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In the phase retrieval problem, an unknown vector is to be recovered given quadratic measurements. This problem has received considerable attention in recent times. In this paper, we present an algorithm to solve a nonconvex formulation of the phase retrieval problem, that we call $\textit{Incremental Truncated Wirtinger Flow}$. Given random Gaussian sensing vectors, we prove that it converges linearly to the solution, with an optimal sample complexity. We also provide stability guarantees of the algorithm under noisy measurements. Performance and comparisons with existing algorithms are illustrated via numerical experiments on simulated and real data, with both random and structured sensing vectors.

The capacity regions of semideterministic multiuser channels, such as the semideterministic relay channel and the multiple access channel with partially cribbing encoders, have been characterized using the idea of partial-decode-forward. However, the requirement to explicitly decode part of the message at intermediate nodes can be restrictive in some settings; for example, when nodes have different side information regarding the state of the channel. In this paper, we generalize this scheme to $\textit{cooperative-bin-forward}$ by building on the observation that explicit recovering of part of the message is not needed to induce cooperation. Instead, encoders can bin their received signals and cooperatively forward the bin index to the decoder. The main advantage of this new scheme is illustrated by considering state-dependent extensions of the aforementioned semideterministic setups. While partial-decode-forward is not applicable in these new setups, cooperative-bin-forward continues to achieve capacity., Comment: submitted to IEEE Transactions on Information Theory, presented in part at ISIT 2015; version-2 contains an illustrative example

The best known inner bound for the 2-user discrete memoryless interference channel is the Han-Kobayashi rate region. The coding schemes that achieve this region are based on rate-splitting and superposition coding. In this paper, we develop a multicoding scheme to achieve the same rate region. A key advantage of the multicoding nature of the proposed coding scheme is that it can be naturally extended to more general settings, such as when encoders have state information or can overhear each other. In particular, we extend our coding scheme to characterize the capacity region of the state-dependent deterministic Z-interference channel when noncausal state information is available at the interfering transmitter. We specialize our results to the case of the linear deterministic model with on/off interference which models a wireless system where a cognitive transmitter is noncausally aware of the times it interferes with a primary transmission. For this special case, we provide an explicit expression for the capacity region and discuss some interesting properties of the optimal strategy. We also extend our multicoding scheme to find the capacity region of the deterministic Z-interference channel when the signal of the interfering transmitter can be overheard at the other transmitter (a.k.a. unidirectional partial cribbing).

Consider a Gaussian relay network where a source node communicates to a destination node with the help of several layers of relays. Recent work has shown that compress-and-forward based strategies can achieve the capacity of this network within an additive gap. Here, the relays quantize their received signals at the noise level and map them to random Gaussian codebooks. The resultant gap to capacity is independent of the SNRs of the channels in the network and the topology but is linear in the total number of nodes. In this paper, we provide an improved lower bound on the rate achieved by compress-and-forward based strategies (noisy network coding in particular) in arbitrary Gaussian relay networks, whose gap to capacity depends on the network not only through the total number of nodes but also through the degrees of freedom of the min cut of the network. We illustrate that for many networks, this refined lower bound can lead to a better approximation of the capacity. In particular, we demonstrate that it leads to a logarithmic rather than linear capacity gap in the total number of nodes for certain classes of layered networks. The improvement comes from quantizing the received signals of the relays at a resolution decreasing with the total number of nodes in the network. This suggests that the rule-of-thumb in literature of quantizing the received signals at the noise level can be highly suboptimal., Comment: to appear in IEEE Transactions on Information Theory

We study a simple question: when are dynamic relaying strategies essential in optimizing the diversity-multiplexing tradeoff (DMT) in half-duplex wireless relay networks? This is motivated by apparently two contrasting results even for a simple 3 node network, with a single half-duplex relay. When all channels are assumed to be i.i.d. fading, a static schedule where the relay listens half the time and transmits half the time combined with quantize-map-forward (QMF) relaying is known to achieve the full-duplex performance. However, when there is no direct link between source and destination, a dynamic-decode-forward (DDF) strategy is needed to achieve the optimal tradeoff. In this case, a static schedule is strictly suboptimal and the optimal tradeoff is significantly worse than the full-duplex performance. In this paper we study the general case when the direct link is neither as strong as the other links nor fully non-existent, and identify regimes where dynamic schedules are necessary and those where static schedules are enough. We identify 4 qualitatively different regimes for the single relay channel where the tradeoff between diversity and multiplexing is significantly different. We show that in all these regimes one of the above two strategies is sufficient to achieve the optimal tradeoff by developing a new upper bound on the best achievable tradeoff under channel state information available only at the receivers. A natural next question is whether these two strategies are sufficient to achieve the DMT of more general half-duplex wireless networks. We propose a generalization of the two existing schemes through a dynamic QMF (DQMF) strategy, where the relay listens for a fraction of time depending on received CSI but not long enough to be able to decode. We show that such a DQMF strategy is needed to achieve the optimal DMT in a parallel channel with two relays., Comment: to appear in IEEE Transactions on Information Theory

Communication networks are often designed and analyzed assuming tight synchronization among nodes. However, in applications that require communication in the energy-efficient regime of low signal-to-noise ratios, establishing tight synchronization among nodes in the network can result in a significant energy overhead. Motivated by a recent result showing that near-optimal energy efficiency can be achieved over the AWGN channel without requiring tight synchronization, we consider the question of whether the potential gains of cooperative communication can be achieved in the absence of synchronization. We focus on the symmetric Gaussian diamond network and establish that cooperative-communication gains are indeed feasible even with unsynchronized nodes. More precisely, we show that the capacity per unit energy of the unsynchronized symmetric Gaussian diamond network is within a constant factor of the capacity per unit energy of the corresponding synchronized network. To this end, we propose a distributed relaying scheme that does not require tight synchronization but nevertheless achieves most of the energy gains of coherent combining., Comment: 20 pages, 4 figures, submitted to IEEE Transactions on Information Theory, presented at IEEE ISIT 2014

Diversity-multiplexing trade-off has been studied extensively to quantify the benefits of different relaying strategies in terms of error and rate performance. However, even in the case of a single half-duplex relay, which seems fully characterized, implications are not clear. When all channels in the system are assumed to be independent and identically fading, a fixed schedule where the relay listens half of the total duration for communication and transmits the second half combined with quantize-map-and-forward relaying (static QMF) is known to achieve the full-duplex performance [1]. However, when there is no direct link between the source and the destination, a dynamic decode-and-forward (DDF) strategy is needed [2]. It is not clear which one of these two conclusions would carry to a less idealized setup, where the direct link can be neither as strong as the other links nor fully non-existent. In this paper, we provide a generalized diversity-multiplexing trade-off for the half-duplex relay channel which accounts for different channel strengths and recovers the two earlier results as two special cases. We show that these two strategies are sufficient to achieve the diversity-multiplexing trade-off across all channel configurations, by characterizing the best achievable trade-off when channel state information (CSI) is only available at the receivers (CSIR). However, for general relay networks we show that a generalization of these two schemes through a dynamic QMF strategy is needed to achieve optimal performance., Comment: to be presented at IEEE International Symposium on Information Theory (ISIT) 2013

Consider a Gaussian relay network where a number of sources communicate to a destination with the help of several layers of relays. Recent work has shown that a compress-and-forward based strategy at the relays can achieve the capacity of this network within an additive gap. In this strategy, the relays quantize their observations at the noise level and map it to a random Gaussian codebook. The resultant capacity gap is independent of the SNR's of the channels in the network but linear in the total number of nodes. In this paper, we show that if the relays quantize their signals at a resolution decreasing with the number of nodes in the network, the additive gap to capacity can be made logarithmic in the number of nodes for a class of layered, time-varying wireless relay networks. This suggests that the rule-of-thumb to quantize the received signals at the noise level used for compress-and-forward in the current literature can be highly suboptimal., Comment: Submitted to ITW 2013