arXiv subjects

Alexandros G. Dimakis

Publications and source records attributed to Alexandros G. Dimakis.

At least 109 records · Page 6Linked to original sources

arXiv2015

Stay on path: PCA along graph paths

We introduce a variant of (sparse) PCA in which the set of feasible support sets is determined by a graph. In particular, we consider the following setting: given a directed acyclic graph $G$ on $p$ vertices corresponding to variables, the non-zero entries of the extracted principal component must coincide with vertices lying along a path in $G$. From a statistical perspective, information on the underlying network may potentially reduce the number of observations required to recover the population principal component. We consider the canonical estimator which optimally exploits the prior knowledge by solving a non-convex quadratic maximization on the empirical covariance. We introduce a simple network and analyze the estimator under the spiked covariance model. We show that side information potentially improves the statistical complexity. We propose two algorithms to approximate the solution of the constrained quadratic maximization, and recover a component with the desired properties. We empirically evaluate our schemes on synthetic and real datasets.

stat.ML↗

arXiv2015

FrogWild! -- Fast PageRank Approximations on Graph Engines

We propose FrogWild, a novel algorithm for fast approximation of high PageRank vertices, geared towards reducing network costs of running traditional PageRank algorithms. Our algorithm can be seen as a quantized version of power iteration that performs multiple parallel random walks over a directed graph. One important innovation is that we introduce a modification to the GraphLab framework that only partially synchronizes mirror vertices. This partial synchronization vastly reduces the network traffic generated by traditional PageRank algorithms, thus greatly reducing the per-iteration cost of PageRank. On the other hand, this partial synchronization also creates dependencies between the random walks used to estimate PageRank. Our main theoretical innovation is the analysis of the correlations introduced by this partial synchronization process and a bound establishing that our approximation is close to the true PageRank vector. We implement our algorithm in GraphLab and compare it against the default PageRank implementation. We show that our algorithm is very fast, performing each iteration in less than one second on the Twitter graph and can be up to 7x faster compared to the standard GraphLab PageRank implementation.

cs.DC↗

arXiv2014

Index Coding with Coded Side-Information

This letter investigates a new class of index coding problems. One sender broadcasts packets to multiple users, each desiring a subset, by exploiting prior knowledge of linear combinations of packets. We refer to this class of problems as index coding with coded side-information. Our aim is to characterize the minimum index code length that the sender needs to transmit to simultaneously satisfy all user requests. We show that the optimal binary vector index code length is equal to the minimum rank (minrank) of a matrix whose elements consist of the sets of desired packet indices and side- information encoding matrices. This is the natural extension of matrix minrank in the presence of coded side information. Using the derived expression, we propose a greedy randomized algorithm to minimize the rank of the derived matrix.

cs.IT↗

arXiv2014

On the Information Theoretic Limits of Learning Ising Models

We provide a general framework for computing lower-bounds on the sample complexity of recovering the underlying graphs of Ising models, given i.i.d samples. While there have been recent results for specific graph classes, these involve fairly extensive technical arguments that are specialized to each specific graph class. In contrast, we isolate two key graph-structural ingredients that can then be used to specify sample complexity lower-bounds. Presence of these structural properties makes the graph class hard to learn. We derive corollaries of our main result that not only recover existing recent results, but also provide lower bounds for novel graph classes not considered previously. We also extend our framework to the random graph setting and derive corollaries for Erdős-Rényi graphs in a certain dense setting.

cs.LG↗

arXiv2014

Sparse Polynomial Learning and Graph Sketching

Let $f:\{-1,1\}^n$ be a polynomial with at most $s$ non-zero real coefficients. We give an algorithm for exactly reconstructing f given random examples from the uniform distribution on $\{-1,1\}^n$ that runs in time polynomial in $n$ and $2s$ and succeeds if the function satisfies the unique sign property: there is one output value which corresponds to a unique set of values of the participating parities. This sufficient condition is satisfied when every coefficient of f is perturbed by a small random noise, or satisfied with high probability when s parity functions are chosen randomly or when all the coefficients are positive. Learning sparse polynomials over the Boolean domain in time polynomial in $n$ and $2s$ is considered notoriously hard in the worst-case. Our result shows that the problem is tractable for almost all sparse polynomials. Then, we show an application of this result to hypergraph sketching which is the problem of learning a sparse (both in the number of hyperedges and the size of the hyperedges) hypergraph from uniformly drawn random cuts. We also provide experimental results on a real world dataset.

cs.LG↗

arXiv2014

Batch Codes through Dense Graphs without Short Cycles

Consider a large database of $n$ data items that need to be stored using $m$ servers. We study how to encode information so that a large number $k$ of read requests can be performed in parallel while the rate remains constant (and ideally approaches one). This problem is equivalent to the design of multiset Batch Codes introduced by Ishai, Kushilevitz, Ostrovsky and Sahai [17]. We give families of multiset batch codes with asymptotically optimal rates of the form $1-1/\text{poly}(k)$ and a number of servers $m$ scaling polynomially in the number of read requests $k$. An advantage of our batch code constructions over most previously known multiset batch codes is explicit and deterministic decoding algorithms and asymptotically optimal fault tolerance. Our main technical innovation is a graph-theoretic method of designing multiset batch codes using dense bipartite graphs with no small cycles. We modify prior graph constructions of dense, high-girth graphs to obtain our batch code results. We achieve close to optimal tradeoffs between the parameters for bipartite graph based batch codes.

cs.IT↗

arXiv2014

Auditing for Distributed Storage Systems

Distributed storage codes have recently received a lot of attention in the community. Independently, another body of work has proposed integrity checking schemes for cloud storage, none of which, however, is customized for coding-based storage or can efficiently support repair. In this work, we bridge the gap between these two currently disconnected bodies of work. We propose NC-Audit, a novel cryptography-based remote data integrity checking scheme, designed specifically for network coding-based distributed storage systems. NC-Audit combines, for the first time, the following desired properties: (i) efficient checking of data integrity, (ii) efficient support for repairing failed nodes, and (iii) protection against information leakage when checking is performed by a third party. The key ingredient of the design of NC-Audit is a novel combination of SpaceMac, a homomorphic message authentication code (MAC) scheme for network coding, and NCrypt, a novel chosen-plaintext attack (CPA) secure encryption scheme that is compatible with SpaceMac. Our evaluation of a Java implementation of NC-Audit shows that an audit costs the storage node and the auditor a modest amount computation time and lower bandwidth than prior work.

cs.CR↗

arXiv2014

Network Codes for Real-Time Applications

We consider the scenario of broadcasting for real-time applications and loss recovery via instantly decodable network coding. Past work focused on minimizing the completion delay, which is not the right objective for real-time applications that have strict deadlines. In this work, we are interested in finding a code that is instantly decodable by the maximum number of users. First, we prove that this problem is NP-Hard in the general case. Then we consider the practical probabilistic scenario, where users have i.i.d. loss probability and the number of packets is linear or polynomial in the number of users. In this scenario, we provide a polynomial-time (in the number of users) algorithm that finds the optimal coded packet. The proposed algorithm is evaluated using both simulation and real network traces of a real-time Android application. Both results show that the proposed coding scheme significantly outperforms the state-of-the-art baselines: an optimal repetition code and a COPE-like greedy scheme.

cs.NI↗

arXiv2014

Sparse PCA through Low-rank Approximations

We introduce a novel algorithm that computes the $k$-sparse principal component of a positive semidefinite matrix $A$. Our algorithm is combinatorial and operates by examining a discrete set of special vectors lying in a low-dimensional eigen-subspace of $A$. We obtain provable approximation guarantees that depend on the spectral decay profile of the matrix: the faster the eigenvalue decay, the better the quality of our approximation. For example, if the eigenvalues of $A$ follow a power-law decay, we obtain a polynomial-time approximation algorithm for any desired accuracy. A key algorithmic component of our scheme is a combinatorial feature elimination step that is provably safe and in practice significantly reduces the running complexity of our algorithm. We implement our algorithm and test it on multiple artificial and real data sets. Due to the feature elimination step, it is possible to perform sparse PCA on data sets consisting of millions of entries in a few minutes. Our experimental evaluation shows that our scheme is nearly optimal while finding very sparse vectors. We compare to the prior state of the art and show that our scheme matches or outperforms previous algorithms in all tested data sets.

stat.ML↗

arXiv2014

Locally Repairable Codes

Distributed storage systems for large-scale applications typically use replication for reliability. Recently, erasure codes were used to reduce the large storage overhead, while increasing data reliability. A main limitation of off-the-shelf erasure codes is their high-repair cost during single node failure events. A major open problem in this area has been the design of codes that {\it i)} are repair efficient and {\it ii)} achieve arbitrarily high data rates. In this paper, we explore the repair metric of {\it locality}, which corresponds to the number of disk accesses required during a {\color{black}single} node repair. Under this metric we characterize an information theoretic trade-off that binds together locality, code distance, and the storage capacity of each node. We show the existence of optimal {\it locally repairable codes} (LRCs) that achieve this trade-off. The achievability proof uses a locality aware flow-graph gadget which leads to a randomized code construction. Finally, we present an optimal and explicit LRC that achieves arbitrarily high data-rates. Our locality optimal construction is based on simple combinations of Reed-Solomon blocks.

cs.IT↗

arXiv2014

Bounding Multiple Unicasts through Index Coding and Locally Repairable Codes

We establish a duality result between linear index coding and Locally Repairable Codes (LRCs). Specifically, we show that a natural extension of LRCs we call Generalized Locally Repairable Codes (GLCRs) are exactly dual to linear index codes. In a GLRC, every node is decodable from a specific set of other nodes and these sets induce a recoverability directed graph. We show that the dual linear subspace of a GLRC is a solution to an index coding instance where the side information graph is this GLRC recoverability graph. We show that the GLRC rate is equivalent to the complementary index coding rate, i.e. the number of transmissions saved by coding. Our second result uses this duality to establish a new upper bound for the multiple unicast network coding problem. In multiple unicast network coding, we are given a directed acyclic graph and r sources that want to send independent messages to r corresponding destinations. Our new upper bound is efficiently computable and relies on a strong approximation result for complementary index coding. We believe that our bound could lead to a logarithmic approximation factor for multiple unicast network coding if a plausible connection we state is verified.

cs.IT↗

arXiv2014

Graph Theory versus Minimum Rank for Index Coding

We obtain novel index coding schemes and show that they provably outperform all previously known graph theoretic bounds proposed so far. Further, we establish a rather strong negative result: all known graph theoretic bounds are within a logarithmic factor from the chromatic number. This is in striking contrast to minrank since prior work has shown that it can outperform the chromatic number by a polynomial factor in some cases. The conclusion is that all known graph theoretic bounds are not much stronger than the chromatic number.

cs.IT↗

arXiv2014

Locality and Availability in Distributed Storage

This paper studies the problem of code symbol availability: a code symbol is said to have $(r, t)$-availability if it can be reconstructed from $t$ disjoint groups of other symbols, each of size at most $r$. For example, $3$-replication supports $(1, 2)$-availability as each symbol can be read from its $t= 2$ other (disjoint) replicas, i.e., $r=1$. However, the rate of replication must vanish like $\frac{1}{t+1}$ as the availability increases. This paper shows that it is possible to construct codes that can support a scaling number of parallel reads while keeping the rate to be an arbitrarily high constant. It further shows that this is possible with the minimum distance arbitrarily close to the Singleton bound. This paper also presents a bound demonstrating a trade-off between minimum distance, availability and locality. Our codes match the aforementioned bound and their construction relies on combinatorial objects called resolvable designs. From a practical standpoint, our codes seem useful for distributed storage applications involving hot data, i.e., the information which is frequently accessed by multiple processes in parallel.

cs.IT↗

arXiv2014

Repairable Fountain Codes

We introduce a new family of Fountain codes that are systematic and also have sparse parities. Given an input of $k$ symbols, our codes produce an unbounded number of output symbols, generating each parity independently by linearly combining a logarithmic number of randomly selected input symbols. The construction guarantees that for any $ε>0$ accessing a random subset of $(1+ε)k$ encoded symbols, asymptotically suffices to recover the $k$ input symbols with high probability. Our codes have the additional benefit of logarithmic locality: a single lost symbol can be repaired by accessing a subset of $O(\log k)$ of the remaining encoded symbols. This is a desired property for distributed storage systems where symbols are spread over a network of storage nodes. Beyond recovery upon loss, local reconstruction provides an efficient alternative for reading symbols that cannot be accessed directly. In our code, a logarithmic number of disjoint local groups is associated with each systematic symbol, allowing multiple parallel reads. Our main mathematical contribution involves analyzing the rank of sparse random matrices with specific structure over finite fields. We rely on establishing that a new family of sparse random bipartite graphs have perfect matchings with high probability.

cs.IT↗

arXiv2013

A Repair Framework for Scalar MDS Codes

Several works have developed vector-linear maximum-distance separable (MDS) storage codes that min- imize the total communication cost required to repair a single coded symbol after an erasure, referred to as repair bandwidth (BW). Vector codes allow communicating fewer sub-symbols per node, instead of the entire content. This allows non trivial savings in repair BW. In sharp contrast, classic codes, like Reed- Solomon (RS), used in current storage systems, are deemed to suffer from naive repair, i.e. downloading the entire stored message to repair one failed node. This mainly happens because they are scalar-linear. In this work, we present a simple framework that treats scalar codes as vector-linear. In some cases, this allows significant savings in repair BW. We show that vectorized scalar codes exhibit properties that simplify the design of repair schemes. Our framework can be seen as a finite field analogue of real interference alignment. Using our simplified framework, we design a scheme that we call clique-repair which provably identifies the best linear repair strategy for any scalar 2-parity MDS code, under some conditions on the sub-field chosen for vectorization. We specify optimal repair schemes for specific (5,3)- and (6,4)-Reed- Solomon (RS) codes. Further, we present a repair strategy for the RS code currently deployed in the Facebook Analytics Hadoop cluster that leads to 20% of repair BW savings over naive repair which is the repair scheme currently used for this code.

cs.IT↗

arXiv2013

Index Coding Problem with Side Information Repositories

To tackle the expected enormous increase in mobile video traffic in cellular networks, an architecture involving a base station along with caching femto stations (referred to as helpers), storing popular files near users, has been proposed [1]. The primary benefit of caching is the enormous increase in downloading rate when a popular file is available at helpers near a user requesting that file. In this work, we explore a secondary benefit of caching in this architecture through the lens of index coding. We assume a system with n users and constant number of caching helpers. Only helpers store files, i.e. have side information. We investigate the following scenario: Each user requests a distinct file that is not found in the set of helpers nearby. Users are served coded packets (through an index code) by an omniscient base station. Every user decodes its desired packet from the coded packets and the side information packets from helpers nearby. We assume that users can obtain any file stored in their neighboring helpers without incurring transmission costs. With respect to the index code employed, we investigate two achievable schemes: 1) XOR coloring based on coloring of the side information graph associated with the problem and 2)Vector XOR coloring based on fractional coloring of the side information graph. We show that the general problem reduces to a canonical problem where every user is connected to exactly one helper under some topological constraints. For the canonical problem, with constant number of helpers (k), we show that the complexity of computing the best XOR/vector XOR coloring schemes are polynomial in the number of users n. The result exploits a special complete bi-partite structure that the side information graphs exhibit for any finite k.

cs.IT↗

arXiv2013

Optimal Locally Repairable Codes and Connections to Matroid Theory

Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly sub-optimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and optimal LRCs that are simple to construct. Our construction is based on grouping Reed-Solomon (RS) coded symbols to obtain RS coded symbols over a larger finite field. We then partition these RS symbols in small groups, and re-encode them using a simple local code that offers low repair locality. For the analysis of the optimality of the code, we derive a new result on the matroid represented by the code generator matrix.

cs.IT↗

arXiv2013

FemtoCaching: Wireless Video Content Delivery through Distributed Caching Helpers

Video on-demand streaming from Internet-based servers is becoming one of the most important services offered by wireless networks today. In order to improve the area spectral efficiency of video transmission in cellular systems, small cells heterogeneous architectures (e.g., femtocells, WiFi off-loading) are being proposed, such that video traffic to nomadic users can be handled by short-range links to the nearest small cell access points (referred to as "helpers"). As the helper deployment density increases, the backhaul capacity becomes the system bottleneck. In order to alleviate such bottleneck we propose a system where helpers with low-rate backhaul but high storage capacity cache popular video files. Files not available from helpers are transmitted by the cellular base station. We analyze the optimum way of assigning files to the helpers, in order to minimize the expected downloading time for files. We distinguish between the uncoded case (where only complete files are stored) and the coded case, where segments of Fountain-encoded versions of the video files are stored at helpers. We show that the uncoded optimum file assignment is NP-hard, and develop a greedy strategy that is provably within a factor 2 of the optimum. Further, for a special case we provide an efficient algorithm achieving a provably better approximation ratio of $1-(1-1/d)^d$, where $d$ is the maximum number of helpers a user can be connected to. We also show that the coded optimum cache assignment problem is convex that can be further reduced to a linear program. We present numerical results comparing the proposed schemes.

cs.NI↗