arXiv subjects

Michael Langberg

Publications and source records attributed to Michael Langberg.

At least 55 records · Page 3Linked to original sources

arXiv2016

Sufficiently Myopic Adversaries are Blind

In this work we consider a communication problem in which a sender, Alice, wishes to communicate with a receiver, Bob, over a channel controlled by an adversarial jammer, James, who is {\em myopic}. Roughly speaking, for blocklength $n$, the codeword $X^n$ transmitted by Alice is corrupted by James who must base his adversarial decisions (of which locations of $X^n$ to corrupt and how to corrupt them) not on the codeword $X^n$ but on $Z^n$, an image of $X^n$ through a noisy memoryless channel. More specifically, our communication model may be described by two channels. A memoryless channel $p(z|x)$ from Alice to James, and an {\it Arbitrarily Varying Channel} from Alice to Bob, $p(y|x,s)$ governed by a state $X^n$ determined by James. In standard adversarial channels the states $S^n$ may depend on the codeword $X^n$, but in our setting $S^n$ depends only on James's view $Z^n$. The myopic channel captures a broad range of channels and bridges between the standard models of memoryless and adversarial (zero-error) channels. In this work we present upper and lower bounds on the capacity of myopic channels. For a number of special cases of interest we show that our bounds are tight. We extend our results to the setting of {\em secure} communication in which we require that the transmitted message remain secret from James. For example, we show that if (i) James may flip at most a $p$ fraction of the bits communicated between Alice and Bob, and (ii) James views $X^n$ through a binary symmetric channel with parameter $q$, then once James is "sufficiently myopic" (in this case, when $q>p$), then the optimal communication rate is that of an adversary who is "blind" (that is, an adversary that does not see $X^n$ at all), which is $1-H(p)$ for standard communication, and $H(q)-H(p)$ for secure communication. A similar phenomenon exists for our general model of communication.

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arXiv2016

The Unbounded Benefit of Encoder Cooperation for the $k$-user MAC

Cooperation strategies allow communication devices to work together to improve network capacity. Consider a network consisting of a $k$-user multiple access channel (MAC) and a node that is connected to all $k$ encoders via rate-limited bidirectional links, referred to as the "cooperation facilitator" (CF). Define the cooperation benefit as the sum-capacity gain resulting from the communication between the encoders and the CF and the cooperation rate as the total rate the CF shares with the encoders. This work demonstrates the existence of a class of $k$-user MACs where the ratio of the cooperation benefit to cooperation rate tends to infinity as the cooperation rate tends to zero. Examples of channels in this class include the binary erasure MAC for $k=2$ and the $k$-user Gaussian MAC for any $k\geq 2$.

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arXiv2016

Can Negligible Cooperation Increase Network Reliability?

In network cooperation strategies, nodes work together with the aim of increasing transmission rates or reliability. This paper demonstrates that enabling cooperation between the transmitters of a two-user multiple access channel, via a cooperation facilitator that has access to both messages, always results in a network whose maximal- and average-error sum-capacities are the same---even when those capacities differ in the absence of cooperation and the information shared with the encoders is negligible. From this result, it follows that if a multiple access channel with no transmitter cooperation has different maximal- and average-error sum-capacities, then the maximal-error sum-capacity of the network consisting of this channel and a cooperation facilitator is not continuous with respect to the output edge capacities of the facilitator. This shows that there exist networks where sharing even a negligible number of bits per channel use with the encoders yields a non-negligible benefit.

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arXiv2016

On the Capacity Advantage of a Single Bit

In this work we study the capacity advantage achieved by adding a single bit of communication -- not a link of capacity 1 but a single bit over all time -- to a memoryless network. Specifically, we present a memoryless network in which adding a single bit of communication strictly increases the capacity region.

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arXiv2016

A Unified Framework for Approximating and Clustering Data

Given a set $F$ of $n$ positive functions over a ground set $X$, we consider the problem of computing $x^*$ that minimizes the expression $\sum_{f\in F}f(x)$, over $x\in X$. A typical application is \emph{shape fitting}, where we wish to approximate a set $P$ of $n$ elements (say, points) by a shape $x$ from a (possibly infinite) family $X$ of shapes. Here, each point $p\in P$ corresponds to a function $f$ such that $f(x)$ is the distance from $p$ to $x$, and we seek a shape $x$ that minimizes the sum of distances from each point in $P$. In the $k$-clustering variant, each $x\in X$ is a tuple of $k$ shapes, and $f(x)$ is the distance from $p$ to its closest shape in $x$. Our main result is a unified framework for constructing {\em coresets} and {\em approximate clustering} for such general sets of functions. To achieve our results, we forge a link between the classic and well defined notion of $\varepsilon$-approximations from the theory of PAC Learning and VC dimension, to the relatively new (and not so consistent) paradigm of coresets, which are some kind of "compressed representation" of the input set $F$. Using traditional techniques, a coreset usually implies an LTAS (linear time approximation scheme) for the corresponding optimization problem, which can be computed in parallel, via one pass over the data, and using only polylogarithmic space (i.e, in the streaming model). We show how to generalize the results of our framework for squared distances (as in $k$-mean), distances to the $q$th power, and deterministic constructions.

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arXiv2016

Communication Efficient Secret Sharing

A secret sharing scheme is a method to store information securely and reliably. Particularly, in a threshold secret sharing scheme, a secret is encoded into $n$ shares, such that any set of at least $t_1$ shares suffice to decode the secret, and any set of at most $t_2 < t_1$ shares reveal no information about the secret. Assuming that each party holds a share and a user wishes to decode the secret by receiving information from a set of parties; the question we study is how to minimize the amount of communication between the user and the parties. We show that the necessary amount of communication, termed "decoding bandwidth", decreases as the number of parties that participate in decoding increases. We prove a tight lower bound on the decoding bandwidth, and construct secret sharing schemes achieving the bound. Particularly, we design a scheme that achieves the optimal decoding bandwidth when $d$ parties participate in decoding, universally for all $t_1 \le d \le n$. The scheme is based on Shamir's secret sharing scheme and preserves its simplicity and efficiency. In addition, we consider secure distributed storage where the proposed communication efficient secret sharing schemes further improve disk access complexity during decoding.

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arXiv2016

The benefit of a 1-bit jump-start, and the necessity of stochastic encoding, in jamming channels

We consider the problem of communicating a message $m$ in the presence of a malicious jamming adversary (Calvin), who can erase an arbitrary set of up to $pn$ bits, out of $n$ transmitted bits $(x_1,\ldots,x_n)$. The capacity of such a channel when Calvin is exactly causal, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends on his observations $(x_1,\ldots,x_i)$ was recently characterized to be $1-2p$. In this work we show two (perhaps) surprising phenomena. Firstly, we demonstrate via a novel code construction that if Calvin is delayed by even a single bit, i.e. Calvin's decision of whether or not to erase bit $x_i$ depends only on $(x_1,\ldots,x_{i-1})$ (and is independent of the "current bit" $x_i$) then the capacity increases to $1-p$ when the encoder is allowed to be stochastic. Secondly, we show via a novel jamming strategy for Calvin that, in the single-bit-delay setting, if the encoding is deterministic (i.e. the transmitted codeword is a deterministic function of the message $m$) then no rate asymptotically larger than $1-2p$ is possible with vanishing probability of error, hence stochastic encoding (using private randomness at the encoder) is essential to achieve the capacity of $1-p$ against a one-bit-delayed Calvin.

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arXiv2016

The Capacity of Online (Causal) $q$-ary Error-Erasure Channels

In the $q$-ary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1,\ldots,q-1\}^n$ symbol by symbol via a channel limited to at most $pn$ errors and/or $p^{*} n$ erasures. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th symbol or not based on its view so far, i.e., its decision depends only on the transmitted symbols $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the corruption is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of $q$-ary online channels for a combined corruption model, in which the channel may impose at most $pn$ {\em errors} and at most $p^{*} n$ {\em erasures} on the transmitted codeword. The online channel (in both the error and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we give a full characterization of the capacity as a function of $q,p$, and $p^{*}$.

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arXiv2016

The Multivariate Covering Lemma and its Converse

The multivariate covering lemma states that given a collection of $k$ codebooks, each of sufficiently large cardinality and independently generated according to one of the marginals of a joint distribution, one can always choose one codeword from each codebook such that the resulting $k$-tuple of codewords is jointly typical with respect to the joint distribution. We give a proof of this lemma for weakly typical sets. This allows achievability proofs that rely on the covering lemma to go through for continuous channels (e.g., Gaussian) without the need for quantization. The covering lemma and its converse are widely used in information theory, including in rate-distortion theory and in achievability results for multi-user channels.

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arXiv2015

Asymmetric Error Correction and Flash-Memory Rewriting using Polar Codes

We propose efficient coding schemes for two communication settings: 1. asymmetric channels, and 2. channels with an informed encoder. These settings are important in non-volatile memories, as well as optical and broadcast communication. The schemes are based on non-linear polar codes, and they build on and improve recent work on these settings. In asymmetric channels, we tackle the exponential storage requirement of previously known schemes, that resulted from the use of large Boolean functions. We propose an improved scheme, that achieves the capacity of asymmetric channels with polynomial computational complexity and storage requirement. The proposed non-linear scheme is then generalized to the setting of channel coding with an informed encoder, using a multicoding technique. We consider specific instances of the scheme for flash memories, that incorporate error-correction capabilities together with rewriting. Since the considered codes are non-linear, they eliminate the requirement of previously known schemes (called polar write-once-memory codes) for shared randomness between the encoder and the decoder. Finally, we mention that the multicoding scheme is also useful for broadcast communication in Marton's region, improving upon previous schemes for this setting.

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arXiv2015

Connecting Multiple-unicast and Network Error Correction: Reduction and Unachievability

We show that solving a multiple-unicast network coding problem can be reduced to solving a single-unicast network error correction problem, where an adversary may jam at most a single edge in the network. Specifically, we present an efficient reduction that maps a multiple-unicast network coding instance to a network error correction instance while preserving feasibility. The reduction holds for both the zero probability of error model and the vanishing probability of error model. Previous reductions are restricted to the zero-error case. As an application of the reduction, we present a constructive example showing that the single-unicast network error correction capacity may not be achievable, a result of separate interest.

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arXiv2015

On the Cost and Benefit of Cooperation (Extended Version)

In a cooperative coding scheme, network nodes work together to achieve higher transmission rates. To obtain a better understanding of cooperation, we consider a model in which two transmitters send rate-limited descriptions of their messages to a "cooperation facilitator", a node that sends back rate-limited descriptions of the pair to each transmitter. This model includes the conferencing encoders model and a prior model from the current authors as special cases. We show that except for a special class of multiple access channels, the gain in sum-capacity resulting from cooperation under this model is quite large. Adding a cooperation facilitator to any such channel results in a network that does not satisfy the edge removal property. An important special case is the Gaussian multiple access channel, for which we explicitly characterize the sum-rate cooperation gain.

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arXiv2014

A characterization of the capacity of online (causal) binary channels

In the binary online (or "causal") channel coding model, a sender wishes to communicate a message to a receiver by transmitting a codeword $\mathbf{x} =(x_1,\ldots,x_n) \in \{0,1\}^n$ bit by bit via a channel limited to at most $pn$ corruptions. The channel is "online" in the sense that at the $i$th step of communication the channel decides whether to corrupt the $i$th bit or not based on its view so far, i.e., its decision depends only on the transmitted bits $(x_1,\ldots,x_i)$. This is in contrast to the classical adversarial channel in which the error is chosen by a channel that has a full knowledge on the sent codeword $\mathbf{x}$. In this work we study the capacity of binary online channels for two corruption models: the {\em bit-flip} model in which the channel may flip at most $pn$ of the bits of the transmitted codeword, and the {\em erasure} model in which the channel may erase at most $pn$ bits of the transmitted codeword. Specifically, for both error models we give a full characterization of the capacity as a function of $p$. The online channel (in both the bit-flip and erasure case) has seen a number of recent studies which present both upper and lower bounds on its capacity. In this work, we present and analyze a coding scheme that improves on the previously suggested lower bounds and matches the previously suggested upper bounds thus implying a tight characterization.

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arXiv2014

Single-Source/Sink Network Error Correction Is as Hard as Multiple-Unicast

We study the problem of communicating over a single-source single-terminal network in the presence of an adversary that may jam a single link of the network. If any one of the edges can be jammed, the capacity of such networks is well understood and follows directly from the connection between the minimum cut and maximum flow in single-source single- terminal networks. In this work we consider networks in which some edges cannot be jammed, and show that determining the network communication capacity is at least as hard as solving the multiple-unicast network coding problem for the error-free case. The latter problem is a long standing open problem.

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arXiv2014

On the Power of Cooperation: Can a Little Help a Lot? (Extended Version)

In this paper, we propose a new cooperation model for discrete memoryless multiple access channels. Unlike in prior cooperation models (e.g., conferencing encoders), where the transmitters cooperate directly, in this model the transmitters cooperate through a larger network. We show that under this indirect cooperation model, there exist channels for which the increase in sum-capacity resulting from cooperation is significantly larger than the rate shared by the transmitters to establish the cooperation. This result contrasts both with results on the benefit of cooperation under prior models and results in the network coding literature, where attempts to find examples in which similar small network modifications yield large capacity benefits have to date been unsuccessful.

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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.

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arXiv2014

Reverse Edge Cut-Set Bounds for Secure Network Coding

We consider the problem of secure communication over a network in the presence of wiretappers. We give a new cut-set bound on secrecy capacity which takes into account the contribution of both forward and backward edges crossing the cut, and the connectivity between their endpoints in the rest of the network. We show the bound is tight on a class of networks, which demonstrates that it is not possible to find a tighter bound by considering only cut set edges and their connectivity.

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arXiv2013

Edge distribution in generalized graph products

Given a graph $G=(V,E)$, an integer $k$, and a function $f_G:V^k \times V^k \to {0,1}$, the $k^{th}$ graph product of $G$ w.r.t $f_G$ is the graph with vertex set $V^k$, and an edge between two vertices $x=(x_1,...,x_k)$ and $y=(y_1,...,y_k)$ iff $f_G(x,y)=1$. Graph products are a basic combinatorial object, widely studied and used in different areas such as hardness of approximation, information theory, etc. We study graph products for functions $f_G$ of the form $f_G(x,y)=1$ iff there are at least $t$ indices $i \in [k]$ s.t. $(x_i,y_i)\in E$, where $t \in [k]$ is a fixed parameter in $f_G$. This framework generalizes the well-known graph tensor-product (obtained for $t=k$) and the graph or-product (obtained for $t=1$). The property that interests us is the edge distribution in such graphs. We show that if $G$ has a spectral gap, then the number of edges connecting "large-enough" sets in $G^k$ is "well-behaved", namely, it is close to the expected value, had the sets been random. We extend our results to bi-partite graph products as well. For a bi-partite graph $G=(X,Y,E)$, the $k^{th}$ bi-partite graph product of $G$ w.r.t $f_G$ is the bi-partite graph with vertex sets $X^k$ and $Y^k$ and edges between $x \in X^k$ and $y \in Y^k$ iff $f_G(x,y)=1$. Finally, for both types of graph products, optimality is asserted using the "Converse to the Expander Mixing Lemma" obtained by Bilu and Linial in 2006. A byproduct of our proof technique is a new explicit construction of a family of co-spectral graphs.

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