A Tight Upper Bound on the Mutual Information of Two Boolean Functions

Pichler, Georg; Matz, Gerald; Piantanida, Pablo

doi:10.1109/itw.2016.7606787

Cited by 8 publications

(8 citation statements)

References 9 publications

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“…where in (10) follows from Proposition 1, in (11) we have used Lemma 1, and ( 12) follows from ( 6) and (7). Proposition 4, stated and proved in the next section, shows that H n 2 (α) = 1 + (n − 1)h(α).…”

Section: Proof Of Theoremmentioning

confidence: 90%

“…Clearly, by the symmetry of the pair (X n , Y n ) we have that for any function I(b(X n ); Y n ) = I(X n ; b(Y n )), so we can equivalently think of the problem at hand as seeking the optimal one-bit quantizer of n outputs of the channel. Despite attempts in various directions [1]- [7], Conjecture 1 remains open in general. However, for the "very noisy" case, where α > 1/2 − δ, for some δ > 0 independent of n, the validity of the conjecture was established by Samorodnitsky [8].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

How to quantize n outputs of a binary symmetric channel to n − 1 bits?

Huleihel¹,

Ordentlich

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Section: Proof Of Theoremmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

How to quantize n outputs of a binary symmetric channel to n − 1 bits?

Huleihel¹,

Ordentlich

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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“…Their work was inspired by a similar problem in computational biology [48]. A weaker form of their conjecture [47, Section IV, 2)], which was solved in [49], corresponds to a zero-rate/one-bit variant of the binary example studied here.…”

Section: E Doubly Symmetric Binary Sourcementioning

confidence: 99%

Distributed Information-Theoretic Clustering

Pichler,

Piantanida,

Matz

2016

Preprint

Self Cite

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We study a novel multi-terminal source coding setup motivated by the biclustering problem. Two separate encoders observe two i.i.d. sources X n and Z n , respectively. The goal is to find rate-limited encodings f (x n ) and g(z n ) that maximize the mutual information I f (X n ); g(Z n ) /n. We provide inner and outer bounds on the achievable region and discuss connections of this problem with hypothesis testing against independence, pattern recognition, the information bottleneck method, and lossy source coding with logarithmic-loss distortion. Improving previous cardinality bounds allows us to thoroughly study the special case of a binary symmetric source and to quantify the gap between the inner and the outer bound in this special case. Furthermore, we generalize our results to the case of more than two i.i.d. sources. As a special case of this generalization we investigate a multiple description (MD) extension of the CEO problem with log-loss distortion. Surprisingly, this MD-CEO problem permits a tight single-letter characterization of the achievable region.

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“…Although there has been some progress in this line of work [ 2 , 3 ], this simple conjecture still remains open. There are also a number of variations of this conjecture.…”

Section: Introductionmentioning

confidence: 99%

Information Geometric Approach on Most Informative Boolean Function Conjecture

2018

Entropy

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Let X n be a memoryless uniform Bernoulli source and Y n be the output of it through a binary symmetric channel. Courtade and Kumar conjectured that the Boolean function f : { 0 , 1 } n → { 0 , 1 } that maximizes the mutual information I ( f ( X n ) ; Y n ) is a dictator function, i.e., f ( x n ) = x i for some i. We propose a clustering problem, which is equivalent to the above problem where we emphasize an information geometry aspect of the equivalent problem. Moreover, we define a normalized geometric mean of measures and interesting properties of it. We also show that the conjecture is true when the arithmetic and geometric mean coincide in a specific set of measures.

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A Tight Upper Bound on the Mutual Information of Two Boolean Functions

Cited by 8 publications

References 9 publications

How to quantize n outputs of a binary symmetric channel to n − 1 bits?

How to quantize n outputs of a binary symmetric channel to n − 1 bits?

Distributed Information-Theoretic Clustering

Information Geometric Approach on Most Informative Boolean Function Conjecture

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