Online Seminar on Thursday

Online Seminar on Thursday

 

05.05.2022, 16:00 (4:00 pm) CET
Nicolas Verzelen: Some recent results on graph and point clustering
Abstract: In this presentation, we consider two prototypical unsupervised learning problems (i) clustering nodes from a graph sampled from a Stochastic Block Model and (ii) clustering points sampled from a Gaussian Mixture Model. In these two models, the statistician aims at estimating an hidden partition (of nodes or points) from the data. I will first introduce suitable notions of distances between the groups in each model. Then, I will survey recent results on the minimal separation distance between the cluster so that a procedure is able to recover the partition with high probability. This will be mostly done through the prism of the K-means criterion and its convex relaxations. Interestingly, these clustering problems seem to exhibit a computational-statistical trade-off: known polynomial-time procedures are shown to recover the hidden partitions under stronger separation conditions than minimax (but exponential time) one, at least when the number of groups is large. Partial computational lower bounds that support the existence of this gap will be discussed at the end of the talk. 

Download slides: [here]

 

02.06.202, 16:00 (4:00 pm) CET
Arnak Dalalyan: Estimating the matching map between two sets of high-dimensional, noisy and corrupted features
Abstract: In this talk, I will present some recent results on finding the matching map between subsets of two sets of size n consisting of d-dimensional noisy feature vectors. The main result shows that, if the signal-to-noise ratio of the feature vectors is of order at least d¼, then it is possible to recover the true matching map exactly with a high probability. A notable feature of this result is that it does not assume the knowledge of the number of feature-vectors in the first set that have their pairs in the second set. We also show that the rate d¼ can not be improved by other procedure. When the number k of matching pairs is known, this rate is achieved by the minimizer of the sum sum squares of distances between matched pairs of feature-vectors. We show how this estimator can be extended to the setting of unknown k. In addition, we show that the resulting optimization problem can be formulated as a minimum-cost flow problem, and thus solved efficiently, with complexity O(k½ n2).

Finally, we will report the result of numerical experiments illustrating our theoretical findings.

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Topic: Online seminar research unit 5381, June 2022
Time: 2.Juni.2022 04:00 PM Rom

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07.07.2022, 16:00 (4:00 pm) CET
Mathias Drton: Identification and Estimation of Graphical Continuous Lyapunov Models
Abstract: Graphical continuous Lyapunov models offer a new perspective on modeling causally interpretable dependence structure in multivariate data by treating each independent observation as a one-time cross-sectional snapshot of a temporal process. Specifically, the models consider multivariate Ornstein-Uhlenbeck processes in equilibrium. This setup leads to Gaussian models in which the covariance matrix is determined by the continuous Lyapunov equation. In this setting, each graphical model assumes a sparse drift matrix with support determined by a directed graph. The talk will discuss identifiability of such sparse drift matrices as well as their regularized estimation.

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Topic: Online seminar research unit 5381, July 2022
Time: 7.Juli.2022 04:00 PM Rom

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