Alternating least squares for Tucker model

The function tucker_als computes the best rank(R1,R2,..,Rn) approximation of tensor X, according to the specified dimensions in vector R. The input X can be a tensor, sptensor, ktensor, or ttensor. The result returned in T is a ttensor.

The method is originally from Tucker (1966) and later revisited in De Lathauwer et al. (2000).

Note: Oftentimes it's better to use hosvd instead.

Contents

Create a data tensor of size [5 4 3]

rng('default'); rng(0); %<-- Set seed for reproducibility
X = sptenrand([5 4 3], 10)
X is a sparse tensor of size 5 x 4 x 3 with 10 nonzeros
	(1,2,3)    0.0759
	(1,3,2)    0.0540
	(2,2,2)    0.5308
	(2,2,3)    0.7792
	(3,1,3)    0.9340
	(3,4,2)    0.1299
	(4,1,2)    0.5688
	(4,4,2)    0.4694
	(5,2,1)    0.0119
	(5,4,3)    0.3371

Create a [2 2 2] approximation

T = tucker_als(X,2)        %<-- best rank(2,2,2) approximation
Tucker Alternating Least-Squares:
 Iter  1: fit = 3.266855e-01 fitdelta = 3.3e-01
 Iter  2: fit = 4.285677e-01 fitdelta = 1.0e-01
 Iter  3: fit = 4.707375e-01 fitdelta = 4.2e-02
 Iter  4: fit = 4.728036e-01 fitdelta = 2.1e-03
 Iter  5: fit = 4.728492e-01 fitdelta = 4.6e-05
T is a ttensor of size 5 x 4 x 3
	T.core is a tensor of size 2 x 2 x 2
		T.core(:,:,1) = 
	    0.9045    0.0007
	   -0.0007    0.8920
		T.core(:,:,2) = 
	    0.2732    0.0006
	    0.0006   -0.2771
	T.U{1} = 
		    0.0666    0.0001
		    0.9978    0.0008
		   -0.0008    1.0000
		   -0.0001    0.0007
		   -0.0001    0.0018
	T.U{2} = 
		   -0.0015    1.0000
		    1.0000    0.0015
		    0.0021    0.0000
		   -0.0001    0.0007
	T.U{3} = 
		   -0.0000   -0.0000
		    0.2971    0.9548
		    0.9548   -0.2971

Create a [2 2 1] approximation

T = tucker_als(X,[2 2 1])  %<-- best rank(2,2,1) approximation
Tucker Alternating Least-Squares:
 Iter  1: fit = 2.941961e-01 fitdelta = 2.9e-01
 Iter  2: fit = 3.996958e-01 fitdelta = 1.1e-01
 Iter  3: fit = 4.229258e-01 fitdelta = 2.3e-02
 Iter  4: fit = 4.304139e-01 fitdelta = 7.5e-03
 Iter  5: fit = 4.325472e-01 fitdelta = 2.1e-03
 Iter  6: fit = 4.330654e-01 fitdelta = 5.2e-04
 Iter  7: fit = 4.331799e-01 fitdelta = 1.1e-04
 Iter  8: fit = 4.332039e-01 fitdelta = 2.4e-05
T is a ttensor of size 5 x 4 x 3
	T.core is a tensor of size 2 x 2 x 1
		T.core(:,:,1) = 
	    0.9293    0.0000
	   -0.0000    0.8920
	T.U{1} = 
		    0.0747   -0.0000
		    0.9972   -0.0000
		    0.0000    0.9575
		   -0.0000    0.2844
		   -0.0000    0.0475
	T.U{2} = 
		    0.0000    0.9911
		    1.0000   -0.0000
		    0.0017   -0.0000
		   -0.0000    0.1333
	T.U{3} = 
		   -0.0000
		    0.4011
		    0.9160

Use a different ordering of the dimensions

T = tucker_als(X,2,struct('dimorder',[3 2 1]))
Tucker Alternating Least-Squares:
 Iter  1: fit = 3.515073e-01 fitdelta = 3.5e-01
 Iter  2: fit = 3.548900e-01 fitdelta = 3.4e-03
 Iter  3: fit = 3.560941e-01 fitdelta = 1.2e-03
 Iter  4: fit = 3.565373e-01 fitdelta = 4.4e-04
 Iter  5: fit = 3.566994e-01 fitdelta = 1.6e-04
 Iter  6: fit = 3.567586e-01 fitdelta = 5.9e-05
T is a ttensor of size 5 x 4 x 3
	T.core is a tensor of size 2 x 2 x 2
		T.core(:,:,1) = 
	    0.8911   -0.2312
	   -0.1522    0.0400
		T.core(:,:,2) = 
	    0.1445    0.1767
	    0.6535    0.2855
	T.U{1} = 
		    0.0011   -0.0003
		    0.0124    0.0067
		    0.9855   -0.1691
		    0.1690    0.9856
		   -0.0000    0.0000
	T.U{2} = 
		    0.9678   -0.2514
		    0.0146    0.0015
		   -0.0000    0.0001
		    0.2513    0.9679
	T.U{3} = 
		   -0.0000    0.0000
		    0.0008    1.0000
		    1.0000   -0.0008

Use the n-vecs initialization method

This initialization is more expensive but generally works very well.

T = tucker_als(X,2,struct('dimorder',[3 2 1],'init','eigs'))
  Computing 2 leading e-vectors for factor 2.
  Computing 2 leading e-vectors for factor 1.

Tucker Alternating Least-Squares:
 Iter  1: fit = 4.726805e-01 fitdelta = 4.7e-01
 Iter  2: fit = 4.728466e-01 fitdelta = 1.7e-04
 Iter  3: fit = 4.728501e-01 fitdelta = 3.5e-06
T is a ttensor of size 5 x 4 x 3
	T.core is a tensor of size 2 x 2 x 2
		T.core(:,:,1) = 
	    0.9045   -0.0000
	    0.0000    0.8918
		T.core(:,:,2) = 
	    0.2731   -0.0000
	   -0.0000   -0.2775
	T.U{1} = 
		    0.0666   -0.0000
		    0.9978   -0.0000
		    0.0000    1.0000
		    0.0000    0.0001
		    0.0000    0.0002
	T.U{2} = 
		    0.0000    1.0000
		    1.0000   -0.0000
		    0.0021   -0.0000
		    0.0000    0.0005
	T.U{3} = 
		    0.0000    0.0000
		    0.2973    0.9548
		    0.9548   -0.2973

Specify the initial guess manually

U0 = {rand(5,2),rand(4,2),[]}; %<-- Initial guess for factors of T
T = tucker_als(X,2,struct('dimorder',[3 2 1],'init',{U0}))
Tucker Alternating Least-Squares:
 Iter  1: fit = 3.843537e-01 fitdelta = 3.8e-01
 Iter  2: fit = 4.493600e-01 fitdelta = 6.5e-02
 Iter  3: fit = 4.721643e-01 fitdelta = 2.3e-02
 Iter  4: fit = 4.728355e-01 fitdelta = 6.7e-04
 Iter  5: fit = 4.728499e-01 fitdelta = 1.4e-05
T is a ttensor of size 5 x 4 x 3
	T.core is a tensor of size 2 x 2 x 2
		T.core(:,:,1) = 
	    0.9047   -0.0006
	    0.0006    0.8917
		T.core(:,:,2) = 
	    0.2727   -0.0002
	   -0.0002   -0.2778
	T.U{1} = 
		    0.0666   -0.0000
		    0.9978   -0.0001
		    0.0001    1.0000
		    0.0000    0.0001
		    0.0000    0.0004
	T.U{2} = 
		    0.0008    1.0000
		    1.0000   -0.0008
		    0.0021   -0.0000
		    0.0000    0.0010
	T.U{3} = 
		    0.0000    0.0000
		    0.2976    0.9547
		    0.9547   -0.2976