The Experts below are selected from a list of 306 Experts worldwide ranked by ideXlab platform
Syed Ali Jafar - One of the best experts on this subject based on the ideXlab platform.
-
Interference Alignment and the Degrees of Freedom of Wireless $X$ Networks
IEEE Transactions on Information Theory, 2009Co-Authors: Viveck R Cadambe, Syed Ali JafarAbstract:We explore the Degrees of Freedom of M times N user wireless X networks, i.e., networks of M transmitters and N receivers where every transmitter has an independent message for every receiver. We derive a general outer bound on the Degrees of Freedom region of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the total number of Degrees of Freedom of the X network is equal to [(MN)/(M+N-1)] per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for X networks that can come arbitrarily close to the outer bound on Degrees of Freedom. For the case where either M=2 or N=2 we find that the Degrees of Freedom characterization also provides a capacity approximation that is accurate to within O(1) . For these cases the Degrees of Freedom outer bound is exactly achievable.
-
Degrees of Freedom of the k user mimo interference channel
Asilomar Conference on Signals Systems and Computers, 2008Co-Authors: Syed Ali JafarAbstract:We provide inner bound and outer bound for the total number of Degrees of Freedom of the K user multiple input multiple output (MIMO) Gaussian interference channel with M antennas at each transmitter and N antennas at each receiver if the channel coefficients are time-varying and drawn from a continuous distribution. The bounds are tight when the ratio max(M, N)/min (M, N) = R is equal to an integer. For this case, we show that the total number of Degrees of Freedom is equal to min(M, N)K if K les R and min(M, N) R/R +1 K if K > R. Achievability is based on interference alignment.
-
On the secure Degrees of Freedom of wireless X networks
2008 46th Annual Allerton Conference on Communication Control and Computing, 2008Co-Authors: Syed Ali JafarAbstract:Previous work showed that the X network with M transmitters, N receivers has MN/M+N-1 Degrees of Freedom. In this work we study the Degrees of Freedom of the X network with secrecy constraints, i.e. the X network where some/all messages are confidential. We consider the M times N network where all messages are secured and show that N(M-1)/M+N-1 Degrees of Freedom can be achieved. Secondly, we show that if messages from only M - 1 transmitters are confidential, then MN/M+N-1 Degrees of Freedom can be achieved meaning that there is no loss of Degrees of Freedom because of secrecy constraints. We also consider the achievable secure Degrees of Freedom under a more conservative secrecy constraint. We require that messages from any subset of transmitters are secure even if other transmitters are compromised, i.e., messages from the compromised transmitter are revealed to the unintended receivers. We also study the achievable secure Degrees of Freedom of the K user Gaussian interference channel under two different secrecy constraints where 1/2 secure Degrees of Freedom per message can be achieved. The achievable scheme in all cases is based on random binning combined with interference alignment.
-
Degrees of Freedom of wireless x networks
International Symposium on Information Theory, 2008Co-Authors: Viveck R Cadambe, Syed Ali JafarAbstract:We study the Degrees of Freedom characterization of wireless X networks, i.e. networks of M distributed single antenna transmitters and N distributed single antenna receivers where every transmitter has an independent message to every receiver. We provide an outerbound on the capacity region of X networks within o(log(SNR)). If the channel co-efficients are time-varying/frequency selective, we show that the total number of Degrees of Freedom is equal to MN/M+N-1 using a coding scheme based on the idea of interference alignment.
-
Degrees of Freedom region of the MIMO X channel
IEEE Transactions on Information Theory, 2008Co-Authors: Syed Ali Jafar, Shlomo ShamaiAbstract:We provide achievability as well as converse results for the Degrees of Freedom region of a MIMO $X$ channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. With M=1 antennas at each node, we find that the total (sum rate) Degrees of Freedom are bounded above and below as $1 \leq\eta_X^\star \leq {4/3}$. If $M>1$ and channel matrices are non-degenerate then the precise Degrees of Freedom $\eta_X^\star = {4/3}M$. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the ${4/3}M$ Degrees of Freedom. With equal number of antennas at all nodes, we explore the increase in Degrees of Freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio. With a cognitive transmitter we show that the number of Degrees of Freedom $\eta = {3/2}M$ (for $M>1$) on the MIMO $X$ channel. The same Degrees of Freedom are obtained on the MIMO $X$ channel with a cognitive receiver as well. In contrast to the $X$ channel result, we show that for the MIMO \emph{interference} channel, the Degrees of Freedom are not increased even if both the transmitter and the receiver of one user know the other user's message. However, the interference channel can achieve the full $2M$ Degrees of Freedom if \emph{each} user has either a cognitive transmitter or a cognitive receiver. Lastly, if the channels vary with time/frequency then the $X$ channel with single antennas $(M=1)$ at all nodes has exactly 4/3 Degrees of Freedom with no shared messages and exactly 3/2 Degrees of Freedom with a cognitive transmitter or a cognitive receiver.
Shlomo Shamai - One of the best experts on this subject based on the ideXlab platform.
-
Degrees of Freedom region of the MIMO X channel
IEEE Transactions on Information Theory, 2008Co-Authors: Syed Ali Jafar, Shlomo ShamaiAbstract:We provide achievability as well as converse results for the Degrees of Freedom region of a MIMO $X$ channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. With M=1 antennas at each node, we find that the total (sum rate) Degrees of Freedom are bounded above and below as $1 \leq\eta_X^\star \leq {4/3}$. If $M>1$ and channel matrices are non-degenerate then the precise Degrees of Freedom $\eta_X^\star = {4/3}M$. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the ${4/3}M$ Degrees of Freedom. With equal number of antennas at all nodes, we explore the increase in Degrees of Freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio. With a cognitive transmitter we show that the number of Degrees of Freedom $\eta = {3/2}M$ (for $M>1$) on the MIMO $X$ channel. The same Degrees of Freedom are obtained on the MIMO $X$ channel with a cognitive receiver as well. In contrast to the $X$ channel result, we show that for the MIMO \emph{interference} channel, the Degrees of Freedom are not increased even if both the transmitter and the receiver of one user know the other user's message. However, the interference channel can achieve the full $2M$ Degrees of Freedom if \emph{each} user has either a cognitive transmitter or a cognitive receiver. Lastly, if the channels vary with time/frequency then the $X$ channel with single antennas $(M=1)$ at all nodes has exactly 4/3 Degrees of Freedom with no shared messages and exactly 3/2 Degrees of Freedom with a cognitive transmitter or a cognitive receiver.
-
Degrees of Freedom Region of the MIMO $X$ Channel
IEEE Transactions on Information Theory, 2008Co-Authors: Syed Ali Jafar, Shlomo ShamaiAbstract:We provide achievability as well as converse results for the Degrees of Freedom region of a multiple-input multiple-output (MIMO) X channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. The inner and outer bounds on the Degrees of Freedom region are tight whenever integer Degrees of Freedom are optimal for each message. With M = 1 antennas at each node, we find that the total (sum rate) Degrees of Freedom are bounded above and below as 1 les eta*x les 4/3. If M > 1 and channel matrices are nondegenerate then the precise Degrees of Freedom eta*x = (4/3)M. Thus, the MIMO X channel has noninteger Degrees of Freedom when M is not a multiple of 3. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the (4/3)M Degrees of Freedom. If the channels vary with time/frequency then the channel with single antennas (M = 1) at all nodes has exactly 4/3 Degrees of Freedom. The key idea for the achievability of the Degrees of Freedom is interference alignment-i.e., signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired. We also explore the increase in Degrees of Freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio.
-
GLOBECOM - Degrees of Freedom of the MIMO X Channel
IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference, 2007Co-Authors: Syed Ali Jafar, Shlomo ShamaiAbstract:We provide achievability as well as converse results for the Degrees of Freedom region of a MIMO X channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. The inner and outerbounds on the Degrees of Freedom region are tight whenever integer Degrees of Freedom are optimal for each message. If all nodes have equal number of antennas M > 1 and channel matrices are non-degenerate then the Degrees of Freedom etaX* = 4/3 M. If the channels vary with time/frequency then the X channel with single antennas (M = 1) at all nodes has 4/3 Degrees of Freedom. Thus, the MIMO X channel has non-integer Degrees of Freedom when M is not a multiple of 3. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the 4/3 M Degrees of Freedom in all cases. The key idea for the achievability of the Degrees of Freedom is interference alignment - i.e., signal spaces are aligned at receivers where they constitute interference while they are separable at receivers where they are desired. With equal number of antennas at all nodes, we also explore the increase in Degrees of Freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio.
-
Degrees of Freedom Region for the MIMO X Channel
arXiv: Information Theory, 2006Co-Authors: Syed Ali Jafar, Shlomo ShamaiAbstract:We provide achievability as well as converse results for the Degrees of Freedom region of a MIMO $X$ channel, i.e., a system with two transmitters, two receivers, each equipped with multiple antennas, where independent messages need to be conveyed over fixed channels from each transmitter to each receiver. With M=1 antennas at each node, we find that the total (sum rate) Degrees of Freedom are bounded above and below as $1 \leq\eta_X^\star \leq {4/3}$. If $M>1$ and channel matrices are non-degenerate then the precise Degrees of Freedom $\eta_X^\star = {4/3}M$. Simple zero forcing without dirty paper encoding or successive decoding, suffices to achieve the ${4/3}M$ Degrees of Freedom. With equal number of antennas at all nodes, we explore the increase in Degrees of Freedom when some of the messages are made available to a transmitter or receiver in the manner of cognitive radio. With a cognitive transmitter we show that the number of Degrees of Freedom $\eta = {3/2}M$ (for $M>1$) on the MIMO $X$ channel. The same Degrees of Freedom are obtained on the MIMO $X$ channel with a cognitive receiver as well. In contrast to the $X$ channel result, we show that for the MIMO \emph{interference} channel, the Degrees of Freedom are not increased even if both the transmitter and the receiver of one user know the other user's message. However, the interference channel can achieve the full $2M$ Degrees of Freedom if \emph{each} user has either a cognitive transmitter or a cognitive receiver. Lastly, if the channels vary with time/frequency then the $X$ channel with single antennas $(M=1)$ at all nodes has exactly 4/3 Degrees of Freedom with no shared messages and exactly 3/2 Degrees of Freedom with a cognitive transmitter or a cognitive receiver.
Viveck R Cadambe - One of the best experts on this subject based on the ideXlab platform.
-
Interference Alignment and the Degrees of Freedom of Wireless $X$ Networks
IEEE Transactions on Information Theory, 2009Co-Authors: Viveck R Cadambe, Syed Ali JafarAbstract:We explore the Degrees of Freedom of M times N user wireless X networks, i.e., networks of M transmitters and N receivers where every transmitter has an independent message for every receiver. We derive a general outer bound on the Degrees of Freedom region of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the total number of Degrees of Freedom of the X network is equal to [(MN)/(M+N-1)] per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for X networks that can come arbitrarily close to the outer bound on Degrees of Freedom. For the case where either M=2 or N=2 we find that the Degrees of Freedom characterization also provides a capacity approximation that is accurate to within O(1) . For these cases the Degrees of Freedom outer bound is exactly achievable.
-
Degrees of Freedom of wireless x networks
International Symposium on Information Theory, 2008Co-Authors: Viveck R Cadambe, Syed Ali JafarAbstract:We study the Degrees of Freedom characterization of wireless X networks, i.e. networks of M distributed single antenna transmitters and N distributed single antenna receivers where every transmitter has an independent message to every receiver. We provide an outerbound on the capacity region of X networks within o(log(SNR)). If the channel co-efficients are time-varying/frequency selective, we show that the total number of Degrees of Freedom is equal to MN/M+N-1 using a coding scheme based on the idea of interference alignment.
-
Degrees of Freedom of wireless x networks
arXiv: Information Theory, 2007Co-Authors: Viveck R Cadambe, Syed Ali JafarAbstract:We explore the Degrees of Freedom of $M\times N$ user wireless $X$ networks, i.e. networks of $M$ transmitters and $N$ receivers where every transmitter has an independent message for every receiver. We derive a general outerbound on the Degrees of Freedom \emph{region} of these networks. When all nodes have a single antenna and all channel coefficients vary in time or frequency, we show that the \emph{total} number of Degrees of Freedom of the $X$ network is equal to $\frac{MN}{M+N-1}$ per orthogonal time and frequency dimension. Achievability is proved by constructing interference alignment schemes for $X$ networks that can come arbitrarily close to the outerbound on Degrees of Freedom. For the case where either M=2 or N=2 we find that the outerbound is exactly achievable. While $X$ networks have significant Degrees of Freedom benefits over interference networks when the number of users is small, our results show that as the number of users increases, this advantage disappears. Thus, for large $K$, the $K\times K$ user wireless $X$ network loses half the Degrees of Freedom relative to the $K\times K$ MIMO outerbound achievable through full cooperation. Interestingly, when there are few transmitters sending to many receivers ($N\gg M$) or many transmitters sending to few receivers ($M\gg N$), $X$ networks are able to approach the $\min(M,N)$ Degrees of Freedom possible with full cooperation on the $M\times N$ MIMO channel. Similar to the interference channel, we also construct an example of a 2 user $X$ channel with propagation delays where the outerbound on Degrees of Freedom is achieved through interference alignment based on a simple TDMA strategy.
-
interference alignment and the Degrees of Freedom for the k user interference channel
arXiv: Information Theory, 2007Co-Authors: Viveck R Cadambe, Syed Ali JafarAbstract:While the best known outerbound for the K user interference channel states that there cannot be more than K/2 Degrees of Freedom, it has been conjectured that in general the constant interference channel with any number of users has only one degree of Freedom. In this paper, we explore the spatial Degrees of Freedom per orthogonal time and frequency dimension for the K user wireless interference channel where the channel coefficients take distinct values across frequency slots but are fixed in time. We answer five closely related questions. First, we show that K/2 Degrees of Freedom can be achieved by channel design, i.e. if the nodes are allowed to choose the best constant, finite and nonzero channel coefficient values. Second, we show that if channel coefficients can not be controlled by the nodes but are selected by nature, i.e., randomly drawn from a continuous distribution, the total number of spatial Degrees of Freedom for the K user interference channel is almost surely K/2 per orthogonal time and frequency dimension. Thus, only half the spatial Degrees of Freedom are lost due to distributed processing of transmitted and received signals on the interference channel. Third, we show that interference alignment and zero forcing suffice to achieve all the Degrees of Freedom in all cases. Fourth, we show that the Degrees of Freedom $D$ directly lead to an $\mathcal{O}(1)$ capacity characterization of the form $C(SNR)=D\log(1+SNR)+\mathcal{O}(1)$ for the multiple access channel, the broadcast channel, the 2 user interference channel, the 2 user MIMO X channel and the 3 user interference channel with M>1 antennas at each node. Fifth, we characterize the degree of Freedom benefits from cognitive sharing of messages on the 3 user interference channel.
Xiaodong Wang - One of the best experts on this subject based on the ideXlab platform.
-
$(n,K)$ -User Interference Channels: Degrees of Freedom
IEEE Transactions on Information Theory, 2012Co-Authors: Ali Tajer, Xiaodong WangAbstract:This paper analyzes the gains of opportunistic communication in multiuser interference channels. Consider a fully connected n-user Gaussian interference channel. At each time instance, only K≤n transmitters are allowed to be communicating with their respective receivers and the remaining (n-K) transmitter-receiver pairs remain inactive. For finite n, if the transmitters can acquire the instantaneous channel realizations and if all channel gains are bounded away from zero and infinity, the seminal results on interference alignment establish that for any K arbitrary active pairs the total number of spatial Degrees of Freedom per orthogonal time and frequency domain is K/2. In dense networks (n → ∞), however, as the size of the network increases, it becomes less likely to sustain the bounding conditions on the channel gains. By exploiting this fact, we show that when n obeys certain scaling laws, by opportunistically and dynamically selecting the K active pairs at each time instance, the number of Degrees of Freedom can exceed K/2 and in fact can be made arbitrarily close to K. More specifically, for single-antenna transmitters and receivers, the network size scaling as n ∈ ω(SNRd⌈d-1⌉) when power allocation is allowed and scaling as n ∈ ω(SNRd(K-1)) without power allocation are sufficient conditions for achieving d ∈ [1, K] Degrees of Freedom. Moreover, for achieving these Degrees of Freedom the transmitters do not require the knowledge of the instantaneous channel realizations. Hence, invoking opportunistic communication in the context of interference channels leads to achieving higher Degrees of Freedom that are not achievable otherwise. We extend the results for multi-antenna Gaussian interference channels.
Ryan J. Tibshirani - One of the best experts on this subject based on the ideXlab platform.
-
Degrees of Freedom and Model Search
arXiv: Statistics Theory, 2014Co-Authors: Ryan J. TibshiraniAbstract:Degrees of Freedom is a fundamental concept in statistical modeling, as it provides a quantitative description of the amount of fitting performed by a given procedure. But, despite this fundamental role in statistics, its behavior not completely well-understood, even in some fairly basic settings. For example, it may seem intuitively obvious that the best subset selection fit with subset size k has Degrees of Freedom larger than k, but this has not been formally verified, nor has is been precisely studied. In large part, the current paper is motivated by this particular problem, and we derive an exact expression for the Degrees of Freedom of best subset selection in a restricted setting (orthogonal predictor variables). Along the way, we develop a concept that we name "search Degrees of Freedom"; intuitively, for adaptive regression procedures that perform variable selection, this is a part of the (total) Degrees of Freedom that we attribute entirely to the model selection mechanism. Finally, we establish a modest extension of Stein's formula to cover discontinuous functions, and discuss its potential role in Degrees of Freedom and search Degrees of Freedom calculations.
-
Degrees of Freedom in lasso problems
Annals of Statistics, 2012Co-Authors: Ryan J. Tibshirani, Jonathan TaylorAbstract:We derive the Degrees of Freedom of the lasso fit, placing no assumptions on the predictor matrix $X$. Like the well-known result of Zou, Hastie and Tibshirani [Ann. Statist. 35 (2007) 2173-2192], which gives the Degrees of Freedom of the lasso fit when $X$ has full column rank, we express our result in terms of the active set of a lasso solution. We extend this result to cover the Degrees of Freedom of the generalized lasso fit for an arbitrary predictor matrix $X$ (and an arbitrary penalty matrix $D$). Though our focus is Degrees of Freedom, we establish some intermediate results on the lasso and generalized lasso that may be interesting on their own.
-
on the Degrees of Freedom of the lasso
Annals of Statistics, 2007Co-Authors: Trevor Hastie, Ryan J. TibshiraniAbstract:We study the effective Degrees of Freedom of the lasso in the framework of Stein’s unbiased risk estimation (SURE). We show that the number of nonzero coefficients is an unbiased estimate forthe Degrees of Freedom of the lasso—a conclusion that requires no special assumption on the predictors. In addition, the unbiased estimator is shown to be asymptotically consistent. With these results on hand, various model selection criteria—Cp, AIC and BIC—are available, which, along with the LARS algorithm, provide a principled and efficient approach to obtaining the optimal lasso fit with the computational effort of a single ordinary least-squares fit.
