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- Thursday, November 5, 2020, 3:15pm, Virtual, Probability Seminar
*Critical One-dimensional Multi-particle DLA*

Danny Nam (Princeton, math)- In multi-particle Diffusion Limited Aggregation (DLA) a sea of particles performs independent random walks until they run into the aggregate and are absorbed. In dimension 1, the rate of growth of the aggregate depends on $\lambda$, the density of the particles. Kesten and Sidoravicius proved that when $\lambda < 1$ the aggregate grows like $t^{1/2}$. They furthermore predicted linear growth when $\lambda > 1$ (subsequently confirmed by Sly) and $t^{2/3}$ growth at the critical density $\lambda = 1$. We address the critical case, confirming the $t^{2/3}$ growth and show that aggregate has a scaling limit whose derivative is a self-similar diffusion. Surprisingly, this contradicts conjectures on the speed in the mildly supercritical regime when $\lambda = 1 + \epsilon$. Email Jim Nolen (nolen@math.duke.edu) for the zoom link.

- Thursday, November 12, 2020, 3:15pm, Virtual, Probability Seminar
*Approximating Quasi-Stationary Distributions with Interacting Reinforced Random Walks*

Adam Waterbury (UNC-Chapel Hill)- We propose two numerical schemes for approximating quasi-stationary distributions (QSD) of finite state Markov chains with absorbing states. Both schemes are described in terms of certain interacting chains in which the interaction is given in terms of the total time occupation measure of all particles in the system. The schemes can be viewed as combining the key features of the two basic simulation-based methods for approximating QSD originating from the works of Fleming and Viot (1979) and Aldous, Flannery, and Palacios (1998), respectively. In this talk I will describe the two schemes, discuss their convergence properties, and present some exploratory numerical results comparing them to other QSD approximation methods. Email Jim Nolen (nolen@math.duke.edu) for the zoom link.

- Monday, November 16, 2020, 3:30pm, Zoom link, Probability Seminar
*TBA*

Kavita Ramnan (Brown, applied math) - Thursday, November 19, 2020, 3:15pm, Virtual, Probability Seminar
*K-means clustering with optimization*

Soledad Villars (NYU)- K-means clustering aims to partition a set of n points into k clusters in such a way that each observation belongs to the cluster with the nearest mean, and such that the sum of squared distances from each point to its nearest mean is minimal. In the worst case, this is a hard optimization problem, a priori requiring an exhaustive search over all possible partitions of the data into k clusters in order to find the optimal clustering. At the same time, fast heuristic algorithms for k-means are widely used for data science applications, despite only being guaranteed to converge to local minimizers of the k-means objective. In this talk, we consider a semidefinite programming relaxation of the k-means optimization problem. We discuss two regimes where the SDP provides an algorithm with improved clustering guarantees compared to previous results in the literature: (a) for points drawn from isotropic distributions supported in separated balls, the SDP recovers the globally optimal k-means clustering under mild separation conditions; (b) for points drawn from mixtures of distributions with bounded variance, the SDP solution can be rounded to a clustering which is guaranteed to classify all but a small fraction of the points correctly. An interesting feature about the theoretical tools developed for proving (approximate) optimality of partitions under models (a) and (b) is that they can also be used to a posteriori certify (approximate) optimality of k-means clustering solutions of real data, no model required. Contact Jim Nolen (nolen@math.duke.edu) for zoom link.

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