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<html xmlns:mwsh="http://www.mathworks.com/namespace/mcode/v1/syntaxhighlight.dtd"> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Generating the leading mode-n vectors</title> <meta name="generator" content="MATLAB 7.2"> <meta name="date" content="2007-01-10"> <meta name="m-file" content="N_nvecs_doc"><style>body { background-color: white; margin:10px;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}/* Make the text shrink to fit narrow windows, but not stretch too far in wide windows. On Gecko-based browsers, the shrink-to-fit doesn't work. */ p,h1,h2,div.content div { /* for MATLAB's browser */ width: 600px; /* for Mozilla, but the "width" tag overrides it anyway */ max-width: 600px; /* for IE */ width:expression(document.body.clientWidth > 620 ? "600px": "auto" );}pre.codeinput { background: #EEEEEE; padding: 10px;}@media print { pre.codeinput {word-wrap:break-word; width:100%;}} span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.codeoutput { color: #666666; padding: 10px;}pre.error { color: red;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;} </style></head> <body> <div class="content"> <h1>Generating the leading mode-n vectors</h1> <introduction> <p>The leading mode-n vectors are those vectors that span the subspace of the mode-n fibers. In other words, the left singular vectors of the n-mode matricization of X. </p> </introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">Using nvecs to calculate the leading mode-n vectors</a></li> <li><a href="#11">Using nvecs for the HOSVD</a></li> </ul> </div> <h2>Using nvecs to calculate the leading mode-n vectors<a name="1"></a></h2> <p>The <tt>nvecs</tt> command efficient computes the leading n-mode vectors. </p><pre class="codeinput">rand(<span class="string">'state'</span>,0);X = sptenrand([4,3,2],6) <span class="comment">%<-- A sparse tensor</span></pre><pre class="codeoutput">X is a sparse tensor of size 4 x 3 x 2 with 6 nonzeros (1,2,1) 0.8385 (2,3,1) 0.5681 (3,2,1) 0.3704 (3,3,1) 0.7027 (4,2,2) 0.5466 (4,3,2) 0.4449</pre><pre class="codeinput">nvecs(X,1,2) <span class="comment">%<-- The 2 leading mode-1 vectors</span></pre><pre class="codeoutput">ans = 0.5810 0.7687 0.3761 -0.5451 0.7219 -0.3347 0.0000 0.0000</pre><pre class="codeinput">nvecs(X,1,3) <span class="comment">% <-- The 3 leading mode-1 vectors</span></pre><pre class="codeoutput">ans = 0.5810 0.7687 0.0000 0.3761 -0.5451 -0.0000 0.7219 -0.3347 -0.0000 0.0000 -0.0000 1.0000</pre><pre class="codeinput">nvecs(full(X),1,3) <span class="comment">%<-- The same thing for a dense tensor</span></pre><pre class="codeoutput">ans = 0.5810 0.7687 0.0000 0.3761 -0.5451 0.0000 0.7219 -0.3347 0.0000 -0.0000 0.0000 1.0000</pre><pre class="codeinput">X = ktensor({rand(3,2),rand(3,2),rand(2,2)}) <span class="comment">%<-- A random ktensor</span></pre><pre class="codeoutput">X is a ktensor of size 3 x 3 x 2 X.lambda = [ 1 1 ] X.U{1} = 0.1365 0.1991 0.0118 0.2987 0.8939 0.6614 X.U{2} = 0.2844 0.9883 0.4692 0.5828 0.0648 0.4235 X.U{3} = 0.5155 0.4329 0.3340 0.2259</pre><pre class="codeinput">nvecs(X,2,1) <span class="comment">%<-- The 1 leading mode-2 vector</span></pre><pre class="codeoutput">ans = 0.7147 0.6480 0.2633</pre><pre class="codeinput">nvecs(full(X),2,1) <span class="comment">%<-- Same thing for a dense tensor</span></pre><pre class="codeoutput">ans = 0.7147 0.6480 0.2633</pre><pre class="codeinput">X = ttensor(tenrand([2,2,2,2]),{rand(3,2),rand(3,2),rand(2,2),rand(2,2)}); <span class="comment">%<-- A random ttensor</span></pre><pre class="codeinput">nvecs(X,4,2) <span class="comment">%<-- The 1 leading mode-2 vector</span></pre><pre class="codeoutput">ans = 0.6725 0.7401 0.7401 -0.6725</pre><pre class="codeinput">nvecs(full(X),4,2) <span class="comment">%<-- Same thing for a dense tensor</span></pre><pre class="codeoutput">ans = 0.6725 0.7401 0.7401 -0.6725</pre><h2>Using nvecs for the HOSVD<a name="11"></a></h2><pre class="codeinput">X = tenrand([4 3 2]) <span class="comment">%<-- Generate data</span></pre><pre class="codeoutput">X is a tensor of size 4 x 3 x 2 X(:,:,1) = 0.0272 0.6831 0.6085 0.3127 0.0928 0.0158 0.0129 0.0353 0.0164 0.3840 0.6124 0.1901 X(:,:,2) = 0.5869 0.7176 0.4418 0.0576 0.6927 0.3533 0.3676 0.0841 0.1536 0.6315 0.4544 0.6756</pre><pre class="codeinput">U1 = nvecs(X,1,4); <span class="comment">%<-- Mode 1</span>U2 = nvecs(X,2,3); <span class="comment">%<-- Mode 2</span>U3 = nvecs(X,3,2); <span class="comment">%<-- Mode 3</span>S = ttm(X,{pinv(U1),pinv(U2),pinv(U3)}); <span class="comment">%<-- Core</span>Y = ttensor(S,{U1,U2,U3}) <span class="comment">%<-- HOSVD of X</span></pre><pre class="codeoutput">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) = 1.9279 0.0684 -0.0009 0.0669 -0.1193 -0.1012 -0.0229 0.3216 0.0848 0.0013 0.0852 -0.0282 Y.core(:,:,2) = 0.0560 -0.2314 -0.0911 -0.2194 0.5059 -0.0932 -0.0393 0.0200 -0.3189 -0.1786 -0.0452 0.0974 Y.U{1} = 0.6809 -0.4132 -0.6031 0.0424 0.3358 0.8973 -0.2230 0.1799 0.1548 -0.1551 0.3452 0.9126 0.6321 0.0064 0.6836 -0.3647 Y.U{2} = 0.4538 0.8780 -0.1521 0.7111 -0.4597 -0.5319 0.5370 -0.1332 0.8330 Y.U{3} = 0.5412 0.8409 0.8409 -0.5412</pre><pre class="codeinput">norm(full(Y) - X) <span class="comment">%<-- Reproduces the same result.</span></pre><pre class="codeoutput">ans = 1.2486e-015</pre><pre class="codeinput">U1 = nvecs(X,1,2); <span class="comment">%<-- Mode 1</span>U2 = nvecs(X,2,2); <span class="comment">%<-- Mode 2</span>U3 = nvecs(X,3,2); <span class="comment">%<-- Mode 3</span>S = ttm(X,{pinv(U1),pinv(U2),pinv(U3)}); <span class="comment">%<-- Core</span>Y = ttensor(S,{U1,U2,U3}) <span class="comment">%<-- Rank-(2,2,2) HOSVD approximation of X</span></pre><pre class="codeoutput">Y is a ttensor of size 4 x 3 x 2 Y.core is a tensor of size 2 x 2 x 2 Y.core(:,:,1) = 1.9279 0.0684 0.0669 -0.1193 Y.core(:,:,2) = 0.0560 -0.2314 -0.2194 0.5059 Y.U{1} = 0.6809 -0.4132 0.3358 0.8973 0.1548 -0.1551 0.6321 0.0064 Y.U{2} = 0.4538 0.8780 0.7111 -0.4597 0.5370 -0.1332 Y.U{3} = 0.5412 0.8409 0.8409 -0.5412</pre><pre class="codeinput">100*(1-norm(full(Y)-X)/norm(X)) <span class="comment">%<-- Percentage explained by approximation</span></pre><pre class="codeoutput">ans = 74.1571</pre><p class="footer"><br> Published with MATLAB® 7.2<br></p> </div> <!--##### SOURCE BEGIN #####%% Generating the leading mode-n vectors
% The leading mode-n vectors are those vectors that span the subspace of
% the mode-n fibers. In other words, the left singular vectors of the
% n-mode matricization of X.
%% Using nvecs to calculate the leading mode-n vectors
% The |nvecs| command efficient computes the leading n-mode vectors.
rand('state',0);
X = sptenrand([4,3,2],6) %<REPLACE_WITH_DASH_DASH A sparse tensor
%%
nvecs(X,1,2) %<REPLACE_WITH_DASH_DASH The 2 leading mode-1 vectors
%%
nvecs(X,1,3) % <REPLACE_WITH_DASH_DASH The 3 leading mode-1 vectors
%%
nvecs(full(X),1,3) %<REPLACE_WITH_DASH_DASH The same thing for a dense tensor
%%
X = ktensor({rand(3,2),rand(3,2),rand(2,2)}) %<REPLACE_WITH_DASH_DASH A random ktensor
%%
nvecs(X,2,1) %<REPLACE_WITH_DASH_DASH The 1 leading mode-2 vector
%%
nvecs(full(X),2,1) %<REPLACE_WITH_DASH_DASH Same thing for a dense tensor
%%
X = ttensor(tenrand([2,2,2,2]),{rand(3,2),rand(3,2),rand(2,2),rand(2,2)}); %<REPLACE_WITH_DASH_DASH A random ttensor
%%
nvecs(X,4,2) %<REPLACE_WITH_DASH_DASH The 1 leading mode-2 vector
%%
nvecs(full(X),4,2) %<REPLACE_WITH_DASH_DASH Same thing for a dense tensor
%% Using nvecs for the HOSVD
X = tenrand([4 3 2]) %<REPLACE_WITH_DASH_DASH Generate data
%%
U1 = nvecs(X,1,4); %<REPLACE_WITH_DASH_DASH Mode 1
U2 = nvecs(X,2,3); %<REPLACE_WITH_DASH_DASH Mode 2
U3 = nvecs(X,3,2); %<REPLACE_WITH_DASH_DASH Mode 3
S = ttm(X,{pinv(U1),pinv(U2),pinv(U3)}); %<REPLACE_WITH_DASH_DASH Core
Y = ttensor(S,{U1,U2,U3}) %<REPLACE_WITH_DASH_DASH HOSVD of X
%%
norm(full(Y) - X) %<REPLACE_WITH_DASH_DASH Reproduces the same result.
%%
U1 = nvecs(X,1,2); %<REPLACE_WITH_DASH_DASH Mode 1
U2 = nvecs(X,2,2); %<REPLACE_WITH_DASH_DASH Mode 2
U3 = nvecs(X,3,2); %<REPLACE_WITH_DASH_DASH Mode 3
S = ttm(X,{pinv(U1),pinv(U2),pinv(U3)}); %<REPLACE_WITH_DASH_DASH Core
Y = ttensor(S,{U1,U2,U3}) %<REPLACE_WITH_DASH_DASH Rank-(2,2,2) HOSVD approximation of X
%%
100*(1-norm(full(Y)-X)/norm(X)) %<REPLACE_WITH_DASH_DASH Percentage explained by approximation##### SOURCE END #####--> </body></html>⌨️ 快捷键说明
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