Today I would like to comment on two arxiv-preprints I stumbled upon:

1. “Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm” – The Elastic Net rediscovered

The paper “Augmented L1 and Nuclear-Norm Models with a Globally Linearly Convergent Algorithm” by Ming-Jun Lai and Wotao Yin is another contribution to a field which is (or was?) probably the fastest growing field in applied mathematics: Algorithms for convex problems with non-smooth ${\ell^1}$-like terms. The “mother problem” here is as follows: Consider a matrix ${A\in{\mathbb R}^{m\times n}}$, ${b\in{\mathbb R}^m}$ try to find a solution of

$\displaystyle \min_{x\in{\mathbb R}^n}\|x\|_1\quad\text{s.t.}\quad Ax=b$

or, for ${\sigma>0}$

$\displaystyle \min_{x\in{\mathbb R}^n}\|x\|_1\quad\text{s.t.}\quad \|Ax-b\|\leq\sigma$

which appeared here on this blog previously. Although this is a convex problem and even has a reformulation as linear program, some instances of this problem are notoriously hard to solve and gained a lot of attention (because their applicability in sparse recovery and compressed sensing). Very roughly speaking, a part of its hardness comes from the fact that the problem is neither smooth nor strictly convex.

The contribution of Lai and Yin is that they analyze a slight perturbation of the problem which makes its solution much easier: They add another term in the objective; for ${\alpha>0}$ they consider

$\displaystyle \min_{x\in{\mathbb R}^n}\|x\|_1 + \frac{1}{2\alpha}\|x\|_2^2\quad\text{s.t.}\quad Ax=b$

or

$\displaystyle \min_{x\in{\mathbb R}^n}\|x\|_1 + \frac{1}{2\alpha}\|x\|_2^2\quad\text{s.t.}\quad \|Ax-b\|\leq\sigma.$

This perturbation does not make the problem smooth but renders it strongly convex (which usually makes the dual more smooth). It turns out that this perturbation makes life with this problem (and related ones) much easier – recovery guarantees still exists and algorithms behave better.

I think it is important to note that the “augmentation” of the ${\ell^1}$ objective with an additional squared ${\ell^2}$-term goes back to Zou and Hastie from the statistics community. There, the motivation was as follows: They observed that the pure ${\ell^1}$ objective tends to “overpromote” sparsity in the sense that if there are two columns in ${A}$ which are almost equally good in explaining some component of ${b}$ then only one of them is used. The “augmented problem”, however, tends to use both of them. They coined the method as “elastic net” (for reasons which I never really got).

I also worked on elastic-net problems for problems in the form

$\displaystyle \min_x \frac{1}{2}\|Ax-b\|^2 + \alpha\|x\|_1 + \frac{\beta}{2}\|x\|_2^2$

in this paper (doi-link). Here it also turns out that the problem gets much easier algorithmically. I found it very convenient to rewrite the elastic-net problem as

$\displaystyle \min_x \frac{1}{2}\|\begin{bmatrix}A\\ \sqrt{\beta} I\end{bmatrix}x-\begin{bmatrix}b\\ 0\end{bmatrix}\|^2 + \alpha\|x\|_1$

which turns the elastic-net problem into just another ${\ell^1}$-penalized problem with a special matrix and right hand side. Quite convenient for analysis and also somehow algorithmically.

2. Towards a Mathematical Theory of Super-Resolution

The second preprint is “Towards a Mathematical Theory of Super-Resolution” by Emmanuel Candes and Carlos Fernandez-Granda.

The idea of super-resolution seems to pretty old and, very roughly speaking, is to extract a higher resolution of a measured quantity (e.g. an image) than the measured data allows. Of course, in this formulation this is impossible. But often one can gain something by additional knowledge of the image. Basically, this also is the idea behind compressed sensing and hence, it does not come as a surprise that the results in compressed sensing are used to try to explain when super-resolution is possible.

The paper by Candes and Fernandez-Granada seems to be pretty close in spirit to Exact Reconstruction using Support Pursuit on which I blogged earlier. They model the sparse signal as a Radon measure, especially as a sum of Diracs. However, different from the support-pursuit-paper they use complex exponentials (in contrast to real polynomials). Their reconstruction method is basically the same as support pursuit: The try to solve

$\displaystyle \min_{x\in\mathcal{M}} \|x\|\quad\text{s.t.}\quad Fx=y, \ \ \ \ \ (1)$

i.e. they minimize over the set of Radon measures ${\mathcal{M}}$ under the constraint that certain measurements ${Fx\in{\mathbb R}^n}$ result in certain given values ${y}$. Moreover, they make a thorough analysis of what is “reconstructable” by their ansatz and obtain a lower bound on the distance of two Diracs (in other words, a lower bound in the Prokhorov distance). I have to admit that I do not share one of their claims from the abstract: “We show that one can super-resolve these point sources with infinite precision—i.e. recover the exact locations and amplitudes—by solving a simple convex program.” My point is that I can not see to what extend the problem (1) is a simple one. Well, it is convex, but it does not seem to be simple.

I want to add that the idea of “continuous sparse modelling” in the space of signed measures is very appealing to me and appeared first in Inverse problems in spaces of measures by Kristian Bredies and Hanna Pikkarainen.