Tucker Tensors
Tucker format is a decomposition of a tensor X as the product of a core tensor G and matrices (e.g., A,B,C) in each dimension. In other words, a tensor X is expressed as:
In MATLAB notation, X=ttm(G,{A,B,C}). The ttensor class stores the components of the tensor X and can perform many operations, e.g., ttm, without explicitly forming the tensor X.
Contents
- Creating a ttensor with a tensor core
- Alternate core formats: sptensor, ktensor, or ttensor
- Creating a one-dimensional ttensor
- Constituent parts of a ttensor
- Creating a ttensor from its constituent parts
- Creating an empty ttensor.
- Use full or tensor to convert a ttensor to a tensor
- Use double to convert a ttensor to a (multidimensional) array
- Use ndims and size to get the size of a ttensor
- Subscripted reference to a ttensor
- Subscripted assignment for a ttensor
- Using end for last index
- Basic operations (uplus, uminus, mtimes) for a ttensor.
- Use permute to reorder the modes of a ttensor
- Displaying a ttensor
Creating a ttensor with a tensor core
core = tensor(rand(3,2,1),[3 2 1]); %<-- The core tensor. U = {rand(5,3), rand(4,2), rand(3,1)}; %<-- The matrices. X = ttensor(core,U) %<-- Create the ttensor.
X is a ttensor of size 5 x 4 x 3
X.core is a tensor of size 3 x 2 x 1
X.core(:,:,1) =
0.6808 0.7942
0.4611 0.0592
0.5678 0.6029
X.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
X.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
X.U{3} =
0.8392
0.6288
0.1338
Alternate core formats: sptensor, ktensor, or ttensor
core1 = sptenrand([3 2 1],3); %<-- Create a 3 x 2 x 1 sptensor. Y = ttensor(core1,U) %<-- Core is a sptensor.
Y is a ttensor of size 5 x 4 x 3
Y.core is a sparse tensor of size 3 x 2 x 1 with 3 nonzeros
(1,1,1) 0.0129
(2,1,1) 0.3840
(2,2,1) 0.6831
Y.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
Y.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
Y.U{3} =
0.8392
0.6288
0.1338
V = {rand(3,2),rand(2,2),rand(1,2)}; %<-- Create some random matrices.
core2 = ktensor(V); %<-- Create a 3 x 2 x 1 ktensor.
Y = ttensor(core2,U) %<-- Core is a ktensor.
Y is a ttensor of size 5 x 4 x 3
Y.core is a ktensor of size 3 x 2 x 1
Y.core.lambda =
1 1
Y.core.U{1} =
0.0928 0.6085
0.0353 0.0158
0.6124 0.0164
Y.core.U{2} =
0.1901 0.0576
0.5869 0.3676
Y.core.U{3} =
0.6315 0.7176
Y.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
Y.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
Y.U{3} =
0.8392
0.6288
0.1338
core3 = ttensor(tensor(1:8,[2 2 2]),V); %<-- Create a 3 x 2 x 1 ttensor. Y = ttensor(core3,U) %<-- Core is a ttensor.
Y is a ttensor of size 5 x 4 x 3
Y.core is a ttensor of size 3 x 2 x 1
Y.core.core is a tensor of size 2 x 2 x 2
Y.core.core(:,:,1) =
1 3
2 4
Y.core.core(:,:,2) =
5 7
6 8
Y.core.U{1} =
0.0928 0.6085
0.0353 0.0158
0.6124 0.0164
Y.core.U{2} =
0.1901 0.0576
0.5869 0.3676
Y.core.U{3} =
0.6315 0.7176
Y.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
Y.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
Y.U{3} =
0.8392
0.6288
0.1338
Creating a one-dimensional ttensor
Z = ttensor(tensor(rand(2,1),2), rand(4,2)) %<-- One-dimensional ttensor.
Z is a ttensor of size 4
Z.core is a tensor of size 2
Z.core(:) =
0.6927
0.0841
Z.U{1} =
0.4544 0.6756
0.4418 0.6992
0.3533 0.7275
0.1536 0.4784
Constituent parts of a ttensor
X.core %<-- Core tensor.
ans is a tensor of size 3 x 2 x 1 ans(:,:,1) = 0.6808 0.7942 0.4611 0.0592 0.5678 0.6029
X.U %<-- Cell array of matrices.
ans =
1×3 cell array
{5×3 double} {4×2 double} {3×1 double}
Creating a ttensor from its constituent parts
Y = ttensor(X.core,X.U) %<-- Recreate a tensor from its parts.
Y is a ttensor of size 5 x 4 x 3
Y.core is a tensor of size 3 x 2 x 1
Y.core(:,:,1) =
0.6808 0.7942
0.4611 0.0592
0.5678 0.6029
Y.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
Y.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
Y.U{3} =
0.8392
0.6288
0.1338
Creating an empty ttensor.
X = ttensor %<-- empty ttensor
X is a ttensor of size [empty tensor] X.core is a tensor of size [empty tensor] X.core = []
Use full or tensor to convert a ttensor to a tensor
X = ttensor(core,U) %<-- Create a tensor
X is a ttensor of size 5 x 4 x 3
X.core is a tensor of size 3 x 2 x 1
X.core(:,:,1) =
0.6808 0.7942
0.4611 0.0592
0.5678 0.6029
X.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
X.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
X.U{3} =
0.8392
0.6288
0.1338
full(X) %<-- Converts to a tensor.
ans is a tensor of size 5 x 4 x 3 ans(:,:,1) = 0.5518 0.5343 0.6407 0.2198 0.7102 0.6990 0.8285 0.2837 0.8821 0.8955 1.0383 0.3544 1.0405 1.1261 1.2485 0.4232 0.6897 0.7450 0.8271 0.2804 ans(:,:,2) = 0.4134 0.4003 0.4800 0.1647 0.5321 0.5237 0.6207 0.2125 0.6609 0.6709 0.7780 0.2655 0.7796 0.8437 0.9354 0.3170 0.5168 0.5582 0.6197 0.2101 ans(:,:,3) = 0.0880 0.0852 0.1021 0.0350 0.1132 0.1114 0.1321 0.0452 0.1406 0.1427 0.1655 0.0565 0.1659 0.1795 0.1990 0.0674 0.1099 0.1188 0.1318 0.0447
tensor(X) %<-- Also converts to a tensor.
ans is a tensor of size 5 x 4 x 3 ans(:,:,1) = 0.5518 0.5343 0.6407 0.2198 0.7102 0.6990 0.8285 0.2837 0.8821 0.8955 1.0383 0.3544 1.0405 1.1261 1.2485 0.4232 0.6897 0.7450 0.8271 0.2804 ans(:,:,2) = 0.4134 0.4003 0.4800 0.1647 0.5321 0.5237 0.6207 0.2125 0.6609 0.6709 0.7780 0.2655 0.7796 0.8437 0.9354 0.3170 0.5168 0.5582 0.6197 0.2101 ans(:,:,3) = 0.0880 0.0852 0.1021 0.0350 0.1132 0.1114 0.1321 0.0452 0.1406 0.1427 0.1655 0.0565 0.1659 0.1795 0.1990 0.0674 0.1099 0.1188 0.1318 0.0447
Use double to convert a ttensor to a (multidimensional) array
double(X) %<-- Converts to a MATLAB array
ans(:,:,1) =
0.5518 0.5343 0.6407 0.2198
0.7102 0.6990 0.8285 0.2837
0.8821 0.8955 1.0383 0.3544
1.0405 1.1261 1.2485 0.4232
0.6897 0.7450 0.8271 0.2804
ans(:,:,2) =
0.4134 0.4003 0.4800 0.1647
0.5321 0.5237 0.6207 0.2125
0.6609 0.6709 0.7780 0.2655
0.7796 0.8437 0.9354 0.3170
0.5168 0.5582 0.6197 0.2101
ans(:,:,3) =
0.0880 0.0852 0.1021 0.0350
0.1132 0.1114 0.1321 0.0452
0.1406 0.1427 0.1655 0.0565
0.1659 0.1795 0.1990 0.0674
0.1099 0.1188 0.1318 0.0447
Use ndims and size to get the size of a ttensor
ndims(X) %<-- Number of dimensions.
ans =
3
size(X) %<-- Row vector of the sizes.
ans =
5 4 3
size(X,2) %<-- Size of the 2nd mode.
ans =
4
Subscripted reference to a ttensor
X.core(1,1,1) %<-- Access an element of the core.
ans =
0.6808
X.U{2} %<-- Extract a matrix.
ans =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
X{2} %<-- Same as above.
ans =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
Subscripted assignment for a ttensor
X.core = tenones(size(X.core)) %<-- Insert a new core.
X is a ttensor of size 5 x 4 x 3
X.core is a tensor of size 3 x 2 x 1
X.core(:,:,1) =
1 1
1 1
1 1
X.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
X.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
X.U{3} =
0.8392
0.6288
0.1338
X.core(2,2,1) = 7 %<-- Change a single element.
X is a ttensor of size 5 x 4 x 3
X.core is a tensor of size 3 x 2 x 1
X.core(:,:,1) =
1 1
1 7
1 1
X.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
X.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
X.U{3} =
0.8392
0.6288
0.1338
X{3}(1:2,1) = [1;1] %<-- Change the matrix for mode 3.
X is a ttensor of size 5 x 4 x 3
X.core is a tensor of size 3 x 2 x 1
X.core(:,:,1) =
1 1
1 7
1 1
X.U{1} =
0.0503 0.7680 0.4983
0.4154 0.9708 0.2140
0.3050 0.9901 0.6435
0.8744 0.7889 0.3200
0.0150 0.4387 0.9601
X.U{2} =
0.7266 0.4399
0.4120 0.9334
0.7446 0.6833
0.2679 0.2126
X.U{3} =
1.0000
1.0000
0.1338
Using end for last index
X{end} %<-- The same as X{3}.
ans =
1.0000
1.0000
0.1338
Basic operations (uplus, uminus, mtimes) for a ttensor.
X = ttensor(tenrand([2 2 2]),{rand(3,2),rand(1,2),rand(2,2)}) %<-- Data.
+X %<-- Calls uplus.
X is a ttensor of size 3 x 1 x 2
X.core is a tensor of size 2 x 2 x 2
X.core(:,:,1) =
0.5548 0.4508
0.1210 0.7159
X.core(:,:,2) =
0.8928 0.2548
0.2731 0.8656
X.U{1} =
0.2324 0.2319
0.8049 0.2393
0.9084 0.0498
X.U{2} =
0.0784 0.6408
X.U{3} =
0.1909 0.1739
0.8439 0.1708
ans is a ttensor of size 3 x 1 x 2
ans.core is a tensor of size 2 x 2 x 2
ans.core(:,:,1) =
0.5548 0.4508
0.1210 0.7159
ans.core(:,:,2) =
0.8928 0.2548
0.2731 0.8656
ans.U{1} =
0.2324 0.2319
0.8049 0.2393
0.9084 0.0498
ans.U{2} =
0.0784 0.6408
ans.U{3} =
0.1909 0.1739
0.8439 0.1708
-X %<-- Calls uminus.
ans is a ttensor of size 3 x 1 x 2
ans.core is a tensor of size 2 x 2 x 2
ans.core(:,:,1) =
-0.5548 -0.4508
-0.1210 -0.7159
ans.core(:,:,2) =
-0.8928 -0.2548
-0.2731 -0.8656
ans.U{1} =
0.2324 0.2319
0.8049 0.2393
0.9084 0.0498
ans.U{2} =
0.0784 0.6408
ans.U{3} =
0.1909 0.1739
0.8439 0.1708
5*X %<-- Calls mtimes.
ans is a ttensor of size 3 x 1 x 2
ans.core is a tensor of size 2 x 2 x 2
ans.core(:,:,1) =
2.7742 2.2538
0.6052 3.5794
ans.core(:,:,2) =
4.4642 1.2738
1.3655 4.3280
ans.U{1} =
0.2324 0.2319
0.8049 0.2393
0.9084 0.0498
ans.U{2} =
0.0784 0.6408
ans.U{3} =
0.1909 0.1739
0.8439 0.1708
Use permute to reorder the modes of a ttensor
permute(X,[3 2 1]) %<-- Reverses the modes of X
ans is a ttensor of size 2 x 1 x 3
ans.core is a tensor of size 2 x 2 x 2
ans.core(:,:,1) =
0.5548 0.4508
0.8928 0.2548
ans.core(:,:,2) =
0.1210 0.7159
0.2731 0.8656
ans.U{1} =
0.1909 0.1739
0.8439 0.1708
ans.U{2} =
0.0784 0.6408
ans.U{3} =
0.2324 0.2319
0.8049 0.2393
0.9084 0.0498
Displaying a ttensor
The tensor displays by displaying the core and each of the component matrices.
disp(X) %<-- Prints out the ttensor.
ans is a ttensor of size 3 x 1 x 2
ans.core is a tensor of size 2 x 2 x 2
ans.core(:,:,1) =
0.5548 0.4508
0.1210 0.7159
ans.core(:,:,2) =
0.8928 0.2548
0.2731 0.8656
ans.U{1} =
0.2324 0.2319
0.8049 0.2393
0.9084 0.0498
ans.U{2} =
0.0784 0.6408
ans.U{3} =
0.1909 0.1739
0.8439 0.1708