IEEE Transactions on Information Theory最新文献

{"title":"Improved Field Size Bounds for Higher Order MDS Codes","authors":"Joshua Brakensiek;Manik Dhar;Sivakanth Gopi","doi":"10.1109/TIT.2024.3449030","DOIUrl":"10.1109/TIT.2024.3449030","url":null,"abstract":"Higher order MDS codes are an interesting generalization of MDS codes recently introduced by Brakensiek et al., (2023). In later works, they were shown to be intimately connected to optimally list-decodable codes and maximally recoverable tensor codes. Therefore (explicit) constructions of higher order MDS codes over small fields is an important open problem. Higher order MDS codes are denoted by \u0000<inline-formula> <tex-math>$rm {MDS}(ell)$ </tex-math></inline-formula>\u0000 where \u0000<inline-formula> <tex-math>$ell $ </tex-math></inline-formula>\u0000 denotes the order of generality, \u0000<inline-formula> <tex-math>$rm {MDS}(2)$ </tex-math></inline-formula>\u0000 codes are equivalent to the usual MDS codes. The best prior lower bound on the field size of an \u0000<inline-formula> <tex-math>${[}n,k{]}$ </tex-math></inline-formula>\u0000-\u0000<inline-formula> <tex-math>$rm {MDS}(ell)$ </tex-math></inline-formula>\u0000 codes is \u0000<inline-formula> <tex-math>$Omega _{ell } (n^{ell -1})$ </tex-math></inline-formula>\u0000, whereas the best known (non-explicit) upper bound is \u0000<inline-formula> <tex-math>$O_{ell } (n^{k(ell -1)})$ </tex-math></inline-formula>\u0000 which is exponential in the dimension. In this work, we nearly close this exponential gap between upper and lower bounds. We show that an \u0000<inline-formula> <tex-math>${[}n,k{]}$ </tex-math></inline-formula>\u0000-\u0000<inline-formula> <tex-math>$rm {MDS}(3)$ </tex-math></inline-formula>\u0000 codes requires a field of size \u0000<inline-formula> <tex-math>$Omega _{k}(n^{k-1})$ </tex-math></inline-formula>\u0000, which is close to the known upper bound. Using the connection between higher order MDS codes and optimally list-decodable codes, we show that even for a list size of 2, a code which meets the optimal list-decoding Singleton bound requires exponential field size; this resolves an open question by Shangguan and Tamo, (2020). We also give explicit constructions of \u0000<inline-formula> <tex-math>${[}n,k{]}$ </tex-math></inline-formula>\u0000-\u0000<inline-formula> <tex-math>$rm {MDS}(ell)$ </tex-math></inline-formula>\u0000 code over fields of size \u0000<inline-formula> <tex-math>$n^{(ell k)^{O(ell k)}}$ </tex-math></inline-formula>\u0000. The smallest non-trivial case where we still do not have optimal constructions is \u0000<inline-formula> <tex-math>${[}n,3{]}$ </tex-math></inline-formula>\u0000-\u0000<inline-formula> <tex-math>$rm {MDS}(3)$ </tex-math></inline-formula>\u0000. In this case, the known lower bound on the field size is \u0000<inline-formula> <tex-math>$Omega (n^{2})$ </tex-math></inline-formula>\u0000 and the best known upper bounds are \u0000<inline-formula> <tex-math>$O(n^{5})$ </tex-math></inline-formula>\u0000 for a non-explicit construction and \u0000<inline-formula> <tex-math>$O(n^{32})$ </tex-math></inline-formula>\u0000 for an explicit construction. In this paper, we give an explicit construction over fields of size \u0000<inline-formula> <tex-math>$O(n^{3})$ </tex-math></inline-formula>\u0000 which comes very close to being optimal.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"6950-6960"},"PeriodicalIF":2.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194879","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}

{"title":"On Confidence Sequences for Bounded Random Processes via Universal Gambling Strategies","authors":"J. Jon Ryu;Alankrita Bhatt","doi":"10.1109/TIT.2024.3448461","DOIUrl":"10.1109/TIT.2024.3448461","url":null,"abstract":"This paper considers the problem of constructing a confidence sequence, which is a sequence of confidence intervals that hold uniformly over time, for estimating the mean of bounded real-valued random processes. This paper revisits the gambling-based approach established in the recent literature from a natural two-horse race perspective, and demonstrates new properties of the resulting algorithm induced by Cover (1991)’s universal portfolio. The main result of this paper is a new algorithm based on a mixture of lower bounds, which closely approximates the performance of Cover’s universal portfolio with constant per-round time complexity. A higher-order generalization of a lower bound on a logarithmic function in (Fan et al., 2015), which is developed as a key technique for the proposed algorithm, may be of independent interest.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"7143-7161"},"PeriodicalIF":2.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194881","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}

{"title":"Achieving the Exactly Optimal Privacy-Utility Trade-Off With Low Communication Cost via Shared Randomness","authors":"Seung-Hyun Nam;Hyun-Young Park;Si-Hyeon Lee","doi":"10.1109/TIT.2024.3448475","DOIUrl":"10.1109/TIT.2024.3448475","url":null,"abstract":"We consider a discrete distribution estimation problem under a local differential privacy (LDP) constraint in the presence of shared randomness. For this problem, we propose a new class of LDP schemes achieving the exactly optimal privacy-utility trade-off (PUT), with the communication cost less than or equal to the size of the input data. Moreover, it is shown as a simple corollary that one-bit communication is sufficient for achieving the exactly optimal PUT for a high privacy regime if the input data size is an even number. The main idea is to decompose a block design scheme proposed by Park et al. (2023), based on the combinatorial concept called resolution. We call the resultant decomposed LDP scheme with shared randomness as a resolution of the original block design scheme. A resolution of a block design scheme has a communication cost less than or equal to that of the original block design scheme. Also, the resolution of a block design scheme is exactly optimal whenever the original block design scheme is exactly optimal. Accordingly, we provide two resolutions of the exactly optimal subset selection scheme proposed by Ye and Barg (2018), called the Baranyai’s resolution and the cyclic shift resolution. We show that the Baranyai’s resolution achieves the minimum communication cost among all exactly optimal resolutions of block design schemes. One drawback of the Baranyai’s resolution is that its explicit structure is unknown in general. In contrast, the cyclic shift resolution has an explicit structure, but its communication cost can be larger than that of the Baranyai’s resolution. To complement this, we also suggest resolutions of other block design schemes achieving the exactly optimal PUT for some input data size and privacy budget. Those require the minimum communication cost as the Baranyai’s resolution and have explicit structures as the cyclic shift resolution.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"7447-7462"},"PeriodicalIF":2.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194883","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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}