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<p class="caption" role="heading"><span class="caption-text">파이토치(PyTorch) 레시피</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../recipes/recipes_index.html">모든 레시피 보기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../prototype/prototype_index.html">모든 프로토타입 레시피 보기</a></li>
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<p class="caption" role="heading"><span class="caption-text">파이토치(PyTorch) 시작하기</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/intro.html">파이토치(PyTorch) 기본 익히기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/quickstart_tutorial.html">빠른 시작(Quickstart)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/tensorqs_tutorial.html">텐서(Tensor)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/data_tutorial.html">Dataset과 DataLoader</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/transforms_tutorial.html">변형(Transform)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/buildmodel_tutorial.html">신경망 모델 구성하기</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>를 사용한 자동 미분</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/basics/optimization_tutorial.html">모델 매개변수 최적화하기</a></li>
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<p class="caption" role="heading"><span class="caption-text">Introduction to PyTorch on YouTube</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt.html">PyTorch 소개 - YouTube 시리즈</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/introyt/tensors_deeper_tutorial.html">Pytorch Tensor 소개</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>
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<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>
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<p class="caption" role="heading"><span class="caption-text">파이토치(PyTorch) 배우기</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/deep_learning_60min_blitz.html">PyTorch로 딥러닝하기: 60분만에 끝장내기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/pytorch_with_examples.html">예제로 배우는 파이토치(PyTorch)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/nn_tutorial.html"><cite>torch.nn</cite> 이 <em>실제로</em> 무엇인가요?</a></li>
<li class="toctree-l1"><a class="reference internal" href="tensorboard_tutorial.html">TensorBoard로 모델, 데이터, 학습 시각화하기</a></li>
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<li class="toctree-l1"><a class="reference internal" href="torchvision_tutorial.html">TorchVision Object Detection Finetuning Tutorial</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/transfer_learning_tutorial.html">컴퓨터 비전(Vision)을 위한 전이학습(Transfer Learning)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/fgsm_tutorial.html">적대적 예제 생성(Adversarial Example Generation)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/dcgan_faces_tutorial.html">DCGAN 튜토리얼</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/vt_tutorial.html">배포를 위해 비전 트랜스포머(Vision Transformer) 모델 최적화하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="tiatoolbox_tutorial.html">Whole Slide Image Classification Using PyTorch and TIAToolbox</a></li>
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<p class="caption" role="heading"><span class="caption-text">오디오</span></p>
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<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>
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<li class="toctree-l1"><a class="reference internal" href="char_rnn_generation_tutorial.html">기초부터 시작하는 NLP: 문자-단위 RNN으로 이름 생성하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="seq2seq_translation_tutorial.html">기초부터 시작하는 NLP: Sequence to Sequence 네트워크와 Attention을 이용한 번역</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/text_sentiment_ngrams_tutorial.html">torchtext 라이브러리로 텍스트 분류하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/translation_transformer.html"><code class="docutils literal notranslate"><span class="pre">nn.Transformer</span></code> 와 torchtext로 언어 번역하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/torchtext_custom_dataset_tutorial.html">Preprocess custom text dataset using Torchtext</a></li>
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<p class="caption" role="heading"><span class="caption-text">백엔드</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/onnx/intro_onnx.html">Introduction to ONNX</a></li>
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<li class="toctree-l1"><a class="reference internal" href="mario_rl_tutorial.html">마리오 게임 RL 에이전트로 학습하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/pendulum.html">Pendulum: Writing your environment and transforms with TorchRL</a></li>
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<p class="caption" role="heading"><span class="caption-text">PyTorch 모델을 프로덕션 환경에 배포하기</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/onnx/intro_onnx.html">Introduction to ONNX</a></li>
<li class="toctree-l1"><a class="reference internal" href="flask_rest_api_tutorial.html">Flask를 사용하여 Python에서 PyTorch를 REST API로 배포하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/Intro_to_TorchScript_tutorial.html">TorchScript 소개</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/cpp_export.html">C++에서 TorchScript 모델 로딩하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/super_resolution_with_onnxruntime.html">(선택) PyTorch 모델을 ONNX으로 변환하고 ONNX 런타임에서 실행하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="realtime_rpi.html">Raspberry Pi 4 에서 실시간 추론(Inference) (30fps!)</a></li>
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<p class="caption" role="heading"><span class="caption-text">PyTorch 프로파일링</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/profiler.html">PyTorch 모듈 프로파일링하기</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../advanced/dispatcher.html">Registering a Dispatched Operator in C++</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/profiler.html">PyTorch 모듈 프로파일링하기</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/hyperparameter_tuning_tutorial.html">Ray Tune을 사용한 하이퍼파라미터 튜닝</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/vt_tutorial.html">배포를 위해 비전 트랜스포머(Vision Transformer) 모델 최적화하기</a></li>
<li class="toctree-l1 current"><a class="current reference internal" href="#">Parametrizations Tutorial</a></li>
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<li class="toctree-l1"><a class="reference internal" href="quantized_transfer_learning_tutorial.html">(베타) 컴퓨터 비전 튜토리얼을 위한 양자화된 전이학습(Quantized Transfer Learning)</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/static_quantization_tutorial.html">(베타) PyTorch에서 Eager Mode를 이용한 정적 양자화</a></li>
<li class="toctree-l1"><a class="reference internal" href="torchserve_with_ipex.html">Grokking PyTorch Intel CPU performance from first principles</a></li>
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<li class="toctree-l1"><a class="reference internal" href="scaled_dot_product_attention_tutorial.html">(Beta) Scaled Dot Product Attention (SDPA)로 고성능 트랜스포머(Transformers) 구현하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="scaled_dot_product_attention_tutorial.html#torch-compile-sdpa"><code class="docutils literal notranslate"><span class="pre">torch.compile</span></code> 과 함께 SDPA 사용하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="scaled_dot_product_attention_tutorial.html#sdpa-atteition-bias">SDPA를 <code class="docutils literal notranslate"><span class="pre">atteition.bias</span></code> 하위 클래스와 사용하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="scaled_dot_product_attention_tutorial.html#id8">결론</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/knowledge_distillation_tutorial.html">Knowledge Distillation Tutorial</a></li>
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<li class="toctree-l1"><a class="reference internal" href="../distributed/home.html">Distributed and Parallel Training Tutorials</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/dist_overview.html">PyTorch Distributed Overview</a></li>
<li class="toctree-l1"><a class="reference internal" href="../beginner/ddp_series_intro.html">Distributed Data Parallel in PyTorch - Video Tutorials</a></li>
<li class="toctree-l1"><a class="reference internal" href="model_parallel_tutorial.html">단일 머신을 사용한 모델 병렬화 모범 사례</a></li>
<li class="toctree-l1"><a class="reference internal" href="ddp_tutorial.html">분산 데이터 병렬 처리 시작하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="dist_tuto.html">PyTorch로 분산 어플리케이션 개발하기</a></li>
<li class="toctree-l1"><a class="reference internal" href="FSDP_tutorial.html">Getting Started with Fully Sharded Data Parallel(FSDP)</a></li>
<li class="toctree-l1"><a class="reference internal" href="FSDP_adavnced_tutorial.html">Advanced Model Training with Fully Sharded Data Parallel (FSDP)</a></li>
<li class="toctree-l1"><a class="reference internal" href="TP_tutorial.html">Large Scale Transformer model training with Tensor Parallel (TP)</a></li>
<li class="toctree-l1"><a class="reference internal" href="process_group_cpp_extension_tutorial.html">Cpp 확장을 사용한 프로세스 그룹 백엔드 사용자 정의</a></li>
<li class="toctree-l1"><a class="reference internal" href="rpc_tutorial.html">Getting Started with Distributed RPC Framework</a></li>
<li class="toctree-l1"><a class="reference internal" href="rpc_param_server_tutorial.html">Implementing a Parameter Server Using Distributed RPC Framework</a></li>
<li class="toctree-l1"><a class="reference internal" href="dist_pipeline_parallel_tutorial.html">Distributed Pipeline Parallelism Using RPC</a></li>
<li class="toctree-l1"><a class="reference internal" href="rpc_async_execution.html">Implementing Batch RPC Processing Using Asynchronous Executions</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/rpc_ddp_tutorial.html">분산 데이터 병렬(DDP)과 분산 RPC 프레임워크 결합</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/ddp_pipeline.html">분산 데이터 병렬 처리와 병렬 처리 파이프라인을 사용한 트랜스포머 모델 학습</a></li>
<li class="toctree-l1"><a class="reference internal" href="../advanced/generic_join.html">Distributed Training with Uneven Inputs Using the Join Context Manager</a></li>
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<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/executorch/stable/tutorials/export-to-executorch-tutorial.html">Exporting to ExecuTorch Tutorial</a></li>
<li class="toctree-l1"><a class="reference external" href=" https://pytorch.org/executorch/stable/running-a-model-cpp-tutorial.html">Running an ExecuTorch Model in C++ Tutorial</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/executorch/stable/tutorials/sdk-integration-tutorial.html">Using the ExecuTorch SDK to Profile a Model</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/executorch/stable/demo-apps-ios.html">Building an ExecuTorch iOS Demo App</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/executorch/stable/demo-apps-android.html">Building an ExecuTorch Android Demo App</a></li>
<li class="toctree-l1"><a class="reference external" href="https://pytorch.org/executorch/stable/examples-end-to-end-to-lower-model-to-delegate.html">Lowering a Model as a Delegate</a></li>
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<p class="caption" role="heading"><span class="caption-text">추천 시스템</span></p>
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<li class="toctree-l1"><a class="reference internal" href="../beginner/flava_finetuning_tutorial.html">TorchMultimodal 튜토리얼: FLAVA 미세조정</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="이 제목에 대한 퍼머링크">¶</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
《<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.nn.utils.weight_norm.html">weight normalization</a>》.
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 learned.
A similar regularization was proposed for GANs under the name of
《<a class="reference external" href="https://pytorch.org/docs/stable/generated/torch.nn.utils.spectral_norm.html">spectral normalization</a>》. 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="이 제목에 대한 퍼머링크">¶</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">Xᵀ</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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor([[0.8823, 0.9150, 0.3829],
[0.9150, 0.3904, 0.6009],
[0.3829, 0.6009, 0.9408]])
</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><p>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
Transformer…</p></li>
<li><p>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.</p></li>
<li><p>It recomputes the parametrization every time 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.</p></li>
</ol>
</div>
<div class="section" id="introduction-to-parametrizations">
<h2>Introduction to parametrizations<a class="headerlink" href="#introduction-to-parametrizations" title="이 제목에 대한 퍼머링크">¶</a></h2>
<p>Parametrizations can solve all these problems as well as others.</p>
<p>Let’s 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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>ParametrizedLinear(
in_features=3, out_features=3, bias=True
(parametrizations): ModuleDict(
(weight): ParametrizationList(
(0): Symmetric()
)
)
)
</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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor([[ 0.2430, 0.5155, 0.3337],
[ 0.5155, 0.3333, 0.1033],
[ 0.3337, 0.1033, -0.5715]], grad_fn=<AddBackward0>)
</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 class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor([[ 0.0000, 0.0457, -0.0311],
[-0.0457, 0.0000, -0.0889],
[ 0.0311, 0.0889, 0.0000]], grad_fn=<SelectBackward0>)
tensor([[ 0.0000, -0.1314, 0.0626],
[ 0.1314, 0.0000, 0.1280],
[-0.0626, -0.1280, 0.0000]], grad_fn=<SelectBackward0>)
</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="이 제목에 대한 퍼머링크">¶</a></h2>
<p>When a module is parametrized, we find that the module has changed in three ways:</p>
<ol class="arabic simple">
<li><p><code class="docutils literal notranslate"><span class="pre">model.weight</span></code> is now a property</p></li>
<li><p>It has a new <code class="docutils literal notranslate"><span class="pre">module.parametrizations</span></code> attribute</p></li>
<li><p>The unparametrized weight has been moved to <code class="docutils literal notranslate"><span class="pre">module.parametrizations.weight.original</span></code></p></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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Unparametrized:
Linear(in_features=3, out_features=3, bias=True)
Parametrized:
ParametrizedLinear(
in_features=3, out_features=3, bias=True
(parametrizations): ModuleDict(
(weight): ParametrizationList(
(0): Symmetric()
)
)
)
</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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>ModuleDict(
(weight): ParametrizationList(
(0): Symmetric()
)
)
ParametrizationList(
(0): Symmetric()
)
</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. Here’s
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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Symmetric()
</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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>{'bias': Parameter containing:
tensor([-0.0730, -0.2283, 0.3217], requires_grad=True), 'parametrizations.weight.original': Parameter containing:
tensor([[-0.4328, 0.3425, 0.4643],
[ 0.0937, -0.1005, -0.5348],
[-0.2103, 0.1470, 0.2722]], requires_grad=True)}
</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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Parameter containing:
tensor([[-0.4328, 0.3425, 0.4643],
[ 0.0937, -0.1005, -0.5348],
[-0.2103, 0.1470, 0.2722]], requires_grad=True)
</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 class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor(0., grad_fn=<DistBackward0>)
</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="이 제목에 대한 퍼머링크">¶</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="이 제목에 대한 퍼머링크">¶</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 class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Computing the Parametrization
Here, layer.weight is recomputed every time we call it
Computing the Parametrization
Computing the Parametrization
Computing the Parametrization
Here, it is computed just the first time layer.weight is called
Computing the Parametrization
</pre></div>
</div>
</div>
<div class="section" id="concatenating-parametrizations">
<h2>Concatenating parametrizations<a class="headerlink" href="#concatenating-parametrizations" title="이 제목에 대한 퍼머링크">¶</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">linalg</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="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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor(2.1073e-08, grad_fn=<DistBackward0>)
</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">linalg</span><span class="o">.</span><span class="n">eigvalsh</span><span class="p">(</span><span class="n">X</span><span class="p">)</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 class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor(1.8492e-07, grad_fn=<DistBackward0>)
tensor(4.2147e-08, grad_fn=<DistBackward0>)
tensor(True)
</pre></div>
</div>
</div>
<div class="section" id="initializing-parametrizations">
<h2>Initializing parametrizations<a class="headerlink" href="#initializing-parametrizations" title="이 제목에 대한 퍼머링크">¶</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>Let’s 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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor(0., grad_fn=<DistBackward0>)
</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, let’s 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">linalg</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="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"># (A - I)(A + I)^{-1}</span>
<span class="k">return</span> <span class="n">torch</span><span class="o">.</span><span class="n">linalg</span><span class="o">.</span><span class="n">solve</span><span class="p">(</span><span class="n">A</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">A</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">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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>tensor(2.2141, grad_fn=<DistBackward0>)
</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 after the initialization 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 class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Initialization matrix:
tensor([[0.3513, 0.3546, 0.7670],
[0.2533, 0.2636, 0.8081],
[0.0643, 0.5611, 0.9417],
[0.5857, 0.6360, 0.2088]])
Initialized weight:
tensor([[0.3513, 0.3546, 0.7670],
[0.2533, 0.0000, 0.8081],
[0.0643, 0.5611, 0.9417],
[0.5857, 0.6360, 0.0000]], grad_fn=<MulBackward0>)
</pre></div>
</div>
</div>
<div class="section" id="removing-parametrizations">
<h2>Removing parametrizations<a class="headerlink" href="#removing-parametrizations" title="이 제목에 대한 퍼머링크">¶</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>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Before:
Linear(in_features=3, out_features=3, bias=True)
Parameter containing:
tensor([[ 0.0669, -0.3112, 0.3017],
[-0.5464, -0.2233, -0.1125],
[-0.4906, -0.3671, -0.0942]], requires_grad=True)
Parametrized:
ParametrizedLinear(
in_features=3, out_features=3, bias=True
(parametrizations): ModuleDict(
(weight): ParametrizationList(
(0): Skew()
)
)
)
tensor([[ 0.0000, -0.3112, 0.3017],
[ 0.3112, 0.0000, -0.1125],
[-0.3017, 0.1125, 0.0000]], grad_fn=<SubBackward0>)
After. Weight has skew-symmetric values but it is unconstrained:
Linear(in_features=3, out_features=3, bias=True)
Parameter containing:
tensor([[ 0.0000, -0.3112, 0.3017],
[ 0.3112, 0.0000, -0.1125],
[-0.3017, 0.1125, 0.0000]], requires_grad=True)
</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>
</div>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Before:
Linear(in_features=3, out_features=3, bias=True)
Parameter containing:
tensor([[-0.3447, -0.3777, 0.5038],
[ 0.2042, 0.0153, 0.0781],
[-0.4640, -0.1928, 0.5558]], requires_grad=True)
Parametrized:
ParametrizedLinear(
in_features=3, out_features=3, bias=True
(parametrizations): ModuleDict(
(weight): ParametrizationList(
(0): Skew()
)
)
)
tensor([[ 0.0000, -0.3777, 0.5038],
[ 0.3777, 0.0000, 0.0781],
[-0.5038, -0.0781, 0.0000]], grad_fn=<SubBackward0>)
After. Same as Before:
Linear(in_features=3, out_features=3, bias=True)
Parameter containing:
tensor([[ 0.0000, -0.3777, 0.5038],
[ 0.0000, 0.0000, 0.0781],
[ 0.0000, 0.0000, 0.0000]], requires_grad=True)
</pre></div>
</div>
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