IEEE Transactions on Information Theory最新文献

{"title":"Self-Orthogonal Codes From p-Divisible Codes","authors":"Xiaoru Li;Ziling Heng","doi":"10.1109/TIT.2024.3449921","DOIUrl":"10.1109/TIT.2024.3449921","url":null,"abstract":"The self-orthogonality and divisibility are two important properties of linear codes. It is interesting to establish relationship between them. By the well-known Gleason-Pierce-Ward Theorem, all self-dual divisible codes have been totally classified. However, the relationship between the self-orthogonality and divisibility of a q-ary linear codes is known only for \u0000<inline-formula> <tex-math>$q=2,3$ </tex-math></inline-formula>\u0000 by Huffman and Pless in 2003. It has remained open for more than 20 years to consider other cases. The purpose of this paper is to settle this open problem under certain conditions and construct new families of self-orthogonal codes. Let q be a power of an odd prime p. Firstly, we prove that any p-divisible code containing the all-1 vector over the finite field \u0000<inline-formula> <tex-math>${mathbb {F}}_{q}$ </tex-math></inline-formula>\u0000 is self-orthogonal. More generally, it is concluded that any p-divisible \u0000<inline-formula> <tex-math>$[n,k]$ </tex-math></inline-formula>\u0000 linear code over \u0000<inline-formula> <tex-math>${mathbb {F}}_{q}$ </tex-math></inline-formula>\u0000 containing codewords of weight n is monomially equivalent to an \u0000<inline-formula> <tex-math>$[n,k]$ </tex-math></inline-formula>\u0000 self-orthogonal code over \u0000<inline-formula> <tex-math>${mathbb {F}}_{q}$ </tex-math></inline-formula>\u0000. This result provides a very efficient way to find self-orthogonal codes from p-divisible codes. Secondly, we apply this result to construct self-orthogonal codes with excellent parameters or nice applications. For one thing, we use this result to study the self-orthogonality of generalized Reed-Muller codes, certain projective two-weight codes, and Griesmer codes. For another thing, by this useful result as well as the extending and augmentation techniques for linear codes, we construct eight new families of self-orthogonal divisible codes. These self-orthogonal codes and their duals contain many optimal or almost optimal codes. Besides, some self-orthogonal codes support combinatorial designs and some of them are proved to be optimal or almost optimal locally recoverable codes.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"8562-8586"},"PeriodicalIF":2.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194877","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":"Approximate Leave-One-Out Cross Validation for Regression With ℓ₁ Regularizers","authors":"Arnab Auddy;Haolin Zou;Kamiar Rahnama Rad;Arian Maleki","doi":"10.1109/TIT.2024.3450002","DOIUrl":"10.1109/TIT.2024.3450002","url":null,"abstract":"The out-of-sample error (OO) is the main quantity of interest in risk estimation and model selection. Leave-one-out cross validation (LO) offers a (nearly) distribution-free yet computationally demanding approach to estimate OO. Recent theoretical work showed that approximate leave-one-out cross validation (ALO) is a computationally efficient and statistically reliable estimate of LO (and OO) for generalized linear models with differentiable regularizers. For problems involving non-differentiable regularizers, despite significant empirical evidence, the theoretical understanding of ALO’s error remains unknown. In this paper, we present a novel theory for a wide class of problems in the generalized linear model family with non-differentiable regularizers. We bound the error \u0000<inline-formula> <tex-math>$|{mathrm { ALO}}-{mathrm { LO}}|$ </tex-math></inline-formula>\u0000 in terms of intuitive metrics such as the size of leave-i-out perturbations in active sets, sample size n, number of features p and regularization parameters. As a consequence, for the \u0000<inline-formula> <tex-math>$ell _{1}$ </tex-math></inline-formula>\u0000-regularized problems, we show that \u0000<inline-formula> <tex-math>$|{mathrm { ALO}}-{mathrm { LO}}| xrightarrow {prightarrow infty } 0$ </tex-math></inline-formula>\u0000 while \u0000<inline-formula> <tex-math>$n/p$ </tex-math></inline-formula>\u0000 and signal-to-noise ratio (SNR) are bounded.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 11","pages":"8040-8071"},"PeriodicalIF":2.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194884","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":"Phase Retrieval With Background Information: Decreased References and Efficient Methods","authors":"Ziyang Yuan;Haoxing Yang;Ningyi Leng;Hongxia Wang","doi":"10.1109/TIT.2024.3449554","DOIUrl":"10.1109/TIT.2024.3449554","url":null,"abstract":"Fourier phase retrieval (PR) is a severely ill-posed inverse problem that arises in various applications. To guarantee a unique solution and relieve the dependence on the initialization, background information can be exploited as a structural prior. However, the requirement for the background information may be challenging when moving to high-resolution imaging. At the same time, the previously proposed projected gradient descent (PGD) method also demands much background information. In this paper, we present an improved theoretical result about the demand for the background information, along with two Douglas Rachford (DR) based methods. Analytically, we demonstrate that the background information required to ensure a unique solution can be decreased by nearly \u0000<inline-formula> <tex-math>$1/2$ </tex-math></inline-formula>\u0000 for the 2-D signals compared to the 1-D signals. By generalizing the results into d-dimension, we show that the length of the background information more than \u0000<inline-formula> <tex-math>$left ({{2^{frac {d+1}{d}}-1}}right)$ </tex-math></inline-formula>\u0000 folds of the signal is sufficient to ensure uniqueness. At the same time, we also analyze the stability and robustness of the model when the measurements and background information are corrupted by noise. Furthermore, two methods called Background Douglas Rachford (BDR) and Convex Background Douglas Rachford (CBDR) are proposed. BDR, which is a kind of non-convex method, is proven to have the local R-linear convergence rate under mild assumptions. Instead, the CBDR method uses the techniques of convexification and can be proven to have a global convergence guarantee as long as the background information is sufficient. To support this, a new property called F-RIP is established. We test the performance of the proposed methods through simulations as well as real experimental measurements, and demonstrate that they achieve a higher recovery rate with less background information compared to the PGD method.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 10","pages":"7498-7520"},"PeriodicalIF":2.2,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194876","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 the Noise Sensitivity of the Randomized SVD","authors":"Elad Romanov","doi":"10.1109/tit.2024.3450412","DOIUrl":"https://doi.org/10.1109/tit.2024.3450412","url":null,"abstract":"","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"15 1","pages":""},"PeriodicalIF":2.5,"publicationDate":"2024-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194878","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":"Repairing Reed-Solomon Codes Over Prime Fields via Exponential Sums","authors":"Roni Con;Noah Shutty;Itzhak Tamo;Mary Wootters","doi":"10.1109/TIT.2024.3449041","DOIUrl":"10.1109/TIT.2024.3449041","url":null,"abstract":"This paper presents two repair schemes for low-rate Reed-Solomon (RS) codes over prime fields that can repair any node by downloading a constant number of bits from each surviving node. The total bandwidth resulting from these schemes is greater than that incurred during trivial repair; however, this is particularly relevant in the context of leakage-resilient secret sharing. In that framework, our results provide attacks showing that k-out-of-n Shamir’s Secret Sharing over prime fields for small k is not leakage-resilient, even when the parties leak only a constant number of bits. To the best of our knowledge, these are the first such attacks. Our results are derived from a novel connection between exponential sums and the repair of RS codes. Specifically, we establish that non-trivial bounds on certain exponential sums imply the existence of explicit nonlinear repair schemes for RS codes over prime fields.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"8587-8594"},"PeriodicalIF":2.2,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194882","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":"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":"Optimal Trace Distance and Fidelity Estimations for Pure Quantum States","authors":"Qisheng Wang","doi":"10.1109/TIT.2024.3447915","DOIUrl":"10.1109/TIT.2024.3447915","url":null,"abstract":"Measuring the distinguishability between quantum states is a basic problem in quantum information theory. In this paper, we develop optimal quantum algorithms that estimate both the trace distance and the (square root) fidelity between pure states to within additive error \u0000<inline-formula> <tex-math>$varepsilon $ </tex-math></inline-formula>\u0000 using \u0000<inline-formula> <tex-math>$Theta (1/varepsilon)$ </tex-math></inline-formula>\u0000 queries to their state-preparation circuits, quadratically improving the long-standing folklore \u0000<inline-formula> <tex-math>$O(1/varepsilon ^{2}) $ </tex-math></inline-formula>\u0000. At the heart of our construction, is an algorithmic tool for quantum square root amplitude estimation, which generalizes the well-known quantum amplitude estimation.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"8791-8805"},"PeriodicalIF":2.2,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10643559","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194950","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":"Improving Explicit Constructions of r-PD-Sets for Zₚs-Linear Generalized Hadamard Codes","authors":"Josep Rifà;Adrián Torres-Martín;Mercè Villanueva","doi":"10.1109/TIT.2024.3448230","DOIUrl":"10.1109/TIT.2024.3448230","url":null,"abstract":"It is known that \u0000<inline-formula> <tex-math>$mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>\u0000-linear codes, which are the Gray map image of \u0000<inline-formula> <tex-math>$mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>\u0000-additive codes (linear codes over \u0000<inline-formula> <tex-math>$mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>\u0000), are systematic and a systematic encoding has been found. This makes \u0000<inline-formula> <tex-math>$mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>\u0000-linear codes suitable to apply the permutation decoding method, based on the existence of r-PD-sets, which are subsets of the permutation automorphism group of the code. Some constructions of r-PD-sets of minimum size \u0000<inline-formula> <tex-math>$r+1$ </tex-math></inline-formula>\u0000 for \u0000<inline-formula> <tex-math>$mathbb {Z}_{p^{s}}$ </tex-math></inline-formula>\u0000-linear generalized Hadamard codes of type \u0000<inline-formula> <tex-math>$(n;t_{1}, {dots },t_{s})$ </tex-math></inline-formula>\u0000 are known. In this paper, for these codes, we present new constructions of r-PD-sets of size \u0000<inline-formula> <tex-math>$r+1$ </tex-math></inline-formula>\u0000, which are suitable for all parameters \u0000<inline-formula> <tex-math>$t_{1}, {dots },t_{s}$ </tex-math></inline-formula>\u0000. These allow us to obtain new r-PD-sets for values of r closer to the theoretical upper bound, improving previous known results.","PeriodicalId":13494,"journal":{"name":"IEEE Transactions on Information Theory","volume":"70 12","pages":"8675-8687"},"PeriodicalIF":2.2,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=10643529","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142194891","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}