arXiv subjects

Ofer Shayevitz

Publications and source records attributed to Ofer Shayevitz.

At least 37 records · Page 2Linked to original sources

arXiv2019

On the Interactive Capacity of Finite-State Protocols

The interactive capacity of a noisy channel is the highest possible rate at which arbitrary interactive protocols can be simulated reliably over the channel. Determining the interactive capacity is notoriously difficult, and the best known lower bounds are far below the associated Shannon capacity, which serves as a trivial (and also generally the best known) upper bound. This paper considers the more restricted setup of simulating finite-state protocols. It is shown that all two-state protocols, as well as rich families of arbitrary finite-state protocols, can be simulated at the Shannon capacity, establishing the interactive capacity for those families of protocols.

cs.IT↗

arXiv2019

Distributed Source Simulation With No Communication

We consider the problem of distributed source simulation with no communication, in which Alice and Bob observe sequences $U^n$ and $V^n$ respectively, drawn from a joint distribution $p_{UV}^{\otimes n}$, and wish to locally generate sequences $X^n$ and $Y^n$ respectively with a joint distribution that is close (in KL divergence) to $p_{XY}^{\otimes n}$. We provide a single-letter condition under which such a simulation is asymptotically possible with a vanishing KL divergence. Our condition is nontrivial only in the case where the Gàcs-Körner (GK) common information between $U$ and $V$ is nonzero, and we conjecture that only scalar Markov chains $X-U-V-Y$ can be simulated otherwise. Motivated by this conjecture, we further examine the case where both $p_{UV}$ and $p_{XY}$ are doubly symmetric binary sources with parameters $p,q\leq 1/2$ respectively. While it is trivial that in this case $p\leq q$ is both necessary and sufficient, we show that when $p$ is close to $q$ then any successful simulation is close to being scalar in the total variation sense.

cs.IT↗

arXiv2019

Communication Complexity of Estimating Correlations

We characterize the communication complexity of the following distributed estimation problem. Alice and Bob observe infinitely many iid copies of $ρ$-correlated unit-variance (Gaussian or $\pm1$ binary) random variables, with unknown $ρ\in[-1,1]$. By interactively exchanging $k$ bits, Bob wants to produce an estimate $\hatρ$ of $ρ$. We show that the best possible performance (optimized over interaction protocol $Π$ and estimator $\hat ρ$) satisfies $\inf_{Π\hatρ}\sup_ρ\mathbb{E} [|ρ-\hatρ|^2] = \tfrac{1}{k} (\frac{1}{2 \ln 2} + o(1))$. Curiously, the number of samples in our achievability scheme is exponential in $k$; by contrast, a naive scheme exchanging $k$ samples achieves the same $Ω(1/k)$ rate but with a suboptimal prefactor. Our protocol achieving optimal performance is one-way (non-interactive). We also prove the $Ω(1/k)$ bound even when $ρ$ is restricted to any small open sub-interval of $[-1,1]$ (i.e. a local minimax lower bound). Our proof techniques rely on symmetric strong data-processing inequalities and various tensorization techniques from information-theoretic interactive common-randomness extraction. Our results also imply an $Ω(n)$ lower bound on the information complexity of the Gap-Hamming problem, for which we show a direct information-theoretic proof.

cs.IT↗

arXiv2019

Self-Predicting Boolean Functions

A Boolean function $g$ is said to be an optimal predictor for another Boolean function $f$, if it minimizes the probability that $f(X^{n})\neq g(Y^{n})$ among all functions, where $X^{n}$ is uniform over the Hamming cube and $Y^{n}$ is obtained from $X^{n}$ by independently flipping each coordinate with probability $δ$. This paper is about self-predicting functions, which are those that coincide with their optimal predictor.

cs.DM↗

arXiv2018

Shannon Capacity is Achievable for Binary Interactive First-Order Markovian Protocols

We address the problem of simulating an arbitrary binary interactive first-order Markovian protocol over a pair of binary symmetric channels with crossover probability $\varepsilon$. We are interested in the achievable rates of reliable simulation, i.e., in characterizing the smallest possible blowup in communications such that a vanishing error probability (in the protocol length) can be attained. Whereas for general interactive protocols the output of each party may depend on all previous outputs of its counterpart, in a (first-order) Markovian protocol this dependence is limited to the last observed output only. Previous works in the field discuss broader families of protocols but assess the achievable rates only at the limit where $\varepsilon\to0$. In this paper, we prove that the one-way Shannon capacity, $1-h(\varepsilon)$, can be achieved for any binary first-order Markovian protocol. This surprising result, is to the best of our knowledge, the first example in which non-trivial interactive protocol can be simulated in the Shannon capacity. We give two capacity achieving coding schemes, which both divide the protocol into independent blocks, and implement vertical block coding. The first scheme is based on a random separation into blocks with variable lengths. The second scheme is based on a deterministic separation into blocks, and efficiently predicting their last transmission. The prediction can be regarded as a binary pointer jumping game, for which we show that the final step can be calculated with $O(\log m)$ bits, where $m$ is the number of rounds in the game. We conclude the work by discussing possible extensions of the result to higher order models.

cs.IT↗

arXiv2018

On the VC-Dimension of Binary Codes

We investigate the asymptotic rates of length-$n$ binary codes with VC-dimension at most $dn$ and minimum distance at least $δn$. Two upper bounds are obtained, one as a simple corollary of a result by Haussler and the other via a shortening approach combining Sauer-Shelah lemma and the linear programming bound. Two lower bounds are given using Gilbert-Varshamov type arguments over constant-weight and Markov-type sets.

cs.IT↗

arXiv2018

Distributed Estimation of Gaussian Correlations

We study a distributed estimation problem in which two remotely located parties, Alice and Bob, observe an unlimited number of i.i.d. samples corresponding to two different parts of a random vector. Alice can send $k$ bits on average to Bob, who in turn wants to estimate the cross-correlation matrix between the two parts of the vector. In the case where the parties observe jointly Gaussian scalar random variables with an unknown correlation $ρ$, we obtain two constructive and simple unbiased estimators attaining a variance of $(1-ρ^2)/(2k\ln 2)$, which coincides with a known but non-constructive random coding result of Zhang and Berger. We extend our approach to the vector Gaussian case, which has not been treated before, and construct an estimator that is uniformly better than the scalar estimator applied separately to each of the correlations. We then show that the Gaussian performance can essentially be attained even when the distribution is completely unknown. This in particular implies that in the general problem of distributed correlation estimation, the variance can decay at least as $O(1/k)$ with the number of transmitted bits. This behavior, however, is not tight: we give an example of a rich family of distributions for which local samples reveal essentially nothing about the correlations, and where a slightly modified estimator attains a variance of $2^{-Ω(k)}$.

math.ST↗

arXiv2018

Reducing Guesswork via an Unreliable Oracle

Alice holds an random variable $X$, and Bob is trying to guess its value by asking questions of the form "is $X=x$?". Alice answers truthfully and the game terminates once Bob guesses correctly. Before the game begins, Bob is allowed to reach out to an oracle, Carole, and ask her any yes/no question, i.e., a question of the form "is $X\in A$?". Carole is known to lie with a given probability $p$. What should Bob ask Carole if he would like to minimize his expected guessing time? When Carole is always truthful ($p=0$), it is not difficult to check that Bob should order the symbol probabilities in descending order, and ask Carole whether the index of $X$ w.r.t this order is even or odd. We show that this strategy is almost optimal for any lying probability $p$, up to a small additive constant upper bounded by a $1/4$. We discuss a connection to the cutoff rate of the BSC with feedback.

cs.IT↗

arXiv2018

Guessing with a Bit of Help

What is the value of a single bit to a guesser? We study this problem in a setup where Alice wishes to guess an i.i.d. random vector, and can procure one bit of information from Bob, who observes this vector through a memoryless channel. We are interested in the guessing efficiency, which we define as the best possible multiplicative reduction in Alice's guessing-moments obtainable by observing Bob's bit. For the case of a uniform binary vector observed through a binary symmetric channel, we provide two lower bounds on the guessing efficiency by analyzing the performance of the Dictator and Majority functions, and two upper bounds via maximum entropy and Fourier-analytic / hypercontractivity arguments. We then extend our maximum entropy argument to give a lower bound on the guessing efficiency for a general channel with a binary uniform input, via the strong data-processing inequality constant of the reverse channel. We compute this bound for the binary erasure channel, and conjecture that Greedy Dictator functions achieve the guessing efficiency.

cs.IT↗

arXiv2018

A Bound on the Shannon Capacity via a Linear Programming Variation

We prove an upper bound on the Shannon capacity of a graph via a linear programming variation. We show that our bound can outperform both the Lovász theta number and the Haemers minimum rank bound. As a by-product, we also obtain a new upper bound on the broadcast rate of Index Coding.

cs.IT↗

arXiv2017

Interactive Coding for Markovian Protocols

We address the problem of simulating an arbitrary Markovian interactive protocol over binary symmetric channels with crossover probability $\varepsilon$. We are interested in the achievable rates of reliable simulation, i.e., in characterizing the smallest possible blowup in communications such that a vanishing error probability (in the protocol length) can be attained. Whereas for general interactive protocols the output of each party may depend on all previous outputs of its counterpart, in a (first order) Markovian protocol this dependence is limited to the last observed output only. In the special case where there is no dependence on previous outputs (no interaction), the maximal achievable rate is given by the (one-way) Shannon capacity $1-h(\varepsilon)$. For Markovian protocols, we first show that a rate of $\frac{2}{3}(1-h(\varepsilon))$ can be trivially achieved. We then describe a more involved coding scheme and provide a closed-form lower bound for its rate at any noise level $\varepsilon$. Specifically, we show that this scheme outperforms the trivial one for any $\varepsilon<0.044$, and achieves a rate higher than $\frac{1-h(\varepsilon)}{1+h(\varepsilon)+h\left(<\varepsilon(2-\varepsilon)>\right)}=1-Θ(h(\varepsilon))$ as $\varepsilon\to 0$, which is order-wise the best possible. This should be juxtaposed with a result of Kol and Raz that shows the capacity for interactive protocols with alternating rounds is lower bounded by $1-O(\sqrt{h(\varepsilon)})$.

cs.IT↗

arXiv2017

Graph Information Ratio

We introduce the notion of information ratio $\text{Ir}(H/G)$ between two (simple, undirected) graphs $G$ and $H$, defined as the supremum of ratios $k/n$ such that there exists a mapping between the strong products $G^k$ to $H^n$ that preserves non-adjacency. Operationally speaking, the information ratio is the maximal number of source symbols per channel use that can be reliably sent over a channel with a confusion graph $H$, where reliability is measured w.r.t. a source confusion graph $G$. Various results are provided, including in particular lower and upper bounds on $\text{Ir}(H/G)$ in terms of different graph properties, inequalities and identities for behavior under strong product and disjoint union, relations to graph cores, and notions of graph criticality. Informally speaking, $\text{Ir}(H/G)$ can be interpreted as a measure of similarity between $G$ and $H$. We make this notion precise by introducing the concept of information equivalence between graphs, a more quantitative version of homomorphic equivalence. We then describe a natural partial ordering over the space of information equivalence classes, and endow it with a suitable metric structure that is contractive under the strong product. Various examples and open problems are discussed.

cs.IT↗

arXiv2017

Quickest Sequence Phase Detection

A phase detection sequence is a length-$n$ cyclic sequence, such that the location of any length-$k$ contiguous subsequence can be determined from a noisy observation of that subsequence. In this paper, we derive bounds on the minimal possible $k$ in the limit of $n\to\infty$, and describe some sequence constructions. We further consider multiple phase detection sequences, where the location of any length-$k$ contiguous subsequence of each sequence can be determined simultaneously from a noisy mixture of those subsequences. We study the optimal trade-offs between the lengths of the sequences, and describe some sequence constructions. We compare these phase detection problems to their natural channel coding counterparts, and show a strict separation between the fundamental limits in the multiple sequence case. Both adversarial and probabilistic noise models are addressed.

cs.IT↗

arXiv2017

The $ρ$-Capacity of a Graph

Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed $ρ$-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the $ρ$-capacity of arbitrary graphs, and provide a Lovász-type upper bound for regular graphs. We study the behavior of the $ρ$-capacity under two graph operations: the strong product and the disjoint union. Finally, we investigate the connection between the structure of a graph and its $ρ$-capacity.

cs.IT↗

arXiv2017

Interactive Schemes for the AWGN Channel with Noisy Feedback

We study the problem of communication over an additive white Gaussian noise (AWGN) channel with an AWGN feedback channel. When the feedback channel is noiseless, the classic Schalkwijk-Kailath (S-K) scheme is known to achieve capacity in a simple sequential fashion, while attaining reliability superior to non-feedback schemes. In this work, we show how simplicity and reliability can be attained even when the feedback is noisy, provided that the feedback channel is sufficiently better than the feedforward channel. Specifically, we introduce a low-complexity low-delay interactive scheme that operates close to capacity for a fixed bit error probability (e.g. $10^{-6}$). We then build on this scheme to provide two asymptotic constructions, one based on high dimensional lattices, and the other based on concatenated coding, that admit an error exponent significantly exceeding the best possible non-feedback exponent. Our approach is based on the interpretation of feedback transmission as a side-information problem, and employs an interactive modulo-lattice solution.

cs.IT↗

arXiv2016

Searching with Measurement Dependent Noise

Consider a target moving at a constant velocity on a unit-circumference circle, starting at an arbitrary location. To acquire the target, any region of the circle can be probed to obtain a noisy measurement of the target's presence, where the noise level increases with the size of the probed region. We are interested in the expected time required to find the target to within some given resolution and error probability. For a known velocity, we characterize the optimal tradeoff between time and resolution, and show that in contrast to the well studied case of constant measurement noise, measurement dependent noise incurs a multiplicative gap in the targeting rate between adaptive and non-adaptive search strategies. Moreover, our adaptive strategy attains the optimal rate-reliability tradeoff. We further show that for optimal non-adaptive search, accounting for an unknown velocity incurs a factor of at least two in the targeting rate.

cs.IT↗

arXiv2016

Mutual Information Bounds via Adjacency Events

The mutual information between two jointly distributed random variables $X$ and $Y$ is a functional of the joint distribution $P_{XY},$ which is sometimes difficult to handle or estimate. A coarser description of the statistical behavior of $(X,Y)$ is given by the marginal distributions $P_X, P_Y$ and the adjacency relation induced by the joint distribution, where $x$ and $y$ are adjacent if $P(x,y)>0$. We derive a lower bound on the mutual information in terms of these entities. The bound is obtained by viewing the channel from $X$ to $Y$ as a probability distribution on a set of possible actions, where an action determines the output for any possible input, and is independently drawn. We also provide an alternative proof based on convex optimization, that yields a generally tighter bound. Finally, we derive an upper bound on the mutual information in terms of adjacency events between the action and the pair $(X,Y)$, where in this case an action $a$ and a pair $(x,y)$ are adjacent if $y=a(x)$. As an example, we apply our bounds to the binary deletion channel and show that for the special case of an i.i.d. input distribution and a range of deletion probabilities, our lower and upper bounds both outperform the best known bounds for the mutual information.

cs.IT↗

arXiv2016

On the Optimal Boolean Function for Prediction under Quadratic Loss

Suppose $Y^{n}$ is obtained by observing a uniform Bernoulli random vector $X^{n}$ through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between $Y^{n}$ and a Boolean function $\mathsf{b}(X^{n})$ could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of $Y^{n}$ under logarithmic loss, given $\mathsf{b}(X^{n})$. In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function - the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than dictator, and that for strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels.

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