École polytechnique learning platform

Machine learning is a scientific discipline that is concerned with the design and development of algorithms that allow computers to learn from data. A major focus of machine learning is to automatically learn complex patterns and to make intelligent decisions based on them.

This course focuses on the methodology underlying supervised and unsupervised learning, with a particular emphasis on the mathematical formulation of algorithms, and the way they can be implemented and used in practice. We will therefore describe some necessary tools from optimization theory, and explain how to use them for machine learning. A glimpse about theoretical guarantees, such as upper bounds on the generalization error, are provided during the last lecture.

The methodology will be the main concern of the lectures while some proofs will be done during the PCs. Practice will be done through a challenge.

Initially known for its successes in telecommunications, signal processing is now part of all domains of data processing that require to analyse, extract and transform numerical information. This course is an introduction to the field of signal processing and as such requires basic knowledge of analysis (Fourier Transform), probabilities (random variables, random process) and linear algebra.

The course begins with a presentation of Fourier analysis and analog filtering with some applicative examples such as modulation and Fourier optics in astronomy. Next we will introduce signal sampling and digital signal filtering that has
become the de-facto standard in practical applications. We will study the very important Fast Fourier Transform (FFT) algorithm and discuss some examples of filtering in image processing. Next we will study the random/stochastic aspects of signals and the optimal linear filtering of signal and noise when modeled as as stochastic processes. The modeling of speech with will also be taken as an example for the study of auto-regressive models. Finally the last part of the course will briefly introduce several signal representations commonly used such as the Discrete Cosine Transform (DCT), and wavelet transforms used in JPEG encoding and image reconstruction. The short time Fourier transform will also be introduced to model non-stationary signals. Finally some recent approaches based on machine learning such as dictionary learning and deep learning signal reconstruction will be presented.

The course will be completed by practical sessions in Python/Numpy that will allow the students to implement the methods seen in the course on practical problems such as audio signal generation and filtering.

This course will be given in french or english depending on the public with lecture material in english.
A working knowledge of Python/Numpy is strongly recommended for the practical sessions.

**Evaluation**: practical session reports and final theoretical+practical exam.

Course Evaluation: 1 or 2 group projects + 1 PC report + a final exam
Course Language: French with all material in English

Director of the option:
Grégoire Allaire
Email: stages-map591@cmap.polytechnique.fr

Secretariat of the Applied Mathematics Department
Tel.: 01 69 33 46 07
Fax: 01 69 33 46 46.
Email: leyla.marzuk@polytechnique.edu

The analysis and automatic processing of the information in signals or images is a major sector of data processing wich applications are significative. There is a wide range of scopes: medical imaging, learning and recognition (images or speaking), computer vision, opposite problems, compression, telecommunications...

Internships are offered in France or abroad, in the industrial sector or in centers of research.
Our foreign partners are in Europe (UK, Germany, Austrich, Switzerland), but also in the United States, Australia...
In any case, the aim is to give students the opportunity to take part in research work or an innovative industrial development, while enriching their knowledge. All these internships include a modeling and implementing part with a ratio depending of the internship.

This option is in the of the MAP555 course of Signal processing, but it is not a prerequisite.

An internship catalog is available online but we are at your disposal to help you buiding your dream internship on this topic.

Course language: French

Professors in charge of the option:
Stefano De Marco
Email: stages-map595@cmap.polytechnique.fr

Pierre Henry-Labordère
Email: stages-map595@cmap.polytechnique.fr

Nizar Touzi
Email: stages-map595@cmap.polytechnique.fr

Secretariat of the Applied Mathematics Department
Assistant of the departement: Leyla Marzuk
T. +33 (0)169334607 – leyla.marzuk@polytechnique.edu

Internships offered in Financial mathematics generally take place in centers of research of banks or other organisms, such as investment funds. For some internships, it is mandatory to have taken the Year-3 course of "Stochastic models in finance". Internships abroad take place either in the bank or in academical centers of research.

Some internships require several interviews with different teams.

Example of internships of the earlier years:

En France

• AXA
  Optimization of pricing strategy
  
  

• CMAP
  Principal-Agent to several agents and to a jump dynamic. Applications to the structuing and pricing of electricity contracts
  
  
  CREDIT SUISSE
  Genetical methods for portfolio optimization
  
  

• KEPLER CHEVREUX
  Dynamics of order-book data and algorithm detection
  
  

• SOCIETE GENERALE
  Pricing de produits structurés très long terme
  
  

• UNIVERSITE PARIS 7
  Financial models with arbitraging, application to long-term asset and liability management

Abroad

• BLOOMBERG LP
  Calibration of a Path Dependent Volatility model to the VIX and S&P markets
  
  

• BNP PARIBAS London Branch
  Models for Overnight indexed swap rates and Libor dynamics, and related Market risk
  
  

• BRITISH PETROLEUM
  Application de techniques de Machine Learning à la méthode de Monte-Carlo des moindres carrés
  
  

• DEUTSCHE BANK Londres
  Pricing and risk management of interest rate derivatives
  
  

• GOLDMAN SACHS
  Predictive flow Analytics & inventory optimization
  
  

• IMC Trading
  Identification of the impact of market participants on the European Futures Market
  
  

• JANE STREET
  The volume synchronized probability of informed trading
  
  

• JP MORGAN
  Capital optimization, funding optimization, derivates clearing businesses, credit value adjustement
  
  

• JUMP TRADING INTERNATIONAL
  Latent order book in the context of market impact and liquidity drought
  
  

• MONASH UNIVERSITY
  Option pricing with linear market impact
  
  

• SQUAREPOINT CAPITAL
  Multi-period portfolio optimization to manage tail risk in equities
  
  

• UNIVERSITY OF OXFORD
  Numerics for the robust pricing and hedging problem in discrete time

The objective of this course is to provide a practical introduction to the field of machine learning. We will discuss the different machine learning problems from unsupervised (dimensionality reduction, clustering and density estimation) to supervised (classification, regression, ranking). In this course we will introduce for each method the problem, provide its modeling as an optimization problem and discuss the algorithms that are used to solve the problem. The practical aspect of each method will also be discussed along with python code and existing implementations.

The course will be completed by practical sessions that will allow the students to implement the methods seen in the course on practical problems such as image classification and time series prediction (biomedical and climate data). The objective of the practical session will be not only to learn to use the methods but also to interpret their models and results with respect to the data and the theoretical models.

Course overview:

• Introduction
• Machine learning problems
• Knowing your data
• Preprocessing
• Unsupervised learning
• Dimensionality reduction and
• Dictionary learning and collaborative filtering
• Clustering and generative modeling
• Generative modeling
• Supervised learning
• Linear models and kernel methods for regression and classification
• Nearest neighbors and bayesian decision
• Trees and ensemble methods
• ML in practice
• Find your problem
• Model selection

This course will be given in english with lecture material in english.

Evaluation : practical session reports and oral

Objectifs du cours :

• Introduction to the concept of data stream processing
• Learning the basics on and how to use Data Stream Management Systems (DSMS)
• Understanding the main sampling techniques used for stream processing : sampling, sketching, etc.
• Understanding and using the main data stream processing algorithms

Syllabus :  

This course deals with the algorithms and softwares commonly used to process large data streams. It aims at understanding the main difficulties and specificities of this type of data, knowing what different types of streams exist, what are the theoretical models and practical algorithms to analyze them, and what are the right tools to process these streams.

After an introduction of what data streams are from a conceptual point of view, this class covers the question of data stream processing from two different angles:

1. A Machine Learning and Data Mining approach to cover the theoretical and algorithmic difficulties of learning from data streams: online learning vs incremental and batch learning, and sampling techniques.
2. A more practical approach with an introduction to the various systems and software that are used to handle these data.

In terms of organization, the course will consist of an alternance of lectures and practical sessions. Finally, during the last class the students will have to present a recent research article of their choice on the subject of data stream processing. 

Prérequis : 

• Basics in SQL language
• Basics in Machine Learning (supervised and unsupervised)
• A knowledge of Java programming is recommended but not mandatory

Évaluation :

• The practical sessions will make ⅔ of the mark
• The research paper presentation will make ⅓ of the mark