1,162 research outputs found
The Holographic Shape of Entanglement and Einstein's Equations
We study shape-deformations of the entanglement entropy and the modular
Hamiltonian for an arbitrary subregion and state (with a smooth dual geometry)
in a holographic conformal field theory. More precisely, we study a
double-deformation comprising of a shape deformation together with a state
deformation, where the latter corresponds to a small change in the bulk
geometry. Using a purely gravitational identity from the Hollands-Iyer-Wald
formalism together with the assumption of equality between bulk and boundary
modular flows for the original, undeformed state and subregion, we rewrite a
purely CFT expression for this double deformation of the entropy in terms of
bulk gravitational variables and show that it precisely agrees with the
Ryu-Takayanagi formula including quantum corrections. As a corollary, this
gives a novel, CFT derivation of the JLMS formula for arbitrary subregions in
the vacuum, without using the replica trick. Finally, we use our results to
give an argument that if a general, asymptotically AdS spacetime satisfies the
Ryu-Takayanagi formula for arbitrary subregions, then it must necessarily
satisfy the non-linear Einstein equation.Comment: 37 pages, 3 figure
Analysis of a Collaborative Filter Based on Popularity Amongst Neighbors
In this paper, we analyze a collaborative filter that answers the simple
question: What is popular amongst your friends? While this basic principle
seems to be prevalent in many practical implementations, there does not appear
to be much theoretical analysis of its performance. In this paper, we partly
fill this gap. While recent works on this topic, such as the low-rank matrix
completion literature, consider the probability of error in recovering the
entire rating matrix, we consider probability of an error in an individual
recommendation (bit error rate (BER)). For a mathematical model introduced in
[1],[2], we identify three regimes of operation for our algorithm (named
Popularity Amongst Friends (PAF)) in the limit as the matrix size grows to
infinity. In a regime characterized by large number of samples and small
degrees of freedom (defined precisely for the model in the paper), the
asymptotic BER is zero; in a regime characterized by large number of samples
and large degrees of freedom, the asymptotic BER is bounded away from 0 and 1/2
(and is identified exactly except for a special case); and in a regime
characterized by a small number of samples, the algorithm fails. We also
present numerical results for the MovieLens and Netflix datasets. We discuss
the empirical performance in light of our theoretical results and compare with
an approach based on low-rank matrix completion.Comment: 47 pages. Submitted to IEEE Transactions on Information Theory
(revised in July 2011). A shorter version would be presented at ISIT 201
Higher Spin Fronsdal Equations from the Exact Renormalization Group
We show that truncating the exact renormalization group equations of free
vector models in the single-trace sector to the linearized level
reproduces the Fronsdal equations on for all higher spin fields,
with the correct boundary conditions. More precisely, we establish canonical
equivalence between the linearized RG equations and the familiar local, second
order differential equations on , namely the higher spin Fronsdal
equations. This result is natural because the second-order bulk equations of
motion on simply report the value of the quadratic Casimir of the
corresponding conformal modules in the CFT. We thus see that the bulk
Hamiltonian dynamics given by the boundary exact RG is in a different but
equivalent canonical frame than that which is most natural from the bulk point
of view.Comment: 34 pages, 4 figures; v2: typos fixed, better abstrac
A Channel Coding Perspective of Collaborative Filtering
We consider the problem of collaborative filtering from a channel coding
perspective. We model the underlying rating matrix as a finite alphabet matrix
with block constant structure. The observations are obtained from this
underlying matrix through a discrete memoryless channel with a noisy part
representing noisy user behavior and an erasure part representing missing data.
Moreover, the clusters over which the underlying matrix is constant are {\it
unknown}. We establish a sharp threshold result for this model: if the largest
cluster size is smaller than (where the rating matrix is of size
), then the underlying matrix cannot be recovered with any
estimator, but if the smallest cluster size is larger than , then
we show a polynomial time estimator with diminishing probability of error. In
the case of uniform cluster size, not only the order of the threshold, but also
the constant is identified.Comment: 32 pages, 1 figure, Submitted to IEEE Transactions on Information
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