Seminars and Colloquia by Series

Friday, November 4, 2016 - 13:05 , Location: Skiles 005 , Aurko Roy , Georgia Tech , Organizer: Marcel Celaya
We study the cost function for hierarchical clusterings introduced by Dasgupta where hierarchies are treated as first-class objects rather than deriving their cost from projections into flat clusters. It was also shown that a top-down algorithm returns a hierarchical clustering of cost at most O (α_n log n) times the cost of the optimal hierarchical clustering, where α_n is the approximation ratio of the Sparsest Cut subroutine used. Thus using the best known approximation algorithm for Sparsest Cut due to Arora-Rao-Vazirani, the top down algorithm returns a hierarchical clustering of cost at most O(log^{3/2} n) times the cost of the optimal solution. We improve this by giving an O(log n)- approximation algorithm for this problem. Our main technical ingredients are a combinatorial characterization of ultrametrics induced by this cost function, deriving an Integer Linear Programming (ILP) formulation for this family of ultrametrics, and showing how to iteratively round an LP relaxation of this formulation by using the idea of sphere growing which has been extensively used in the context of graph partitioning. We also prove that our algorithm returns an O(log n)-approximate hierarchical clustering for a generalization of this cost function also studied in Dasgupta. This joint work with Sebastian Pokutta is to appear in NIPS 2016 (oral presentation).
Friday, October 28, 2016 - 13:05 , Location: Skiles 005 , Kevin Lai , College of Computing, Georgia Tech , Organizer: Marcel Celaya
We consider the problem of estimating the mean and covariance of a distribution from iid samples in R^n in the presence of an η fraction of malicious noise; this is in contrast to much recent work where the noise itself is assumed to be from a distribution of known type. This agnostic learning problem includes many interesting special cases, e.g., learning the parameters of a single Gaussian (or finding the best-fit Gaussian) when η fraction of data is adversarially corrupted, agnostically learning a mixture of Gaussians, agnostic ICA, etc. We present polynomial-time algorithms to estimate the mean and covariance with error guarantees in terms of information-theoretic lower bounds. We also give an agnostic algorithm for estimating the 2-norm of the covariance matrix of a Gaussian. This joint work with Santosh Vempala and Anup Rao appeared in FOCS 2016.
Friday, October 14, 2016 - 13:00 , Location: Skiles 005 , Matthew Fahrbach , College of Computing, Georgia Tech , Organizer: Marcel Celaya
Graded posets are partially ordered sets equipped with a unique rank function that respects the partial order and such that neighboring elements in the Hasse diagram have ranks that differ by one. We frequently find them throughout combinatorics, including the canonical partial order on Young diagrams and plane partitions, where their respective rank functions are the area and volume under the configuration. We ask when it is possible to efficiently sample elements with a fixed rank in a graded poset. We show that for certain classes of posets, a biased Markov chain that connects elements in the Hasse diagram allows us to approximately generate samples from any fixed rank in expected polynomial time. While varying a bias parameter to increase the likelihood of a sample of a desired size is common in statistical physics, one typically needs properties such as log-concavity in the number of elements of each size to generate desired samples with sufficiently high probability. Here we do not even require unimodality in order to guarantee that the algorithm succeeds in generating samples of the desired rank efficiently. This joint work with Prateek Bhakta, Ben Cousins, and Dana Randall will appear at SODA 2017. 
Friday, September 23, 2016 - 13:05 , Location: Skiles 005 , Richard Peng , College of Computing, Georgia Tech , Organizer: Marcel Celaya
Parallel algorithms study ways of speeding up sequential algorithms by splitting work onto multiple processors. Theoretical studies of parallel algorithms often focus on performing a small number of operations, but assume more generous models of communication. Recent progresses led to parallel algorithms for many graph optimization problems that have proven to be difficult to parallelize. In this talk I will survey routines at the core of these results: low diameter decompositions, random sampling, and iterative methods.
Friday, September 16, 2016 - 13:05 , Location: Skiles 005 , Sarah Cannon , Georgia Tech , sarah.cannon@gatech.edu , Organizer: Marcel Celaya
I will present work on a new application of Markov chains to distributed computing. Motivated by programmable matter and the behavior of biological distributed systems such as ant colonies, the geometric amoebot model abstracts these processes as self-organizing particle systems where particles with limited computational power move on the triangular lattice. Previous algorithms developed in this setting have relied heavily on leader election, tree structures that are not robust to failures, and persistent memory. We developed a distributed algorithm for the compression problem, where all particles want to gather together as tightly as possible, that is based on a Markov chain and is simple, robust, and oblivious. Tools from Markov chain analysis enable rigorous proofs about its behavior, and we show compression will occur with high probability. This joint work with Joshua J. Daymude, Dana Randall, and Andrea Richa appeared at PODC 2016. I will also present some more recent extensions of this approach to other problems, which is joint work with Marta Andres Arroyo as well.
Friday, April 22, 2016 - 13:05 , Location: Skiles 005 , David Durfee , Georgia Tech , Organizer: Yan Wang
We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three main results are as follows. First, we give a fully dynamic algorithm for maintaining a $(1 \pm \epsilon)$-spectral sparsifier with amortized update time $poly(\log{n},\epsilon^{-1})$. Second, we give a fully dynamic algorithm for maintaining a $(1 \pm \epsilon)$-cut sparsifier with worst-case update time $poly(\log{n},\epsilon^{-1})$. Third, we apply our dynamic sparsifier algorithm to obtain a fully dynamic algorithm for maintaining a $(1 + \epsilon)$-approximate minimum cut in an unweighted, undirected, bipartite graph with amortized update time $poly(\log{n},\epsilon^{-1})$.Joint work with Ittai Abraham, Ioannis Koutis, Sebastian Krinninger, and Richard Peng
Friday, April 15, 2016 - 13:05 , Location: Skiles 005 , Daniel Blado , Georgia Tech , Organizer: Yan Wang
We examine a variant of the knapsack problem in which item sizes are random according to an arbitrary but known distribution. In each iteration, an item size is realized once the decision maker chooses and attempts to insert an item. With the aim of maximizing the expected profit, the process ends when either all items are successfully inserted or a failed insertion occurs. We investigate the strength of a particular dynamic programming based LP bound by examining its gap with the optimal adaptive policy. Our new relaxation is based on a quadratic value function approximation which introduces the notion of diminishing returns by encoding interactions between remaining items. We compare the bound to previous bounds in literature, including the best known pseudopolynomial bound, and contrast their corresponding policies with two natural greedy policies. Our main conclusion is that the quadratic bound is theoretically more efficient than the pseudopolyomial bound yet empirically comparable to it in both value and running time.
Friday, April 8, 2016 - 13:05 , Location: Skiles 005 , Samantha Petti , Georgia Tech , Organizer: Yan Wang
Motivated by neurally feasible computation, we study Boolean functions of an arbitrary number of input variables that can be realized by recursively applying a small set of functions with a constant number of inputs each. This restricted type of construction is neurally feasible since it uses little global coordination or control. Valiant’s construction of a majority function can be realized in this manner and, as we show, can be generalized to any uniform threshold function. We study the rate of convergence, finding that while linear convergence to the correct function can be achieved for any threshold using a fixed set of primitives, for quadratic convergence, the size of the primitives must grow as the threshold approaches 0 or 1. We also study finite realizations of this process, and show that the constructions realized are accurate outside a small interval near the target threshold, where the size of the construction at each level grows as the inverse square of the interval width. This phenomenon, that errors are higher closer to thresholds (and thresholds closer to the boundary are harder to represent), is also a well-known cognitive finding. Finally, we give a neurally feasible algorithm that uses recursive constructions to learn threshold functions. This is joint work with Christos Papadimitriou and Santosh Vempala.
Friday, April 1, 2016 - 13:05 , Location: Skiles 005 , Hao Huang , Emory University , Organizer: Yan Wang
A Ruzsa-Szemeredi graph is a graph on n vertices whose edge set can be partitioned into induced matchings of size cn. The study of these graphs goes back more than 35 years and has connections with number theory, combinatorics, complexity theory and information theory. In this talk we will discuss the history and some recent developments in this area. In particular, we show that when c>1/4, there can be only constantly many matchings. On the other hand, for c=1/4, the maximum number of induced matchings is logarithmic in n. This is joint work with Jacob Fox and Benny Sudakov.
Friday, March 18, 2016 - 13:05 , Location: Skiles 005 , Chi Ho Yuen , Georgia Tech , Organizer: Yan Wang
This talk aims to give a glimpse into the theory of divisors on graphs in tropical geometry, and its recent application in bijective combinatorics. I will start by introducing basic notions and results of the subject. Then I will mention some of its connections with other fields in math. Finally I will talk about my own work on how tropical geometry leads to an unexpectedly simple class of bijections between spanning trees of a graph and its sandpile group.

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