Continuity of channel parameters and operations under various DMC topologies

Nasser, Rajai

doi:10.1109/isit.2017.8007117

Cited by 3 publications

(5 citation statements)

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“…Similarly, we can extend the definition of any channel parameter or operation which is continuous in the noisiness/weak- * topology (such as the symmetric capacity, Arıkan's polar transformations, etc. [13]) to MP b (X ).…”

Section: E the Noisiness/weak- * Topologymentioning

confidence: 99%

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On the Convergence of the Polarization Process in the Noisiness/Weak-∗ Topology

Nasser

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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Let W be a channel where the input alphabet is endowed with an Abelian group operation, and let (Wn) n≥0 be Arıkan's channel-valued polarization process that is obtained from W using this operation. We prove that the process (Wn) n≥0 converges almost surely to deterministic homomorphism channels in the noisiness/weak- * topology. This provides a simple proof of multilevel polarization for a large family of channels, containing among others, discrete memoryless channels (DMC), and channels with continuous output alphabets. This also shows that any continuous channel functional converges almost surely (even if the functional does not induce a submartingale or a supermartingale).

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Section: E the Noisiness/weak- * Topologymentioning

confidence: 99%

“…where (a) follows from the fact that Since the symmetric capacity and the − transformation are continuous in the noisiness/weak- * topology (see [13]), the function f is also continuous in the same topology.…”

Section: Convergence Of the Polarization Processmentioning

confidence: 99%

On the Convergence of the Polarization Process in the Noisiness/Weak-∗ Topology

Nasser

2019

2019 IEEE International Symposium on Information Theory (ISIT)

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“…For example, one may require the continuity of all mappings that are relevant to information theory such as capacity, mutual information, probability of error of any fixed code, optimal probability of error of a given rate and blocklength, channel sums and products, etc. The continuity of these mappings under different topologies on

is studied in [ 8 ].…”

Section: Space Of Equivalent Channelsmentioning

confidence: 99%

“…The continuity (under the topologies introduced here) of mappings that are relevant to information theory (such as capacity, mutual information, Bhattacharyya parameter, probability of error of a fixed code, optimal probability of error of a given rate and blocklength, channel sums and products, etc, …) is studied in [ 8 ].…”

Section: Introductionmentioning

confidence: 99%

Topological Structures on DMC Spaces †

Nasser

2018

Entropy

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Two channels are said to be equivalent if they are degraded from each other. The space of equivalent channels with input alphabet X and output alphabet Y can be naturally endowed with the quotient of the Euclidean topology by the equivalence relation. A topology on the space of equivalent channels with fixed input alphabet X and arbitrary but finite output alphabet is said to be natural if and only if it induces the quotient topology on the subspaces of equivalent channels sharing the same output alphabet. We show that every natural topology is σ-compact, separable and path-connected. The finest natural topology, which we call the strong topology, is shown to be compactly generated, sequential and T 4. On the other hand, the strong topology is not first-countable anywhere, hence it is not metrizable. We introduce a metric distance on the space of equivalent channels which compares the noise levels between channels. The induced metric topology, which we call the noisiness topology, is shown to be natural. We also study topologies that are inherited from the space of meta-probability measures by identifying channels with their Blackwell measures.

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“…This paper is an extended version of our paper that is published in the International Symposium on Information Theory 2017 (ISIT 2017) [ 1 ].…”

Section: Introductionmentioning

confidence: 99%