48   64   64   64   64   64   64   64   16

total_elements =  596
compression_ratio_percentage =  99.996
approx_error_tt =    3.6922e-04

Results for parallel tensor train approximation with LSR approach
Core tensor size = 1 X 4 X 4
Core tensor size = 4 X 4 X 12
Core tensor size = 12 X 4 X 6
Core tensor size = 6 X 4 X 15
Core tensor size = 15 X 4 X 7
Core tensor size = 7 X 4 X 4
Core tensor size = 4 X 4 X 16
Core tensor size = 16 X 4 X 22
Core tensor size = 22 X 4 X 7
Core tensor size = 7 X 4 X 15
Core tensor size = 15 X 4 X 4
Core tensor size = 4 X 4 X 1
ne =

     16    192    288    360    420    112    256   1408    616    420    240     16

total_elements =  4344
compression_ratio_percentage =  99.974
approx_error_ptt_h1 =    7.4966e-05

Results for parallel tensor train approximation with SLSB approach
Core tensor size = 1 X 4 X 4
Core tensor size = 4 X 4 X 6
Core tensor size = 6 X 4 X 5
Core tensor size = 5 X 4 X 11
Core tensor size = 11 X 4 X 7
Core tensor size = 7 X 4 X 4
Core tensor size = 4 X 4 X 9
Core tensor size = 9 X 4 X 8
Core tensor size = 8 X 4 X 6
Core tensor size = 6 X 4 X 8
Core tensor size = 8 X 4 X 4
Core tensor size = 4 X 4 X 1
ne =

    16    96   120   220   308   112   144   288   192   192   128    16

total_elements =  1832
compression_ratio_percentage =  99.989
approx_error_ptt_h2 =    7.0418e-05

Results for parallel tensor train approximation with LSB approach
Core tensor size = 1 X 4 X 3
Core tensor size = 3 X 4 X 3
Core tensor size = 3 X 4 X 3
Core tensor size = 3 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 4
Core tensor size = 4 X 4 X 1
ne =

   12   36   36   48   64   64   64   64   64   64   64   16

total_elements =  596
compression_ratio_percentage =  99.996
approx_error_ptt_h3 =    3.6922e-04

/***** Implementations based on lapack and scalapack calls are in progress ******/

Requirements (C++ sequential code)

• LAPACK
• BLAS

Requirements (C++ parallel code)

• Scalapack