The rate region of the task-encoding problem for two correlated sources is characterized using a novel parametric family of dependence measures. The converse uses a new expression for the $rho$-th moment of the list size, which is derived using the relative $alpha$-entropy.
The secrecy of a distributed-storage system for passwords is studied. The encoder, Alice, observes a length-n password and describes it using two hints, which she stores in different locations. The legitimate receiver, Bob, observes both hints. In on
e scenario the requirement is that the expected number of guesses it takes Bob to guess the password approach one as n tends to infinity, and in the other that the expected size of the shortest list that Bob must form to guarantee that it contain the password approach one. The eavesdropper, Eve, sees only one of the hints. Assuming that Alice cannot control which hints Eve observes, the largest normalized (by n) exponent that can be guaranteed for the expected number of guesses it takes Eve to guess the password is characterized for each scenario. Key to the proof are new results on Arikans guessing and Bunte and Lapidoths task-encoding problem; in particular, the paper establishes a close relation between the two problems. A rate-distortion version of the model is also discussed, as is a generalization that allows for Alice to produce {delta} (not necessarily two) hints, for Bob to observe { u} (not necessarily two) of the hints, and for Eve to observe {eta} (not necessarily one) of the hints. The generalized model is robust against {delta} - { u} disk failures.
The zero-error feedback capacity of the Gelfand-Pinsker channel is established. It can be positive even if the channels zero-error capacity is zero in the absence of feedback. Moreover, the error-free transmission of a single bit may require more tha
n one channel use. These phenomena do not occur when the state is revealed to the transmitter causally, a case that is solved here using Shannon strategies. Cost constraints on the channel inputs or channel states are also discussed, as is the scenario where---in addition to the message---also the state sequence must be recovered.
The identification (ID) capacity region of the two-receiver broadcast channel (BC) is shown to be the set of rate-pairs for which, for some distribution on the channel input, each receivers ID rate does not exceed the mutual information between the c
hannel input and the channel output that it observes. Moreover, the capacity regions interior is achieved by codes with deterministic encoders. The results are obtained under the average-error criterion, which requires that each receiver reliably identify its message whenever the message intended for the other receiver is drawn at random. They hold also for channels whose transmission capacity region is to-date unknown. Key to the proof is a new ID code construction for the single-user channel. Extensions to the BC with one-sided feedback and the three-receiver BC are also discussed: inner bounds on their ID capacity regions are obtained, and those are shown to be in some cases tight.
The capacity of the semideterministic discrete memoryless broadcast channel (SD-BC) with partial message side-information (P-MSI) at the receivers is established. In the setting without a common message, it is shown that P-MSI to the stochastic recei
ver alone can increase capacity, whereas P-MSI to the deterministic receiver can only increase capacity if also the stochastic receiver has P-MSI. The latter holds only for the setting without a common message: if the encoder also conveys a common message, then P-MSI to the deterministic receiver alone can increase capacity. These capacity results are used to show that feedback from the stochastic receiver can increase the capacity of the SD-BC without P-MSI and the sum-rate capacity of the SD-BC with P-MSI at the deterministic receiver. The link between P-MSI and feedback is a feedback code, which---roughly speaking---turns feedback into P-MSI at the stochastic receiver and hence helps the stochastic receiver mitigate experienced interference. For the case where the stochastic receiver has full MSI (F-MSI) and can thus fully mitigate experienced interference also in the absence of feedback, it is shown that feedback cannot increase capacity.
We study the secrecy of a distributed storage system for passwords. The encoder, Alice, observes a length-n password and describes it using two hints, which she then stores in different locations. The legitimate receiver, Bob, observes both hints. Th
e eavesdropper, Eve, sees only one of the hints; Alice cannot control which. We characterize the largest normalized (by n) exponent that we can guarantee for the number of guesses it takes Eve to guess the password subject to the constraint that either the number of guesses it takes Bob to guess the password or the size of the list that Bob must form to guarantee that it contain the password approach 1 as n tends to infinity.
We consider the problem of clustering a graph $G$ into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables $Y=B_G x oplus Z$, where $B_G$ is the incidence matrix of a
graph $G$, $x$ is the vector of unknown vertex variables (with a uniform prior) and $Z$ is a noise vector with Bernoulli$(varepsilon)$ i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of $x$ is possible if and only the graph $G$ is connected, with a sharp threshold at the edge probability $log(n)/n$ for ErdH{o}s-Renyi random graphs. The first goal of this paper is to determine how the edge probability $p$ needs to scale to allow exact recovery in the presence of noise. Defining the degree (oversampling) rate of the graph by $alpha =np/log(n)$, it is shown that exact recovery is possible if and only if $alpha >2/(1-2varepsilon)^2+ o(1/(1-2varepsilon)^2)$. In other words, $2/(1-2varepsilon)^2$ is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph $G$, defining the degree rate as $alpha=d/log(n)$, where $d$ is the minimum degree of the graph, it is shown that the proposed method achieves the rate $alpha> 4((1+lambda)/(1-lambda)^2)/(1-2varepsilon)^2+ o(1/(1-2varepsilon)^2)$, where $1-lambda$ is the spectral gap of the graph $G$.
