RubenRanval/TensorTrainTools | Paclet Repository

RubenRanval`TensorTrainTools`

TensorTrainDecomposition

TensorTrainDecomposition [t]
	decomposes the numeric array tensor t into a tensor train, a list of rank-3 cores { A 1 , A 2 …, A n }

Details and Options

▪

TensorTrainDecomposition decomposes a numerical array of any rank into a Tensor Train (also known as a Matrix Product State) representation.

▪

The result is a list of rank-3 arrays

{

…,

}

where

is the number of dimensions of the input tensor.

▪

The bond dimension refers to the size of the internal contracted indices connecting adjacent tensor cores. It effectively dictates the rank and compression level of the Tensor Train.

▪

Each core

has dimensions

{

k-1

}

where

is the size of the

dimension of the input tensor, and

is the bond dimension between cores k and k+1.

▪

The first core has

and the last core has

▪

The following options can be given:

"MaxBondDimension"	Infinity	maximum bond dimension χ at each core
Tolerance	0	truncation threshold ε
Method	"SVD"	decomposition method

▪

With

Method

→ "QR", the function ignores "MaxBondDimension" and

Tolerance

and returns the exact decomposition.

QRDecomposition

does not provide singular values, so truncation is not available.

▪

With the default settings, TensorTrainDecomposition returns an exact decomposition with no truncation.

▪

Setting only

"MaxBondDimension"

controls memory: bond dimensions are capped at the specified value.

▪

Setting only

Tolerance

controls accuracy: bond dimensions adapt automatically to meet the error target.

▪

With

Method

→ "SVD", singular values are discarded until the sum of the squared discarded values reaches

, where

is the value specified by

Tolerance

▪

For more mathematical background on this algorithm, see the Wikipedia articles on Tensor Network and Matrix Product State.

Examples

(7)

Basic Examples

(1)

Decompose a simple 2×3×4 tensor into a Tensor Train with exact mathematical precision:

In[1]:=

tensor=RandomReal[1,{2,3,4}];tensor//MatrixForm

Out[1]//MatrixForm=

0.847338

0.21448

0.937994

0.988433

0.511721

0.343519

0.159504

0.10954

0.293368

0.981938

0.0382194

0.826166

0.9938

0.681778

0.721473

0.0140569

0.788157

0.218408

0.751931

0.606123

0.729557

0.677594

0.414111

0.145786

In[2]:=

TensorTrainDecomposition[tensor]

Out[2]=

TensorTrain

Cores: 3

Max χ: 4

Tensor dims: {2,3,4}



The underlying cores can be accessed with the "Cores" property:

In[3]:=

TensorTrain

Cores: 3

Max χ: 4

Tensor dims: {2,3,4}

["Cores"]

Out[3]=

{{{{-0.801956,0.597382},{-0.597382,-0.801956}}},{{{-0.460303,-0.133718,-0.489801,0.468487},{-0.594543,0.356838,-0.201702,-0.626958},{-0.657099,-0.155319,0.537927,0.268915}},{{0.0307311,0.582246,-0.435329,0.333059},{0.039545,0.615667,0.441332,0.36397},{-0.0187889,-0.33557,-0.213918,0.267817}}},{{{1.26522},{1.20165},{1.09235},{1.65644}},{{0.0209867},{-0.814381},{0.283008},{0.388124}},{{0.523402},{-0.135013},{-0.1399},{-0.209582}},{{-0.000402348},{-0.00521084},{-0.215149},{0.145969}}}}

Inspect the dimensions of the underlying cores forming the Tensor Train:

In[4]:=

Dimensions/@%

Out[4]=