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Register a new userOn Two-Stage Guessing
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Added by
Igal Sason
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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Stationary memoryless sources produce two correlated random sequences $X^n$ and $Y^n$. A guesser seeks to recover $X^n$ in two stages, by first guessing $Y^n$ and then $X^n$. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in $n$) of any positive $rho$-th moment of the total number of guesses when $Y^n$ is obtained by applying a deterministic function $f$ component-wise to $X^n$. We prove that, depending on $f$, the least exponential growth rate in the two-stage setup is lower than when guessing $X^n$ directly. We further propose a simple Huffman code-based construction of a function $f$ that is a viable candidate for the minimization of the least exponential growth rate in the two-stage guessing setup. (2) We characterize the least achievable exponential growth rate of the $rho$-th moment of the total number of guesses required to recover $X^n$ when Stage 1 need not end with a correct guess of $Y^n$ and without assumptions on the stationary memoryless sources producing $X^n$ and $Y^n$.
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We study four problems namely, Campbells source coding problem, Arikans guessing problem, Huieihel et al.s memoryless guessing problem, and Bunte and Lapidoths task partitioning problem. We observe a close relationship among these problems. In all these problems, the objective is to minimize moments of some functions of random variables, and Renyi entropy and Sundaresans divergence arise as optimal solutions. This motivates us to establish a connection among these four problems. In this paper, we study a more general problem and show that R{e}nyi and Shannon entropies arise as its solution. We show that the problems on source coding, guessing and task partitioning are particular instances of this general optimization problem, and derive the lower bounds using this framework. We also refine some known results and present new results for mismatched version of these problems using a unified approach. We strongly feel that this generalization would, in addition to help in understanding the similarities and distinctiveness of these problems, also help to solve any new problem that falls in this framework.
This paper provides tight bounds on the Renyi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one. To that end, a tight lower bound on the Renyi entropy of a discrete random variable with a finite support is derived as a function of the size of the support, and the ratio of the maximal to minimal probability masses. This work was inspired by the recently published paper by Cicalese et al., which is focused on the Shannon entropy, and it strengthens and generalizes the results of that paper to Renyi entropies of arbitrary positive orders. In view of these generalized bounds and the works by Arikan and Campbell, non-asymptotic bounds are derived for guessing moments and lossless data compression of discrete memoryless sources.
This paper establishes a close relationship among the four information theoretic problems, namely Campbell source coding, Arikan guessing, Huleihel et al. memoryless guessing and Bunte and Lapidoth tasks partitioning problems. We first show that the aforementioned problems are mathematically related via a general moment minimization problem whose optimum solution is given in terms of Renyi entropy. We then propose a general framework for the mismatched version of these problems and establish all the asymptotic results using this framework. Further, we study an ordered tasks partitioning problem that turns out to be a generalisation of Arikans guessing problem. Finally, with the help of this general framework, we establish an equivalence among all these problems, in the sense that, knowing an asymptotically optimal solution in one problem helps us find the same in all other problems.
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 one 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.
A source sequence is to be guessed with some fidelity based on a rate-limited description of an observed sequence with which it is correlated. The trade-off between the description rate and the exponential growth rate of the least power mean of the number of guesses is characterized.
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