Topological structures on DMC spaces

2017 IEEE International Symposium on Information Theory (ISIT)

2017

Self Cite

A channel W is said to be input-degraded from another channel W ′ if W can be simulated from W ′ by randomization at the input. We provide a necessary and sufficient condition for a channel to be input-degraded from another one. We show that any decoder that is good for W ′ is also good for W . We provide two characterizations for input-degradedness, one of which is similar to the Blackwell-Sherman-Stein theorem. We say that two channels are input-equivalent if they are input-degraded from each other. We study the topologies that can be constructed on the space of input-equivalent channels, and we investigate their properties. Moreover, we study the continuity of several channel parameters and operations under these topologies. 1 of the results in [3] and [4] can be replicated (with some variation) for the space of input-equivalent channels.In Section II, we introduce the preliminaries for this paper. In Section III, we introduce and study the input-degradedness ordering. Various operational implications and characterizations of input-degradedness are provided in Section IV. The quotient topology of the space of input-equivalent channels with fixed input and output alphabets is studied in Section V. The space of input-equivalent channels with fixed output alphabet and arbitrary but finite input alphabet is defined in Section VI. A topology on this space is said to be natural if it induces the quotient topology on the subspaces of input-equivalent channels with fixed input alphabet. In Section VI, we investigate the properties of natural topologies. The finest natural topology, which we call the strong topology, is studied in Section VII. The similarity metric on the space of input-equivalent channels is introduced in Section VIII. We study the continuity of various channel parameters and operations under the strong and similarity topologies in Section IX. Finally, we show that the Borel σ-algebra is the same for all Hausdorff natural topologies. II. PRELIMINARIESWe assume that the reader is familiar with the basic concepts of general topology. The main concepts and theorems that we need can be found in the preliminaries section of [3]. A. Measure theoretic notationsThe set of probability measures on a measurable space (M, Σ) is denoted as P(M, Σ). For every P 1 , P 2 ∈ P(M, Σ), the total variation distance between P 1 and P 2 is defined as:Let P be a probability measure on (M, Σ), and let f : M → M ′ be a measurable mapping from (M, Σ) to another measurable space (M ′ , Σ ′ ). The push-forward probability measure of P by f is the probability measure f # P on (M ′ , Σ ′ ) defined as (f # P )(A ′ ) = P (f −1 (A ′ )) for every A ′ ∈ Σ ′ . If A is a subset of P(M, Σ), we define its push-forward by f as f # (A) = {f # P : P ∈ A}.We denote the product of two measurable spaces (M 1 , Σ 1 ) and (M 2 , Σ 2 ) as (M 1 × M 2 , Σ 1 ⊗ Σ 2 ). If P 1 ∈ P(M 1 , Σ 1 ) and P 2 ∈ P(M 2 , Σ 2 ), we denote the product of P 1 and P 2 as P 1 × P 2 . Let A 1 and A 2 be two subsets of P(M 1 , Σ 1 ) and P(M 2 , Σ 2 ) respectively. We defi...

“…where (a) follows from (3). Since this is true for every R 1 ∈ R(co(Ŵ 1 ), co(Ŵ ′ 1 )) and every R 2 ∈ R(co(Ŵ 2 ), co(Ŵ ′ 2 )), we conclude that d…”

Section: Appendix D Proof Of Proposition 19mentioning

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Section: Definition 4 We Define the Strong Topology Tmentioning

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Section: Introductionmentioning

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See 3 more Smart Citations

On the input-degradedness and input-equivalence between channels

2017 IEEE International Symposium on Information Theory (ISIT)

2017

Self Cite

Continuity of Channel Parameters and Operations under Various DMC Topologies

2018

Entropy

Abstract:We study the continuity of many channel parameters and operations under various topologies on the space of equivalent discrete memoryless channels (DMC). We show that mutual information, channel capacity, Bhattacharyya parameter, probability of error of a fixed code and optimal probability of error for a given code rate and block length are continuous under various DMC topologies. We also show that channel operations such as sums, products, interpolations and Arıkan-style transformations are continuous.

Topological Structures on DMC Spaces †

2018

Entropy

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.