Collapsing and Scaling Tensors
The tensor and sptensor classes support the notion of collapsing and scaling dimensions, and the tensor class also supports centering (see below).
Contents
Examples of collapsing a tensor
rng('default'); %<-- Make this reproducible. X = tenrand([4 3 2]) %<-- Generate some data.
X is a tensor of size 4 x 3 x 2 X(:,:,1) = 0.8147 0.6324 0.9575 0.9058 0.0975 0.9649 0.1270 0.2785 0.1576 0.9134 0.5469 0.9706 X(:,:,2) = 0.9572 0.4218 0.6557 0.4854 0.9157 0.0357 0.8003 0.7922 0.8491 0.1419 0.9595 0.9340
Y = collapse(X,[2 3]) %<-- Sum of entries in each mode-1 slice.
Y is a tensor of size 4 Y(:) = 4.4393 3.4050 3.0047 4.4662
Y = collapse(X,-1) %<-- Same as above.
Y is a tensor of size 4 Y(:) = 4.4393 3.4050 3.0047 4.4662
Z = collapse(X,2) %<-- Sum of entries in each row fiber.
Z is a tensor of size 4 x 2 Z(:,:) = 2.4046 2.0347 1.9682 1.4368 0.5631 2.4416 2.4309 2.0354
collapse(X,1:3) %<-- Sum of all entries.
ans = 15.3152
Alternate accumulation functions for tensor
Y = collapse(X,[1 2],@max) %<-- Max entry in each mode-3 slice.
Y is a tensor of size 2 Y(:) = 0.9706 0.9595
Z = collapse(X,-3,@mean) %<-- Average entry in each mode-3 slice.
Z is a tensor of size 2 Z(:) = 0.6139 0.6624
Examples of collapsing a sptensor
X = sptenrand([4 3 2],6) %<-- Generate some data.
X is a sparse tensor of size 4 x 3 x 2 with 6 nonzeros (1,3,1) 0.7655 (2,1,1) 0.7952 (3,1,1) 0.1869 (3,1,2) 0.4898 (3,3,2) 0.4456 (4,1,1) 0.6463
Y = collapse(X,[2 3]) %<-- Sum of entries in each mode-1 slice.
Y =
0.7655
0.7952
1.1222
0.6463
Y = collapse(X,-1) %<-- Same as above.
Y =
0.7655
0.7952
1.1222
0.6463
Z = collapse(X,2) %<-- Sum of entries in each row fiber.
Z is a sparse tensor of size 4 x 2 with 5 nonzeros (1,1) 0.7655 (2,1) 0.7952 (3,1) 0.1869 (3,2) 0.9354 (4,1) 0.6463
collapse(X,1:3) %<-- Sum of all entries.
ans =
3.3293
Center tensor fibers for a dense tensor.
Suppose that we want to center a tensor's fiber in mode-2 so that they each have mean zero. To do so, we can use scale function in an unusual way, passing it a function handle to do the differences. (Note that sptensor does not support centering because it would cause the tensor to become dense.)
X = tensor(1:24,[4 3 2]) %<-- Generate some data.
X is a tensor of size 4 x 3 x 2 X(:,:,1) = 1 5 9 2 6 10 3 7 11 4 8 12 X(:,:,2) = 13 17 21 14 18 22 15 19 23 16 20 24
Calculate the means in mode 2
mu = collapse(X,2,@mean)
mu is a tensor of size 4 x 2 mu(:,:) = 5 17 6 18 7 19 8 20
Show the means of the mode-2 fibers
Y = scale(X,mu,[1 3],@(x,y) x-y); mu_new = collapse(Y,2,@mean)
mu_new is a tensor of size 4 x 2 mu_new(:,:) = 0 0 0 0 0 0 0 0
Alternate accumulation functions for sptensor
Y = collapse(X,[1 2],@min) %<-- Min *nonzero* entry in each mode-3 slice.
Y is a tensor of size 2 Y(:) = 1 13
Z = collapse(X,-3,@mean) %<-- Average *nonzero* entry in each mode-3 slice.
Z is a tensor of size 2 Z(:) = 6.5000 18.5000
Scaling a tensor in different modes
X = tenones([3,4,5]); %<-- Generate data S = 10 * [1:5]'; Y = scale(X,S,3) %<-- Scale in mode-3
Y is a tensor of size 3 x 4 x 5 Y(:,:,1) = 10 10 10 10 10 10 10 10 10 10 10 10 Y(:,:,2) = 20 20 20 20 20 20 20 20 20 20 20 20 Y(:,:,3) = 30 30 30 30 30 30 30 30 30 30 30 30 Y(:,:,4) = 40 40 40 40 40 40 40 40 40 40 40 40 Y(:,:,5) = 50 50 50 50 50 50 50 50 50 50 50 50
S = tensor(10 * [1:5]',5); Y = scale(X,S,3) %<-- First argument is a tensor.
Y is a tensor of size 3 x 4 x 5 Y(:,:,1) = 10 10 10 10 10 10 10 10 10 10 10 10 Y(:,:,2) = 20 20 20 20 20 20 20 20 20 20 20 20 Y(:,:,3) = 30 30 30 30 30 30 30 30 30 30 30 30 Y(:,:,4) = 40 40 40 40 40 40 40 40 40 40 40 40 Y(:,:,5) = 50 50 50 50 50 50 50 50 50 50 50 50
S = tensor(1:12,[3 4]); Y = scale(X,S,[1 2]) %<-- Scale in two modes.
Y is a tensor of size 3 x 4 x 5 Y(:,:,1) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,2) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,3) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,4) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,5) = 1 4 7 10 2 5 8 11 3 6 9 12
S = tensor(1:12,[3 4]); Y = scale(X,S,-3) %<-- Same as above.
Y is a tensor of size 3 x 4 x 5 Y(:,:,1) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,2) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,3) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,4) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,5) = 1 4 7 10 2 5 8 11 3 6 9 12
S = tensor(1:60,[3 4 5]); Y = scale(X,S,1:3) %<-- Scale in every mode.
Y is a tensor of size 3 x 4 x 5 Y(:,:,1) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,2) = 13 16 19 22 14 17 20 23 15 18 21 24 Y(:,:,3) = 25 28 31 34 26 29 32 35 27 30 33 36 Y(:,:,4) = 37 40 43 46 38 41 44 47 39 42 45 48 Y(:,:,5) = 49 52 55 58 50 53 56 59 51 54 57 60
Y = S .* X %<-- Same as above.
Y is a tensor of size 3 x 4 x 5 Y(:,:,1) = 1 4 7 10 2 5 8 11 3 6 9 12 Y(:,:,2) = 13 16 19 22 14 17 20 23 15 18 21 24 Y(:,:,3) = 25 28 31 34 26 29 32 35 27 30 33 36 Y(:,:,4) = 37 40 43 46 38 41 44 47 39 42 45 48 Y(:,:,5) = 49 52 55 58 50 53 56 59 51 54 57 60
Scaling a sptensor in different modes
X = ones(sptenrand([3 4 5], 10)) %<-- Generate data.
X is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 1 (1,4,1) 1 (1,4,5) 1 (2,1,3) 1 (2,1,5) 1 (3,1,2) 1 (3,2,3) 1 (3,2,5) 1 (3,4,1) 1 (3,4,2) 1
S = 10 * [1:5]'; Y = scale(X,S,3) %<-- Scale in one mode.
Y is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 10 (1,4,1) 10 (1,4,5) 50 (2,1,3) 30 (2,1,5) 50 (3,1,2) 20 (3,2,3) 30 (3,2,5) 50 (3,4,1) 10 (3,4,2) 20
S = tensor(10 * [1:5]',5); Y = scale(X,S,3) %<-- Same as above.
Y is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 10 (1,4,1) 10 (1,4,5) 50 (2,1,3) 30 (2,1,5) 50 (3,1,2) 20 (3,2,3) 30 (3,2,5) 50 (3,4,1) 10 (3,4,2) 20
S = tensor(1:12,[3 4]); Y = scale(X,S,[1 2]) %<-- Scale in two modes.
Y is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 1 (1,4,1) 10 (1,4,5) 10 (2,1,3) 2 (2,1,5) 2 (3,1,2) 3 (3,2,3) 6 (3,2,5) 6 (3,4,1) 12 (3,4,2) 12
S = tensor(1:12,[3 4]); Y = scale(X,S,-3) %<-- Same as above.
Y is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 1 (1,4,1) 10 (1,4,5) 10 (2,1,3) 2 (2,1,5) 2 (3,1,2) 3 (3,2,3) 6 (3,2,5) 6 (3,4,1) 12 (3,4,2) 12
Z = scale(X,Y,1:3) %<-- Scale by a sparse tensor.
Z is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 1 (1,4,1) 10 (1,4,5) 10 (2,1,3) 2 (2,1,5) 2 (3,1,2) 3 (3,2,3) 6 (3,2,5) 6 (3,4,1) 12 (3,4,2) 12
X .* Y %<-- Same as above.
ans is a sparse tensor of size 3 x 4 x 5 with 10 nonzeros (1,1,1) 1 (1,4,1) 10 (1,4,5) 10 (2,1,3) 2 (2,1,5) 2 (3,1,2) 3 (3,2,3) 6 (3,2,5) 6 (3,4,1) 12 (3,4,2) 12