TTTensor

Tensor-Train representation of multilinear MTI models.

Source: src/tensor/@TTTensor/TTTensor.m

Syntax

obj = TTTensor()
obj = TTTensor(cores)
obj = TTTensor.fromRawCores('cores', cores)
obj = TTTensor.fromFullTensor('tensor', F)
obj = TTTensor.fromFullParameterMatrix('matrix', M)

Description

Use TTTensor to create a Tensor Train (TT) representation of a multilinear MTI tensor.

A Tensor Train decomposes a high-dimensional tensor into a sequence of low-rank tensor cores. Instead of storing the full tensor explicitly, only the TT cores are stored, often resulting in significant reductions in memory consumption and computational effort.

A tensor $T(i_1, i_2, \cdots, i_d)$ is represented as

$T(i_1, i_2, \cdots, i_d) = G_1(i_1) G_2(i_2) \cdots G_d(i_d)$

where each $G_k$ denotes a TT core

For multilinear MTI systems, Tensor Train representations can be used as an alternative to CP-based representations such as CPNTensor while preserving compatibility with mss and mdss models.

The TTTensor class supports: - Construction from TT cores - Construction from full tensors - Construction from multilinear functional matrices - Exact conversion from CP representations - Conversion back to explicit CP structure and multilinear functional matrices and then to TT

For further detail see the references.

Input Arguments

cores - TTcores

Cell array containing Tensor Train cores. The $k^{th}$ TT core has dimensions $ r_k n_K r_{k+1} $

where: - $n_k$ is the mode dimension - $r_k$ and $r_{k+1}$ are TT ranks

tensor - Full tensor

Full tensor represented as a MATLAB multidimensional array. The tensor is decomposed into TT format using tt-svd.

matrix - Full parameter matrix

Multilinear parameter matrix with dimensions $n \times 2^r$ corresponds to the MTI parameter matrix representation.

The matrix is internally reshaped into $[n, 2, 2, \cdots, 2]$ and converted into TT format using tt-svd.

Output Arguments

obj - TTTensor object

Tensor Train tensor object returned as a TTTensor object

Parameters

cores

Cell array containing all TT cores.

Form	Result
`TTTensor()`	An empty Tensor-Train tensor object.
`TTTensor(cores)`	A Tensor-Train tensor from a cell array of TT cores. Each core must be a numeric or logical three-dimensional array with dimensions `r(k) x n(k) x r(k+1)`.
`TTTensor.fromRawCores('cores', cores)`	A Tensor-Train tensor directly from a validated set of TT cores.
`TTTensor.fromFullTensor('tensor', F)`	Converts a full tensor `F` into Tensor-Train format using the TT-SVD decomposition.
`TTTensor.fromFullParameterMatrix('matrix', M)`	Converts a multilinear parameter matrix `M` (dimensions `n`-by-`2^r`, per the MTI multilinear model representation) into Tensor-Train format.

Construction of TT tensor

A simple empty TTTensor can be created by

obj = TTTensor();

If you have the parameter matrix of the model, the TTTensor could be constructed from this matrix by

M = randn(2,16);
tt_obj = TTTensor.fromFullParameterMatrix('matrix', M);

Providing the existing TT-cores would also enable the creation of the TTTensor

F1 = randn(1,3,2);
F2 = randn(2,2,2);
F3 = randn(2,2,1);

T = TTTensor({F1, F2, F3});

If the full tensor of the model is available which represents the model parameters, the TTTensor can be constructed by

fulltensor = randn(3,2,2,2);
obj2 = TTTensor.fromFullTensor(tensor=fulltensor);

Finally, if the ‘mtiTensor’ is a CPNTensor, it can be converted to the TTTensor by

%                   x1   x1*u1
structureMatrix = [  1     1 ;   % x1
                     0     1 ];  % u1

parameterMatrix = [ 0.9   0.1 ;  % x1(k+1)
                    2     0   ]; % y1

cpn_obj = CPNTensor(structureMatrix, parameterMatrix); 
ttobj = CPNTensor.CPN2TT(cpn_obj);

References

J.Cherian, E.UhIenberg, H.G.S.Maregowda, L.S.VaIIejos, T.Warnecke, and G. Lichtenberg (2026). Tensor train based explicit multilinear modeling and control of heating systems. $12^{th}$ CoDIT Conference. https://doi.org/10.1109/CoDIT70676.2026.11631280
Oseledets, I. V. (2011). Tensor-Train Decomposition. SIAM Journal on Scientific Computing. https://doi.org/10.1137/090752286
Pangalos, G., Eichler, A., and Lichtenberg, G. (2013). Tensor Systems: Multilinear Modeling and Applications.