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Commit b020b7b

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1 parent 60bf2ca commit b020b7b

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-12
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‎DifferentiatingPerceptron.ipynb‎

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"views": {}
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},
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"kernelspec": {
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"display_name": "Python 3",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.6.7"
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"version": "3.12.1"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 4
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‎InformationTheoryOptimization.ipynb‎

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"\\end{align*}\n",
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"where $p_1$ and $p_2$ are probability mass functions and $\\lambda \\in [0,1]$\n",
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"\n",
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"Proof: Let $X$ be a discrete random variable with possible outcomes $\\mathcal{X} := {x_i, i \\in 0,1,\\dots N-1}$ and let $u(x)$ be the probability mass function of a discrete uniform distribution on $X \\in \\mathcal{X}$. Then, the entropy of an arbitrary probability mass function $p(x)$ can be rewritten as\n",
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"Proof: Let $X$ be a discrete random variable with possible outcomes $\\mathcal{X} := {x_i, i \\in 0,1,\\dots N-1}$ and let $u(x)$ be the probability mass function of a discrete uniform distribution on $X \\in \\mathcal{X}$, ie $u(x_i)=\\frac{1}{N}$. Then, the entropy of an arbitrary probability mass function $p(x)$ can be rewritten as\n",
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"\n",
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"\\begin{align*} \\tag{1.2}\n",
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" H(X) &= - \\sum_{i=0}^{N-1} p(x_i)log(p(x_i)) \\\\\n",

‎OptimalTransportWasserteinDistance.ipynb‎

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"metadata": {},
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"source": [
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"### OT and statistical concepts\n",
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"Some of the basics to understand the following statements can be found in the notebook \"InformationTheoryOptimization\" this part is also partly a direct reproduction of Marco Cuturi famous article \"Sinkhorn Distances: Lightspeed Computation of Optimal Transport\"\n",
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"Some of the basics to understand the following statements can be found in the notebook \"InformationTheoryOptimization\", this part is also partly a direct reproduction of Marco Cuturi famous article \"Sinkhorn Distances: Lightspeed Computation of Optimal Transport\"\n",
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"\n",
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"I would like to stop and mention that as we now interpret $P$ as a joint probability matrix, we can define its entropy, the marginal probabiilty entropy, and KL-divergence between two different transportation matrix. These takes the form of\n",
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"\n",
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"\\begin{align*} \\tag{1.5}\n",
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" \\forall r,c \\in \\Sigma_d, \\forall P \\in U(r,c), h(P) \\leq h(r) + h(c)\n",
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"\\end{align*}\n",
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"ie, by using log-sum inequality, proved in the notebook called InformationTheoryOptimization\n",
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"ie, by using log-sum inequality, we proved in the notebook called InformationTheoryOptimization\n",
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"\\begin{align*}\\tag{1.6}\n",
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" \\sum_{i=0}^{N-1} a_i log\\left(\\frac{a_i}{b_i}\\right) &\\geq \\left(\\sum_{i=0}^{N-1} a_i\\right) log\\left(\\frac{\\sum_{i=0}^{N-1}a_i}{\\sum_{i=0}^{N-1}b_i}\\right)\n",
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"\\end{align*}\n",

‎RegularizationByDenoising.ipynb‎

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"\n",
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"## Introduction\n",
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"\n",
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"This notebook intends to show what are the next steps in terms of regularized image reconstruction. We will try to focus especially in a framework that allows the introduction of deep learning in a proper mathematical framework that allows for prior and data fitting mitigation called: regularization by denoising (RED).\n",
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"This notebook intends to show what are the next steps in terms of regularized image reconstruction. We will try to focus especially in a framework that allows the introduction of deep learning in a proper mathematical framework that allows for prior and data fitting mitigation called: regularization by denoising (RED) but not only.\n",
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"\n",
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"The following paper guided us to write this notebook:\n",
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"\n",
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"https://ieeexplore.ieee.org/document/9107406\n",
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"\n",
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"* Recovery Analysis for Plug-and-Play Priors using the Restricted Eigenvalue Condition\n",
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"https://arxiv.org/abs/2106.03668"
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"https://arxiv.org/abs/2106.03668\n",
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"\n",
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"* Deep inverse\n",
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"https://github.com/deepinv/deepinv"
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]
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{
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Python 3",
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"display_name": "Python 3 (ipykernel)",
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"language": "python",
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"name": "python3"
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.10"
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"version": "3.12.1"
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}
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"nbformat": 4,
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"nbformat_minor": 4
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‎bayesian_ab_testing.ipynb‎

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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.8.6"
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"version": "3.12.1"
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}
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},
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"nbformat": 4,

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