21 research outputs found
On some low distortion metric Ramsey problems
In this note, we consider the metric Ramsey problem for the normed spaces
l_p. Namely, given some 1=1, and an integer n, we ask
for the largest m such that every n-point metric space contains an m-point
subspace which embeds into l_p with distortion at most alpha. In
[arXiv:math.MG/0406353] it is shown that in the case of l_2, the dependence of
on alpha undergoes a phase transition at alpha=2. Here we consider this
problem for other l_p, and specifically the occurrence of a phase transition
for p other than 2. It is shown that a phase transition does occur at alpha=2
for every p in the interval [1,2]. For p>2 we are unable to determine the
answer, but estimates are provided for the possible location of such a phase
transition. We also study the analogous problem for isometric embedding and
show that for every 1<p<infinity there are arbitrarily large metric spaces, no
four points of which embed isometrically in l_p.Comment: 14 pages, to be published in Discrete and Computational Geometr
Investigating Holography: Traversable Wormholes and Closed Universes
In this thesis we study three recent and overlapping developments in
the subject of holography: traversable wormholes, quantum extremal islands, and
holographic models of closed universes.
We construct a traversable wormhole from a charged AdS black hole by adding
a coupling between the two boundary theories. We investigate how the effect of this
deformation behaves in the extremal limit of the black hole and show that under
certain conditions the wormhole can be made traversable even in the extremal limit.
Next, we use braneworlds in three-dimensional multiboundary wormhole geometries as a model to study the appearance of entanglement islands when a closed
universe with gravity is entangled with two non-gravitating quantum systems. We
show that the entropy of the mixed state in the closed universe is bounded by half
of the coarse-grained entropy of the effective theory on the braneworld.
For large values of the tension T, the worldvolume of a constant-tension brane
inside a Schwarzschild-AdS_{d+1} black hole is a closed FRW cosmology. However, for
d > 2, having a smooth Euclidean solution where the brane does not self-intersect
limits the brane tension to T < T_β, preventing us from realising a separation of
scales between the brane and bulk curvature scales. We show that adding interface
branes to this model does not relax the condition on the brane tension
Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game
In the past 20 years, increasing complexity in real world data has lead to
the study of higher-order data models based on partitioning hypergraphs.
However, hypergraph partitioning admits multiple formulations as hyperedges can
be cut in multiple ways. Building upon a class of hypergraph partitioning
problems introduced by Li & Milenkovic, we study the problem of minimizing
ratio-cut objectives over hypergraphs given by a new class of cut functions,
monotone submodular cut functions (mscf's), which captures hypergraph expansion
and conductance as special cases.
We first define the ratio-cut improvement problem, a family of local
relaxations of the minimum ratio-cut problem. This problem is a natural
extension of the Andersen & Lang cut improvement problem to the hypergraph
setting. We demonstrate the existence of efficient algorithms for approximately
solving this problem. These algorithms run in almost-linear time for the case
of hypergraph expansion, and when the hypergraph rank is at most .
Next, we provide an efficient -approximation algorithm for finding
the minimum ratio-cut of . We generalize the cut-matching game framework of
Khandekar et. al. to allow for the cut player to play unbalanced cuts, and
matching player to route approximate single-commodity flows. Using this
framework, we bootstrap our algorithms for the ratio-cut improvement problem to
obtain approximation algorithms for minimum ratio-cut problem for all mscf's.
This also yields the first almost-linear time -approximation
algorithms for hypergraph expansion, and constant hypergraph rank.
Finally, we extend a result of Louis & Makarychev to a broader set of
objective functions by giving a polynomial time -approximation algorithm for the minimum ratio-cut problem based on
rounding -metric embeddings.Comment: Comments and feedback welcom
Membranes, holography, and quantum information
In this thesis, I study Interface Conformal Field Theories (ICFT) and their
holographic dual, which is composed of two asymptotically Anti-de-Sitter (AdS)
spaces glued through a thin gravitating membrane. I restrict the study to
simple minimal models, which allow for analytic control while providing
universally applicable results. The analysis is set in 2D ICFT/3D gravity, but
I expect much of the results to be generalizable to higher dimensions. I first
consider this system at equilibrium and at finite temperature. By solving the
equations of motion in the bulk, I find the allowable solution landscape.
Classifying the rich set of solutions among 3 thermodynamical phases, I draw
the phase diagram outlining the nature of the various phase transitions. I then
examine a simple out-of-equilibrium situation arising from connecting at an
interface two spatially infinite CFTs at different temperatures. Then a
"Non-Equilibrium Steady State" (NESS) describes the growing region where the
interaction has settled into a stationary phase. I determine the holographic
dual of this region, composed of two spinning planar black holes glued at the
membrane. I find an expression for the deformed out-of-equilibrium event
horizon. This geometry suggests that the field theory interface acts as a
perfect scrambler, a property that until now seemed unique to black hole
horizons. Finally, I study the entanglement structure of the aforementioned
geometries by means of the Ryu-Takayanagi prescription. After reviewing a
construction in the vacuum state, I present partial results for more general
geometries at finite temperature. For this purpose, I need to introduce
numerical methods. I outline the main difficulties in their application and
conclude by mentioning the Quantum Null Energy Condition (QNEC), an inequality
linking entanglement entropy and energy, that can be used to test the
consistency of the models.Comment: 159 pages, 65 figure
Compression bounds for Lipschitz maps from the Heisenberg group to
We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg
group with its Carnot-Carath\'eodory metric and apply it to give a lower bound
on the integrality gap of the Goemans-Linial semidefinite relaxation of the
Sparsest Cut problem
Input Sparsity and Hardness for Robust Subspace Approximation
In the subspace approximation problem, we seek a k-dimensional subspace F of
R^d that minimizes the sum of p-th powers of Euclidean distances to a given set
of n points a_1, ..., a_n in R^d, for p >= 1. More generally than minimizing
sum_i dist(a_i,F)^p,we may wish to minimize sum_i M(dist(a_i,F)) for some loss
function M(), for example, M-Estimators, which include the Huber and Tukey loss
functions. Such subspaces provide alternatives to the singular value
decomposition (SVD), which is the p=2 case, finding such an F that minimizes
the sum of squares of distances. For p in [1,2), and for typical M-Estimators,
the minimizing gives a solution that is more robust to outliers than that
provided by the SVD. We give several algorithmic and hardness results for these
robust subspace approximation problems.
We think of the n points as forming an n x d matrix A, and letting nnz(A)
denote the number of non-zero entries of A. Our results hold for p in [1,2). We
use poly(n) to denote n^{O(1)} as n -> infty. We obtain: (1) For minimizing
sum_i dist(a_i,F)^p, we give an algorithm running in O(nnz(A) +
(n+d)poly(k/eps) + exp(poly(k/eps))), (2) we show that the problem of
minimizing sum_i dist(a_i, F)^p is NP-hard, even to output a
(1+1/poly(d))-approximation, answering a question of Kannan and Vempala, and
complementing prior results which held for p >2, (3) For loss functions for a
wide class of M-Estimators, we give a problem-size reduction: for a parameter
K=(log n)^{O(log k)}, our reduction takes O(nnz(A) log n + (n+d) poly(K/eps))
time to reduce the problem to a constrained version involving matrices whose
dimensions are poly(K eps^{-1} log n). We also give bicriteria solutions, (4)
Our techniques lead to the first O(nnz(A) + poly(d/eps)) time algorithms for
(1+eps)-approximate regression for a wide class of convex M-Estimators.Comment: paper appeared in FOCS, 201
Approximation Algorithms for Semi-random Graph Partitioning Problems
In this paper, we propose and study a new semi-random model for graph
partitioning problems. We believe that it captures many properties of
real--world instances. The model is more flexible than the semi-random model of
Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and
Sipser.
We develop a general framework for solving semi-random instances and apply it
to several problems of interest. We present constant factor bi-criteria
approximation algorithms for semi-random instances of the Balanced Cut,
Multicut, Min Uncut, Sparsest Cut and Small Set Expansion problems. We also
show how to almost recover the optimal solution if the instance satisfies an
additional expanding condition. Our algorithms work in a wider range of
parameters than most algorithms for previously studied random and semi-random
models.
Additionally, we study a new planted algebraic expander model and develop
constant factor bi-criteria approximation algorithms for graph partitioning
problems in this model.Comment: To appear at the 44th ACM Symposium on Theory of Computing (STOC
2012