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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>Tensors</title> <meta name="generator" content="MATLAB 7.2"> <meta name="date" content="2007-01-10"> <meta name="m-file" content="A1_tensor_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>Tensors</h1> <introduction> <p>Tensors are extensions of multidimensional arrays with additional operations defined on them. Here we explain the basics of creating and working with tensors. </p> </introduction> <h2>Contents</h2> <div> <ul> <li><a href="#1">Creating a tensor from an array</a></li> <li><a href="#3">Creating a one-dimensional tensor</a></li> <li><a href="#5">Specifying trailing singleton dimensions in a tensor</a></li> <li><a href="#8">The constituent parts of a tensor</a></li> <li><a href="#10">Creating a tensor from its constituent parts</a></li> <li><a href="#11">Creating an empty tensor</a></li> <li><a href="#12">Use tenone to create a tensor of all ones</a></li> <li><a href="#13">Use tenzeros to create a tensor of all zeros</a></li> <li><a href="#14">Use tenrand to create a random tensor</a></li> <li><a href="#15">Use squeeze to remove singleton dimensions from a tensor</a></li> <li><a href="#16">Use double to convert a tensor to a (multidimensional) array</a></li> <li><a href="#18">Use ndims and size to get the size of a tensor</a></li> <li><a href="#21">Subscripted reference for a tensor</a></li> <li><a href="#29">Subscripted assignment for a tensor</a></li> <li><a href="#34">Using end for the last array index.</a></li> <li><a href="#37">Use find for subscripts of nonzero elements of a tensor</a></li> <li><a href="#41">Basic operations (plus, minus, and, or, etc.) on a tensor</a></li> <li><a href="#63">Using tenfun for elementwise operations on one or more tensors</a></li> <li><a href="#66">Use permute to reorder the modes of a tensor</a></li> <li><a href="#69">Displaying a tensor</a></li> </ul> </div> <h2>Creating a tensor from an array<a name="1"></a></h2> <p>The <tt>tensor</tt> command converts a (multidimensional) array to a tensor object. </p><pre class="codeinput">M = ones(4,3,2); <span class="comment">%<-- A 4 x 3 x 2 array.</span>X = tensor(M) <span class="comment">%<-- Convert to a tensor object.</span></pre><pre class="codeoutput">X is a tensor of size 4 x 3 x 2 X(:,:,1) = 1 1 1 1 1 1 1 1 1 1 1 1 X(:,:,2) = 1 1 1 1 1 1 1 1 1 1 1 1</pre><p>Optionally, you can specify a different shape for the tensor, so long as the input array has the right number of elements.</p><pre class="codeinput">X = tensor(M,[2 3 4]) <span class="comment">%<-- M has 24 elements.</span></pre><pre class="codeoutput">X is a tensor of size 2 x 3 x 4 X(:,:,1) = 1 1 1 1 1 1 X(:,:,2) = 1 1 1 1 1 1 X(:,:,3) = 1 1 1 1 1 1 X(:,:,4) = 1 1 1 1 1 1</pre><h2>Creating a one-dimensional tensor<a name="3"></a></h2> <p>The tensor class explicitly supports order-one tensors as well as trailing singleton dimensions, but the size must be explicit in the constructor. By default, a column array produces a 2-way tensor. </p><pre class="codeinput">X = tensor(rand(5,1)) <span class="comment">%<-- Creates a 2-way tensor.</span></pre><pre class="codeoutput">X is a tensor of size 5 x 1 X(:,:) = 0.9473 0.8133 0.9238 0.1990 0.6743</pre><p>This is fixed by specifying the size explicitly.</p><pre class="codeinput">X = tensor(rand(5,1),5) <span class="comment">%<-- Creates a 1-way tensor.</span></pre><pre class="codeoutput">X is a tensor of size 5 X(:) = 0.9271 0.3438 0.5945 0.6155 0.0034</pre><h2>Specifying trailing singleton dimensions in a tensor<a name="5"></a></h2> <p>Likewise, trailing singleton dimensions must be explictly specified.</p><pre class="codeinput">Y = tensor(rand(4,3,1)) <span class="comment">%<-- Creates a 2-way tensor.</span></pre><pre class="codeoutput">Y is a tensor of size 4 x 3 Y(:,:) = 0.9820 0.7010 0.1121 0.8995 0.6097 0.2916 0.6928 0.2999 0.0974 0.4397 0.8560 0.3974</pre><pre class="codeinput">Y = tensor(rand(4,3,1),[4 3 1]) <span class="comment">%<-- Creates a 3-way tensor.</span></pre><pre class="codeoutput">Y is a tensor of size 4 x 3 x 1 Y(:,:,1) = 0.3333 0.0429 0.8068 0.9442 0.0059 0.6376 0.8386 0.5744 0.2513 0.2584 0.7439 0.1443</pre><p>Unfortunately, the <tt>whos</tt> command does not report the size of 1D objects correctly (last checked for MATLAB 2006a). </p><pre class="codeinput">whos <span class="string">X</span> <span class="string">Y</span> <span class="comment">%<-- Doesn't report the right size for X!</span></pre><pre class="codeoutput"> Name Size Bytes Class X 1x1 296 tensor object Y 4x3x1 368 tensor objectGrand total is 25 elements using 664 bytes</pre><h2>The constituent parts of a tensor<a name="8"></a></h2><pre class="codeinput">X = tenrand([4 3 2]); <span class="comment">%<-- Create data.</span>X.data <span class="comment">%<-- The array.</span></pre><pre class="codeoutput">ans(:,:,1) = 0.6516 0.3099 0.2110 0.9461 0.2688 0.2168 0.8159 0.5365 0.6518 0.9302 0.1633 0.0528ans(:,:,2) = 0.2293 0.7207 0.1252 0.6674 0.9544 0.1662 0.3109 0.1311 0.9114 0.3066 0.0683 0.1363</pre><pre class="codeinput">X.size <span class="comment">%<-- The size.</span></pre><pre class="codeoutput">ans = 4 3 2</pre><h2>Creating a tensor from its constituent parts<a name="10"></a></h2><pre class="codeinput">Y = tensor(X.data,X.size) <span class="comment">%<-- Copies X.</span></pre><pre class="codeoutput">Y is a tensor of size 4 x 3 x 2 Y(:,:,1) = 0.6516 0.3099 0.2110 0.9461 0.2688 0.2168 0.8159 0.5365 0.6518 0.9302 0.1633 0.0528 Y(:,:,2) = 0.2293 0.7207 0.1252 0.6674 0.9544 0.1662 0.3109 0.1311 0.9114 0.3066 0.0683 0.1363</pre><h2>Creating an empty tensor<a name="11"></a></h2> <p>An empty constructor exists, primarily to support loading previously saved data in MAT-files.</p><pre class="codeinput">X = tensor <span class="comment">%<-- Creates an empty tensor.</span></pre><pre class="codeoutput">X is a tensor of size [empty tensor] X = []</pre><h2>Use tenone to create a tensor of all ones<a name="12"></a></h2><pre class="codeinput">X = tenones([3 4 2]) <span class="comment">%<-- Creates a 3 x 4 x 2 tensor of ones.</span></pre><pre class="codeoutput">X is a tensor of size 3 x 4 x 2 X(:,:,1) = 1 1 1 1 1 1 1 1 1 1 1 1 X(:,:,2) = 1 1 1 1 1 1 1 1 1 1 1 1</pre><h2>Use tenzeros to create a tensor of all zeros<a name="13"></a></h2><pre class="codeinput">X = tenzeros([1 4 2]) <span class="comment">%<-- Creates a 1 x 4 x 2 tensor of zeros.</span></pre><pre class="codeoutput">X is a tensor of size 1 x 4 x 2 X(:,:,1) = 0 0 0 0 X(:,:,2) = 0 0 0 0</pre><h2>Use tenrand to create a random tensor<a name="14"></a></h2><pre class="codeinput">X = tenrand([5 4 2]) <span class="comment">%<-- Creates a random 5 x 4 x 2 tensor.</span></pre><pre class="codeoutput">X is a tensor of size 5 x 4 x 2 X(:,:,1) = 0.6170 0.9413 0.8802 0.9562 0.2690 0.3299 0.7496 0.1962 0.2207 0.7045 0.3796 0.7762 0.7129 0.9434 0.7256 0.6133 0.5490 0.5816 0.1628 0.1623 X(:,:,2) = 0.0311 0.9585 0.2154 0.0178 0.2886 0.6799 0.1824 0.8779 0.9711 0.0550 0.0768 0.3525 0.9505 0.5998 0.0074 0.7221 0.2280 0.3931 0.7888 0.9685</pre><h2>Use squeeze to remove singleton dimensions from a tensor<a name="15"></a></h2><pre class="codeinput">squeeze(Y) <span class="comment">%<-- Removes singleton dimensions.</span></pre><pre class="codeoutput">ans is a tensor of size 4 x 3 x 2 ans(:,:,1) = 0.6516 0.3099 0.2110 0.9461 0.2688 0.2168 0.8159 0.5365 0.6518 0.9302 0.1633 0.0528 ans(:,:,2) = 0.2293 0.7207 0.1252 0.6674 0.9544 0.1662 0.3109 0.1311 0.9114 0.3066 0.0683 0.1363</pre><h2>Use double to convert a tensor to a (multidimensional) array<a name="16"></a></h2><pre class="codeinput">double(Y) <span class="comment">%<-- Converts Y to a standard MATLAB array.</span></pre><pre class="codeoutput">ans(:,:,1) = 0.6516 0.3099 0.2110 0.9461 0.2688 0.2168 0.8159 0.5365 0.6518 0.9302 0.1633 0.0528ans(:,:,2) = 0.2293 0.7207 0.1252 0.6674 0.9544 0.1662 0.3109 0.1311 0.9114 0.3066 0.0683 0.1363</pre><pre class="codeinput">Y.data <span class="comment">%<-- Same thing.</span></pre><pre class="codeoutput">ans(:,:,1) = 0.6516 0.3099 0.2110 0.9461 0.2688 0.2168 0.8159 0.5365 0.6518 0.9302 0.1633 0.0528ans(:,:,2) = 0.2293 0.7207 0.1252 0.6674 0.9544 0.1662 0.3109 0.1311 0.9114 0.3066 0.0683 0.1363</pre><h2>Use ndims and size to get the size of a tensor<a name="18"></a></h2><pre class="codeinput">ndims(Y) <span class="comment">%<-- Number of dimensions (or ways).</span></pre><pre class="codeoutput">ans = 3</pre><pre class="codeinput">size(Y) <span class="comment">%<-- Row vector with the sizes of all dimension.</span></pre><pre class="codeoutput">ans = 4 3 2</pre><pre class="codeinput">size(Y,3) <span class="comment">%<-- Size of a single dimension.</span></pre><pre class="codeoutput">ans = 2</pre><h2>Subscripted reference for a tensor<a name="21"></a></h2><pre class="codeinput">X = tenrand([3 4 2 1]); <span class="comment">%<-- Create a 3 x 4 x 2 x 1 random tensor.</span>X(1,1,1,1) <span class="comment">%<-- Extract a single element.</span></pre><pre class="codeoutput">ans = 0.1557</pre><p>It is possible to extract a subtensor that contains a single element. Observe that singleton dimensions are <b>not</b> dropped unless they are specifically specified, e.g., as above. </p><pre class="codeinput">X(1,1,1,:) <span class="comment">%<-- Produces a tensor of order 1 and size 1.</span></pre><pre class="codeoutput">ans is a tensor of size 1 ans(:) = 0.1557</pre><p>In general, specified dimensions are dropped from the result. Here we specify the second and third dimension.</p><pre class="codeinput">X(:,1,1,:) <span class="comment">%<-- Produces a tensor of size 3 x 1.</span></pre><pre class="codeoutput">ans is a tensor of size 3 x 1 ans(:,:) = 0.1557 0.1630 0.3134</pre><p>Moreover, the subtensor is automatically renumbered/resized in the same way that MATLAB works for arrays except that singleton⌨️ 快捷键说明
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