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diff --git a/notes/formalisation.tex b/notes/formalisation.tex index bdd6aab..00eb891 100644 --- a/notes/formalisation.tex +++ b/notes/formalisation.tex @@ -154,6 +154,31 @@ $$\vec w\sim B(1 - e^{M \theta_{*,i}})$$ where the operation $f: \vec v \rightarrow 1 - e^{\vec v}$ is done element-wise and $B(\vec p)$ is a vector whose entries are bernoulli variables of parameter the entries of $\vec p$. +\subsection{Maximum likelihood problem} + +The maximum likelihood problem takes the following form: +\begin{displaymath} + \begin{split} + \max_\theta &\sum_{k,t}\left(b_{k,t}\log\left(1-\exp\inprod{A_k^t, \theta}\right) + + (1-b_{k,t})\log\exp\inprod{A_k^t,\theta}\right)\\ + \text{s.t. }& \|\theta\|_1 \leq k + \end{split} +\end{displaymath} +which we can rewrite as: +\begin{displaymath} + \begin{split} + \max_\theta &\sum_{k,t}b_{k,t}\log\left(\exp\left(-\inprod{A_k^t, \theta}\right)-1\right) + + \sum_{k,t}\inprod{A_k^t,\theta}\\ + \text{s.t. }& \|\theta\|_1 \leq k + \end{split} +\end{displaymath} +In this form it is easy to see that this is a convex program: the objective +function is the sum of a linear function and a concave function. + +\paragraph{Questions} to check: is the program also convex in the $p_{j,i}$'s? +What is the theory of sparse recovery for maximum likelihood problems? (TO DO: +read about Bregman divergence, I think it could be related). + \subsection{Results under strong assumptions} What can we hope to have etc. (If M has full column rank or if same sets S occur frequently). |