arXiv subjects

Netanel Raviv

Publications and source records attributed to Netanel Raviv.

54 records · Page 3Linked to original sources

arXiv2019

On Coding over Sliced Information

The interest in channel models in which the data is sent as an unordered set of binary strings has increased lately, due to emerging applications in DNA storage, among others. In this paper we analyze the minimal redundancy of binary codes for this channel under substitution errors, and provide several constructions, some of which are shown to be asymptotically optimal up to constants. The surprising result in this paper is that while the information vector is sliced into a set of unordered strings, the amount of redundant bits that are required to correct errors is order-wise equivalent to the amount required in the classical error correcting paradigm.

cs.IT↗

arXiv2019

Hierarchical Erasure Correction of Linear Codes

Linear codes over finite extension fields have widespread applications in theory and practice. In some scenarios, the decoder has a sequential access to the codeword symbols, giving rise to a hierarchical erasure structure. In this paper we develop a mathematical framework for hierarchical erasures over extension fields, provide several bounds and constructions, and discuss potential applications in distributed storage and flash memories. Our results show intimate connection to Universally Decodable Matrices, as well as to Reed-Solomon and Gabidulin codes.

cs.IT↗

arXiv2019

Gradient Coding from Cyclic MDS Codes and Expander Graphs

Gradient coding is a technique for straggler mitigation in distributed learning. In this paper we design novel gradient codes using tools from classical coding theory, namely, cyclic MDS codes, which compare favorably with existing solutions, both in the applicable range of parameters and in the complexity of the involved algorithms. Second, we introduce an approximate variant of the gradient coding problem, in which we settle for approximate gradient computation instead of the exact one. This approach enables graceful degradation, i.e., the $\ell_2$ error of the approximate gradient is a decreasing function of the number of stragglers. Our main result is that normalized adjacency matrices of expander graphs yield excellent approximate gradient codes, which enable significantly less computation compared to exact gradient coding, and guarantee faster convergence than trivial solutions under standard assumptions. We experimentally test our approach on Amazon EC2, and show that the generalization error of approximate gradient coding is very close to the full gradient while requiring significantly less computation from the workers.

cs.IT↗

arXiv2019

Private Polynomial Computation from Lagrange Encoding

Private computation is a generalization of private information retrieval, in which a user is able to compute a function on a distributed dataset without revealing the identity of that function to the servers. In this paper it is shown that Lagrange encoding, a powerful technique for encoding Reed-Solomon codes, enables private computation in many cases of interest. In particular, we present a scheme that enables private computation of polynomials of any degree on Lagrange encoded data, while being robust to Byzantine and straggling servers, and to servers colluding to attempt to deduce the identities of the functions to be evaluated. Moreover, incorporating ideas from the well-known Shamir secret sharing scheme allows the data itself to be concealed from the servers as well. Our results extend private computation to high degree polynomials and to data-privacy, and reveal a tight connection between private computation and coded computation.

cs.IT↗

arXiv2019

Lagrange Coded Computing: Optimal Design for Resiliency, Security and Privacy

We consider a scenario involving computations over a massive dataset stored distributedly across multiple workers, which is at the core of distributed learning algorithms. We propose Lagrange Coded Computing (LCC), a new framework to simultaneously provide (1) resiliency against stragglers that may prolong computations; (2) security against Byzantine (or malicious) workers that deliberately modify the computation for their benefit; and (3) (information-theoretic) privacy of the dataset amidst possible collusion of workers. LCC, which leverages the well-known Lagrange polynomial to create computation redundancy in a novel coded form across workers, can be applied to any computation scenario in which the function of interest is an arbitrary multivariate polynomial of the input dataset, hence covering many computations of interest in machine learning. LCC significantly generalizes prior works to go beyond linear computations. It also enables secure and private computing in distributed settings, improving the computation and communication efficiency of the state-of-the-art. Furthermore, we prove the optimality of LCC by showing that it achieves the optimal tradeoff between resiliency, security, and privacy, i.e., in terms of tolerating the maximum number of stragglers and adversaries, and providing data privacy against the maximum number of colluding workers. Finally, we show via experiments on Amazon EC2 that LCC speeds up the conventional uncoded implementation of distributed least-squares linear regression by up to $13.43\times$, and also achieves a $2.36\times$-$12.65\times$ speedup over the state-of-the-art straggler mitigation strategies.

cs.IT↗

arXiv2019

Private Information Retrieval in Graph Based Replication Systems

In a Private Information Retrieval (PIR) protocol, a user can download a file from a database without revealing the identity of the file to each individual server. A PIR protocol is called $t$-private if the identity of the file remains concealed even if $t$ of the servers collude. Graph based replication is a simple technique, which is prevalent in both theory and practice, for achieving erasure robustness in storage systems. In this technique each file is replicated on two or more storage servers, giving rise to a (hyper-)graph structure. In this paper we study private information retrieval protocols in graph based replication systems. The main interest of this work is maximizing the parameter $t$, and in particular, understanding the structure of the colluding sets which emerge in a given graph. Our main contribution is a $2$-replication scheme which guarantees perfect privacy from acyclic sets in the graph, and guarantees partial-privacy in the presence of cycles. Furthermore, by providing an upper bound, it is shown that the PIR rate of this scheme is at most a factor of two from its optimal value for an important family of graphs. Lastly, we extend our results to larger replication factors and to graph-based coding, which is a similar technique with smaller storage overhead and larger PIR rate.

cs.IT↗

arXiv2018

Two Deletion Correcting Codes from Indicator Vectors

Construction of capacity achieving deletion correcting codes has been a baffling challenge for decades. A recent breakthrough by Brakensiek $et~al$., alongside novel applications in DNA storage, have reignited the interest in this longstanding open problem. In spite of recent advances, the amount of redundancy in existing codes is still orders of magnitude away from being optimal. In this paper, a novel approach for constructing binary two-deletion correcting codes is proposed. By this approach, parity symbols are computed from indicator vectors (i.e., vectors that indicate the positions of certain patterns) of the encoded message, rather than from the message itself. Most interestingly, the parity symbols and the proof of correctness are a direct generalization of their counterparts in the Varshamov-Tenengolts construction. Our techniques require $7\log(n)+o(\log(n)$ redundant bits to encode an~$n$-bit message, which is near-optimal.

cs.IT↗

arXiv2017

Rank modulation codes for DNA storage

Synthesis of DNA molecules offers unprecedented advances in storage technology. Yet, the microscopic world in which these molecules reside induces error patterns that are fundamentally different from their digital counterparts. Hence, to maintain reliability in reading and writing, new coding schemes must be developed. In a reading technique called shotgun sequencing, a long DNA string is read in a sliding window fashion, and a profile vector is produced. It was recently suggested by Kiah et al. that such a vector can represent the permutation which is induced by its entries, and hence a rank-modulation scheme arises. Although this interpretation suggests high error tolerance, it is unclear which permutations are feasible, and how to produce a DNA string whose profile vector induces a given permutation. In this paper, by observing some necessary conditions, an upper bound for the number of feasible permutations is given. Further, a technique for deciding the feasibility of a permutation is devised. By using insights from this technique, an algorithm for producing a considerable number of feasible permutations is given, which applies to any alphabet size and any window length.

cs.IT↗

arXiv2017

LDPC Codes over the q-ary Multi-Bit Channel

In this paper, we introduce a new channel model we term the q-ary multi-bit channel (QMBC). This channel models a memory device, where q-ary symbols (q=2^s) are stored in the form of current/voltage levels. The symbols are read in a measurement process, which provides a symbol bit in each measurement step, starting from the most significant bit. An error event occurs when not all the symbol bits are known. To deal with such error events, we use GF(q) low-density parity-check (LDPC) codes and analyze their decoding performance. We start with iterative-decoding threshold analysis, and derive optimal edge-label distributions for maximizing the decoding threshold. We later move to finite-length iterative-decoding analysis and propose an edge-labeling algorithm for improved decoding performance. We then provide finite-length maximum-likelihood decoding analysis for both the standard non-binary random ensemble and LDPC ensembles. Finally, we demonstrate by simulations that the proposed edge-labeling algorithm improves finite-length decoding performance by orders of magnitude.

cs.IT↗

arXiv2017

Construction of Sidon spaces with applications to coding

A subspace of a finite extension field is called a Sidon space if the product of any two of its elements is unique up to a scalar multiplier from the base field. Sidon spaces were recently introduced by Bachoc et al. as a means to characterize multiplicative properties of subspaces, and yet no explicit constructions were given. In this paper, several constructions of Sidon spaces are provided. In particular, in some of the constructions the relation between $k$, the dimension of the Sidon space, and $n$, the dimension of the ambient extension field, is optimal. These constructions are shown to provide cyclic subspace codes, which are useful tools in network coding schemes. To the best of the authors' knowledge, this constitutes the first set of constructions of non-trivial cyclic subspace codes in which the relation between $k$ and $n$ is polynomial, and in particular, linear. As a result, a conjecture by Trautmann et al. regarding the existence of non-trivial cyclic subspace codes is resolved for most parameters, and multi-orbit cyclic subspace codes are attained, whose cardinality is within a constant factor (close to $1/2$) from the sphere-packing bound for subspace codes.

cs.IT↗

arXiv2016

Some Gabidulin Codes cannot be List Decoded Efficiently at any Radius

Gabidulin codes can be seen as the rank-metric equivalent of Reed-Solomon codes. It was recently proven, using subspace polynomials, that Gabidulin codes cannot be list decoded beyond the so-called Johnson radius. In another result, cyclic subspace codes were constructed by inspecting the connection between subspaces and their subspace polynomials. In this paper, these subspace codes are used to prove two bounds on the list size in decoding certain Gabidulin codes. The first bound is an existential one, showing that exponentially-sized lists exist for codes with specific parameters. The second bound presents exponentially-sized lists explicitly, for a different set of parameters. Both bounds rule out the possibility of efficiently list decoding several families of Gabidulin codes for any radius beyond half the minimum distance. Such a result was known so far only for non-linear rank-metric codes, and not for Gabidulin codes. Using a standard operation called lifting, identical results also follow for an important class of constant dimension subspace codes.

cs.IT↗

arXiv2016

Asymptotically Optimal Regenerating Codes Over Any Field

The study of regenerating codes has advanced tremendously in recent years. However, most known constructions require large field size, and hence may be hard to implement in practice. By using notions from the theory of extension fields, we obtain two explicit constructions of regenerating codes. These codes approach the cut-set bound as the reconstruction degree increases, and may be realized over any given field if the file size is large enough. Since distributed storage systems are the main purpose of regenerating codes, this file size restriction is trivially satisfied in most conceivable scenarios. The first construction attains the cut-set bound at the MBR point asymptotically for all parameters, whereas the second one attains the cut-set bound at the MSR point asymptotically for low-rate parameters.

cs.IT↗

arXiv2016

Coding for Locality in Reconstructing Permutations

The problem of storing permutations in a distributed manner arises in several common scenarios, such as efficient updates of a large, encrypted, or compressed data set. This problem may be addressed in either a combinatorial or a coding approach. The former approach boils down to presenting large sets of permutations with \textit{locality}, that is, any symbol of the permutation can be computed from a small set of other symbols. In the latter approach, a permutation may be coded in order to achieve locality. This paper focuses on the combinatorial approach. We provide upper and lower bounds for the maximal size of a set of permutations with locality, and provide several simple constructions which attain the upper bound. In cases where the upper bound is not attained, we provide alternative constructions using Reed-Solomon codes, permutation polynomials, and multi-permutations.

cs.IT↗

arXiv2016

Constructions of High-Rate MSR Codes over Small Fields

A novel technique for construction of minimum storage regenerating (MSR) codes is presented. Based on this technique, three explicit constructions of MSR codes are given. The first two constructions provide access-optimal MSR codes, with two and three parities, respectively, which attain the sub-packetization bound for access-optimal codes. The third construction provides longer MSR codes with three parities, which are not access-optimal, and do not necessarily attain the sub-packetization bound. In addition to a minimum storage in a node, all three constructions allow the entire data to be recovered from a minimal number of storage nodes. That is, given storage $\ell$ in each node, the entire stored data can be recovered from any $2\log_2 \ell$ for 2 parity nodes, and either $3\log_3\ell$ or $4\log_3\ell$ for 3 parity nodes. Second, in the first two constructions, a helper node accesses the minimum number of its symbols for repair of a failed node (access-optimality). The generator matrix of these codes is based on perfect matchings of complete graphs and hypergraphs, and on a rational canonical form of matrices. The goal of this paper is to provide a construction of such optimal codes over the smallest possible finite fields. For two parities, the field size is reduced by a factor of two for access-optimal codes compared to previous constructions. For three parities, in the first construction the field size is $6\log_3 \ell+1$ (or $3\log_3 \ell+1$ for fields with characteristic 2), and in the second construction the field size is larger, yet linear in $\log_3\ell$. Both constructions with 3 parities provide a significant improvement over existing previous works, since only non-explicit constructions with exponential field size (in $\log_3\ell$) were known so far.

cs.IT↗

arXiv2015

Equidistant Codes in the Grassmannian

Equidistant codes over vector spaces are considered. For $k$-dimensional subspaces over a large vector space the largest code is always a sunflower. We present several simple constructions for such codes which might produce the largest non-sunflower codes. A novel construction, based on the Plücker embedding, for 1-intersecting codes of $k$-dimensional subspaces over $\F_q^n$, $n \geq \binom{k+1}{2}$, where the code size is $\frac{q^{k+1}-1}{q-1}$ is presented. Finally, we present a related construction which generates equidistant constant rank codes with matrices of size $n \times \binom{n}{2}$ over $\F_q$, rank $n-1$, and rank distance $n-1$.

math.CO↗

arXiv2015

Subspace Polynomials and Cyclic Subspace Codes

Subspace codes have received an increasing interest recently due to their application in error-correction for random network coding. In particular, cyclic subspace codes are possible candidates for large codes with efficient encoding and decoding algorithms. In this paper we consider such cyclic codes and provide constructions of optimal codes for which their codewords do not have full orbits. We further introduce a new way to represent subspace codes by a class of polynomials called subspace polynomials. We present some constructions of such codes which are cyclic and analyze their parameters.

cs.IT↗

arXiv2014

Distributed Storage Systems based on Equidistant Subspace Codes

Distributed storage systems based on equidistant constant dimension codes are presented. These equidistant codes are based on the Plücker embedding, which is essential in the repair and the reconstruction algorithms. These systems posses several useful properties such as high failure resilience, minimum bandwidth, low storage, simple algebraic repair and reconstruction algorithms, good locality, and compatibility with small fields.

cs.IT↗

arXiv2013

Truth Table Minimization of Computational Models

Complexity theory offers a variety of concise computational models for computing boolean functions - branching programs, circuits, decision trees and ordered binary decision diagrams to name a few. A natural question that arises in this context with respect to any such model is this: Given a function f:{0,1}^n \to {0,1}, can we compute the optimal complexity of computing f in the computational model in question? (according to some desirable measure). A critical issue regarding this question is how exactly is f given, since a more elaborate description of f allows the algorithm to use more computational resources. Among the possible representations are black-box access to f (such as in computational learning theory), a representation of f in the desired computational model or a representation of f in some other model. One might conjecture that if f is given as its complete truth table (i.e., a list of f's values on each of its 2^n possible inputs), the most elaborate description conceivable, then any computational model can be efficiently computed, since the algorithm computing it can run poly(2^n) time. Several recent studies show that this is far from the truth - some models have efficient and simple algorithms that yield the desired result, others are believed to be hard, and for some models this problem remains open. In this thesis we will discuss the computational complexity of this question regarding several common types of computational models. We shall present several new hardness results and efficient algorithms, as well as new proofs and extensions for known theorems, for variants of decision trees, formulas and branching programs.

cs.CC↗