arXiv subjects

Michael Gastpar

Publications and source records attributed to Michael Gastpar.

At least 109 records · Page 6Linked to original sources

arXiv2012

Relaxing the Gaussian AVC

The arbitrarily varying channel (AVC) is a conservative way of modeling an unknown interference, and the corresponding capacity results are pessimistic. We reconsider the Gaussian AVC by relaxing the classical model and thereby weakening the adversarial nature of the interference. We examine three different relaxations. First, we show how a very small amount of common randomness between transmitter and receiver is sufficient to achieve the rates of fully randomized codes. Second, akin to the dirty paper coding problem, we study the impact of an additional interference known to the transmitter. We provide partial capacity results that differ significantly from the standard AVC. Third, we revisit a Gaussian MIMO AVC in which the interference is arbitrary but of limited dimension.

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arXiv2012

The Sampling Rate-Distortion Tradeoff for Sparsity Pattern Recovery in Compressed Sensing

Recovery of the sparsity pattern (or support) of an unknown sparse vector from a limited number of noisy linear measurements is an important problem in compressed sensing. In the high-dimensional setting, it is known that recovery with a vanishing fraction of errors is impossible if the measurement rate and the per-sample signal-to-noise ratio (SNR) are finite constants, independent of the vector length. In this paper, it is shown that recovery with an arbitrarily small but constant fraction of errors is, however, possible, and that in some cases computationally simple estimators are near-optimal. Bounds on the measurement rate needed to attain a desired fraction of errors are given in terms of the SNR and various key parameters of the unknown vector for several different recovery algorithms. The tightness of the bounds, in a scaling sense, as a function of the SNR and the fraction of errors, is established by comparison with existing information-theoretic necessary bounds. Near optimality is shown for a wide variety of practically motivated signal models.

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arXiv2012

Ergodic Interference Alignment

This paper develops a new communication strategy, ergodic interference alignment, for the K-user interference channel with time-varying fading. At any particular time, each receiver will see a superposition of the transmitted signals plus noise. The standard approach to such a scenario results in each transmitter-receiver pair achieving a rate proportional to 1/K its interference-free ergodic capacity. However, given two well-chosen time indices, the channel coefficients from interfering users can be made to exactly cancel. By adding up these two observations, each receiver can obtain its desired signal without any interference. If the channel gains have independent, uniform phases, this technique allows each user to achieve at least 1/2 its interference-free ergodic capacity at any signal-to-noise ratio. Prior interference alignment techniques were only able to attain this performance as the signal-to-noise ratio tended to infinity. Extensions are given for the case where each receiver wants a message from more than one transmitter as well as the "X channel" case (with two receivers) where each transmitter has an independent message for each receiver. Finally, it is shown how to generalize this strategy beyond Gaussian channel models. For a class of finite field interference channels, this approach yields the ergodic capacity region.

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arXiv2012

Approximate Feedback Capacity of the Gaussian Multicast Channel

We characterize the capacity region to within log{2(M-1)} bits/s/Hz for the M-transmitter K-receiver Gaussian multicast channel with feedback where each receiver wishes to decode every message from the M transmitters. Extending Cover-Leung's achievable scheme intended for (M,K)=(2,1), we show that this generalized scheme achieves the cutset-based outer bound within log{2(M-1)} bits per transmitter for all channel parameters. In contrast to the capacity in the non-feedback case, the feedback capacity improves upon the naive intersection of the feedback capacities of K individual multiple access channels. We find that feedback provides unbounded multiplicative gain at high signal-to-noise ratios as was shown in the Gaussian interference channel. To complement the results, we establish the exact feedback capacity of the Avestimehr-Diggavi-Tse (ADT) deterministic model, from which we make the observation that feedback can also be beneficial for function computation.

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arXiv2012

Minimum Cost Multicast with Decentralized Sources

In this paper we study the multisource multicast problem where every sink in a given directed acyclic graph is a client and is interested in a common file. We consider the case where each node can have partial knowledge about the file as a side information. Assuming that nodes can communicate over the capacity constrained links of the graph, the goal is for each client to gain access to the file, while minimizing some linear cost function of number of bits transmitted in the network. We consider three types of side-information settings:(ii) side information in the form of linearly correlated packets; and (iii) the general setting where the side information at the nodes have an arbitrary (i.i.d.) correlation structure. In this work we 1) provide a polynomial time feasibility test, i.e., whether or not all the clients can recover the file, and 2) we provide a polynomial-time algorithm that finds the optimal rate allocation among the links of the graph, and then determines an explicit transmission scheme for cases (i) and (ii).

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arXiv2012

Reduced-Dimension Linear Transform Coding of Correlated Signals in Networks

A model, called the linear transform network (LTN), is proposed to analyze the compression and estimation of correlated signals transmitted over directed acyclic graphs (DAGs). An LTN is a DAG network with multiple source and receiver nodes. Source nodes transmit subspace projections of random correlated signals by applying reduced-dimension linear transforms. The subspace projections are linearly processed by multiple relays and routed to intended receivers. Each receiver applies a linear estimator to approximate a subset of the sources with minimum mean squared error (MSE) distortion. The model is extended to include noisy networks with power constraints on transmitters. A key task is to compute all local compression matrices and linear estimators in the network to minimize end-to-end distortion. The non-convex problem is solved iteratively within an optimization framework using constrained quadratic programs (QPs). The proposed algorithm recovers as special cases the regular and distributed Karhunen-Loeve transforms (KLTs). Cut-set lower bounds on the distortion region of multi-source, multi-receiver networks are given for linear coding based on convex relaxations. Cut-set lower bounds are also given for any coding strategy based on information theory. The distortion region and compression-estimation tradeoffs are illustrated for different communication demands (e.g. multiple unicast), and graph structures.

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arXiv2012

Data Exchange Problem with Helpers

In this paper we construct a deterministic polynomial time algorithm for the problem where a set of users is interested in gaining access to a common file, but where each has only partial knowledge of the file. We further assume the existence of another set of terminals in the system, called helpers, who are not interested in the common file, but who are willing to help the users. Given that the collective information of all the terminals is sufficient to allow recovery of the entire file, the goal is to minimize the (weighted) sum of bits that these terminals need to exchange over a noiseless public channel in order achieve this goal. Based on established connections to the multi-terminal secrecy problem, our algorithm also implies a polynomial-time method for constructing the largest shared secret key in the presence of an eavesdropper. We consider the following side-information settings: (i) side-information in the form of uncoded packets of the file, where the terminals' side-information consists of subsets of the file; (ii) side-information in the form of linearly correlated packets, where the terminals have access to linear combinations of the file packets; and (iii) the general setting where the the terminals' side-information has an arbitrary (i.i.d.) correlation structure. We provide a polynomial-time algorithm (in the number of terminals) that finds the optimal rate allocations for these terminals, and then determines an explicit optimal transmission scheme for cases (i) and (ii).

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arXiv2011

Optimal Deterministic Polynomial-Time Data Exchange for Omniscience

We study the problem of constructing a deterministic polynomial time algorithm that achieves omniscience, in a rate-optimal manner, among a set of users that are interested in a common file but each has only partial knowledge about it as side-information. Assuming that the collective information among all the users is sufficient to allow the reconstruction of the entire file, the goal is to minimize the (possibly weighted) amount of bits that these users need to exchange over a noiseless public channel in order for all of them to learn the entire file. Using established connections to the multi-terminal secrecy problem, our algorithm also implies a polynomial-time method for constructing a maximum size secret shared key in the presence of an eavesdropper. We consider the following types of side-information settings: (i) side information in the form of uncoded fragments/packets of the file, where the users' side-information consists of subsets of the file; (ii) side information in the form of linearly correlated packets, where the users have access to linear combinations of the file packets; and (iii) the general setting where the the users' side-information has an arbitrary (i.i.d.) correlation structure. Building on results from combinatorial optimization, we provide a polynomial-time algorithm (in the number of users) that, first finds the optimal rate allocations among these users, then determines an explicit transmission scheme (i.e., a description of which user should transmit what information) for cases (i) and (ii).

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arXiv2011

On the Role of Diversity in Sparsity Estimation

A major challenge in sparsity pattern estimation is that small modes are difficult to detect in the presence of noise. This problem is alleviated if one can observe samples from multiple realizations of the nonzero values for the same sparsity pattern. We will refer to this as "diversity". Diversity comes at a price, however, since each new realization adds new unknown nonzero values, thus increasing uncertainty. In this paper, upper and lower bounds on joint sparsity pattern estimation are derived. These bounds, which improve upon existing results even in the absence of diversity, illustrate key tradeoffs between the number of measurements, the accuracy of estimation, and the diversity. It is shown, for instance, that diversity introduces a tradeoff between the uncertainty in the noise and the uncertainty in the nonzero values. Moreover, it is shown that the optimal amount of diversity significantly improves the behavior of the estimation problem for both optimal and computationally efficient estimators.

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arXiv2011

Compute-and-Forward: Harnessing Interference through Structured Codes

Interference is usually viewed as an obstacle to communication in wireless networks. This paper proposes a new strategy, compute-and-forward, that exploits interference to obtain significantly higher rates between users in a network. The key idea is that relays should decode linear functions of transmitted messages according to their observed channel coefficients rather than ignoring the interference as noise. After decoding these linear equations, the relays simply send them towards the destinations, which given enough equations, can recover their desired messages. The underlying codes are based on nested lattices whose algebraic structure ensures that integer combinations of codewords can be decoded reliably. Encoders map messages from a finite field to a lattice and decoders recover equations of lattice points which are then mapped back to equations over the finite field. This scheme is applicable even if the transmitters lack channel state information.

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arXiv2011

A Compressed Sensing Wire-Tap Channel

A multiplicative Gaussian wire-tap channel inspired by compressed sensing is studied. Lower and upper bounds on the secrecy capacity are derived, and shown to be relatively tight in the large system limit for a large class of compressed sensing matrices. Surprisingly, it is shown that the secrecy capacity of this channel is nearly equal to the capacity without any secrecy constraint provided that the channel of the eavesdropper is strictly worse than the channel of the intended receiver. In other words, the eavesdropper can see almost everything and yet learn almost nothing. This behavior, which contrasts sharply with that of many commonly studied wiretap channels, is made possible by the fact that a small number of linear projections can make a crucial difference in the ability to estimate sparse vectors.

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arXiv2011

Reliable Physical Layer Network Coding

When two or more users in a wireless network transmit simultaneously, their electromagnetic signals are linearly superimposed on the channel. As a result, a receiver that is interested in one of these signals sees the others as unwanted interference. This property of the wireless medium is typically viewed as a hindrance to reliable communication over a network. However, using a recently developed coding strategy, interference can in fact be harnessed for network coding. In a wired network, (linear) network coding refers to each intermediate node taking its received packets, computing a linear combination over a finite field, and forwarding the outcome towards the destinations. Then, given an appropriate set of linear combinations, a destination can solve for its desired packets. For certain topologies, this strategy can attain significantly higher throughputs over routing-based strategies. Reliable physical layer network coding takes this idea one step further: using judiciously chosen linear error-correcting codes, intermediate nodes in a wireless network can directly recover linear combinations of the packets from the observed noisy superpositions of transmitted signals. Starting with some simple examples, this survey explores the core ideas behind this new technique and the possibilities it offers for communication over interference-limited wireless networks.

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arXiv2010

"Compressed" Compressed Sensing

The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that in certain cases (e.g. discrete valued vectors or large distortions) the number of samples can be decreased. Interestingly though, it is also shown that in many cases no reduction is possible.

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arXiv2009

State constraints and list decoding for the AVC

List decoding for arbitrarily varying channels (AVCs) under state constraints is investigated. It is shown that rates within $ε$ of the randomized coding capacity of AVCs with input-dependent state can be achieved under maximal error with list decoding using lists of size $O(1/ε)$. Under average error an achievable rate region and converse bound are given for lists of size $L$. These bounds are based on two different notions of symmetrizability and do not coincide in general. An example is given that shows that for list size $L$ the capacity may be positive but strictly smaller than the randomized coding capacity. This behavior is different than the situation without state constraints.

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arXiv2009

Rateless codes for AVC models

The arbitrarily varying channel (AVC) is a channel model whose state is selected maliciously by an adversary. Fixed-blocklength coding assumes a worst-case bound on the adversary's capabilities, which leads to pessimistic results. This paper defines a variable-length perspective on this problem, for which achievable rates are shown that depend on the realized actions of the adversary. Specifically, rateless codes are constructed which require a limited amount of common randomness. These codes are constructed for two kinds of AVC models. In the first the channel state cannot depend on the channel input, and in the second it can. As a byproduct, the randomized coding capacity of the AVC with state depending on the transmitted codeword is found and shown to be achievable with a small amount of common randomness. The results for this model are proved using a randomized strategy based on list decoding.

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arXiv2009

Zero-rate feedback can achieve the empirical capacity

The utility of limited feedback for coding over an individual sequence of DMCs is investigated. This study complements recent results showing how limited or noisy feedback can boost the reliability of communication. A strategy with fixed input distribution $P$ is given that asymptotically achieves rates arbitrarily close to the mutual information induced by $P$ and the state-averaged channel. When the capacity achieving input distribution is the same over all channel states, this achieves rates at least as large as the capacity of the state averaged channel, sometimes called the empirical capacity.

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arXiv2008

Bits through ARQs

A fundamental problem in dynamic frequency reuse is that the cognitive radio is ignorant of the amount of interference it inflicts on the primary license holder. A model for such a situation is proposed and analyzed. The primary sends packets across an erasure channel and employs simple ACK/NAK feedback (ARQs) to retransmit erased packets. Furthermore, its erasure probabilities are influenced by the cognitive radio's activity. While the cognitive radio does not know these interference characteristics, it can eavesdrop on the primary's ARQs. The model leads to strategies in which the cognitive radio adaptively adjusts its input based on the primary's ARQs thereby guaranteeing the primary exceeds a target packet rate. A relatively simple strategy whereby the cognitive radio transmits only when the primary's empirical packet rate exceeds a threshold is shown to have interesting universal properties in the sense that for unknown time-varying interference characteristics, the primary is guaranteed to meet its target rate. Furthermore, a more intricate version of this strategy is shown to be capacity-achieving for the cognitive radio when the interference characteristics are time-invariant.

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arXiv2008

The pre-log of Gaussian broadcast with feedback can be two

A generic intuition says that the pre-log, or multiplexing gain, cannot be larger than the minimum of the number of transmit and receive dimensions. This suggests that for the scalar broadcast channel, the pre-log cannot exceed one. By contrast, in this note, we show that when the noises are anti-correlated and feedback is present, then a pre-log of two can be attained. In other words, in this special case, in the limit of high SNR, the scalar Gaussian broadcast channel turns into two parallel AWGN channels. Achievability is established via a coding strategy due to Schalkwijk, Kailath, and Ozarow.

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