This post is based on two papers (one and two). The task is to quickly solve typical instances of a given problem, and to quickly produce a certificate of that solution. Generally, problems of interest are NP-hard, and so we consider a random distribution on problem instances with the philosophy that real-world instances might mimic this distribution. In my community, it is common to consider NP-hard optimization problems:
minimize subject to . (1)
In some cases, is convex but is not, and so one might relax accordingly:
minimize subject to , (2)
where is some convex set. If the minimizer of (2) happens to be a member of , then it’s also a minimizer of (1) — when this happens, we say the relaxation is tight. For some problems (and distributions on instances), the relaxation is typically tight, which means that (1) can be typically solved by instead solving (2); for example, this phenomenon occurs in phase retrieval, in community detection, and in geometric clustering. Importantly, strong duality ensures that solving the dual of the convex relaxation provides a certificate of optimality.
Continue reading Probably certifiably correct algorithms
A couple of weeks ago, I attended the “Sparse Representations, Numerical Linear Algebra, and Optimization Workshop.” It was my first time at Banff, and I was thoroughly impressed by the weather, the facility, and the workshop organization. A few of the talks were recorded and are available here. Check out this good-looking group of participants:
I wanted to briefly outline some of the problems that were identified throughout the workshop.
Continue reading Sparse Representations, Numerical Linear Algebra, and Optimization Workshop
A couple of months ago, I attended a workshop at Oberwolfach (my first!) called “Mathematical Physics meets Sparse Recovery.” I had a great time. I was asked to give the first talk of the week to get everyone on the same page with respect to sparse recovery. Here are the slides from my talk. What follows is an extended abstract (I added hyperlinks throughout for easy navigation):
Compressed sensing has been an exciting subject of research over the last decade, and the purpose of this talk was to provide a brief overview of the subject. First, we considered certain related topics (namely image compression and denoising) which led up to the rise of compressed sensing. In particular, wavelets provide a useful model for images, as natural images tend to be approximated by linear combinations of particularly few wavelets. This sparsity model has enabled JPEG2000 to provide particularly efficient image compression with negligible distortion. Additionally, this model has been leveraged to remove random noise from natural images.
Considering natural images enjoy such a useful model, one may ask whether the model can be leveraged to decrease the number of measurements necessary to completely determine an image. For example, an MRI scan might require up to 2 hours of exposure time, and then the image might be compressed with JPEG2000 after the fact, meaning most of the measurements can be effectively ignored. So is it possible to simply measure the important parts of the image and not waste time in the image acquisition process? This is the main idea underlying compressed sensing, as introduced by Candes, Romberg and Tao and Donoho.
Continue reading Compressed sensing: Variations on a theme
In a previous post, I described a paper I wrote with Afonso and Yutong about how to design masked illuminations that enable efficient phase retrieval of -dimensional signals for X-ray crystallography and related applications. This masked-illumination methodology was originally proposed in this paper, and our phase retrieval guarantee was based on a recovery method known as polarization. This week, the following paper was posted online, and it gives the first guarantee of this kind for a more popular recovery method called PhaseLift:
Phase retrieval from coded diffraction patterns
Emmanuel J. Candes, Xiaodong Li, Mahdi Soltanolkotabi
In particular, this paper shows that masked illuminations enable PhaseLift recovery, and their result actually holds for a wide assortment of masks. Since this paper is so related to my research, I decided to interview one of the authors (Mahdi Soltanolkotabi). I’ve lightly edited his responses for formatting and hyperlinks:
Continue reading Phase retrieval from coded diffraction patterns
Before the AIM meeting last month, I posted a paper on the arXiv that I wrote with Matt Fickus, Aaron Nelson and Yang Wang. This paper provides some very interesting results for phase retrieval in the setting where you wish to use as few intensity measurements as possible. There are really three different types of results in this paper, and they are partitioned into three different sections accordingly.
— 4M-4 injective intensity measurements —
Recall that Bodmann and Hammen provide a construction of vectors in which yield injective intensity measurements. Such an explicit construction is rather interesting because is conjectured to be the smallest for which this is even possible. From my perspective, any insight into the structure of such ensembles could lead to a deeper understanding of what it takes to be injective in the complex case, and so such constructions are particularly important.
In our paper, we provide a second construction of injective intensity measurements, and our method for proving injectivity was very different from the Bodmann-Hammen approach. Specifically, it is not clear if the Bodmann-Hammen construction has an efficient reconstruction algorithm, whereas for our construction, injectivity is essentially proved by applying a particular reconstruction algorithm that we designed in concert with the ensemble. We used several key ideas in our design, and I’ll briefly discuss them here.
Continue reading Phase retrieval from very few measurements