Continuity of Channel Parameters and Operations under Various DMC Topologies

Nasser, Rajai

doi:10.3390/e20050330

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%

“…The next lemma shows that the Blackwell measures of channels with a small |I(W − ) − I(W )| are close (in the noisiness metric sense) to measures in POL G . 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: Now Define the Setmentioning

confidence: 99%

“…where (a) follows from[13, Proposition 10], (b) follows from the definition of the (−, +)-convolution (see Page 20 of[13]), (c) follows from the properties of the push-forward probability measure, and (d) follows from the definition of the C −,+ map (see Page 20 of[13]) and Fubini's theorem.The lemma now follows from the fact that |I(W − ) − I(W )| = I(W ) − I(W − ).…”

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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%

Section: Now Define the Setmentioning

confidence: 99%

mentioning

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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“…X , * is studied in [7]. In this paper, we focus on one particular requirement that we consider the most basic property required from any "acceptable" topology on DMC…”

Section: A Natural Topologies On Dmcmentioning

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 [7]. We also study the polarization process of Arıkan [8] and its convergence under various topologies in [9].…”

Section: Introductionmentioning

confidence: 99%

Topological structures on DMC spaces

Nasser

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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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. We show that this topology is compact, path-connected and metrizable. 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. On the other hand, if |X | ≥ 2, a Hausdorff natural topology is not Baire and it is not locally compact anywhere. This implies that no natural topology can be completely metrized if |X | ≥ 2. 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 show that in the strong topology, a subspace is compact if and only if it is rank-bounded and strongly-closed. We provide a necessary and sufficient condition for a sequence of channels to converge in the strong topology. 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. We show that the weak- * topology is exactly the same as the noisiness topology and hence it is natural. We prove that if |X | ≥ 2, the total variation topology is not natural nor Baire, hence it is not completely metrizable. Moreover, it is not locally compact anywhere. Finally, we show that the Borel σ-algebra is the same for all Hausdorff natural topologies.

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Communicating Correlated Sources Over MAC and Interference Channels I: Separation-Based Schemes

Padakandla

2020

IEEE Trans. Inform. Theory

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We consider the two scenarios of communicating a pair S 1 , S 2 of distributed correlated sources over 2−user multiple access (MAC) and interference channels (IC) respectively. While in the MAC problem, the receiver intends to reconstruct both sources losslessly, in the IC problem, receiver j intends to reconstruct S j losslessly. We undertake a Shannon theoretic study and focus on achievability, i.e., characterizing sufficient conditions. In the absence of a Gaćs-Körner-Witsenhausen common part, the current known single-letter (S-L) coding schemes are constrained to choosing X jt -the symbol input on the channel by encoder j at time t -based only S jt -the source symbol observed by it, at time t, resulting in the pmf p X1X2 of the inputs X 1 , X 2 constrained to the S-L long Markov Chain (LMC) X 1 − S 1 − S 2 − X 2 . Taking the lead of Dueck's example [1], we recognize that the latter constraint is debilitating, leading to sub-optimality of S-L coding schemes. The goal of our work is to design a coding scheme wherein (i) the choice of X jt is based on multiple source symbols S l j , and is yet ii) amenable to performance characterization via S-L expressions. In this article, we present the first part of our findings. We propose a new separation-based coding scheme comprising of (i) a fixed block-length (B-L) code that enables choice of X jt based on a generic number l of source symbols, thus permitting correlation of the input symbols X 1 , X 2 through a multiletter LMC X 1 − S l 1 − S l 2 − X 2 , (ii) arbitrarily large B-L codes superimposed on multiple sub-blocks of the fixed B-L code that communicate the rest of the information necessary for source reconstruction at the decoder(s), and (iii) a multiplexing unit based on the interleaving technique [2] that ensures the latter codes of arbitrarily large B-L experience a memoryless channel. This careful stitching of S-L coding techniques enables us to devise a multi-letter coding scheme that permits characterization of sufficient conditions via a S-L expression. We prove that the derived inner bound is strictly larger than the current known largest inner bounds for both the MAC and IC problems.Since the proposed coding scheme is inherently separation based, the derived inner bound does not subsume the current known largest. In the second part of our work, we propose to enlarge the inner bound derived in this article by incorporating the technique of inducing source correlation onto channel inputs [3].

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Continuity of Channel Parameters and Operations under Various DMC Topologies

Cited by 3 publications

References 23 publications

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

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

Topological structures on DMC spaces

Communicating Correlated Sources Over MAC and Interference Channels I: Separation-Based Schemes

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