Showing papers by "Chao Tian published in 2016"

Symmetry, demand types and outer bounds in caching systems

Abstract: The fundamental limit of coded caching is considered in this work. We started with an investigation of the symmetry structure in the caching problem, which is used to show the existence of optimal symmetric solutions, as well as to motivate the notion of demand types. With a combination of analytical analysis using the symmetry and a computational approach developed earlier for the regenerating code problem, we obtain the following results on the memory-rate tradeoff: (1) A complete solution for any systems with K = 2 users; (2) A complete solution for the case with N = 2 files and K = 3 users; (3) The best outer bound for the cases (N, K) = (2, 4) and (N, K) = (3, 3) in the literature, which are tight in certain regimes. These results provide new insights to the problem, and also help to identify the main coding challenges for different system parameters.

Caching and delivery via interference elimination

Abstract: We propose a new caching scheme where linear combinations of the file segments are cached at the users, for the scenarios where the number of files is no greater than the number of users. When a user requests a certain file in the delivery phase, the other file segments in the cached linear combinations can be viewed as interferences. The proposed scheme combines rank metric codes and maximum distance separable codes to facilitate the decoding and elimination of these interferences, and also to simultaneously deliver useful contents to the intended users. The performance of the proposed scheme can be explicitly evaluated, and we show that the new scheme can strictly improve existing tradeoff inner bounds in the literature; for certain cases, the new tradeoff points are in fact optimal.

Multilevel Diversity Coding With Regeneration

Abstract: Digital contents distributed storage systems may have different reliability and access delay requirements, and erasure codes with different strengths can provide the best storage efficiency in these systems. At the same time, in such large-scale distributed storage systems, nodes fail on a regular basis, and the contents stored on them need to be regenerated from the data downloaded from the remaining nodes. The efficiency of this repair process is an important factor that affects the overall quality of service. In this paper, we formulate the problem of multilevel diversity coding with regeneration to address these considerations, for which the storage versus repair-bandwidth tradeoff is investigated. We show that the extreme point on the optimal tradeoff curve that corresponds to the minimum possible storage can be achieved by a simple coding scheme, in which contents with different reliability requirements are encoded separately with individual regenerating codes without any mixing. On the other hand, we establish the complete storage-repair-bandwidth tradeoff for the case of four storage nodes, which reveals that codes mixing different contents can, in general, strictly improve the optimal tradeoff over the separate-coding solution.

Symmetry, Outer Bounds, and Code Constructions: A Computer-Aided Investigation on the Fundamental Limits of Caching

Abstract: We illustrate how computer-aided methods can be used to investigate the fundamental limits of the caching systems, which are significantly different from the conventional analytical approach usually seen in the information theory literature. The linear programming (LP) outer bound of the entropy space serves as the starting point of this approach; however, our effort goes significantly beyond using it to prove information inequalities. We first identify and formalize the symmetry structure in the problem, which enables us to show the existence of optimal symmetric solutions. A symmetry-reduced linear program is then used to identify the boundary of the memory-transmission-rate tradeoff for several small cases, for which we obtain a set of tight outer bounds. General hypotheses on the optimal tradeoff region are formed from these computed data, which are then analytically proven. This leads to a complete characterization of the optimal tradeoff for systems with only two users, and certain partial characterization for systems with only two files. Next, we show that by carefully analyzing the joint entropy structure of the outer bounds for certain cases, a novel code construction can be reverse-engineered, which eventually leads to a general class of codes. Finally, we show that outer bounds can be computed through strategically relaxing the LP in different ways, which can be used to explore the problem computationally. This allows us firstly to deduce generic characteristic of the converse proof, and secondly to compute outer bounds for larger problem cases, despite the seemingly impossible computation scale.

Caching and Delivery via Interference Elimination

Abstract: We propose a new caching scheme where linear combinations of the file segments are cached at the users, for the cases where the number of files is no greater than the number of users. When a user requests a certain file in the delivery phase, the other file segments in the cached linear combinations can be viewed as interferences. The proposed scheme combines rank metric codes and maximum distance separable codes to facilitate the decoding and elimination of these interferences, and also to simultaneously deliver useful contents to the intended users. The performance of the proposed scheme can be explicitly evaluated, and we show that the tradeoff points achieved by this scheme can strictly improve known tradeoff inner bounds in the literature; for certain special cases, the new tradeoff points can be shown to be optimal.

Multilevel diversity coding with regeneration: Separate coding achieves the MBR point

Abstract: The problem of multilevel diversity coding with regeneration is considered in this work. Two new outer bounds on the optimal tradeoffs between the normalized storage capacity and repair bandwidth are established, by which the optimality of separate coding at the minimum-bandwidth-regeneration (MBR) point follows immediately. This resolves a question left open in a previous work by Tian and Liu.

Enabling All-Node-Repair in Minimum Storage Regenerating Codes.

Abstract: We consider the problem of constructing exact-repair minimum storage regenerating (MSR) codes, for which both the systematic nodes and parity nodes can be repaired optimally. Although there exist several recent explicit high-rate MSR code constructions (usually with certain restrictions on the coding parameters), quite a few constructions in the literature only allow the optimal repair of systematic nodes. This phenomenon suggests that there might be a barrier between explicitly constructing codes that can only optimally repair systematic nodes and those that can optimally repair both systematic nodes and parity nodes. In the work, we show that this barrier can be completely circumvented by providing a generic transformation that is able to convert any non-binary linear maximum distance separable (MDS) storage codes that can optimally repair only systematic nodes into new MSR codes that can optimally repair all nodes. This transformation does not increase the alphabet size of the original codes, and only increases the sub-packetization by a factor that is equal to the number of parity nodes. Furthermore, the resultant MSR codes also have the optimal access property for all nodes if the original MDS storage codes have the optimal access property for systematic nodes.

Symmetry reduction of information inequalities

Abstract: Information inequalities can be used to bound the fundamental limits of communication systems and data storage systems. Information inequalities, particularly Shannon-type inequalities, and the problem-specific constraints are usually either linear equalities or inequalities of joint entropies, and thus outer bounding the fundamental limit can be viewed and solved as a linear program (LP). However, for many practical engineering problems, the resultant LP is very large. It was shown previously that symmetry in these problems can be used to reduce the scale of the LP, however the precise amount of reduction was not well understood. In this work, we provide a generic method to pinpoint this reduction. In particular, three problems are studied: extremal pairwise cyclic entropy inequalities, the regenerating code problem, and the caching problem. By viewing the symmetry as an induced permutation group on certain set, Polya counting theorem can be applied, which however requires identifying the cycle index of the induced permutation.

Orbit-entropy cones and extremal pairwise orbit-entropy inequalities

Abstract: The notion of orbit-entropy cone is introduced. Specifically, orbit-entropy cone equation is the projection of equation induced by G, where equation is the closure of entropy region for n random variables and G is a permutation group over {0; 1;...; n−1}. For symmetric group S n (with arbitrary n) and cyclic group C n (with n ≤ 5), the associated orbit-entropy cones are shown to be characterized by the Shannon type inequalities. Moreover, the extremal pairwise relationship between orbit-entropies is determined completely for partitioned symmetric groups and partially for cyclic groups.

Cyclically symmetric entropy inequalities

Abstract: A cyclically symmetric entropy inequality is of the form h O ≥ch O′ , where h O and h O′ are two cyclic orbit entropy terms. A computational approach is formulated for bounding the extremal value of c, which is denoted by c O,O′ . For two non-empty orbits O and O′ of a cyclic group, it is said that O dominates O′ if c O,O′ = 1. Special attention is paid to characterizing such dominance relationship, and a graphical method is developed for that purpose.

Selective Sampling Based Efficient Classifier Representation in Distributed Learning

Abstract: Communication is a bottleneck for many distributed machine learning problems with large data sets. One effective approach is to exchange the set of good classifiers among the learners. The key challenge in such an approach is how to represent the set of good classifiers, which can be viewed as a source coding problem in communication systems. A nonuniform sampling framework is proposed for this source coding problem. A metric that describes the quality of the sampled data for the target hypothesis space is proposed. An optimization problem is formulated by minimizing the upper bound of the proposed metric, and an efficient weight based algorithm is provided to solve the optimization problem. Numerical simulations on synthetic and real world data set show that the learning performance based on the proposed sampling framework is close to the global optimum with high confidence, while requiring less samples than baseline random sampling algorithms.