Pub Date : 2024-07-29DOI: 10.1109/TIT.2024.3435414
Lei Yu;Hao Wu
In this paper, we generalize the log-Sobolev inequalities to Rényi–Sobolev inequalities by replacing the entropy with the two-parameter entropy, which is a generalized version of entropy and closely related to Rényi divergences. We derive the sharp nonlinear dimension-free version of this kind of inequalities. Interestingly, the resultant inequalities show a transition phenomenon depending on the parameters. We then connect Rényi–Sobolev inequalities to contractive and data-processing inequalities, concentration inequalities, and spectral graph theory. Our proofs in this paper are based on the information-theoretic characterization of the Rényi–Sobolev inequalities, as well as the method of types.
{"title":"Rényi–Sobolev Inequalities and Connections to Spectral Graph Theory","authors":"Lei Yu;Hao Wu","doi":"10.1109/TIT.2024.3435414","DOIUrl":"10.1109/TIT.2024.3435414","url":null,"abstract":"In this paper, we generalize the log-Sobolev inequalities to Rényi–Sobolev inequalities by replacing the entropy with the two-parameter entropy, which is a generalized version of entropy and closely related to Rényi divergences. We derive the sharp nonlinear dimension-free version of this kind of inequalities. Interestingly, the resultant inequalities show a transition phenomenon depending on the parameters. We then connect Rényi–Sobolev inequalities to contractive and data-processing inequalities, concentration inequalities, and spectral graph theory. Our proofs in this paper are based on the information-theoretic characterization of the Rényi–Sobolev inequalities, as well as the method of types.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"6809-6822"},"PeriodicalIF":2.2,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141864888","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-25DOI: 10.1109/TIT.2024.3433550
Ching-Yi Lai;Pin-Chieh Tseng;Wei-Hsuan Yu
In this paper, we explore the application of semidefinite programming to the realm of quantum codes, specifically focusing on codeword stabilized (CWS) codes with entanglement assistance. Notably, we utilize the isotropic subgroup of the CWS group and the set of word operators of a CWS-type quantum code to derive an upper bound on the minimum distance. Furthermore, this characterization can be incorporated into the associated distance enumerators, enabling us to construct semidefinite constraints that lead to SDP bounds on the minimum distance or size of CWS-type quantum codes. We illustrate several instances where SDP bounds outperform LP bounds, and there are even cases where LP fails to yield meaningful results, while SDP consistently provides tighter and relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme weight enumerators and shadow enumerators for codeword stabilized codes, enhancing our understanding of quantum codes.
{"title":"Semidefinite Programming Bounds on the Size of Entanglement-Assisted Codeword Stabilized Quantum Codes","authors":"Ching-Yi Lai;Pin-Chieh Tseng;Wei-Hsuan Yu","doi":"10.1109/TIT.2024.3433550","DOIUrl":"10.1109/TIT.2024.3433550","url":null,"abstract":"In this paper, we explore the application of semidefinite programming to the realm of quantum codes, specifically focusing on codeword stabilized (CWS) codes with entanglement assistance. Notably, we utilize the isotropic subgroup of the CWS group and the set of word operators of a CWS-type quantum code to derive an upper bound on the minimum distance. Furthermore, this characterization can be incorporated into the associated distance enumerators, enabling us to construct semidefinite constraints that lead to SDP bounds on the minimum distance or size of CWS-type quantum codes. We illustrate several instances where SDP bounds outperform LP bounds, and there are even cases where LP fails to yield meaningful results, while SDP consistently provides tighter and relevant bounds. Finally, we also provide interpretations of the Shor-Laflamme weight enumerators and shadow enumerators for codeword stabilized codes, enhancing our understanding of quantum codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7867-7881"},"PeriodicalIF":2.2,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-23DOI: 10.1109/TIT.2024.3432658
Gabriel Lacerda;Sergio Romaña
Given a compact smooth boundaryless manifold with dimension greater than one endowed with a locally positive non-atomic measure $mu $ , we prove that typical $mu $ -preserving homeomorphisms have upper metric mean dimension, with respect to the Riemannian distance, equal to the dimension of the manifold. Moreover, we prove that $mu $ is a measure of maximal metric mean dimension, with respect to the variational principle established by Velozo and Velozo.
{"title":"Typical Conservative Homeomorphisms Have Total Metric Mean Dimension","authors":"Gabriel Lacerda;Sergio Romaña","doi":"10.1109/TIT.2024.3432658","DOIUrl":"10.1109/TIT.2024.3432658","url":null,"abstract":"Given a compact smooth boundaryless manifold with dimension greater than one endowed with a locally positive non-atomic measure \u0000<inline-formula> <tex-math>$mu $ </tex-math></inline-formula>\u0000, we prove that typical \u0000<inline-formula> <tex-math>$mu $ </tex-math></inline-formula>\u0000-preserving homeomorphisms have upper metric mean dimension, with respect to the Riemannian distance, equal to the dimension of the manifold. Moreover, we prove that \u0000<inline-formula> <tex-math>$mu $ </tex-math></inline-formula>\u0000 is a measure of maximal metric mean dimension, with respect to the variational principle established by Velozo and Velozo.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7664-7672"},"PeriodicalIF":2.2,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780041","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-23DOI: 10.1109/tit.2024.3432573
Ligong Wang, Gregory W. Wornell
{"title":"Communication Over Discrete Channels Subject to State Obfuscation","authors":"Ligong Wang, Gregory W. Wornell","doi":"10.1109/tit.2024.3432573","DOIUrl":"https://doi.org/10.1109/tit.2024.3432573","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"245 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780040","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-23DOI: 10.1109/TIT.2024.3432822
Guodong Li;Ningning Wang;Sihuang Hu;Min Ye
An <inline-formula> <tex-math>$(n, k, ell)$ </tex-math></inline-formula> array code has k information coordinates and <inline-formula> <tex-math>$r = n - k$ </tex-math></inline-formula> parity coordinates, where each coordinate is a vector in <inline-formula> <tex-math>$mathbb {F}_{q}^{ell }$ </tex-math></inline-formula> for some finite field <inline-formula> <tex-math>$mathbb {F}_{q}$ </tex-math></inline-formula> . An <inline-formula> <tex-math>$(n, k, ell)$ </tex-math></inline-formula> MDS array code has the additional property that any k out of n coordinates suffice to recover the whole codeword. Dimakis et al. considered the problem of repairing the erasure of a single coordinate and proved a lower bound on the amount of data transmission that is needed for the repair. A minimum storage regenerating (MSR) code with repair degree d is an MDS array code that achieves this lower bound for the repair of any single erased coordinate from any d out of <inline-formula> <tex-math>$n-1$ </tex-math></inline-formula> remaining coordinates. An MSR code has the optimal access property if the amount of accessed data is the same as the amount of transmitted data in the repair procedure. The sub-packetization <inline-formula> <tex-math>$ell $ </tex-math></inline-formula> and the field size q are of paramount importance in MSR code constructions. For optimal-access MSR codes, Balaji et al. proved that <inline-formula> <tex-math>$ell geq s^{left lceil {{ n/s }}right rceil }$ </tex-math></inline-formula> , where <inline-formula> <tex-math>$s = d-k+1$ </tex-math></inline-formula> . Rawat et al. showed that this lower bound is attainable for all admissible values of d when the field size is exponential in n. After that, tremendous efforts have been devoted to reducing the field size. However, so far, reduction to a linear field size is only available for <inline-formula> <tex-math>$din {k+1,k+2,k+3}$ </tex-math></inline-formula> and <inline-formula> <tex-math>$d=n-1$ </tex-math></inline-formula> . In this paper, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization <inline-formula> <tex-math>$ell = s^{left lceil {{ n/s }}right rceil }$ </tex-math></inline-formula> for all d between <inline-formula> <tex-math>$k+1$ </tex-math></inline-formula> and <inline-formula> <tex-math>$n-1$ </tex-math></inline-formula> , resolving an open problem in the survey (Ramkumar et al., Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with an even smaller sub-packetization <inline-formula> <tex-math>$s^{left lceil {{ n/(s+1)}}right rceil }$ </tex-math></inline-formula> for all admissible values of d, making significant progress on another open problem in the survey. Previously, MSR codes with <inline-formula> <tex-math>$ell =s^{left lceil {{ n/(s+1)}}right rceil }$ </tex-math></inline-formul
{"title":"MSR Codes With Linear Field Size and Smallest Sub-Packetization for Any Number of Helper Nodes","authors":"Guodong Li;Ningning Wang;Sihuang Hu;Min Ye","doi":"10.1109/TIT.2024.3432822","DOIUrl":"10.1109/TIT.2024.3432822","url":null,"abstract":"An \u0000<inline-formula> <tex-math>$(n, k, ell)$ </tex-math></inline-formula>\u0000 array code has k information coordinates and \u0000<inline-formula> <tex-math>$r = n - k$ </tex-math></inline-formula>\u0000 parity coordinates, where each coordinate is a vector in \u0000<inline-formula> <tex-math>$mathbb {F}_{q}^{ell }$ </tex-math></inline-formula>\u0000 for some finite field \u0000<inline-formula> <tex-math>$mathbb {F}_{q}$ </tex-math></inline-formula>\u0000. An \u0000<inline-formula> <tex-math>$(n, k, ell)$ </tex-math></inline-formula>\u0000 MDS array code has the additional property that any k out of n coordinates suffice to recover the whole codeword. Dimakis et al. considered the problem of repairing the erasure of a single coordinate and proved a lower bound on the amount of data transmission that is needed for the repair. A minimum storage regenerating (MSR) code with repair degree d is an MDS array code that achieves this lower bound for the repair of any single erased coordinate from any d out of \u0000<inline-formula> <tex-math>$n-1$ </tex-math></inline-formula>\u0000 remaining coordinates. An MSR code has the optimal access property if the amount of accessed data is the same as the amount of transmitted data in the repair procedure. The sub-packetization \u0000<inline-formula> <tex-math>$ell $ </tex-math></inline-formula>\u0000 and the field size q are of paramount importance in MSR code constructions. For optimal-access MSR codes, Balaji et al. proved that \u0000<inline-formula> <tex-math>$ell geq s^{left lceil {{ n/s }}right rceil }$ </tex-math></inline-formula>\u0000, where \u0000<inline-formula> <tex-math>$s = d-k+1$ </tex-math></inline-formula>\u0000. Rawat et al. showed that this lower bound is attainable for all admissible values of d when the field size is exponential in n. After that, tremendous efforts have been devoted to reducing the field size. However, so far, reduction to a linear field size is only available for \u0000<inline-formula> <tex-math>$din {k+1,k+2,k+3}$ </tex-math></inline-formula>\u0000 and \u0000<inline-formula> <tex-math>$d=n-1$ </tex-math></inline-formula>\u0000. In this paper, we construct the first class of explicit optimal-access MSR codes with the smallest sub-packetization \u0000<inline-formula> <tex-math>$ell = s^{left lceil {{ n/s }}right rceil }$ </tex-math></inline-formula>\u0000 for all d between \u0000<inline-formula> <tex-math>$k+1$ </tex-math></inline-formula>\u0000 and \u0000<inline-formula> <tex-math>$n-1$ </tex-math></inline-formula>\u0000, resolving an open problem in the survey (Ramkumar et al., Foundations and Trends in Communications and Information Theory: Vol. 19: No. 4). We further propose another class of explicit MSR code constructions (not optimal-access) with an even smaller sub-packetization \u0000<inline-formula> <tex-math>$s^{left lceil {{ n/(s+1)}}right rceil }$ </tex-math></inline-formula>\u0000 for all admissible values of d, making significant progress on another open problem in the survey. Previously, MSR codes with \u0000<inline-formula> <tex-math>$ell =s^{left lceil {{ n/(s+1)}}right rceil }$ </tex-math></inline-formul","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7790-7806"},"PeriodicalIF":2.2,"publicationDate":"2024-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780043","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-22DOI: 10.1109/TIT.2024.3431695
Holger Boche;Volker Pohl;H. Vincent Poor
This paper investigates the complexity of computing the minimum mean square prediction error for wide-sense stationary stochastic processes. It is shown that if the spectral density of the stationary process is a strictly positive, computable continuous function then the minimum mean square error (MMSE) is always a computable number. Nevertheless, we also show that the computation of the MMSE is a $# P_{1}$ complete problem on the set of strictly positive, polynomial-time computable, continuous spectral densities. This means that if, as widely assumed, $FP_{1} neq # P_{1}$ , then there exist strictly positive, polynomial-time computable continuous spectral densities for which the computation of the MMSE is not polynomial-time computable. These results show in particular that under the widely accepted assumptions of complexity theory, the computation of the MMSE is generally much harder than an $NP_{1}$ complete problem.
{"title":"Characterization of the Complexity of Computing the Minimum Mean Square Error of Causal Prediction","authors":"Holger Boche;Volker Pohl;H. Vincent Poor","doi":"10.1109/TIT.2024.3431695","DOIUrl":"10.1109/TIT.2024.3431695","url":null,"abstract":"This paper investigates the complexity of computing the minimum mean square prediction error for wide-sense stationary stochastic processes. It is shown that if the spectral density of the stationary process is a strictly positive, computable continuous function then the minimum mean square error (MMSE) is always a computable number. Nevertheless, we also show that the computation of the MMSE is a \u0000<inline-formula> <tex-math>$# P_{1}$ </tex-math></inline-formula>\u0000 complete problem on the set of strictly positive, polynomial-time computable, continuous spectral densities. This means that if, as widely assumed, \u0000<inline-formula> <tex-math>$FP_{1} neq # P_{1}$ </tex-math></inline-formula>\u0000, then there exist strictly positive, polynomial-time computable continuous spectral densities for which the computation of the MMSE is not polynomial-time computable. These results show in particular that under the widely accepted assumptions of complexity theory, the computation of the MMSE is generally much harder than an \u0000<inline-formula> <tex-math>$NP_{1}$ </tex-math></inline-formula>\u0000 complete problem.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 9","pages":"6627-6638"},"PeriodicalIF":2.2,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10605985","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141780044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-19DOI: 10.1109/TIT.2024.3430842
Nicolas Resch;Chen Yuan;Yihan Zhang
In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of $qgeq 2$ . A code is called $(p,L)_{q}$ -list-decodable if every radius pn Hamming ball contains less than L codewords; $(p,ell ,L)_{q}$ -list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length $ell $ and again stipulate that there be less than L codewords. Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate $(p,ell ,L)_{q}$ -list-recoverable codes, the quantity we call the zero-rate threshold. Denoting this value by $p_{*}$ , we in fact show that codes correcting a $p_{*}+varepsilon $ fraction of errors must have size $O_{varepsilon }(1)$ , i.e., independent of n. Such a result is typically referred to as a “Plotkin bound.” To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a $p_{*}-varepsilon $ fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery. Technically, proving the Plotkin bound boils down to demonstrating the Schur convexity of a certain function defined on the q-simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q-ary list-decoding; however, we point out that this earlier proof is flawed.
{"title":"Zero-Rate Thresholds and New Capacity Bounds for List-Decoding and List-Recovery","authors":"Nicolas Resch;Chen Yuan;Yihan Zhang","doi":"10.1109/TIT.2024.3430842","DOIUrl":"10.1109/TIT.2024.3430842","url":null,"abstract":"In this work we consider the list-decodability and list-recoverability of arbitrary q-ary codes, for all integer values of \u0000<inline-formula> <tex-math>$qgeq 2$ </tex-math></inline-formula>\u0000. A code is called \u0000<inline-formula> <tex-math>$(p,L)_{q}$ </tex-math></inline-formula>\u0000-list-decodable if every radius pn Hamming ball contains less than L codewords; \u0000<inline-formula> <tex-math>$(p,ell ,L)_{q}$ </tex-math></inline-formula>\u0000-list-recoverability is a generalization where we place radius pn Hamming balls on every point of a combinatorial rectangle with side length \u0000<inline-formula> <tex-math>$ell $ </tex-math></inline-formula>\u0000 and again stipulate that there be less than L codewords. Our main contribution is to precisely calculate the maximum value of p for which there exist infinite families of positive rate \u0000<inline-formula> <tex-math>$(p,ell ,L)_{q}$ </tex-math></inline-formula>\u0000-list-recoverable codes, the quantity we call the zero-rate threshold. Denoting this value by \u0000<inline-formula> <tex-math>$p_{*}$ </tex-math></inline-formula>\u0000, we in fact show that codes correcting a \u0000<inline-formula> <tex-math>$p_{*}+varepsilon $ </tex-math></inline-formula>\u0000 fraction of errors must have size \u0000<inline-formula> <tex-math>$O_{varepsilon }(1)$ </tex-math></inline-formula>\u0000, i.e., independent of n. Such a result is typically referred to as a “Plotkin bound.” To complement this, a standard random code with expurgation construction shows that there exist positive rate codes correcting a \u0000<inline-formula> <tex-math>$p_{*}-varepsilon $ </tex-math></inline-formula>\u0000 fraction of errors. We also follow a classical proof template (typically attributed to Elias and Bassalygo) to derive from the zero-rate threshold other tradeoffs between rate and decoding radius for list-decoding and list-recovery. Technically, proving the Plotkin bound boils down to demonstrating the Schur convexity of a certain function defined on the q-simplex as well as the convexity of a univariate function derived from it. We remark that an earlier argument claimed similar results for q-ary list-decoding; however, we point out that this earlier proof is flawed.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 9","pages":"6211-6238"},"PeriodicalIF":2.2,"publicationDate":"2024-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141746152","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-16DOI: 10.1109/TIT.2024.3428325
Zhan Yu;Jun Fan;Zhongjie Shi;Ding-Xuan Zhou
In recent years, different types of distributed and parallel learning schemes have received increasing attention for their strong advantages in handling large-scale data information. In the information era, to face the big data challenges that stem from functional data analysis very recently, we propose a novel distributed gradient descent functional learning (DGDFL) algorithm to tackle functional data across numerous local machines (processors) in the framework of reproducing kernel Hilbert space. Based on integral operator approaches, we provide the first theoretical understanding of the DGDFL algorithm in many different aspects of the literature. On the way of understanding DGDFL, firstly, a data-based gradient descent functional learning (GDFL) algorithm associated with a single-machine model is proposed and comprehensively studied. Under mild conditions, confidence-based optimal learning rates of DGDFL are obtained without the saturation boundary on the regularity index suffered in previous works in functional regression. We further provide a semi-supervised DGDFL approach to weaken the restriction on the maximal number of local machines to ensure optimal rates. To our best knowledge, the DGDFL provides the first divide-and-conquer iterative training approach to functional learning based on data samples of intrinsically infinite-dimensional random functions (functional covariates) and enriches the methodologies for functional data analysis.
{"title":"Distributed Gradient Descent for Functional Learning","authors":"Zhan Yu;Jun Fan;Zhongjie Shi;Ding-Xuan Zhou","doi":"10.1109/TIT.2024.3428325","DOIUrl":"10.1109/TIT.2024.3428325","url":null,"abstract":"In recent years, different types of distributed and parallel learning schemes have received increasing attention for their strong advantages in handling large-scale data information. In the information era, to face the big data challenges that stem from functional data analysis very recently, we propose a novel distributed gradient descent functional learning (DGDFL) algorithm to tackle functional data across numerous local machines (processors) in the framework of reproducing kernel Hilbert space. Based on integral operator approaches, we provide the first theoretical understanding of the DGDFL algorithm in many different aspects of the literature. On the way of understanding DGDFL, firstly, a data-based gradient descent functional learning (GDFL) algorithm associated with a single-machine model is proposed and comprehensively studied. Under mild conditions, confidence-based optimal learning rates of DGDFL are obtained without the saturation boundary on the regularity index suffered in previous works in functional regression. We further provide a semi-supervised DGDFL approach to weaken the restriction on the maximal number of local machines to ensure optimal rates. To our best knowledge, the DGDFL provides the first divide-and-conquer iterative training approach to functional learning based on data samples of intrinsically infinite-dimensional random functions (functional covariates) and enriches the methodologies for functional data analysis.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 9","pages":"6547-6571"},"PeriodicalIF":2.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-16DOI: 10.1109/TIT.2024.3429277
Zicheng Ye;Yuan Li;Huazi Zhang;Jun Wang;Guiying Yan;Zhiming Ma
The automorphism ensemble (AE) decoding framework for polar codes attracts much attention recently. It decodes multiple permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to successive cancellation list (SCL) decoding. However, the AE decoding framework is ineffective for permutations falling into the lower-triangular affine (LTA) automorphism group, as they are invariant under SC decoding. Therefore, the block lower-triangular affine (BLTA) group was discovered to achieve better AE decoding performance. However, the equivalence of the BLTA group and the complete affine automorphism group was unresolved. Additionally, some automorphisms in BLTA group are also SC-invariant, thus are redundant in AE decoding. In this paper, we prove that BLTA group coincides with the complete automorphisms of decreasing polar codes that can be formulated as affine transformations. Also, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, We present an algorithm that efficiently identifies all SC-invariant affine automorphisms under specific constructions.
{"title":"Affine Automorphism Group of Polar Codes","authors":"Zicheng Ye;Yuan Li;Huazi Zhang;Jun Wang;Guiying Yan;Zhiming Ma","doi":"10.1109/TIT.2024.3429277","DOIUrl":"10.1109/TIT.2024.3429277","url":null,"abstract":"The automorphism ensemble (AE) decoding framework for polar codes attracts much attention recently. It decodes multiple permuted codewords with successive cancellation (SC) decoders in parallel and hence has lower latency compared to successive cancellation list (SCL) decoding. However, the AE decoding framework is ineffective for permutations falling into the lower-triangular affine (LTA) automorphism group, as they are invariant under SC decoding. Therefore, the block lower-triangular affine (BLTA) group was discovered to achieve better AE decoding performance. However, the equivalence of the BLTA group and the complete affine automorphism group was unresolved. Additionally, some automorphisms in BLTA group are also SC-invariant, thus are redundant in AE decoding. In this paper, we prove that BLTA group coincides with the complete automorphisms of decreasing polar codes that can be formulated as affine transformations. Also, we find a necessary and sufficient condition related to the block lower-triangular structure of transformation matrices to identify SC-invariant automorphisms. Furthermore, We present an algorithm that efficiently identifies all SC-invariant affine automorphisms under specific constructions.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 9","pages":"6280-6296"},"PeriodicalIF":2.2,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141719805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-15DOI: 10.1109/TIT.2024.3421753
{"title":"IEEE Transactions on Information Theory Information for Authors","authors":"","doi":"10.1109/TIT.2024.3421753","DOIUrl":"https://doi.org/10.1109/TIT.2024.3421753","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 8","pages":"C3-C3"},"PeriodicalIF":2.2,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10599341","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141624158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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