# Benoît Müller

## About me

Master student in applied mathematics at EPFL. Specialized in numerical analysis: continuous optimization, computational linear algebra, differential equations, and others.

## Usefull links

- My GitHub Profile,
- My Linkedin profile,
- My EPFL profile,

## Some of my projects

*Computational Optimal Transport: A Comparison of Two Algorithms*

Open PDF in browser – GitHub repository

**Abstract:** We present here two formulations of the Monge optimal transport problem. The first one is a discretization leading to the linear sum assignment problem, which we solve with the Hungarian algorithm. The second is the dynamical formulation of Benamou-Brenier and leads to a func- tional saddle problem, which we solve by an augmented Lagrangian method. Both methods are implemented, numerically analyzed, and compared.

*Low-rank Optimal Transport and Application to Color Transfer*

Open PDF in browser – GitHub repository

**Abstract:** In this project, we aim to solve an entropy regularized version of the Kantorovich optimal transport problem, by using a low-rank approximation technique on Sinkhorn algorithm. We apply the method to color transfer for images. We obtain newly colored images, and by using the low-rank approximation technique with a rank of only 4% of the dimension, we obtain visually similar images with a time computation reduced by half.

*Fairness of Decision Algorithm in Machine Learning*

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**Abstract:**
In this report, we present a definition of fairness associated to a predictor function in a decision problem of machine learning. We then propose a post-processing step that can be used for a binary classification, that satisfy the notion of fairness. Lastly, we apply this method on a real data set about credit card default prediction and show the benefits of this method.

*Predicting if two frames are part of the same video*

Open PDF in browser – GitHub repository

**Abstract:**
In this project, we tackle the problem of pre- dicting whether two frames come from the same video or not. The motivation is to assess how similar two images are, which could find applications, for instance, for story visualization or video generation. To do so, we use a dataset of videos, we define a class to dynamically extract a pair of frames from the same video or from two different ones. Then, we use the CLIP image encoder (vision transformer) to extract meaningful image features. We then implement two classification methods, a cosine similarity based approach and a neural network with two hidden layers. They take as input our pair of features, and they output the classification prediction, i.e. whether the two frames belong to the same video or not. We find that the cosine similarity method and the neural network trained for 2 epochs have approximately the same accuracy of 88%.

*Machine learning for the Higgs Boson Challenge*

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**Abstract:**
Using machine learning on a dataset from the CERN, we aimed at predicting if a collision event between two protons resulted in the creation of a Higgs boson. To do so, we first cleaned and modified our data and then tried different models and methods. In this report, we present gradient descent, stochastic gradient descent, Ridge regression, logistic regression with stochastic gradient descent (not regularized) and regularized logistic regression with gradient descent. In order to optimize our different methods, we used cross-validation to choose our hyper-parameters. In the end, we found that our best model is Ridge regression, with an accuracy of 0.776.

*Stability of linear dynamical systems and application to Lotka-Volterra equations and epidemiology*

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**Abstract:**
The goal of this project is to study the stability of some linear and nonlinear autonomous ordinary differential systems including the “predator–prey” based Lotka-Volterra models, and to further find applications to epidemiology such as the Covid-19 evolution.
The first chapter is devoted to the investigation of the stability of linear autonomous ODE systems, where a complete classification is given thanks to spectrum information. We refine these results to give a stability decomposition of the solution space, and show that in general, some stability properties are preserved to related nonlinear systems thanks to linearization arguments.
Then, we turn in the second chapter to a more specific form of ODE systems, say Lotka-Volterra models. Instead of considering classical Lotka-Volterra systems, we look at a more general form. Because at some special critical points the linearization stability theory is not always applicable, we introduced the Lyapunov approach, which can even lead to global stability results.
Finally, the third chapter is an application of the preceding chapters, where we try to model Covid- 19 systems concerning patients and vaccinations, and to predict the evolution of the pandemic via stability investigation.

## Some other Mini-Projects/Homeworks:

Those are projects not formated in publication style, they dont systematically have a GitHub associated, ask me for more informations. See Mini-Projects