Deterministic Compression with Uncertain Priors

Haramaty, Elad; Sudan, Madhu

doi:10.1007/s00453-015-0107-6

Cited by 7 publications

(23 citation statements)

References 13 publications

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“…Third, we do not have any better lower bounds for the deterministic or imperfect-randomness settings, respectively from Haramaty and Sudan [4] and Canonne et al [10]. The known upper bounds in these settings are much weaker, featuring redundancy that grows at least linearly in the entropy of the source distribution, and in the case of the deterministic codes of Haramaty and Sudan, some kind of dependence on the size of the distribution's support.…”

Section: Suggestions For Future Workmentioning

confidence: 75%

“…We briefly note that the definition of Haramaty and Sudan [4] differs in two basic ways. First, their definition does not include a common random string since they were primarily interested in deterministic coding schemes.…”

Section: The Model and Prior State Of The Artmentioning

confidence: 99%

“…We now turn to the setting where Alice and Bob are allowed to fail at the communication task with positive probability ǫ. Haramaty and Sudan [4] introduced this setting, but only considered deterministic schemes for this problem.…”

Section: The Price Of Uncertainty When Errors Are Allowedmentioning

confidence: 99%

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The Price of Uncertain Priors in Source Coding

Braverman

Juba

2019

IEEE Trans. Inform. Theory

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We consider the problem of one-way communication when the recipient does not know exactly the distribution that the messages are drawn from, but has a "prior" distribution that is known to be close to the source distribution, a problem first considered by Juba et al. We consider the question of how much longer the messages need to be in order to cope with the uncertainty about the receiver's prior and the source distribution, respectively, as compared to the standard source coding problem. We consider two variants of this uncertain priors problem: the original setting of Juba et al. in which the receiver is required to correctly recover the message with probability 1, and a setting introduced by Haramaty and Sudan, in which the receiver is permitted to fail with some probability ǫ. In both settings, we obtain lower bounds that are tight up to logarithmically smaller terms. In the latter setting, we furthermore present a variant of the coding scheme of Juba et al. with an overhead of log α + log 1/ǫ + 1 bits, thus also establishing the nearly tight upper bound. *

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Section: Suggestions For Future Workmentioning

confidence: 75%

Section: The Model and Prior State Of The Artmentioning

confidence: 99%

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The Price of Uncertain Priors in Source Coding

Braverman

Juba

2019

IEEE Trans. Inform. Theory

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“…The purpose in [5] of determining the chromatic number of the graphs described above was to solve the following compression problem: for some finite universe U, suppose that Alice is operating under the belief that a message m is chosen from U according to the probability distribution P , and that Bob operates under the belief that m is chosen according to the distribution Q. Both Alice and Bob know that their distributions P, Q are "close" in the sense that they know of some ∆ ≥ 0 such that max m∈U max log 2 P (m) Q(m) , log 2 Q(m) P (m) ≤ ∆.…”

Section: Motivationmentioning

confidence: 99%

“…Moreover, log * x denotes the minimum j such that log (j) x ≤ 1. Haramaty and Sudan [5] showed that for any k, s ∈ N the set of all chains of length 2k and size at most s can be colored with 2 6(s+1) · log (k) N colors so that for chains A, A ′ in this set with S 1 (A) ∩ S 1 (A ′ ) = ∅ and A 0 = A ′ 0 , A and A ′ receive different colors. As we will show in Section 3, this result is equivalent to the…”

Section: Introductionmentioning

confidence: 99%

Coloring Chains for Compression with Uncertain Priors

Golowich

2018

Electron. J. Combin.

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Haramaty and Sudan considered the problem of transmitting a message between two people, Alice and Bob, when Alice's and Bob's priors on the message are allowed to differ by at most a given factor. To find a deterministic compression scheme for this problem, they showed that it is sufficient to obtain an upper bound on the chromatic number of a graph, denoted U (N, s, k) for parameters N, s, k, whose vertices are nested sequences of subsets and whose edges are between vertices that have similar sequences of sets. In turn, there is a close relationship between the problem of determining the chromatic number of U (N, s, k) and a local graph coloring problem considered by Erdős et al. We generalize the results of Erdős et al. by finding bounds on the chromatic numbers of graphs H and G when there is a homomorphism φ : H → G that satisfies a nice property. We then use these results to improve upper and lower bounds on χ(U (N, s, k)).

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Communication with Contextual Uncertainty

et al. 2017

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We introduce a simple model illustrating the role of context in communication and the challenge posed by uncertainty of knowledge of context. We consider a variant of distributional communication complexity where Alice gets some information x and Bob gets y, where (x, y) is drawn from a known distribution, and Bob wishes to compute some function g(x, y) (with high probability over (x, y)). In our variant, Alice does not know g, but only knows some function f which is an approximation of g. Thus, the function being computed forms the context for the communication, and knowing it imperfectly models (mild) uncertainty in this context.A naive solution would be for Alice and Bob to first agree on some common function h that is close to both f and g and then use a protocol for h to compute h(x, y). We show that any such agreement leads to a large overhead in communication ruling out such a universal solution.In contrast, we show that if g has a one-way communication protocol with complexity k in the standard setting, then it has a communication protocol with complexity O(k · (1 + I)) in the uncertain setting, where I denotes the mutual information between x and y. In the particular case where the input distribution is a product distribution, the protocol in the uncertain setting only incurs a constant factor blow-up in communication and error.Furthermore, we show that the dependence on the mutual information I is required. Namely, we construct a class of functions along with a non-product distribution over (x, y) for which the communication complexity is a single bit in the standard setting but at least Ω( √ n) bits in the uncertain setting.

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Deterministic Compression with Uncertain Priors

Cited by 7 publications

References 13 publications

The Price of Uncertain Priors in Source Coding

The Price of Uncertain Priors in Source Coding

Coloring Chains for Compression with Uncertain Priors

Communication with Contextual Uncertainty

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