Jason Corso and I have been awarded a DARPA D3M grant. Our project is called SPIDER: Subspace Primitives that are Interpretable and DivERse. We will be contributing machine learning software primitives for a system that helps domain experts perform a wide variety of automated data analysis on their datasets. The project has 24 teams and is already off to a great start — We look forward to final system developed by this team over the next few years!

Last Friday I gave the MIDAS weekly seminar. You can find the description here, along with the link directly to the recording. I talked about two recent problems I have been working on: First I talked about my work with Ravi Ganti and Rebecca Willett on learning a low-rank matrix that is observed through a monotonic function from partial measurements. This is common in calibration and quantization problems. Follow up work with Nikhil Rao and Rob Nowak in addition generalized this to learning structured single index models. Second, I talked about the work of my student David Hong, co-advised by Jeff Fessler, on the asymptotic performance of PCA with heteroscedastic data. This is common in problems like sensor networks or medical imaging, where different measurements of the same phenomenon are taken with different quality sensing (eg high or low radiation). David has recently posted his paper on arxiv showing predictions of the asymptotic performance; exploiting the structure of these expressions we also showed that asymptotic recovery for a fixed average noise variance is maximized when the noise variances are equal (i.e., when the noise is in fact homoscedastic). Average noise variance is often a practically convenient measure for the overall quality of data, but our results show that it gives an overly optimistic estimate of the performance of PCA for heteroscedastic data.

This week I head to the bay area for two presentations: My collaborator Ravi Ganti will be presenting our work on estimating high dimensional structured single index models.  The week following I’ll be at the Simons workshop on Interactive Learning presenting my work with student John Lipor on active labeling from union of subspace data. I’m looking forward to sharing our work and hearing about all the other interesting work going on in these areas! (Edit: See here for my Simons talk on active labeling for union of subspace data.)

My students and I have posted three very exciting papers in the last few months!

“Towards a Theoretical Analysis of PCA for Heteroscedastic Data” with David Hong and Jeff Fessler studies the behavior of PCA with data points that have different noise variances. We published that work at the Allerton conference.

“Convergence of a Grassmannian Gradient Descent Algorithm for Subspace Estimation From Undersampled Data” with Dejiao Zhang proves expected convergence rates for incremental gradient subspace estimation constrained to the Grassmannian. Despite this being a non-convex problem, our results show locally linear convergence rates when the observed vectors have either missing data or compressively sampled data.

“Leveraging Union of Subspace Structure to Improve Constrained Clustering” with John Lipor demonstrates how the assumption of subspace structure can dramatically help active clustering algorithms. By querying users intelligently for pairwise clustering constraints, we drive clustering error on several benchmark datasets down near zero.

This spring I’ve had a couple opportunities to share work with Ravi Sastry Ganti and Rebecca Willett on Matrix completion under monotonic single index model observations that we had at NIPS last December. In a week I’ll be at AI stats with my student Dejiao Zhang presenting our work on Global convergence of a Grassmannian gradient descent algorithm for subspace learning — we now have a proof that the GROUSE subspace learning algorithm converges globally from any random initialization to the global minimizer, or to a ball around the generating subspace in the case of noisy data. And last but never least, I had a great set of grad students in my Estimation, Detection, and Filtering class this year; they did some great projects on data science and statistical inference and did very well in the class overall.

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I just returned from a great CAMSAP meeting, where I learned about a probabilistic theory of deep learning, new work in tensor completion, and an importance sampling strategy for non-convex block coordinate descent. My student John and I also presented our new work on active labeling in the subspace clustering problem, where we showed that a small amount of label information helps tremendously in clustering noisy data when each cluster lies near a low-dimensional subspace. Thanks to the organizers for putting together such a great meeting!