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		    0.6381   -0.0241
		    0.7699    0.9587
		    0.0000   -0.0000
		    0.0000    0.0000
	P.U{2} = 
		    0.0000   -0.0000
		    0.9388   -0.0899
		    0.2834    0.7000
		   -0.1957    0.7085
	P.U{3} = 
		    0.0487    0.8893
		    0.0000    0.0000
		    0.9988    0.4573
</pre><h2>Alternating least squares for Tucker model<a name="6"></a></h2>
         <p>The function <tt>tucker_als</tt> 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.
         </p><pre class="codeinput">X = sptenrand([5 4 3], 10)
</pre><pre class="codeoutput">X is a sparse tensor of size 5 x 4 x 3 with 10 nonzeros
	(1,3,1)    0.7400
	(3,1,2)    0.4319
	(3,2,1)    0.6343
	(3,3,2)    0.8030
	(4,1,2)    0.0839
	(4,2,1)    0.9455
	(4,4,2)    0.9159
	(4,4,3)    0.6020
	(5,3,3)    0.2536
	(5,4,3)    0.8735
</pre><pre class="codeinput">T = tucker_als(X,2)        <span class="comment">%&lt;-- best rank(2,2,2) approximation</span>
</pre><pre class="codeoutput">
Alternating Least-Squares:
 Iter  1: fit = 2.810591e-001 fitdelta = 2.8e-001
 Iter  2: fit = 3.474829e-001 fitdelta = 6.6e-002
 Iter  3: fit = 3.628582e-001 fitdelta = 1.5e-002
 Iter  4: fit = 3.700452e-001 fitdelta = 7.2e-003
 Iter  5: fit = 3.727897e-001 fitdelta = 2.7e-003
 Iter  6: fit = 3.737295e-001 fitdelta = 9.4e-004
 Iter  7: fit = 3.740582e-001 fitdelta = 3.3e-004
 Iter  8: fit = 3.741751e-001 fitdelta = 1.2e-004
 Iter  9: fit = 3.742168e-001 fitdelta = 4.2e-005
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) = 
	    1.1796   -0.0116
	    0.4219   -0.0175
		T.core(:,:,2) = 
	    0.0098    1.0308
	   -0.0191   -0.4827
	T.U{1} = 
		    0.0069   -0.0204
		   -0.0000    0.0000
		    0.2980   -0.6769
		    0.8904   -0.0567
		    0.3439    0.7336
	T.U{2} = 
		    0.0439    0.0018
		    0.0204    0.9997
		    0.1129    0.0109
		    0.9924   -0.0219
	T.U{3} = 
		    0.0109    0.9999
		    0.6015   -0.0016
		    0.7988   -0.0124
</pre><pre class="codeinput">T = tucker_als(X,[2 2 1])  <span class="comment">%&lt;-- best rank(2,2,1) approximation</span>
</pre><pre class="codeoutput">
Alternating Least-Squares:
 Iter  1: fit = 1.812756e-001 fitdelta = 1.8e-001
 Iter  2: fit = 2.272937e-001 fitdelta = 4.6e-002
 Iter  3: fit = 2.412379e-001 fitdelta = 1.4e-002
 Iter  4: fit = 2.436064e-001 fitdelta = 2.4e-003
 Iter  5: fit = 2.444688e-001 fitdelta = 8.6e-004
 Iter  6: fit = 2.449320e-001 fitdelta = 4.6e-004
 Iter  7: fit = 2.451964e-001 fitdelta = 2.6e-004
 Iter  8: fit = 2.453474e-001 fitdelta = 1.5e-004
 Iter  9: fit = 2.454331e-001 fitdelta = 8.6e-005
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) = 
	    1.1975   -0.0004
	   -0.0001    0.7710
	T.U{1} = 
		    0.0024    0.0387
		    0.0000    0.0000
		    0.0728    0.9885
		    0.9137   -0.1170
		    0.3999    0.0872
	T.U{2} = 
		    0.0760    0.4549
		    0.0347    0.0306
		    0.0869    0.8828
		    0.9927   -0.1131
	T.U{3} = 
		    0.0343
		    0.8414
		    0.5394
</pre><pre class="codeinput">T = tucker_als(X,2,struct(<span class="string">'dimorder'</span>,[3 2 1]))
</pre><pre class="codeoutput">
Alternating Least-Squares:
 Iter  1: fit = 3.268831e-001 fitdelta = 3.3e-001
 Iter  2: fit = 3.604384e-001 fitdelta = 3.4e-002
 Iter  3: fit = 3.708956e-001 fitdelta = 1.0e-002
 Iter  4: fit = 3.731357e-001 fitdelta = 2.2e-003
 Iter  5: fit = 3.738515e-001 fitdelta = 7.2e-004
 Iter  6: fit = 3.741016e-001 fitdelta = 2.5e-004
 Iter  7: fit = 3.741906e-001 fitdelta = 8.9e-005
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) = 
	    1.1797   -0.0054
	    0.4208   -0.0338
		T.core(:,:,2) = 
	    0.0015    1.0306
	   -0.0375   -0.4818
	T.U{1} = 
		    0.0069   -0.0208
		         0         0
		    0.2981   -0.6769
		    0.8904   -0.0566
		    0.3439    0.7336
	T.U{2} = 
		    0.0440    0.0028
		    0.0323    0.9992
		    0.1134    0.0181
		    0.9921   -0.0347
	T.U{3} = 
		    0.0298    0.9994
		    0.6017   -0.0051
		    0.7982   -0.0335
</pre><pre class="codeinput">T = tucker_als(X,2,struct(<span class="string">'dimorder'</span>,[3 2 1],<span class="string">'init'</span>,<span class="string">'eigs'</span>))
</pre><pre class="codeoutput">  Computing 2 leading e-vectors for factor 2.
  Computing 2 leading e-vectors for factor 1.

Alternating Least-Squares:
 Iter  1: fit = 3.726300e-001 fitdelta = 3.7e-001
 Iter  2: fit = 3.741337e-001 fitdelta = 1.5e-003
 Iter  3: fit = 3.742335e-001 fitdelta = 1.0e-004
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) = 
	    1.1798    0.0000
	    0.4220   -0.0000
		T.core(:,:,2) = 
	   -0.0000    1.0311
	   -0.0000   -0.4828
	T.U{1} = 
		    0.0000         0
		         0    0.0000
		    0.2970   -0.6795
		    0.8913   -0.0548
		    0.3426    0.7316
	T.U{2} = 
		    0.0427    0.0000
		   -0.0000    1.0000
		    0.1082    0.0000
		    0.9932    0.0000
	T.U{3} = 
		    0.0000    1.0000
		    0.6045   -0.0000
		    0.7966   -0.0000
</pre><pre class="codeinput">U0 = {rand(5,2),rand(4,2),[]}; <span class="comment">%&lt;-- Initial guess for factors of T</span>
T = tucker_als(X,2,struct(<span class="string">'dimorder'</span>,[3 2 1],<span class="string">'init'</span>,{U0}))
</pre><pre class="codeoutput">
Alternating Least-Squares:
 Iter  1: fit = 3.647914e-001 fitdelta = 3.6e-001
 Iter  2: fit = 3.722524e-001 fitdelta = 7.5e-003
 Iter  3: fit = 3.735753e-001 fitdelta = 1.3e-003
 Iter  4: fit = 3.740042e-001 fitdelta = 4.3e-004
 Iter  5: fit = 3.741559e-001 fitdelta = 1.5e-004
 Iter  6: fit = 3.742100e-001 fitdelta = 5.4e-005
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) = 
	    1.1797   -0.0042
	    0.4214   -0.0265
		T.core(:,:,2) = 
	    0.0012    1.0308
	   -0.0293   -0.4823
	T.U{1} = 
		    0.0054   -0.0162
		    0.0000         0
		    0.2980   -0.6769
		    0.8904   -0.0567
		    0.3439    0.7337
	T.U{2} = 
		    0.0440    0.0022
		    0.0253    0.9995
		    0.1131    0.0141
		    0.9923   -0.0272
	T.U{3} = 
		    0.0233    0.9997
		    0.6016   -0.0040
		    0.7985   -0.0262
</pre><p class="footer"><br>
            Published with MATLAB&reg; 7.2<br></p>
      </div>
      <!--
##### SOURCE BEGIN #####
%% Algorithms for computing tensor decompositions

%% Alternating least squares for PARAFAC/CANDECOMP
% The function |parafac_als| computes an estimate of the best rank-R
% PARAFAC model of a tensor X using an alternating least-squares
% algorithm.  The input X can be a tensor, sptensor, ktensor, or
% ttensor. The result P is a ktensor.
rand('state',0);
X = sptenrand([5 4 3], 10)
%%
P = parafac_als(X,2)
%%
P = parafac_als(X,2,struct('dimorder',[3 2 1]))
%%
P = parafac_als(X,2,struct('dimorder',[3 2 1],'init','nvecs'))
%%
U0 = {rand(5,2),rand(4,2),[]}; %<REPLACE_WITH_DASH_DASH Initial guess for factors of P
P = parafac_als(X,2,struct('dimorder',[3 2 1],'init',{U0}))
%% 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.
X = sptenrand([5 4 3], 10)
%%
T = tucker_als(X,2)        %<REPLACE_WITH_DASH_DASH best rank(2,2,2) approximation 
%%
T = tucker_als(X,[2 2 1])  %<REPLACE_WITH_DASH_DASH best rank(2,2,1) approximation 
%%
T = tucker_als(X,2,struct('dimorder',[3 2 1]))
%%
T = tucker_als(X,2,struct('dimorder',[3 2 1],'init','eigs'))
%%
U0 = {rand(5,2),rand(4,2),[]}; %<REPLACE_WITH_DASH_DASH Initial guess for factors of T
T = tucker_als(X,2,struct('dimorder',[3 2 1],'init',{U0}))

##### SOURCE END #####
-->
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