Identities and relationships of tensors

There are many mathematical relationships, identities, and connections among tensors. These identities are presented here and show the versatility of the Tensor Toolbox. The propositions indicated below are references to the following report:

T.G. Kolda, "Multilinear operators for higher-order decompositions", Tech. Rep. SAND2006-2081, Sandia National Laboratories, April 2006, http://csmr.ca.sandia.gov/~tgkolda/pubs/index.html#SAND2006-2081.

Contents

N-mode product properties

Create some data.

Y = tenrand([4 3 2]);
A = rand(3,4);
B = rand(3,3);

Prop 3.4(a): The order of the multiplication in different modes is irrelevant.

$$(Y \times_1 A) \times_2 B = (Y \times_2 B) \times_1 A$$

X1 = ttm( ttm(Y,A,1), B, 2); %<-- Y x_1 A x_2 B
X2 = ttm( ttm(Y,B,2), A, 1); %<-- Y x_2 B x_1 A
norm(X1 - X2) %<-- difference is zero
ans =

   4.6278e-16

N-mode product and matricization

Generate some data to work with.

Y = tenrand([5 4 3]);
A = rand(4,5); B = rand(3,4); C = rand(2,3); U = {A,B,C};

Prop. 3.7a: N-mode multiplication can be expressed in terms of matricized tensors.

$$X = Y \times_n U \Leftrightarrow  X_{(n)} = UY_{(n)} $$

for n = 1:ndims(Y)
  X = ttm(Y,U,n); %<-- X = Y x_n U{n}
  Xn = U{n} * tenmat(Y,n); %<-- Xn = U{n} * Yn
  norm(tenmat(X,n) - Xn)  % <-- should be zero
end
ans =

     0


ans =

     0


ans =

     0

Prop. 3.7b: We can do matricizations in various ways and still be equivalent.

X = ttm(Y,U); %<-- X = Y x_1 A x_2 B x_3 C
Xm1 = kron(B,A)*tenmat(Y,[1 2])*C';  %<-- Kronecker product version
Xm2 = tenmat(X,[1 2]); %<-- Matriczed version
norm(Xm1 - Xm2)  % <-- should be zero
Xm1 = B * tenmat(Y,2,[3 1]) * kron(A,C)'; %<-- Kronecker product version
Xm2 = tenmat(X,2,[3 1]); %<-- Matricized version
norm(Xm1 - Xm2) % <-- should be zero
Xm1 = tenmat(Y,[],[1 2 3]) * kron(kron(C,B),A)'; %<-- Vectorized via Kronecker
Xm2 = tenmat(X,[],[1 2 3]); %<-- Vectorized via matricize
norm(Xm1 - Xm2)
ans =

   1.6616e-15


ans =

   2.1756e-15


ans =

   2.8087e-15

Norm of difference between two tensors

Prop. 3.9: For tensors X and Y, we have:

$$\|X-Y\|^2 = \|X\|^2 + \|Y\|^2 - 2<X,Y> $$

X = tenrand([5 4 3]); Y = tenrand([5 4 3]);
% The following 2 results should be equal
norm(X-Y)
sqrt(norm(X)^2 - 2*innerprod(X,Y) + norm(Y)^2)
ans =

    2.9316


ans =

    2.9316

This relationship makes it more convenient to compare the norm of the difference between two different tensor objects. Imagine if we have a sptensor and a ktensor and we want the norm of the difference, which may be needed to check for convergence, for example, but which is very expensive to convert to a full (dense) tensor. Because innerprod and norm are defined for all types of tensor objects, this is a handy formula.

X = sptensor(X);
Y = ktensor({[1:5]',[1:4]',[1:3]'});
% The following 2 results should be equal
norm(full(X)-full(Y))
sqrt(norm(X)^2 - 2*innerprod(X,Y) + norm(Y)^2)
ans =

  148.5941


ans =

  148.5941

Tucker tensor properties

The properties of the Tucker operator follow directly from the properties of n-mode multiplication.

% Initialize data
Y = tensor(1:24,[4 3 2]);
A1 = reshape(1:20,[5 4]);
A2 = reshape(1:12,[4 3]);
A3 = reshape(1:6,[3 2]);
A = {A1,A2,A3};
B1 = reshape(1:20,[4 5]);
B2 = reshape(1:12,[3 4]);
B3 = reshape(1:6,[2 3]);
B = {B1,B2,B3};

Proposition 4.2a

X = ttensor(ttensor(Y,A),B)
X is a ttensor of size 4 x 3 x 2
	X.core is a ttensor of size 5 x 4 x 3
		X.core.core is a tensor of size 4 x 3 x 2
			X.core.core(:,:,1) = 
	     1     5     9
	     2     6    10
	     3     7    11
	     4     8    12
			X.core.core(:,:,2) = 
	    13    17    21
	    14    18    22
	    15    19    23
	    16    20    24
		X.core.U{1} = 
		     1     6    11    16
		     2     7    12    17
		     3     8    13    18
		     4     9    14    19
		     5    10    15    20
		X.core.U{2} = 
		     1     5     9
		     2     6    10
		     3     7    11
		     4     8    12
		X.core.U{3} = 
		     1     4
		     2     5
		     3     6
	X.U{1} = 
		     1     5     9    13    17
		     2     6    10    14    18
		     3     7    11    15    19
		     4     8    12    16    20
	X.U{2} = 
		     1     4     7    10
		     2     5     8    11
		     3     6     9    12
	X.U{3} = 
		     1     3     5
		     2     4     6
AB = {B1*A1, B2*A2, B3*A3};
Y = ttensor(Y,AB)
Y is a ttensor of size 4 x 3 x 2
	Y.core is a tensor of size 4 x 3 x 2
		Y.core(:,:,1) = 
	     1     5     9
	     2     6    10
	     3     7    11
	     4     8    12
		Y.core(:,:,2) = 
	    13    17    21
	    14    18    22
	    15    19    23
	    16    20    24
	Y.U{1} = 
		         175         400         625         850
		         190         440         690         940
		         205         480         755        1030
		         220         520         820        1120
	Y.U{2} = 
		    70   158   246
		    80   184   288
		    90   210   330
	Y.U{3} = 
		    22    49
		    28    64
norm(full(X)-full(Y))  %<-- should be zero
ans =

     0

Proposition 4.2b

Y = tensor(1:24,[4 3 2]);
X = ttensor(Y,A);
Apinv = {pinv(A1),pinv(A2),pinv(A3)};
Y2 = ttensor(full(X),Apinv);
norm(full(Y)-full(Y2))  %<-- should be zero
ans =

   3.8576e-13

Proposition 4.2c

Y = tensor(1:24,[4 3 2]);
rand('state',0);
Q1 = orth(rand(5,4));
Q2 = orth(rand(4,3));
Q3 = orth(rand(3,2));
Q = {Q1,Q2,Q3};
X = ttensor(Y,Q)
X is a ttensor of size 5 x 4 x 3
	X.core is a tensor of size 4 x 3 x 2
		X.core(:,:,1) = 
	     1     5     9
	     2     6    10
	     3     7    11
	     4     8    12
		X.core(:,:,2) = 
	    13    17    21
	    14    18    22
	    15    19    23
	    16    20    24
	X.U{1} = 
		   -0.4727    0.5608    0.0275   -0.3954
		   -0.4394   -0.4243    0.3178    0.5707
		   -0.4659   -0.6116   -0.1037   -0.5982
		   -0.4209    0.3259    0.5458    0.1138
		   -0.4350    0.1587   -0.7679    0.3837
	X.U{2} = 
		   -0.2570    0.1257    0.8908
		   -0.3751   -0.2111    0.2636
		   -0.4640   -0.7988   -0.1591
		   -0.7602    0.5492   -0.3341
	X.U{3} = 
		   -0.3907   -0.0625
		   -0.8045   -0.4616
		   -0.4473    0.8849
Qt = {Q1',Q2',Q3'};
Y2 = ttensor(full(X),Qt)
norm(full(Y)-full(Y2))  %<-- should be zero
Y2 is a ttensor of size 4 x 3 x 2
	Y2.core is a tensor of size 5 x 4 x 3
		Y2.core(:,:,1) = 
	    1.4195   -0.0317   -1.4127   -0.3848
	   -0.7708    0.2767    1.3323    0.4969
	    8.6788   -0.0536   -8.3316   -2.1970
	   -3.0735    0.1529    3.2424    0.9267
	    2.9652    0.0889   -2.6130   -0.6316
		Y2.core(:,:,2) = 
	    4.6979   -0.3995   -5.3170   -1.6004
	   -2.2186    0.8006    3.8440    1.4349
	   28.9023   -2.1266  -31.9901   -9.4788
	  -10.0638    1.0545   11.8230    3.6490
	   10.0122   -0.4854  -10.5344   -3.0047
		Y2.core(:,:,3) = 
	   -3.4733    0.9238    5.3001    1.8807
	    0.9310   -0.3464   -1.6360   -0.6138
	  -21.7522    5.7319   33.0762   11.7190
	    7.2102   -1.9498  -11.0723   -3.9398
	   -7.8266    2.0225   11.8142    4.1724
	Y2.U{1} = 
		   -0.4727   -0.4394   -0.4659   -0.4209   -0.4350
		    0.5608   -0.4243   -0.6116    0.3259    0.1587
		    0.0275    0.3178   -0.1037    0.5458   -0.7679
		   -0.3954    0.5707   -0.5982    0.1138    0.3837
	Y2.U{2} = 
		   -0.2570   -0.3751   -0.4640   -0.7602
		    0.1257   -0.2111   -0.7988    0.5492
		    0.8908    0.2636   -0.1591   -0.3341
	Y2.U{3} = 
		   -0.3907   -0.8045   -0.4473
		   -0.0625   -0.4616    0.8849

ans =

   3.8190e-14

Tucker operator and matricized tensors

The Tucker operator also has various epressions in terms of matricized tensors and the Kronecker product. Proposition 4.3a

Y = tensor(1:24,[4 3 2]);
A1 = reshape(1:20,[5 4]);
A2 = reshape(1:12,[4 3]);
A3 = reshape(1:6,[3 2]);
A = {A1,A2,A3};
X = ttensor(Y,A)
for n = 1:ndims(Y)
  rdims = n;
  cdims = setdiff(1:ndims(Y),rdims);
  Xn = A{n} * tenmat(Y,rdims,cdims) * kron(A{cdims(2)}, A{cdims(1)})';
  norm(tenmat(full(X),rdims,cdims) - Xn)  % <-- should be zero
end
X is a ttensor of size 5 x 4 x 3
	X.core is a tensor of size 4 x 3 x 2
		X.core(:,:,1) = 
	     1     5     9
	     2     6    10
	     3     7    11
	     4     8    12
		X.core(:,:,2) = 
	    13    17    21
	    14    18    22
	    15    19    23
	    16    20    24
	X.U{1} = 
		     1     6    11    16
		     2     7    12    17
		     3     8    13    18
		     4     9    14    19
		     5    10    15    20
	X.U{2} = 
		     1     5     9
		     2     6    10
		     3     7    11
		     4     8    12
	X.U{3} = 
		     1     4
		     2     5
		     3     6

ans =

     0


ans =

     0


ans =

     0

Orthogonalization of Tucker factors

Proposition 4.4

Y = tensor(1:24,[4 3 2]);
A1 = rand(5,4);
A2 = rand(4,3);
A3 = rand(3,2);
A = {A1,A2,A3};
X = ttensor(Y,A)
X is a ttensor of size 5 x 4 x 3
	X.core is a tensor of size 4 x 3 x 2
		X.core(:,:,1) = 
	     1     5     9
	     2     6    10
	     3     7    11
	     4     8    12
		X.core(:,:,2) = 
	    13    17    21
	    14    18    22
	    15    19    23
	    16    20    24
	X.U{1} = 
		    0.2026    0.3795    0.3046    0.5417
		    0.6721    0.8318    0.1897    0.1509
		    0.8381    0.5028    0.1934    0.6979
		    0.0196    0.7095    0.6822    0.3784
		    0.6813    0.4289    0.3028    0.8600
	X.U{2} = 
		    0.8537    0.8216    0.3420
		    0.5936    0.6449    0.2897
		    0.4966    0.8180    0.3412
		    0.8998    0.6602    0.5341
	X.U{3} = 
		    0.7271    0.5681
		    0.3093    0.3704
		    0.8385    0.7027
[Q1,R1] = qr(A1);
[Q2,R2] = qr(A2);
[Q3,R3] = qr(A3);
R = {R1,R2,R3};
Z = ttensor(Y,R);
norm(X) - norm(Z)  %<-- should be zero
ans =

     0

Kruskal operator properties

Proposition 5.2

A1 = reshape(1:10,[5 2]);
A2 = reshape(1:8,[4 2]);
A3 = reshape(1:6,[3 2]);
K = ktensor({A1,A2,A3});
B1 = reshape(1:20,[4 5]);
B2 = reshape(1:12,[3 4]);
B3 = reshape(1:6,[2 3]);
X = ttensor(K,{B1,B2,B3})

Y = ktensor({B1*A1, B2*A2, B3*A3});
norm(full(X) - full(Y))  %<-- should be zero
X is a ttensor of size 4 x 3 x 2
	X.core is a ktensor of size 5 x 4 x 3
		X.core.lambda = [ 1  1 ]
		X.core.U{1} = 
		     1     6
		     2     7
		     3     8
		     4     9
		     5    10
		X.core.U{2} = 
		     1     5
		     2     6
		     3     7
		     4     8
		X.core.U{3} = 
		     1     4
		     2     5
		     3     6
	X.U{1} = 
		     1     5     9    13    17
		     2     6    10    14    18
		     3     7    11    15    19
		     4     8    12    16    20
	X.U{2} = 
		     1     4     7    10
		     2     5     8    11
		     3     6     9    12
	X.U{3} = 
		     1     3     5
		     2     4     6

ans =

     0

Proposition 5.3a (second part)

A1 = reshape(1:10,[5 2]);
A2 = reshape(1:8,[4 2]);
A3 = reshape(1:6,[3 2]);
A = {A1,A2,A3};
X = ktensor(A);
rdims = 1:ndims(X);
Z = double(tenmat(full(X), rdims, []));
Xn = khatrirao(A{rdims},'r') * ones(length(X.lambda),1);
norm(Z - Xn)  % <-- should be zero
ans =

     0

cdims = 1:ndims(X);
Z = double(tenmat(full(X), [], cdims));
Xn = ones(length(X.lambda),1)' * khatrirao(A{cdims},'r')';
norm(Z - Xn)  % <-- should be zero
ans =

     0

Proposition 5.3b

A1 = reshape(1:10,[5 2]);
A2 = reshape(1:8,[4 2]);
A3 = reshape(1:6,[3 2]);
A = {A1,A2,A3};
X = ktensor(A);
for n = 1:ndims(X)
  rdims = n;
  cdims = setdiff(1:ndims(X),rdims);
  Xn = khatrirao(A{rdims}) * khatrirao(A{cdims},'r')';
  Z = double(tenmat(full(X),rdims,cdims));
  norm(Z - Xn)  % <-- should be zero
end
ans =

     0


ans =

     0


ans =

     0

Proposition 5.3a (first part)

X = ktensor(A);
for n = 1:ndims(X)
  cdims = n;
  rdims = setdiff(1:ndims(X),cdims);
  Xn = khatrirao(A{rdims},'r') * khatrirao(A{cdims})';
  Z = double(tenmat(full(X),rdims,cdims));
  norm(Z - Xn)  % <-- should be zero
end
ans =

     0


ans =

     0


ans =

     0

Norm of Kruskal operator

The norm of a ktensor has a special form because it can be reduced to summing the entries of the Hadamard product of N matrices of size R x R. Proposition 5.4

A1 = reshape(1:10,[5 2]);
A2 = reshape(1:8,[4 2]);
A3 = reshape(1:6,[3 2]);
A = {A1,A2,A3};
X = ktensor(A);
M = ones(size(A{1},2), size(A{1},2));
for i = 1:numel(A)
  M = M .* (A{i}'*A{i});
end
norm(X) - sqrt(sum(M(:)))  %<-- should be zero
ans =

     0

Inner product of Kruskal operator with a tensor

The inner product of a ktensor with a tensor yields Proposition 5.5

X = tensor(1:60,[5 4 3]);
A1 = reshape(1:10,[5 2]);
A2 = reshape(2:9,[4 2]);
A3 = reshape(3:8,[3 2]);
A = {A1,A2,A3};
K = ktensor(A);
v = khatrirao(A,'r') * ones(size(A{1},2),1);
% The following 2 results should be equal
double(tenmat(X,1:ndims(X),[]))' * v
innerprod(X,K)
ans =

      935340


ans =

      935340