An $(n, k, d; r)_{q}$ -locally repairable code (LRC) is called a Singleton-optimal LRC if it achieves the Singleton-type bound. Analogous to the classical MDS conjecture, the maximal length problem of Singleton-optimal LRCs has attracted a lot of attention in recent years. In this paper, we give an improved upper bound for the length of q-ary Singleton-optimal LRCs with disjoint repair groups such that $(r+1)mid n$ based on the parity-check matrix approach. In particular, for any Singleton-optimal $(n, k, d; r)_{q}$ -LRCs, we show that: 1) $nle q+d-4$ , when $r=2$ and $d=3e+8$ with $ege 0$ ; 2) $nleq (r+1)left lfloor {{frac {2(q^{2}+q+1)}{r(r+1)} +e+1}}right rfloor $ , when $dge 8$ and $max left {{{3,frac {d-e-6}{e+1}}}right }le rle frac {d-e-3}{e+1}$ for any $0le ele left lfloor {{frac {d-6}{4} }}right rfloor $ . Furthermore, we establish equivalent connections between the existence of Singleton-optimal $(n,k,d;r)_{q}$ -LRCs for $d=6, r=3$ and $d=7, r=2$ with disjoint repair groups and some subsets of lines in finite projective space with certain properties. Consequently, we prove that the length of q-ary Singleton-optimal LRCs with minimum distance $d=6$ and locality $r=3$ is upper bounded by $O(q^{1.5})$ . We construct Singleton-optimal $(8le nle q+1,k,d=6,r=3)_{q}$ -LRC with disjoint repair groups such that $4mid n$ and determine the exact value of the maximum code length for some specific q. We also prove the existence of $(n, k, d=7; r=2)_{q}$ -Singleton-optimal LRCs for $n approx sqrt {2}q$ .
{"title":"Bounds and Constructions of Singleton-Optimal Locally Repairable Codes With Small Localities","authors":"Weijun Fang;Ran Tao;Fang-Wei Fu;Bin Chen;Shu-Tao Xia","doi":"10.1109/TIT.2024.3448265","DOIUrl":"10.1109/TIT.2024.3448265","url":null,"abstract":"An \u0000<inline-formula> <tex-math>$(n, k, d; r)_{q}$ </tex-math></inline-formula>\u0000-locally repairable code (LRC) is called a Singleton-optimal LRC if it achieves the Singleton-type bound. Analogous to the classical MDS conjecture, the maximal length problem of Singleton-optimal LRCs has attracted a lot of attention in recent years. In this paper, we give an improved upper bound for the length of q-ary Singleton-optimal LRCs with disjoint repair groups such that \u0000<inline-formula> <tex-math>$(r+1)mid n$ </tex-math></inline-formula>\u0000 based on the parity-check matrix approach. In particular, for any Singleton-optimal \u0000<inline-formula> <tex-math>$(n, k, d; r)_{q}$ </tex-math></inline-formula>\u0000-LRCs, we show that: 1) \u0000<inline-formula> <tex-math>$nle q+d-4$ </tex-math></inline-formula>\u0000, when \u0000<inline-formula> <tex-math>$r=2$ </tex-math></inline-formula>\u0000 and \u0000<inline-formula> <tex-math>$d=3e+8$ </tex-math></inline-formula>\u0000 with \u0000<inline-formula> <tex-math>$ege 0$ </tex-math></inline-formula>\u0000; 2) \u0000<inline-formula> <tex-math>$nleq (r+1)left lfloor {{frac {2(q^{2}+q+1)}{r(r+1)} +e+1}}right rfloor $ </tex-math></inline-formula>\u0000, when \u0000<inline-formula> <tex-math>$dge 8$ </tex-math></inline-formula>\u0000 and \u0000<inline-formula> <tex-math>$max left {{{3,frac {d-e-6}{e+1}}}right }le rle frac {d-e-3}{e+1}$ </tex-math></inline-formula>\u0000 for any \u0000<inline-formula> <tex-math>$0le ele left lfloor {{frac {d-6}{4} }}right rfloor $ </tex-math></inline-formula>\u0000. Furthermore, we establish equivalent connections between the existence of Singleton-optimal \u0000<inline-formula> <tex-math>$(n,k,d;r)_{q}$ </tex-math></inline-formula>\u0000-LRCs for \u0000<inline-formula> <tex-math>$d=6, r=3$ </tex-math></inline-formula>\u0000 and \u0000<inline-formula> <tex-math>$d=7, r=2$ </tex-math></inline-formula>\u0000 with disjoint repair groups and some subsets of lines in finite projective space with certain properties. Consequently, we prove that the length of q-ary Singleton-optimal LRCs with minimum distance \u0000<inline-formula> <tex-math>$d=6$ </tex-math></inline-formula>\u0000 and locality \u0000<inline-formula> <tex-math>$r=3$ </tex-math></inline-formula>\u0000 is upper bounded by \u0000<inline-formula> <tex-math>$O(q^{1.5})$ </tex-math></inline-formula>\u0000. We construct Singleton-optimal \u0000<inline-formula> <tex-math>$(8le nle q+1,k,d=6,r=3)_{q}$ </tex-math></inline-formula>\u0000-LRC with disjoint repair groups such that \u0000<inline-formula> <tex-math>$4mid n$ </tex-math></inline-formula>\u0000 and determine the exact value of the maximum code length for some specific q. We also prove the existence of \u0000<inline-formula> <tex-math>$(n, k, d=7; r=2)_{q}$ </tex-math></inline-formula>\u0000-Singleton-optimal LRCs for \u0000<inline-formula> <tex-math>$n approx sqrt {2}q$ </tex-math></inline-formula>\u0000.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"6842-6856"},"PeriodicalIF":2.2,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194886","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 : 2024-08-21DOI: 10.1109/tit.2024.3447552
Evan Scope Crafts, Xianyang Zhang, Bo Zhao
{"title":"Bayesian Cramér-Rao Bound Estimation with Score-Based Models","authors":"Evan Scope Crafts, Xianyang Zhang, Bo Zhao","doi":"10.1109/tit.2024.3447552","DOIUrl":"https://doi.org/10.1109/tit.2024.3447552","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"160 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194902","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 : 2024-08-21DOI: 10.1109/TIT.2024.3447224
Minh-Toan Nguyen
The I-MMSE formula connects two important quantities in information theory and estimation theory: the mutual information and the minimum mean-squared error (MMSE). It states that in a scalar Gaussian channel, the derivative of the mutual information with respect to the signal-to-noise ratio (SNR) is one-half of the MMSE. Although any derivative at a fixed order can be computed in principle, a general formula for all the derivatives is still unknown. In this paper, we derive this general formula for vector Gaussian channels. The obtained result is remarkably similar to the classic cumulant-moment relation in statistical theory.
{"title":"Derivatives of Mutual Information in Gaussian Channels","authors":"Minh-Toan Nguyen","doi":"10.1109/TIT.2024.3447224","DOIUrl":"10.1109/TIT.2024.3447224","url":null,"abstract":"The I-MMSE formula connects two important quantities in information theory and estimation theory: the mutual information and the minimum mean-squared error (MMSE). It states that in a scalar Gaussian channel, the derivative of the mutual information with respect to the signal-to-noise ratio (SNR) is one-half of the MMSE. Although any derivative at a fixed order can be computed in principle, a general formula for all the derivatives is still unknown. In this paper, we derive this general formula for vector Gaussian channels. The obtained result is remarkably similar to the classic cumulant-moment relation in statistical theory.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7525-7531"},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194889","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 : 2024-08-21DOI: 10.1109/TIT.2024.3442211
Ankit Pratap Singh;Namrata Vaswani
This work considers two related learning problems in a federated attack-prone setting – federated principal components analysis (PCA) and federated low rank column-wise sensing (LRCS). The node attacks are assumed to be Byzantine which means that the attackers are omniscient and can collude. We introduce a novel provably Byzantine-resilient communication-efficient and sample-efficient algorithm, called Subspace-Median, that solves the PCA problem and is a key part of the solution for the LRCS problem. We also study the most natural Byzantine-resilient solution for federated PCA, a geometric median based modification of the federated power method, and explain why it is not useful. Our second main contribution is a complete alternating gradient descent (GD) and minimization (altGDmin) algorithm for Byzantine-resilient horizontally federated LRCS and sample and communication complexity guarantees for it. Extensive simulation experiments are used to corroborate our theoretical guarantees. The ideas that we develop for LRCS are easily extendable to other LR recovery problems as well.
{"title":"Byzantine-Resilient Federated PCA and Low-Rank Column-Wise Sensing","authors":"Ankit Pratap Singh;Namrata Vaswani","doi":"10.1109/TIT.2024.3442211","DOIUrl":"10.1109/TIT.2024.3442211","url":null,"abstract":"This work considers two related learning problems in a federated attack-prone setting – federated principal components analysis (PCA) and federated low rank column-wise sensing (LRCS). The node attacks are assumed to be Byzantine which means that the attackers are omniscient and can collude. We introduce a novel provably Byzantine-resilient communication-efficient and sample-efficient algorithm, called Subspace-Median, that solves the PCA problem and is a key part of the solution for the LRCS problem. We also study the most natural Byzantine-resilient solution for federated PCA, a geometric median based modification of the federated power method, and explain why it is not useful. Our second main contribution is a complete alternating gradient descent (GD) and minimization (altGDmin) algorithm for Byzantine-resilient horizontally federated LRCS and sample and communication complexity guarantees for it. Extensive simulation experiments are used to corroborate our theoretical guarantees. The ideas that we develop for LRCS are easily extendable to other LR recovery problems as well.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"8001-8025"},"PeriodicalIF":2.2,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194885","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 : 2024-08-21DOI: 10.1109/tit.2024.3447050
Liyang Lu, Zhaocheng Wang, Zhen Gao, Sheng Chen, H. Vincent Poor
{"title":"Block-Sparse Tensor Recovery","authors":"Liyang Lu, Zhaocheng Wang, Zhen Gao, Sheng Chen, H. Vincent Poor","doi":"10.1109/tit.2024.3447050","DOIUrl":"https://doi.org/10.1109/tit.2024.3447050","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"63 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194890","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 : 2024-08-20DOI: 10.1109/TIT.2024.3442005
{"title":"IEEE Transactions on Information Theory Information for Authors","authors":"","doi":"10.1109/TIT.2024.3442005","DOIUrl":"https://doi.org/10.1109/TIT.2024.3442005","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 9","pages":"C3-C3"},"PeriodicalIF":2.2,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10640357","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142013447","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 : 2024-08-20DOI: 10.1109/TIT.2024.3442003
{"title":"IEEE Transactions on Information Theory Publication Information","authors":"","doi":"10.1109/TIT.2024.3442003","DOIUrl":"https://doi.org/10.1109/TIT.2024.3442003","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 9","pages":"C2-C2"},"PeriodicalIF":2.2,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10642978","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142045105","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 : 2024-08-20DOI: 10.1109/TIT.2024.3446614
Salman Beigi;Marco Tomamichel
We introduce a new quantum decoder based on a variant of the pretty good measurement, but defined via an alternative matrix quotient. We then use this novel decoder to derive new lower bounds on the error exponent both in the one-shot and asymptotic regimes for the classical-quantum and the entanglement-assisted channel coding problems. Our bounds are expressed in terms of measured (for the one-shot bounds) and sandwiched (for the asymptotic bounds) channel Rényi mutual information of order between 1/2 and 1. The bounds are not comparable with some previously established bounds for general channels, yet they are tight (for rates close to capacity) when the channel is classical. Finally, we also use our new decoder to rederive Cheng’s recent tight bound on the decoding error probability, which implies that most existing asymptotic results also hold for the new decoder.
{"title":"Lower Bounds on Error Exponents via a New Quantum Decoder","authors":"Salman Beigi;Marco Tomamichel","doi":"10.1109/TIT.2024.3446614","DOIUrl":"https://doi.org/10.1109/TIT.2024.3446614","url":null,"abstract":"We introduce a new quantum decoder based on a variant of the pretty good measurement, but defined via an alternative matrix quotient. We then use this novel decoder to derive new lower bounds on the error exponent both in the one-shot and asymptotic regimes for the classical-quantum and the entanglement-assisted channel coding problems. Our bounds are expressed in terms of measured (for the one-shot bounds) and sandwiched (for the asymptotic bounds) channel Rényi mutual information of order between 1/2 and 1. The bounds are not comparable with some previously established bounds for general channels, yet they are tight (for rates close to capacity) when the channel is classical. Finally, we also use our new decoder to rederive Cheng’s recent tight bound on the decoding error probability, which implies that most existing asymptotic results also hold for the new decoder.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"7882-7891"},"PeriodicalIF":2.2,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142517948","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}
We consider a version of the classical group testing problem motivated by PCR testing for COVID-19. In the so-called tropical group testing model, the outcome of a test is the lowest cycle threshold (Ct) level of the individuals pooled within it, rather than a simple binary indicator variable. We introduce the tropical counterparts of three classical non-adaptive algorithms (COMP, DD and SCOMP), and analyse their behaviour through both simulations and bounds on error probabilities. By comparing the results of the tropical and classical algorithms, we gain insight into the extra information provided by learning the outcomes (Ct levels) of the tests. We show that in a limiting regime the tropical COMP algorithm requires as many tests as its classical counterpart, but that for sufficiently dense problems tropical DD can recover more information with fewer tests, and can be viewed as essentially optimal in certain regimes.
{"title":"Small Error Algorithms for Tropical Group Testing","authors":"Vivekanand Paligadu;Oliver Johnson;Matthew Aldridge","doi":"10.1109/TIT.2024.3445271","DOIUrl":"10.1109/TIT.2024.3445271","url":null,"abstract":"We consider a version of the classical group testing problem motivated by PCR testing for COVID-19. In the so-called tropical group testing model, the outcome of a test is the lowest cycle threshold (Ct) level of the individuals pooled within it, rather than a simple binary indicator variable. We introduce the tropical counterparts of three classical non-adaptive algorithms (COMP, DD and SCOMP), and analyse their behaviour through both simulations and bounds on error probabilities. By comparing the results of the tropical and classical algorithms, we gain insight into the extra information provided by learning the outcomes (Ct levels) of the tests. We show that in a limiting regime the tropical COMP algorithm requires as many tests as its classical counterpart, but that for sufficiently dense problems tropical DD can recover more information with fewer tests, and can be viewed as essentially optimal in certain regimes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"7232-7250"},"PeriodicalIF":2.2,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194897","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 : 2024-08-19DOI: 10.1109/TIT.2024.3445998
Michael X. Cao;Navneeth Ramakrishnan;Mario Berta;Marco Tomamichel
We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate-deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.
{"title":"Channel Simulation: Finite Blocklengths and Broadcast Channels","authors":"Michael X. Cao;Navneeth Ramakrishnan;Mario Berta;Marco Tomamichel","doi":"10.1109/TIT.2024.3445998","DOIUrl":"10.1109/TIT.2024.3445998","url":null,"abstract":"We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate-deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"6780-6808"},"PeriodicalIF":2.2,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194892","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：