For some irrelevant reasons, I decided to move my website (and my library) to be hosted on Codeberg. So, I am going to write a little bit about how I did the migration. First, I created an account on Codeberg. There are tons of useful tutorials and documentations on the web that explain how to migrate repositories from GitHub to Codeberg. It’s important to note that on GitHub, you usually have to name your repository in a specific way to make GitHub Pages work. In Codeberg, you technically can do the same (with the pages repository name), but I found it easier to configure everything with a general repository name. So, I recommend you to name your website’s repository whatever (mine is named website). ...
Summary This post serves as a broad overview of many optimization methods and algorithms. For specifics, please refer to the post of each algorithm. It is advised to read the post about Functional Analysis to better understand the notation used here. Setup We are going to consider a function $$ f = g + h. $$ Depending on the different hypothesis of $f$, $g$ and $h$ we may be able to use different optimization algorithms. ...
The study of optimization of general functions often rely on many Functional Analysis tools, particularly Hilbert spaces. One can think about Hilbert spaces as a generalization of Euclidean spaces. This allows us to define most algorithms in a generalized fashion, without worrying about the underlying structure of the problem’s domain. Of course, most applications consider an Euclidean space $\mathbb{R}^d$, but it is just a special case of a Hilbert space. ...
Discrete Measures Definition: Shannon-Boltzmann entropy Let $P \in U(a, b)$ be a coupling matrix for discrete measures with vectors $a$ and $b$. Then, the Shannon-Boltzmann entropy is $$ H(P) = - \sum_{i, j} P_{i, j} \log{(P_{i, j})}, $$ where $0 = \log{0}$. Note that $$ \nabla^2 H(P) = - \diag(P_{i, j}^{-1}). $$ So, $H$ is strictly concave. Let us add a regularization term to the discrete Kantorovich problem. $\varepsilon$ will be our regularization weight and it works as a kind of “temperature”. ...
Summary Up to this point we have studied Monge and Kantorovich problems. Duality of Kantorovich and the Wasserstein metric. Slicing OT as a way to lower bound $\W_p$. Now, we are interested in solving the Kantorovich problem in an alternative way. The goal here is to find a condition that is sufficient to retrieve a map from $\mathcal{X}$ to $\mathcal{Y}$. Then, the optimal transport will be given by the objects that minimize a certain criterion over this condition. ...