I recently updated my working hardware and now use a tablet pc for work (namely a Nexus 10). In consequence, I also updated the software I used to have things more synchronized across devices. For my RSS feeds I now use feedly and the gReader app. However, I was not that happy with the method to store and mark paper I found but found the sharing interfaces between the apps pretty handy. I adopted the workflow that when I see a paper that I want to remember I sent them to my Evernote account where I tag them. Then, from time to time I go over the papers I marked and have a more detailed look. If I think, they deserve to be kept for future reference, they get a small entry here. Here’s the first take with just two papers from the last weeks (there are more in my backlog…):

On the convergence rate improvement of a primal-dual splitting algorithm for solving monotone inclusion problems by Radu Ioan Boţ, Ernö Robert Csetnek, André Heinrich, Christopher Hendrich (Math Prog): As first sight, I found this work pretty inaccessible but the title sounded interesting. I was a bit scared by the formula for the kind of problems they investigated: Solve the following inclusion for ${x}$

$\displaystyle 0 \in z + Ax + \sum_{i=1}^m L_i^*((B_i\square D_i)(L_ix -r_i)) + Cx$

where ${A}$, ${B_i}$ and ${D_i}$ are maximally monotone, ${D_i}$ also ${\nu_i}$ strongly monotone, ${C}$ is ${\eta}$-coercive, ${L_i}$ are linear and bounded and ${\square}$ denotes the parallel sum, i.e. ${A\square B = (A^{-1}+B^{-1})^{-1}}$. Also the proposed algorithm looked a bit like a monster. Then, on later pager, things became a bit more familiar. As an application, they considered the optimization problem

$\displaystyle \min_x f(x) + \sum_{i=1}^m (g_i\square l_i)(L_ix - r_i) + h(x) - \langle x,z\rangle$

with convex ${f}$, ${g_i}$, ${l_i}$ (${l_i}$ ${\nu_i^{-1}}$ strongly convex), ${h}$ convex with ${\eta}$-Lipschitz gradient and ${L_i}$ as above. By noting that the parallel sum is related to the infimal convolution of convex functions, things became clearer. Also, the algorithm looks more familiar now (Algorithm 18 in the paper – I’m too lazy to write it down here). They have an analysis of the algorithms that allow to deduce convergence rates for the iterates (usually ${\mathcal{O}(1/n)}$) but I haven’t checked the details yet.

Sparse Regularization: Convergence Of Iterative Jumping Thresholding Algorithm by Jinshan Zeng, Shaobo Lin, Zongben Xu: At first I was excited but then I realized that they simple tackled

$\displaystyle \min F + \lambda \Phi$

with smooth ${F}$ and non-smooth, non-convex ${\Phi}$ by “iterative thresholding”, i.e.

$\displaystyle x^{n+1} = \mathrm{prox}_{\mu\lambda\Phi}(x^n - \mu \nabla F(x^n)).$

The paper really much resembles what Kristian and I did in the paper Minimization of non-smooth, non-convex functionals by iterative thresholding (at least I couldn’t figure out the improvements…).