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<p class="caption"><span class="caption-text">ํ์ดํ ์น(PyTorch) ๋ ์ํผ</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../recipes/recipes_index.html">๋ชจ๋ ๋ ์ํผ ๋ณด๊ธฐ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../prototype/prototype_index.html">๋ชจ๋ ํ๋กํ ํ์
๋ ์ํผ ๋ณด๊ธฐ</a></li>
</ul>
<p class="caption"><span class="caption-text">ํ์ดํ ์น(PyTorch) ์์ํ๊ธฐ</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/intro.html">ํ์ดํ ์น(PyTorch) ๊ธฐ๋ณธ ์ตํ๊ธฐ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/quickstart_tutorial.html">๋น ๋ฅธ ์์(Quickstart)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/tensorqs_tutorial.html">ํ
์(Tensor)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/data_tutorial.html">Dataset๊ณผ DataLoader</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/transforms_tutorial.html">๋ณํ(Transform)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/buildmodel_tutorial.html">์ ๊ฒฝ๋ง ๋ชจ๋ธ ๊ตฌ์ฑํ๊ธฐ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/autogradqs_tutorial.html"><code class="docutils literal notranslate"><span class="pre">torch.autograd</span></code>๋ฅผ ์ฌ์ฉํ ์๋ ๋ฏธ๋ถ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/optimization_tutorial.html">๋ชจ๋ธ ๋งค๊ฐ๋ณ์ ์ต์ ํํ๊ธฐ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/saveloadrun_tutorial.html">๋ชจ๋ธ ์ ์ฅํ๊ณ ๋ถ๋ฌ์ค๊ธฐ</a></li>
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<p class="caption"><span class="caption-text">Introduction to PyTorch on YouTube</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt.html">Introduction to PyTorch - YouTube Series</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/introyt1_tutorial.html">Introduction to PyTorch</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/tensors_deeper_tutorial.html">Introduction to PyTorch Tensors</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/autogradyt_tutorial.html">The Fundamentals of Autograd</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/modelsyt_tutorial.html">Building Models with PyTorch</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/tensorboardyt_tutorial.html">PyTorch TensorBoard Support</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/trainingyt.html">Training with PyTorch</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/captumyt.html">Model Understanding with Captum</a></li>
</ul>
<p class="caption"><span class="caption-text">ํ์ดํ ์น(PyTorch) ๋ฐฐ์ฐ๊ธฐ</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../beginner/deep_learning_60min_blitz.html">PyTorch๋ก ๋ฅ๋ฌ๋ํ๊ธฐ: 60๋ถ๋ง์ ๋์ฅ๋ด๊ธฐ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/pytorch_with_examples.html">์์ ๋ก ๋ฐฐ์ฐ๋ ํ์ดํ ์น(PyTorch)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/nn_tutorial.html"><cite>torch.nn</cite> ์ด <em>์ค์ ๋ก</em> ๋ฌด์์ธ๊ฐ์?</a></li>
<li class="toctree-l1"><a class="reference internal" href="tensorboard_tutorial.html">TensorBoard๋ก ๋ชจ๋ธ, ๋ฐ์ดํฐ, ํ์ต ์๊ฐํํ๊ธฐ</a></li>
</ul>
<p class="caption"><span class="caption-text">์ด๋ฏธ์ง/๋น๋์ค</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="torchvision_tutorial.html">TorchVision ๊ฐ์ฒด ๊ฒ์ถ ๋ฏธ์ธ์กฐ์ (Finetuning) ํํ ๋ฆฌ์ผ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/transfer_learning_tutorial.html">์ปดํจํฐ ๋น์ (Vision)์ ์ํ ์ ์ดํ์ต(Transfer Learning)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/fgsm_tutorial.html">์ ๋์ ์์ ์์ฑ(Adversarial Example Generation)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/dcgan_faces_tutorial.html">DCGAN ํํ ๋ฆฌ์ผ</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/vt_tutorial.html">๋ฐฐํฌ๋ฅผ ์ํ ๋น์ ํธ๋์คํฌ๋จธ(Vision Transformer) ๋ชจ๋ธ ์ต์ ํํ๊ธฐ</a></li>
</ul>
<p class="caption"><span class="caption-text">์ค๋์ค</span></p>
<ul>
<li class="toctree-l1"><a class="reference internal" href="../beginner/audio_io_tutorial.html">Audio I/O</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/audio_resampling_tutorial.html">Audio Resampling</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/audio_data_augmentation_tutorial.html">Audio Data Augmentation</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/audio_feature_extractions_tutorial.html">Audio Feature Extractions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/audio_feature_augmentation_tutorial.html">Audio Feature Augmentation</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/audio_datasets_tutorial.html">Audio Datasets</a></li>
<li class="toctree-l1"><a class="reference internal" href="speech_recognition_pipeline_tutorial.html">Speech Recognition with Wav2Vec2</a></li>
<li class="toctree-l1"><a class="reference internal" href="speech_command_classification_with_torchaudio_tutorial.html">Speech Command Classification with torchaudio</a></li>
<li class="toctree-l1"><a class="reference internal" href="text_to_speech_with_torchaudio.html">Text-to-speech with torchaudio</a></li>
<li class="toctree-l1"><a class="reference internal" href="forced_alignment_with_torchaudio_tutorial.html">Forced Alignment with Wav2Vec2</a></li>
</ul>
<p class="caption"><span class="caption-text">ํ
์คํธ</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/transformer_tutorial.html">nn.Transformer ์ TorchText ๋ก ์ํ์ค-ํฌ-์ํ์ค(Sequence-to-Sequence) ๋ชจ๋ธ๋งํ๊ธฐ</a></li>
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<div class="sphx-glr-example-title section" id="parametrizations-tutorial">
<span id="sphx-glr-intermediate-parametrizations-py"></span><h1>Parametrizations Tutorial<a class="headerlink" href="#parametrizations-tutorial" title="Permalink to this headline">ยถ</a></h1>
<p><strong>Author</strong>: <a class="reference external" href="https://github.com/lezcano">Mario Lezcano</a></p>
<p>Regularizing deep-learning models is a surprisingly challenging task.
Classical techniques such as penalty methods often fall short when applied
on deep models due to the complexity of the function being optimized.
This is particularly problematic when working with ill-conditioned models.
Examples of these are RNNs trained on long sequences and GANs. A number
of techniques have been proposed in recent years to regularize these
models and improve their convergence. On recurrent models, it has been
proposed to control the singular values of the recurrent kernel for the
RNN to be well-conditioned. This can be achieved, for example, by making
the recurrent kernel <a class="reference external" href="https://en.wikipedia.org/wiki/Orthogonal_matrix">orthogonal</a>.
Another way to regularize recurrent models is via
โ<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.nn.utils.weight_norm.html">weight normalization</a>โ.
This approach proposes to decouple the learning of the parameters from the
learning of their norms. To do so, the parameter is divided by its
<a class="reference external" href="https://en.wikipedia.org/wiki/Matrix_norm#Frobenius_norm">Frobenius norm</a>
and a separate parameter encoding its norm is learnt.
A similar regularization was proposed for GANs under the name of
โ<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.nn.utils.spectral_norm.html">spectral normalization</a>โ. This method
controls the Lipschitz constant of the network by dividing its parameters by
their <a class="reference external" href="https://en.wikipedia.org/wiki/Matrix_norm#Special_cases">spectral norm</a>,
rather than their Frobenius norm.</p>
<p>All these methods have a common pattern: they all transform a parameter
in an appropriate way before using it. In the first case, they make it orthogonal by
using a function that maps matrices to orthogonal matrices. In the case of weight
and spectral normalization, they divide the original parameter by its norm.</p>
<p>More generally, all these examples use a function to put extra structure on the parameters.
In other words, they use a function to constrain the parameters.</p>
<p>In this tutorial, you will learn how to implement and use this pattern to put
constraints on your model. Doing so is as easy as writing your own <code class="docutils literal notranslate"><span class="pre">nn.Module</span></code>.</p>
<p>Requirements: <code class="docutils literal notranslate"><span class="pre">torch>=1.9.0</span></code></p>
<div class="section" id="implementing-parametrizations-by-hand">
<h2>Implementing parametrizations by hand<a class="headerlink" href="#implementing-parametrizations-by-hand" title="Permalink to this headline">ยถ</a></h2>
<p>Assume that we want to have a square linear layer with symmetric weights, that is,
with weights <code class="docutils literal notranslate"><span class="pre">X</span></code> such that <code class="docutils literal notranslate"><span class="pre">X</span> <span class="pre">=</span> <span class="pre">Xแต</span></code>. One way to do so is
to copy the upper-triangular part of the matrix into its lower-triangular part</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">torch</span>
<span class="kn">import</span> <span class="nn">torch.nn</span> <span class="k">as</span> <span class="nn">nn</span>
<span class="kn">import</span> <span class="nn">torch.nn.utils.parametrize</span> <span class="k">as</span> <span class="nn">parametrize</span>
<span class="k">def</span> <span class="nf">symmetric</span><span class="p">(</span><span class="n">X</span><span class="p">):</span>
<span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">triu</span><span class="p">()</span> <span class="o">+</span> <span class="n">X</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">symmetric</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="c1"># A is symmetric</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="c1"># Quick visual check</span>
</pre></div>
</div>
<p>We can then use this idea to implement a linear layer with symmetric weights</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">LinearSymmetric</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n_features</span><span class="p">):</span>
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Parameter</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">n_features</span><span class="p">,</span> <span class="n">n_features</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">x</span><span class="p">):</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">symmetric</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
<span class="k">return</span> <span class="n">x</span> <span class="o">@</span> <span class="n">A</span>
</pre></div>
</div>
<p>The layer can be then used as a regular linear layer</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">LinearSymmetric</span><span class="p">(</span><span class="mi">3</span><span class="p">)</span>
<span class="n">out</span> <span class="o">=</span> <span class="n">layer</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">3</span><span class="p">))</span>
</pre></div>
</div>
<p>This implementation, although correct and self-contained, presents a number of problems:</p>
<ol class="arabic simple">
<li>It reimplements the layer. We had to implement the linear layer as <code class="docutils literal notranslate"><span class="pre">x</span> <span class="pre">@</span> <span class="pre">A</span></code>. This is
not very problematic for a linear layer, but imagine having to reimplement a CNN or a
Transformerโฆ</li>
<li>It does not separate the layer and the parametrization. If the parametrization were
more difficult, we would have to rewrite its code for each layer that we want to use it
in.</li>
<li>It recomputes the parametrization everytime we use the layer. If we use the layer
several times during the forward pass, (imagine the recurrent kernel of an RNN), it
would compute the same <code class="docutils literal notranslate"><span class="pre">A</span></code> every time that the layer is called.</li>
</ol>
</div>
<div class="section" id="introduction-to-parametrizations">
<h2>Introduction to parametrizations<a class="headerlink" href="#introduction-to-parametrizations" title="Permalink to this headline">ยถ</a></h2>
<p>Parametrizations can solve all these problems as well as others.</p>
<p>Letโs start by reimplementing the code above using <code class="docutils literal notranslate"><span class="pre">torch.nn.utils.parametrize</span></code>.
The only thing that we have to do is to write the parametrization as a regular <code class="docutils literal notranslate"><span class="pre">nn.Module</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Symmetric</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="k">return</span> <span class="n">X</span><span class="o">.</span><span class="n">triu</span><span class="p">()</span> <span class="o">+</span> <span class="n">X</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
</pre></div>
</div>
<p>This is all we need to do. Once we have this, we can transform any regular layer into a
symmetric layer by doing</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Symmetric</span><span class="p">())</span>
</pre></div>
</div>
<p>Now, the matrix of the linear layer is symmetric</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span>
<span class="k">assert</span> <span class="n">torch</span><span class="o">.</span><span class="n">allclose</span><span class="p">(</span><span class="n">A</span><span class="p">,</span> <span class="n">A</span><span class="o">.</span><span class="n">T</span><span class="p">)</span> <span class="c1"># A is symmetric</span>
<span class="nb">print</span><span class="p">(</span><span class="n">A</span><span class="p">)</span> <span class="c1"># Quick visual check</span>
</pre></div>
</div>
<p>We can do the same thing with any other layer. For example, we can create a CNN with
<a class="reference external" href="https://en.wikipedia.org/wiki/Skew-symmetric_matrix">skew-symmetric</a> kernels.
We use a similar parametrization, copying the upper-triangular part with signs
reversed into the lower-triangular part</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Skew</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">A</span> <span class="o">-</span> <span class="n">A</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="n">cnn</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Conv2d</span><span class="p">(</span><span class="n">in_channels</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="n">out_channels</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span> <span class="n">kernel_size</span><span class="o">=</span><span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">cnn</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="c1"># Print a few kernels</span>
<span class="nb">print</span><span class="p">(</span><span class="n">cnn</span><span class="o">.</span><span class="n">weight</span><span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">])</span>
<span class="nb">print</span><span class="p">(</span><span class="n">cnn</span><span class="o">.</span><span class="n">weight</span><span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">])</span>
</pre></div>
</div>
</div>
<div class="section" id="inspecting-a-parametrized-module">
<h2>Inspecting a parametrized module<a class="headerlink" href="#inspecting-a-parametrized-module" title="Permalink to this headline">ยถ</a></h2>
<p>When a module is parametrized, we find that the module has changed in three ways:</p>
<ol class="arabic simple">
<li><code class="docutils literal notranslate"><span class="pre">model.weight</span></code> is now a property</li>
<li>It has a new <code class="docutils literal notranslate"><span class="pre">module.parametrizations</span></code> attribute</li>
<li>The unparametrized weight has been moved to <code class="docutils literal notranslate"><span class="pre">module.parametrizations.weight.original</span></code></li>
</ol>
<div class="line-block">
<div class="line"><br /></div>
</div>
<p>After parametrizing <code class="docutils literal notranslate"><span class="pre">weight</span></code>, <code class="docutils literal notranslate"><span class="pre">layer.weight</span></code> is turned into a
<a class="reference external" href="https://docs.python.org/3/library/functions.html#property">Python property</a>.
This property computes <code class="docutils literal notranslate"><span class="pre">parametrization(weight)</span></code> every time we request <code class="docutils literal notranslate"><span class="pre">layer.weight</span></code>
just as we did in our implementation of <code class="docutils literal notranslate"><span class="pre">LinearSymmetric</span></code> above.</p>
<p>Registered parametrizations are stored under a <code class="docutils literal notranslate"><span class="pre">parametrizations</span></code> attribute within the module.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Unparametrized:</span><span class="se">\n</span><span class="si">{</span><span class="n">layer</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Symmetric</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Parametrized:</span><span class="se">\n</span><span class="si">{</span><span class="n">layer</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
</pre></div>
</div>
<p>This <code class="docutils literal notranslate"><span class="pre">parametrizations</span></code> attribute is an <code class="docutils literal notranslate"><span class="pre">nn.ModuleDict</span></code>, and it can be accessed as such</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">parametrizations</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">parametrizations</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
</pre></div>
</div>
<p>Each element of this <code class="docutils literal notranslate"><span class="pre">nn.ModuleDict</span></code> is a <code class="docutils literal notranslate"><span class="pre">ParametrizationList</span></code>, which behaves like an
<code class="docutils literal notranslate"><span class="pre">nn.Sequential</span></code>. This list will allow us to concatenate parametrizations on one weight.
Since this is a list, we can access the parametrizations indexing it. Hereโs
where our <code class="docutils literal notranslate"><span class="pre">Symmetric</span></code> parametrization sits</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">parametrizations</span><span class="o">.</span><span class="n">weight</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span>
</pre></div>
</div>
<p>The other thing that we notice is that, if we print the parameters, we see that the
parameter <code class="docutils literal notranslate"><span class="pre">weight</span></code> has been moved</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="nb">dict</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">named_parameters</span><span class="p">()))</span>
</pre></div>
</div>
<p>It now sits under <code class="docutils literal notranslate"><span class="pre">layer.parametrizations.weight.original</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">parametrizations</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">original</span><span class="p">)</span>
</pre></div>
</div>
<p>Besides these three small differences, the parametrization is doing exactly the same
as our manual implementation</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">symmetric</span> <span class="o">=</span> <span class="n">Symmetric</span><span class="p">()</span>
<span class="n">weight_orig</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">parametrizations</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">original</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">,</span> <span class="n">symmetric</span><span class="p">(</span><span class="n">weight_orig</span><span class="p">)))</span>
</pre></div>
</div>
</div>
<div class="section" id="parametrizations-are-first-class-citizens">
<h2>Parametrizations are first-class citizens<a class="headerlink" href="#parametrizations-are-first-class-citizens" title="Permalink to this headline">ยถ</a></h2>
<p>Since <code class="docutils literal notranslate"><span class="pre">layer.parametrizations</span></code> is an <code class="docutils literal notranslate"><span class="pre">nn.ModuleList</span></code>, it means that the parametrizations
are properly registered as submodules of the original module. As such, the same rules
for registering parameters in a module apply to register a parametrization.
For example, if a parametrization has parameters, these will be moved from CPU
to CUDA when calling <code class="docutils literal notranslate"><span class="pre">model</span> <span class="pre">=</span> <span class="pre">model.cuda()</span></code>.</p>
</div>
<div class="section" id="caching-the-value-of-a-parametrization">
<h2>Caching the value of a parametrization<a class="headerlink" href="#caching-the-value-of-a-parametrization" title="Permalink to this headline">ยถ</a></h2>
<p>Parametrizations come with an inbuilt caching system via the context manager
<code class="docutils literal notranslate"><span class="pre">parametrize.cached()</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">NoisyParametrization</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Computing the Parametrization"</span><span class="p">)</span>
<span class="k">return</span> <span class="n">X</span>
<span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">NoisyParametrization</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Here, layer.weight is recomputed every time we call it"</span><span class="p">)</span>
<span class="n">foo</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span> <span class="o">+</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">T</span>
<span class="n">bar</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
<span class="k">with</span> <span class="n">parametrize</span><span class="o">.</span><span class="n">cached</span><span class="p">():</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Here, it is computed just the first time layer.weight is called"</span><span class="p">)</span>
<span class="n">foo</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span> <span class="o">+</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">T</span>
<span class="n">bar</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="o">.</span><span class="n">sum</span><span class="p">()</span>
</pre></div>
</div>
</div>
<div class="section" id="concatenating-parametrizations">
<h2>Concatenating parametrizations<a class="headerlink" href="#concatenating-parametrizations" title="Permalink to this headline">ยถ</a></h2>
<p>Concatenating two parametrizations is as easy as registering them on the same tensor.
We may use this to create more complex parametrizations from simpler ones. For example, the
<a class="reference external" href="https://en.wikipedia.org/wiki/Cayley_transform#Matrix_map">Cayley map</a>
maps the skew-symmetric matrices to the orthogonal matrices of positive determinant. We can
concatenate <code class="docutils literal notranslate"><span class="pre">Skew</span></code> and a parametrization that implements the Cayley map to get a layer with
orthogonal weights</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">CayleyMap</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s2">"Id"</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="c1"># (I + X)(I - X)^{-1}</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Id</span> <span class="o">+</span> <span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Id</span> <span class="o">-</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">solution</span>
<span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">CayleyMap</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">layer</span><span class="o">.</span><span class="n">weight</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span> <span class="c1"># X is orthogonal</span>
</pre></div>
</div>
<p>This may also be used to prune a parametrized module, or to reuse parametrizations. For example,
the matrix exponential maps the symmetric matrices to the Symmetric Positive Definite (SPD) matrices
But the matrix exponential also maps the skew-symmetric matrices to the orthogonal matrices.
Using these two facts, we may reuse the parametrizations before to our advantage</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">MatrixExponential</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">matrix_exp</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="n">layer_orthogonal</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer_orthogonal</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer_orthogonal</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">MatrixExponential</span><span class="p">())</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">layer_orthogonal</span><span class="o">.</span><span class="n">weight</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="o">@</span> <span class="n">X</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="mi">3</span><span class="p">)))</span> <span class="c1"># X is orthogonal</span>
<span class="n">layer_spd</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer_spd</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Symmetric</span><span class="p">())</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer_spd</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">MatrixExponential</span><span class="p">())</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">layer_spd</span><span class="o">.</span><span class="n">weight</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span><span class="p">))</span> <span class="c1"># X is symmetric</span>
<span class="nb">print</span><span class="p">((</span><span class="n">torch</span><span class="o">.</span><span class="n">symeig</span><span class="p">(</span><span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">eigenvalues</span> <span class="o">></span> <span class="mf">0.</span><span class="p">)</span><span class="o">.</span><span class="n">all</span><span class="p">())</span> <span class="c1"># X is positive definite</span>
</pre></div>
</div>
</div>
<div class="section" id="intializing-parametrizations">
<h2>Intializing parametrizations<a class="headerlink" href="#intializing-parametrizations" title="Permalink to this headline">ยถ</a></h2>
<p>Parametrizations come with a mechanism to initialize them. If we implement a method
<code class="docutils literal notranslate"><span class="pre">right_inverse</span></code> with signature</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">right_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">:</span> <span class="n">Tensor</span><span class="p">)</span> <span class="o">-></span> <span class="n">Tensor</span>
</pre></div>
</div>
<p>it will be used when assigning to the parametrized tensor.</p>
<p>Letโs upgrade our implementation of the <code class="docutils literal notranslate"><span class="pre">Skew</span></code> class to support this</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">Skew</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="n">A</span> <span class="o">=</span> <span class="n">X</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
<span class="k">return</span> <span class="n">A</span> <span class="o">-</span> <span class="n">A</span><span class="o">.</span><span class="n">transpose</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="o">-</span><span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">right_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="c1"># We assume that A is skew-symmetric</span>
<span class="c1"># We take the upper-triangular elements, as these are those used in the forward</span>
<span class="k">return</span> <span class="n">A</span><span class="o">.</span><span class="n">triu</span><span class="p">(</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<p>We may now initialize a layer that is parametrized with <code class="docutils literal notranslate"><span class="pre">Skew</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">X</span> <span class="o">-</span> <span class="n">X</span><span class="o">.</span><span class="n">T</span> <span class="c1"># X is now skew-symmetric</span>
<span class="n">layer</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">X</span> <span class="c1"># Initialize layer.weight to be X</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span> <span class="c1"># layer.weight == X</span>
</pre></div>
</div>
<p>This <code class="docutils literal notranslate"><span class="pre">right_inverse</span></code> works as expected when we concatenate parametrizations.
To see this, letโs upgrade the Cayley parametrization to also support being initialized</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">CayleyMap</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
<span class="bp">self</span><span class="o">.</span><span class="n">register_buffer</span><span class="p">(</span><span class="s2">"Id"</span><span class="p">,</span> <span class="n">torch</span><span class="o">.</span><span class="n">eye</span><span class="p">(</span><span class="n">n</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="c1"># Assume X skew-symmetric</span>
<span class="c1"># (I + X)(I - X)^{-1}</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">Id</span> <span class="o">+</span> <span class="n">X</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Id</span> <span class="o">-</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">solution</span>
<span class="k">def</span> <span class="nf">right_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="c1"># Assume A orthogonal</span>
<span class="c1"># See https://en.wikipedia.org/wiki/Cayley_transform#Matrix_map</span>
<span class="c1"># (X - I)(X + I)^{-1}</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">X</span> <span class="o">-</span> <span class="bp">self</span><span class="o">.</span><span class="n">Id</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">Id</span> <span class="o">+</span> <span class="n">X</span><span class="p">)</span><span class="o">.</span><span class="n">solution</span>
<span class="n">layer_orthogonal</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer_orthogonal</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer_orthogonal</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">CayleyMap</span><span class="p">(</span><span class="mi">3</span><span class="p">))</span>
<span class="c1"># Sample an orthogonal matrix with positive determinant</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">empty</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="n">nn</span><span class="o">.</span><span class="n">init</span><span class="o">.</span><span class="n">orthogonal_</span><span class="p">(</span><span class="n">X</span><span class="p">)</span>
<span class="k">if</span> <span class="n">X</span><span class="o">.</span><span class="n">det</span><span class="p">()</span> <span class="o"><</span> <span class="mf">0.</span><span class="p">:</span>
<span class="n">X</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span><span class="o">.</span><span class="n">neg_</span><span class="p">()</span>
<span class="n">layer_orthogonal</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">X</span>
<span class="nb">print</span><span class="p">(</span><span class="n">torch</span><span class="o">.</span><span class="n">dist</span><span class="p">(</span><span class="n">layer_orthogonal</span><span class="o">.</span><span class="n">weight</span><span class="p">,</span> <span class="n">X</span><span class="p">))</span> <span class="c1"># layer_orthogonal.weight == X</span>
</pre></div>
</div>
<p>This initialization step can be written more succinctly as</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer_orthogonal</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">init</span><span class="o">.</span><span class="n">orthogonal_</span><span class="p">(</span><span class="n">layer_orthogonal</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
</pre></div>
</div>
<p>The name of this method comes from the fact that we would often expect
that <code class="docutils literal notranslate"><span class="pre">forward(right_inverse(X))</span> <span class="pre">==</span> <span class="pre">X</span></code>. This is a direct way of rewriting that
the forward afer the initalization with value <code class="docutils literal notranslate"><span class="pre">X</span></code> should return the value <code class="docutils literal notranslate"><span class="pre">X</span></code>.
This constraint is not strongly enforced in practice. In fact, at times, it might be of
interest to relax this relation. For example, consider the following implementation
of a randomized pruning method:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">class</span> <span class="nc">PruningParametrization</span><span class="p">(</span><span class="n">nn</span><span class="o">.</span><span class="n">Module</span><span class="p">):</span>
<span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">,</span> <span class="n">p_drop</span><span class="o">=</span><span class="mf">0.2</span><span class="p">):</span>
<span class="nb">super</span><span class="p">()</span><span class="o">.</span><span class="fm">__init__</span><span class="p">()</span>
<span class="c1"># sample zeros with probability p_drop</span>
<span class="n">mask</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">full_like</span><span class="p">(</span><span class="n">X</span><span class="p">,</span> <span class="mf">1.0</span> <span class="o">-</span> <span class="n">p_drop</span><span class="p">)</span>
<span class="bp">self</span><span class="o">.</span><span class="n">mask</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">bernoulli</span><span class="p">(</span><span class="n">mask</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">forward</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">X</span><span class="p">):</span>
<span class="k">return</span> <span class="n">X</span> <span class="o">*</span> <span class="bp">self</span><span class="o">.</span><span class="n">mask</span>
<span class="k">def</span> <span class="nf">right_inverse</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">A</span><span class="p">):</span>
<span class="k">return</span> <span class="n">A</span>
</pre></div>
</div>
<p>In this case, it is not true that for every matrix A <code class="docutils literal notranslate"><span class="pre">forward(right_inverse(A))</span> <span class="pre">==</span> <span class="pre">A</span></code>.
This is only true when the matrix <code class="docutils literal notranslate"><span class="pre">A</span></code> has zeros in the same positions as the mask.
Even then, if we assign a tensor to a pruned parameter, it will comes as no surprise
that tensor will be, in fact, pruned</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="n">X</span> <span class="o">=</span> <span class="n">torch</span><span class="o">.</span><span class="n">rand_like</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Initialization matrix:</span><span class="se">\n</span><span class="si">{</span><span class="n">X</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">PruningParametrization</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">))</span>
<span class="n">layer</span><span class="o">.</span><span class="n">weight</span> <span class="o">=</span> <span class="n">X</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Initialized weight:</span><span class="se">\n</span><span class="si">{</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
</pre></div>
</div>
</div>
<div class="section" id="removing-parametrizations">
<h2>Removing parametrizations<a class="headerlink" href="#removing-parametrizations" title="Permalink to this headline">ยถ</a></h2>
<p>We may remove all the parametrizations from a parameter or a buffer in a module
by using <code class="docutils literal notranslate"><span class="pre">parametrize.remove_parametrizations()</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Before:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Parametrized:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">remove_parametrizations</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">After. Weight has skew-symmetric values but it is unconstrained:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
</pre></div>
</div>
<p>When removing a parametrization, we may choose to leave the original parameter (i.e. that in
<code class="docutils literal notranslate"><span class="pre">layer.parametriations.weight.original</span></code>) rather than its parametrized version by setting
the flag <code class="docutils literal notranslate"><span class="pre">leave_parametrized=False</span></code></p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">layer</span> <span class="o">=</span> <span class="n">nn</span><span class="o">.</span><span class="n">Linear</span><span class="p">(</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Before:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">register_parametrization</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">Skew</span><span class="p">())</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">Parametrized:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
<span class="n">parametrize</span><span class="o">.</span><span class="n">remove_parametrizations</span><span class="p">(</span><span class="n">layer</span><span class="p">,</span> <span class="s2">"weight"</span><span class="p">,</span> <span class="n">leave_parametrized</span><span class="o">=</span><span class="kc">False</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="se">\n</span><span class="s2">After. Same as Before:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">layer</span><span class="o">.</span><span class="n">weight</span><span class="p">)</span>
</pre></div>
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