init

.gitignore

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+/.DS_Store
+/.vscode
+/blog/_build/
+/poster/*pptx
+/6s989_proposal.pdf

blog/_config.yml

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+# Book settings
+# Learn more at https://jupyterbook.org/customize/config.html
+
+title: Demistiyfing E(3)-equivariant NN
+author: Killian Sheriff, Yifan Cao
+logo: square.png
+
+# Force re-execution of notebooks on each build.
+# See https://jupyterbook.org/content/execute.html
+execute:
+  execute_notebooks: force
+
+# Define the name of the latex output file for PDF builds
+latex:
+  latex_documents:
+    targetname: book.tex
+
+# Add a bibtex file so that we can create citations
+bibtex_bibfiles:
+  - references.bib
+
+# Information about where the book exists on the web
+repository:
+  url: https://github.com/killiansheriff/blog_e3nn/  # Online location of your book
+  path_to_book: docs  # Optional path to your book, relative to the repository root
+  branch: master  # Which branch of the repository should be used when creating links (optional)
+
+# Add GitHub buttons to your book
+# See https://jupyterbook.org/customize/config.html#add-a-link-to-your-repository
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+  use_repository_button: true
+
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+  config:
+    html_js_files:
+    - https://cdnjs.cloudflare.com/ajax/libs/require.js/2.3.4/require.min.js

blog/_toc.yml

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+# Table of contents
+# Learn more at https://jupyterbook.org/customize/toc.html
+
+format: jb-book
+root: index
+chapters:
+- file: notebooks/group_representation
+- file: notebooks/irreducible_representations
+- file: notebooks/transformation_under_e3
+- file: notebooks/spherical_harmonics
+- file: notebooks/equivariance_and_polynomials
+- file: notebooks/tensor_product
+- file: notebooks/data_efficiency
+

blog/index.md

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+# Index
+
+## Demistifying E(3)-equivariant neural networks!
+
+Welcome to our blog post on demistifying E(3)-equivariant neural networks! This blog aim to introduces the following concepts, in order to understand the mathematical tools leading to learnable equivariant operations:
+
+```{tableofcontents}
+```
+## Poster 
+
+![poster]('../../../poster/shared/poster.png)
+
+## Disclaimer
+
+This post serves as an explanation of the paper by ``Geiger et al [1]``. It was written as a final project for the ``Fall 2022 Final Project`` section of the ``6.S898: Deep Learning`` course at the ``Massachusetts Institute of Technology``. Needless to say that the intuition presented herein are heavily influenced by the explanations of Geiger et al.
+
+## References 
+
+1. Geiger, M. & Smidt, T. e3nn: Euclidean Neural Networks. Arxiv (2022).
+
+## Citation
+
+If used, please cite: 
+
+```citation
+@software{killian_sheriff_2022_7430281,
+  author       = {Killian Sheriff and
+                  Yifan Cao},
+  title        = {killiansheriff/blog\_e3nn: blog\_e3nn},
+  month        = dec,
+  year         = 2022,
+  publisher    = {Zenodo},
+  version      = {blog\_e3nn},
+  doi          = {10.5281/zenodo.7430281},
+  url          = {https://doi.org/10.5281/zenodo.7430281}
+}
+```
+

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blog/notebooks/.ipynb_checkpoints/Untitled3-checkpoint.ipynb

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blog/notebooks/equivariance_and_polynomials.ipynb

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+{
+ "cells": [
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Equivariance and Polynomials\n",
+    "\n",
+    "In this section we will define what it means for a function to be ``equivariant``, and what tools are available to create ``equivariant polynomials``. This latter is of great importance as almost any functions can be approximated by polynomials, and that the role of our neural network is to estimate an (equivariant) function."
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Equivariance\n",
+    "We say that a function $f: V \\rightarrow V'$ is equivariant to the action of the symmetry element $g$, represented by $D(g)$ on it's input $x$, if it gives rise to a similar action $D'(g)$ of its output, i.e:\n",
+    "\n",
+    "$$\n",
+    "f(D(g)x) = D'(g)f(x).\n",
+    "$$\n",
+    "\n",
+    "This principle is illustrated in figure 1.\n",
+    "\n",
+    "For a concreate example, consider a 3D rotation of $\\theta$ degrees around the x axis:\n",
+    "\n",
+    "$$\n",
+    "R_{x}(\\theta )=\\begin{bmatrix}1&0&0\\\\0&\\cos \\theta &-\\sin \\theta \\\\[3pt]0&\\sin \\theta &\\cos \\theta \\\\[3pt]\\end{bmatrix},\n",
+    "$$\n",
+    "\n",
+    "and the following functions function:\n",
+    "\n",
+    "$$\n",
+    "f: \\begin{bmatrix}\n",
+    "           x \\\\\n",
+    "           y \\\\\n",
+    "           z\n",
+    "         \\end{bmatrix} \n",
+    "\\rightarrow\n",
+    "\\begin{bmatrix}\n",
+    "           x^2+y^2+z^2\n",
+    "        \n",
+    "         \\end{bmatrix} \n",
+    "$$\n",
+    "\n",
+    "Then $f$ is equivariant to rotation as:\n",
+    "\n",
+    "$$\n",
+    "f(R_x(\\theta) \\vec{x}) = f(\\begin{bmatrix}\n",
+    "           x \\\\\n",
+    "           cos(\\theta)y-sin(\\theta)z \\\\\n",
+    "           sin(\\theta)y+cos(\\theta)z\n",
+    "         \\end{bmatrix} ) = \\begin{bmatrix}\n",
+    "           x^2+y^2+z^2\n",
+    "        \n",
+    "         \\end{bmatrix}  = R'_x(\\theta) f(x) = f(x).\n",
+    "$$\n",
+    "\n",
+    "especialy in this case the function is actually invariant to rotations: $R'_x(\\theta) =1  \\space \\forall \\theta \\in \\mathcal{R}$.\n",
+    "\n",
+    "An example of a non equivariant polynomials would be $p$:\n",
+    "\n",
+    "$$\n",
+    "p : \\begin{bmatrix}\n",
+    "           x \\\\\n",
+    "           y \\\\\n",
+    "           z\n",
+    "         \\end{bmatrix} \n",
+    "\\rightarrow\n",
+    "\\begin{bmatrix}\n",
+    "           x^2+2(y-3z) \\\\\n",
+    "           y^2x \\\\\n",
+    "        \n",
+    "         \\end{bmatrix}.\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\n",
+    "| ![square](../equiv.png) |\n",
+    "|:--:|\n",
+    "| <b>Fig.1 - Principle of equivariance.</b>|"
+   ]
+  },
+  {
+   "attachments": {},
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Equivariant polynomials\n",
+    "\n",
+    "In this section we will discuss the tools to create equivariant polynomials.\n",
+    "\n",
+    "We will first show that the compositon of 2 equivariant polynomial is equivariant: \n",
+    "\n",
+    "Consider two equivariant function:\n",
+    "\n",
+    "$$ \n",
+    "f: V_1 \\rightarrow V_3,\n",
+    "$$ \n",
+    "\n",
+    "$$ \n",
+    "g: V_2 \\rightarrow V_3,\n",
+    "$$ \n",
+    "\n",
+    "then: \n",
+    "\n",
+    "### ``Addition`` of equivariant polynomials is equivariant:\n",
+    "\n",
+    "Indeed, $f+g$ is equivariant as:\n",
+    "\n",
+    "$$\n",
+    "D_3(g)(f(x)+g(x)) = D_1(g)f(x)+D_2(g)g(x) = f(D_1(g)x)+g(D_2(g)x).\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "### ``Composition`` of equivariant polynomials is equivariant:\n",
+    "\n",
+    "Indeed, $f \\circ g$ is equivariant as:\n",
+    "\n",
+    "$$\n",
+    "D_3(g)(f(g(x))) =f( D_2(g) g(x)) = f(g(D_1(g)x)).\n",
+    "$$\n",
+    "\n",
+    "### The ``Tensor Product`` of two polynomial is equivariant:\n",
+    "It is denoted $f \\otimes g$. The ``Tensor Product`` section is dedicated to it."
+   ]
+  },
+  {
+   "cell_type": "markdown",
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