I am honored to have received the NSF CAREER award for a proposal on optimization methods and theory for the joint formulation of dimension reduction and clustering. You can read about the award here in the UM press release and also here on the NSF website. Dimension reduction and clustering are arguably the two most critical problems in unsupervised machine learning; they are used universally for data exploration and understanding. Often dimension reduction is used before clustering (or vice versa) to lend tractability to the modeling algorithm. It’s more typical in real data to see clusters each with their own low-dimensional structure, and so a joint formulation is of great interest. I look forward to working toward this end in the next stage of my career.

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Army Young Investigator

I am very excited that my project “Mathematics for Learning Nonlinear Generalizations of Subspace Models in High Dimensions” has won the Army Young Investigator award! Subspace models are widely used due to simplicity and ease of analysis. However, while these linear models are very powerful in many high-dimensional data contexts, they also often miss out on important nonlinearities in real data. This project aims to extend recent advances in signal processing to the single-index model and the nonlinear variety model. Read the department’s announcement here.

Postdoc Opportunity at the University of Michigan

to begin in spring 2019.


Please email Laura Balzano <girasole@umich.edu> with the subject “Joining the Balzano lab — postdoc 2019” if you are interested.

We are seeking a postdoc who is interested in applying machine learning techniques to real-time dynamic data analysis. While machine learning has advanced significantly over the last decade, its application to dynamic time-varying data is still in its infancy. This project will focus on three ML areas: online learning, stochastic gradient methods, and streaming PCA. We will work on theory to understand how the standard approaches behave when the data are time-varying, develop appropriate models for time-varying data, and develop novel approaches along with convergence theory. Our main applications focus will be power systems engineering and computer vision. In power systems, we will develop methodologies to infer the real-time behavior of aggregations of distributed energy resources from hierarchical, heterogeneous, and incomplete measurements of power system quantities. In computer vision, we will develop real-time algorithms for object tracking and activity recognition in video.

Optimally Weighted PCA for High-dimensional Heteroscedastic Data

Today I had the opportunity to speak about very recent results by my student David Hong (joint work also with Jeff Fessler) in analyzing asymptotic recovery guarantees for weighted PCA for high-dimensional heteroscedastic data. In the paper we recently posted online, we have asymptotic analysis (as both the number of samples and dimension of the problem grow to infinity, but converge to a fixed constant) of the recovery for weighted PCA components, amplitudes, and scores. Those recovery expressions allow us to find weights that give optimal recovery, and the weights turn out to be a very simple expression involving only the noise variance and the PCA amplitudes. To learn more, watch my talk here, and let us know if you have any questions!

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AFOSR Young Investigator

I have great news that my AFOSR Young Investigator proposal was accepted for funding. My proposal was focused on time-varying low-rank factorization models, and various ways of solving a variety of related non-convex problem formulations.  Read more about it here.   I look forward to the contributions we will be able to make with the support of AFOSR.


Ensemble K-Subspaces

Yesterday I gave a talk on Subspace Clustering using Ensemble methods at the Simons Institute. See the video here!

This is work with John Lipor, David Hong, and Yan Shuo Tan. Our related paper has been just updated on the arxiv. Our key observation was that, while K-Subspaces (KSS) works poorly overall and depends heavily on initialization, it still seems to give partially good clustering information. We therefore use it as a “weak clusterer” and combine ensembles of KSS (EKSS) by averaging the co-association/affinity matrices. This works extremely well, both in simulation and on real data, and also in theory. We were able to show that EKSS gives correct clustering in a variety of common cases: e.g. for subspaces with bounded affinity, and with noisy data and missing data. Our theory generalizes theory of the Thresholded Subspace Clustering algorithm to show that any algorithm that produces an affinity matrix that is an approximation to a monotonic function of absolute inner products will give correct clustering. This general theory should be broadly applicable to many geometric approaches to subspace clsutering.

Improving K-Subspaces via Coherence Pursuit

John Lipor, Andrew Gitlin, Bioashuai Tao, and I have a new paper, “Improving K-Subspaces via Coherence Pursuit,” to be published in the Journal of Special Topics in Signal Processing issue “Robust Subspace Learning and Tracking: Theory, Algorithms, and Applications.” In it we present a new subspace clustering algorithm, Coherence Pursuit – K-Subspaces (CoP-KSS). Here is the code for CoP-KSS and for our figures. Our paper considers specifically the PCA step in K-Subspaces, where a best-fit subspace estimate is determined from a (possibly incorrect) clustering. When a given cluster has points from multiple low-rank subspaces, PCA is not a robust approach. We replace that step with Coherence Pursuit, a new algorithm for Robust PCA. We prove that Coherence Pursuit indeed can recover the “majority” subspace when data from other low-rank subspaces is contaminating the cluster. In this paper we also prove — to the best of our knowledge, for the first time — that the K-Subspaces problem is NP-hard, and indeed even NP-hard to approximate within any finite factor for large enough subspace rank.

In turn, roulette, which is now one of the most popular casino table games all over the world, appeared in the 18th century in France, in the gambling houses of Paris. Translated https://pin-ups-casino.com from French, the word “roulette” means “small wheel”, it became incredibly popular throughout Europe and after some time came to the United States, where it became one of the favorite entertainment of Americans.

The origins of poker are also not completely clear, because, like many other games of chance, poker, most likely, also developed over several centuries, taking shape from different card games. Some argue that poker-like gambling was invented in the 17th century in Persia, while others say that the famous game of today was inspired by the French game Poque. The popularity of this game grew rather slowly until the 70s. of the last century, no world poker tournaments were held in Las Vegas. But the greatest recognition of this game was provided by the opportunity to gamble on the Internet when online poker appeared.


Streaming PCA Review Article

The Proceedings of IEEE posted our review article today on Streaming PCA and Subspace Tracking with Missing Data. It was a great experience to work with Yuejie Chi and Yue Lu on this survey. You can also find a less pretty version on the arxiv.

The oldest casino gambling that is still played today is Baccarat, the earliest version of which was first mentioned at the beginning of the 15th century, when it migrated from Italy to France. The exact origin of baccarat has not yet been clarified. The most popular version says that baccarat was https://casinopinups.com invented in Italy and was first played in the Middle Ages using tarot cards. Later, in 1490, this game appeared in France.

New paper in Journal of Multivariate Analysis

Congratulations to my student David Hong (and his co-advisor Jeff Fessler) for our published article in the Journal of Multivariate Analysis, titled “Asymptotic performance of PCA for high-dimensional heteroscedastic data.” Heteroscedastic data, where different data points are of differing quality (precisely, have different noise variance), are common in so many interesting big data problems. Sensor network data, medical imaging using historical data, and astronomical imaging are just a few examples. PCA is known to be the maximum likelihood estimate for data with additive Gaussian noise of a single variance across all the data points. This work investigates the performance of PCA when that homoscedastic noise assumption is violated. We give precise predictions for the recovery of subspaces and singular values in a spiked/planted model, and show that vanilla PCA (perhaps unsurprisingly) has suboptimal subspace recovery when the data are heteroscedastic. 

There is also no clear answer to the question of the origin of one of the most popular modern casino table games – blackjack – most often France is considered to be its country of origin. It is believed that the 17th century French game vingt-et-un is the direct ancestor of blackjack, and it appeared in the United States along with the first colonists from France. The name blackjack is of American origin, and it was associated with special promotions that were held at casinos in Nevada in the 30s. XX century. In order to attract new customers, higher https://pinup-wiki.com/ odds were paid if the player won with a combination that included a jack of black suit (clubs or spades) and an ace of spades. In English, the jack of black suit (clubs or spades) is called “black Jack of Clubs” or “black Jack of Spades”.

Group OWL Regularization for Deep Nets

My student Dejiao Zhang’s code for our paper Learning to Share: Simultaneous Parameter Tying and Sparsification in Deep Nets can be found at this link. We demonstrated that regularizing the weights in a deep network using the Group OWL norm allows for simultaneous enforcement of sparsity (meaning unimportant weights are eliminated) and parameter tying (meaning co-adapted or highly correlated weights are tied together). This is an exciting technique for learning compressed deep net architectures from data.

Interestingly, the first casinos appeared in the 17th century in Venice, Italy, and at first they were not associated with gambling. At the beginning of their existence, casinos were used as public halls for music and dancing, but there they also gambled. The first famous European gambling house, which, incidentally https://casino-pinups.com/, was not called a “casino”, although it did fit the modern definition of a casino, was Ridotto, which opened in Venice in 1638 to ensure control over gambling during the carnival. created throughout continental Europe in the 19th century, while more informal fashion was in vogue in the United States