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Series: ACO Student Seminar

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).

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.