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\noindent\underline{\sc Supervisor}: Aur\'elien Decelle \& Mich\`ele Sebag\\
\underline{\sc Labs}: TAU $-$ CNRS $-$ INRIA $-$ Paris-Sud\\
\underline{\sc Mail}: decelle, sebag@lri.fr\\
\underline{\sc URL}: {\tt http://www.lri.fr/\string~sebag/Stages/Stage\_NoZ\_19.pdf}\\
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{\Large Internship 2018 }\\~\\
{\Large\bf \TITRE}\\~\\
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\ENTET~\\
\noindent\underline{\sc Abstract} :\\
Identifying the joint distribution underlying a finite sample set has many applications, ranging from 
generative learning to feature selection. 
Among the most prominent approaches are Restricted Boltzman Machines \cite{RBM}, defined as a bipartite 
graph involving visible features $v_i$ and hidden features $h_j$, and characterizing a distribution $\hat P$ with:
\[ \hat{P}(v_1, \ldots v_D) = \frac{1}{Z} \sum_{(h_1,\ldots, h_d)} exp \left (- \sum_{i,j} W_{i,j} h_i v_j - \sum_i A_i v_i - \sum_j B_j H_j \right ) \]
where $Z$ is the normalization term ensuring that $P$ actually is a probability (summing to 1), involving a double summation on the $v$ and the $h$.
A main source of difficulty for training an RBM is to take into account the $Z$ term in medium to large-dimensional spaces. Contrastive divergence \cite{CD} and its many variants aim to address this difficulty. \\

\section*{Goal}
The goal of the internship is to to train an RBM (i.e. identify parameters $W, A$ and $B$) and side-step the computation of the $Z$ factor, considering two settings: i) the target distribution $P$ is known; ii) an iid sample generated from target $P$, is available. 

\section*{Proposed approach}
The proposed approach takes inspiration from Energy-based learning \cite{LeCun} and adversarial learning \cite{GAN}. 
The conjecture is that learning a function $f: X \mapsto \RR^+$ monotonous w.r.t. $P$ (i.e. $P(x) > P(x') + \eta \Rightarrow f(x) > f(x') + \epsilon$) will suffice 
from the instance , conjecturing that in many cases a 
is considered.




The internship requires excellent programming skills (C++ or Python, based on existing libraries) and excellent mathematical understanding of the formal background (distributions, learning to rank).\\



\noindent{\bf References}\\

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