Eyal Lubetzky

Broadcasting with Side Information

A sender holds a word x consisting of n blocks xi, each of t bits, and wishes to broadcast a codeword to m receivers, R1,...,Rm. Each receiver Ri is interested in one block, and has prior side information consisting of some subset of the other blocks. Let betat be the minimum number of bits that has to be transmitted when each block is of length t, and let beta be the limit beta=limtrarrinfinbetat/t. Informally, beta is the average communication cost per bit in each block (for long blocks). Finding the coding rate beta, for such an informed broadcast setting, generalizes several coding theoretic parameters related to Informed Source Coding on Demand, Index Coding and Network Coding. In this work we show that usage of large data blocks may strictly improve upon the trivial encoding which treats each bit in the block independently. To this end, we provide general bounds on betat, and prove that for any constant C there is an explicit broadcast setting in which beta = 2 but beta1> C. One of these examples answers a question of . In addition, we provide examples with the following counterintuitive direct-sum phenomena. Consider a union of several mutually independent broadcast settings. The optimal code for the combined setting may yield a significant saving in communication over concatenating optimal encodings for the individual settings. This result also provides new non-linear coding schemes which improve upon the largest known gap between linear and non-linear Network Coding, thus improving the results of. The proofs are based on a relation between this problem and results in the study of Witsenhausens rate, OR graph products, colorings of Cayley graphs, and the chromatic numbers of Kneser graphs.

Non-Linear Index Coding Outperforming the Linear Optimum

The following source coding problem was introduced by Birk and Kol: a sender holds a word x epsi {0,1}n, and wishes to broadcast a codeword to n receivers, R1,..., Rnmiddot. The receiver Ri is interested in x;, and has prior side information comprising some subset of the n bits. This corresponds to a directed graph G on n vertices, where ij is an edge iff Ri knows the bit xj . An index code for G is an encoding scheme which enables each Ri to always reconstruct Xj, given his side information. The minimal word length of an index code was studied by Bar-Yossef Birk, Jay ram and Kol. Thev introduced a graph parameter, minrk2(G), which completely characterizes the length of an optimal linear index code for G. The authors of (Z. Bar-Yossef, 2006) showed that in various cases linear codes attain the optimal word length, and conjectured that linear index coding is in fact always optimal. In this work, we disprove the main conjecture of (Z. Bar-Yossef, 2006) in the following strong sense: for any epsiv > 0 and sufficiently large n, there is an n-vertex graph G so that evety linear index code for G requires codewords of length at least n1-epsiv and yet a non-linear index code for G has a word length of nepsiv. This is achieved by an explicit construction, which extends Alons variant of the celebrated Ramsey construction of Frankl and Wilson.

NON-BACKTRACKING RANDOM WALKS MIX FASTER

We compute the mixing rate of a non-backtracking random walk on a regular expander. Using some properties of Chebyshev polynomials of the second kind, we show that this rate may be up to twice as fast as the mixing rate of the simple random walk. The closer the expander is to a Ramanujan graph, the higher the ratio between the above two mixing rates is. As an application, we show that if G is a high-girth regular expander on n vertices, then a typical non-backtracking random walk of length n on G does not visit a vertex more than times, and this result is tight. In this sense, the multi-set of visited vertices is analogous to the result of throwing n balls to n bins uniformly, in contrast to the simple random walk on G, which almost surely visits some vertex Ω(log n) times.

Lexicographic Products and the Power of Non-linear Network Coding

We introduce a technique for establishing and amplifying gaps between parameters of network coding and index coding problems. The technique uses linear programs to establish separations between combinatorial and coding-theoretic parameters and applies hyper graph lexicographic products to amplify these separations. This entails combining the dual solutions of the lexicographic multiplicands and proving that this is a valid dual solution of the product. Our result is general enough to apply to a large family of linear programs. This blend of linear programs and lexicographic products gives a recipe for constructing hard instances in which the gap between combinatorial or coding-theoretic parameters is polynomially large. We find polynomial gaps in cases in which the largest previously known gaps were only small constant factors or entirely unknown. Most notably, we show a polynomial separation between linear and non-linear network coding rates. This involves exploiting a connection between matroids and index coding to establish a previously unknown separation between linear and non-linear index coding rates. We also construct index coding problems with a polynomial gap between the broadcast rate and the trivial lower bound for which no gap was previously known.

Broadcasting With Side Information: Bounding and Approximating the Broadcast Rate

Index coding has received considerable attention recently motivated in part by applications such as fast video-on-demand and efficient communication in wireless networks and in part by its connection to network coding. Optimal encoding schemes and efficient heuristics were studied in various settings, while also leading to new results for network coding such as improved gaps between linear and non-linear capacity as well as hardness of approximation. The problem of broadcasting with side information, a generalization of the index coding problem, begins with a sender and sets of users and messages. Each user possesses a subset of the messages and desires an additional message from the set. The sender wishes to broadcast a message so that on receipt of the broadcast each user can compute her desired message. The fundamental parameter of interest is the broadcast rate, β, the average communication cost for sufficiently long broadcasts. Though there have been many new nontrivial bounds on β by Bar-Yossef (2006), Lubetzky and Stav (2007), Alon (2008), and Blasiak (2011) there was no known polynomial-time algorithm for approximating β within a nontrivial factor, and the exact value of β remained unknown for all nontrivial instances. Using the information theoretic linear program introduced in Blasiak (2011), we give a polynomial-time algorithm for recognizing instances with β = 2 and pinpoint β precisely for various classes of graphs (e.g., various Cayley graphs of cyclic groups). Further, extending ideas from Ramsey theory, we give a polynomial-time algorithm with a nontrivial approximation ratio for computing β. Finally, we provide insight into the quality of previous bounds by giving constructions showing separations between β and the respective bounds. In particular, we construct graphs where β is uniformly bounded while its upper bound derived from the naïve encoding scheme is polynomially worse.

Coarse to over-fine optical flow estimation

We present a readily applicable way to go beyond the accuracy limits of current optical flow estimators. Modern optical flow algorithms employ the coarse to fine approach. We suggest to upgrade this class of algorithms, by adding over-fine interpolated levels to the pyramid. Theoretical analysis of the coarse to over-fine approach explains its advantages in handling flow-field discontinuities and simulations show its benefit for sub-pixel motion. By applying the suggested technique to various multi-scale optical flow algorithms, we reduced the estimation error by 10-30% on the common test sequences. Using the coarse to over-fine technique, we obtain optical flow estimation results that are currently the best for benchmark sequences.

On replica symmetry of large deviations in random graphs

The following question is due to Chatterjee and Varadhan 2011. Fix 0<p<r<1 and take G~Gn,p, the Erdi¾?s-Renyi random graph with edge density p, conditioned to have at least as many triangles as the typical Gn,r. Is G close in cut-distance to a typical Gn,r? Via a beautiful new framework for large deviation principles in Gn,p, Chatterjee and Varadhan gave bounds on the replica symmetric phase, the region of p,r where the answer is positive. They further showed that for any small enough p there are at least two phase transitions as r varies. We settle this question by identifying the replica symmetric phase for triangles and more generally for any fixed d-regular graph. By analyzing the variational problem arising from the framework of Chatterjee and Varadhan we show that the replica symmetry phase consists of all p,r such that rd,hpr lies on the convex minorant of xi¾?hpx1/d where hp is the rate function of a binomial with parameter p. In particular, the answer for triangles involves hpx rather than the natural guess of hpx1/3 where symmetry was previously known. Analogous results are obtained for linear hypergraphs as well as the setting where the largest eigenvalue of G~Gn,p is conditioned to exceed the typical value of the largest eigenvalue of Gn,r. Building on the work of Chatterjee and Diaconis 2012 we obtain additional results on a class of exponential random graphs including a new range of parameters where symmetry breaking occurs. En route we give a short alternative proof of a graph homomorphism inequality due to Kahn 2001 and Galvin and Tetali 2004.

Anatomy of a young giant component in the random graph

We provide a complete description of the giant component of the Erdős-Renyi random graph \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{G}}(n,p)\end{align*} \end{document} **image** as soon as it emerges from the scaling window, i.e., for p = (1+e)/n where e3n →∞ and e = o(1). Our description is particularly simple for e = o(n-1/4), where the giant component \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** is contiguous with the following model (i.e., every graph property that holds with high probability for this model also holds w.h.p. for \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** ). Let Z be normal with mean \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\frac{2}{3} \varepsilon^3 n\end{align*} \end{document} **image** and variance e3n, and let \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{K}\end{align*} \end{document} **image** be a random 3-regular graph on \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}2\left\lfloor Z\right\rfloor\end{align*} \end{document} **image** vertices. Replace each edge of \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{K}\end{align*} \end{document} **image** by a path, where the path lengths are i.i.d. geometric with mean 1/e. Finally, attach an independent Poisson( 1-e )-Galton-Watson tree to each vertex. A similar picture is obtained for larger e = o(1), in which case the random 3-regular graph is replaced by a random graph with Nk vertices of degree k for k ≥ 3, where Nk has mean and variance of order ekn. This description enables us to determine fundamental characteristics of the supercritical random graph. Namely, we can infer the asymptotics of the diameter of the giant component for any rate of decay of e, as well as the mixing time of the random walk on \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath} \pagestyle{empty} \begin{document} \begin{align*}{\mathcal{C}_1}\end{align*} \end{document} **image** .

On the variational problem for upper tails in sparse random graphs

What is the probability that the number of triangles in Gn,p, the Erdi¾?s-Renyi random graph with edge density p, is at least twice its mean? Writing it as exp[-rn,p], already the order of the rate function rn, p was a longstanding open problem when p=o1, finally settled in 2012 by Chatterjee and by DeMarco and Kahn, who independently showed that rn,pi¾?n2p2log1/p for pi¾?lognn; the exact asymptotics of rn, p remained unknown. The following variational problem can be related to this large deviation question at pi¾?lognn: for i¾?>0 fixed, what is the minimum asymptotic p-relative entropy of a weighted graph on n vertices with triangle density at least 1+i¾?p3? A beautiful large deviation framework of Chatterjee and Varadhan 2011 reduces upper tails for triangles to a limiting version of this problem for fixed p. A very recent breakthrough of Chatterjee and Dembo extended its validity to n-αi¾?pi¾?1 for an explicit α>0, and plausibly it holds in all of the above sparse regime.

Cutoff phenomena for random walks on random regular graphs

The cutoff phenomenon describes a sharp transition in the convergence of a family of ergodic finite Markov chains to equilibrium. Many natural families of chains are believed to exhibit cutoff, and yet establishing this fact is often extremely challenging. An important such family of chains is the random walk on