Merge pull request #13 from NicolaBernini/gdl_20190507_1332_1

Gdl 20190507 1332 1

Geometric_Deep_Learning/CNN/Gauge_Equivariant_Convolutional_Networks_and_the_Icosahedral_CNN/readme.md

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+# Paper Summary - Gauge Equivariant Convolutional Networks and the Icosahedral CNN 
+
+Original Paper: [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://arxiv.org/abs/1902.04615)
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+Note 
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+- As there is some math pleas activate a Browser Side Tex Rendering Plugin like [Tex all the things](https://chrome.google.com/webstore/detail/tex-all-the-things/cbimabofgmfdkicghcadidpemeenbffn) 
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+# Intro 
+
+Main Elements 
+
+- **Convolutional Filter** is a Filter mapping from a Domain to a Co-domain 
+  - $f : \mathcal{D} \rightarrow \mathcal{C}$
+  - In CNN context, it maps a feature map into another feature map 
+
+- **Equivariance** is a property regarding both a filter and its domain of application: 
+  - when a transformation is applied to the filter domain and it fully reflects on the filter output then the filter is called **equivariant** with respect to that transformation 
+    - $f(g(x)) = g(f(x))$ or equivalently 
+    - $g^{-1}(f(g(x))) = f(x)$ (we will use this specific definition later )
+  - if the transformation gets fully absorbed by the filter, so that there is no trace of its application in the filter output, then the filter is called **invariant** with respect to that transformation 
+    - $f(g(x)) = f(x)$
+
+- **Symmetries** is a set of transformations related to a specific space only (so it does not depend on any filter) which map the space in itself without altering its structure 
+
+- The underlying idea between the Classical CNN Layer acting on an Image or generic Planar Feature Map, is to be able to **slide** it over the plane or equivalently to **slide** the plane under it hence more formally to use one of the **plane symmetries** to transform the plane in itself before processing it again and again with Convolutional Filter 
+  - Generalizing this procedure on a generic space, we would like 
+  1. something the works the same way hence transforming the space in itself for continuous processing 
+  2. the Convolutional Filter result should not depend on the specific position (this is typically called **parameters sharing** and it makes CNN Processing **position independent**)
+
+
+- In terms of space generalization, let's consider a **Manifold** $M$ which is a space which is locally homeomorphic to an Euclidean Space 
+  - Hence given a point $p \in M$ we have a local tangent space $T_{p}M$ where we can define an **invertible linear mapping** between this Tangent Space and the Eucliden 
+
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+- Unfortunately we can not simply apply the manifold equivalent for the plane shift (transformation member of plane symmetries set) because a generic manifold does not provide any guarantee to possess any global symmetry 
+  - This means with a manifold we can not approach the problem at the global level hence we need to work at a local level, hence considering a point $p$ its neighborhood is a Tangent Space $T_{p}M$ which has the beautiful property of being homeomorphic to the Euclidean Space 
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+- Let's then introduce the **Gauge** which is a **Position Dependent Invertible Local Linear Mapping** between the Manifold Tangent Space $T_{p}M$ and the Euclidean Space $\mathbb{R}^{d}$  
+  - $w_{p} : \mathbb{R}^{d} \rightarrow T_{p}M$ and $\exists w_{p}^{-1}$
+
+- As a result of this, a Gauge also defines a Local Reference Frame in $T_{p}M$ (mapping the Euclidean Reference Frame) 
+  - In fact if $\{e_{i}\}$ is basis for $\mathbb{R}^{d}$ then $\{w_{p}(e_{i})\}$ is the corresponding Gauge basis for $T_{p}M$
+
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+- A **Gauge Transformation** is a **Position Dependent Invertible Change of the Local Frame** 
+  - Mathematically if $w_{p}$ is a Gauge, then $g_{p} \in GL(d, \mathbb{R})$ is a Gauge Transformation so that $w_{p} \rightarrow w_{p}g_{p}$
+  - With $GL(d, \mathbb{R})$ group of invertible $d \times d$ matrix 
+
+- In order for the **filter sliding** to work properly we need to the Filter to be equivariant with respect to gauge transformation 
+  - Let's call $v_{p}$ the filter output feature map described with respect to the Gauge $w_{p}$
+  - In order for the filter to be equivariant to the Gauge Transformation $g_{p}$ according to the above definition we need 
+
+$$ (w_{p}g_{p})(g_{p}^{-1}v_{p}) = v_{p} $$
+
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+- The underlying idea is hence to build a filter with this property, depending on the specific manifold it has to process 
+
+Work in progress 
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Geometric_Deep_Learning/CNN/readme.md

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+# Overview 
+
+Geometric Deep Learning performed with CNN 
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+

Geometric_Deep_Learning/readme.md

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+# Overview 
+
+Added a Section about [Geometric Deep Learning](http://geometricdeeplearning.com/) and the related papers summaries 
+
+