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 $U(N)$ vector models in the single-trace sector to the linearized level reproduces the Fronsdal equations on $AdS_{d+1}$ 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 $AdS_{d+1}$, namely the higher spin Fronsdal equations. This result is natural because the second-order bulk equations of motion on $AdS$ 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 $C_1 \log(mn)$ (where the rating matrix is of size $m \times n$), then the underlying matrix cannot be recovered with any estimator, but if the smallest cluster size is larger than $C_2 \log(mn)$, 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 Theor