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121
An interiorpoint method for largescale l1regularized logistic regression
 Journal of Machine Learning Research
, 2007
"... Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interiorpoint method for solving largescale ℓ1regularized logistic regression problems. Small problems with up to a thousand ..."
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Cited by 284 (8 self)
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Logistic regression with ℓ1 regularization has been proposed as a promising method for feature selection in classification problems. In this paper we describe an efficient interiorpoint method for solving largescale ℓ1regularized logistic regression problems. Small problems with up to a thousand or so features and examples can be solved in seconds on a PC; medium sized problems, with tens of thousands of features and examples, can be solved in tens of seconds (assuming some sparsity in the data). A variation on the basic method, that uses a preconditioned conjugate gradient method to compute the search step, can solve very large problems, with a million features and examples (e.g., the 20 Newsgroups data set), in a few minutes, on a PC. Using warmstart techniques, a good approximation of the entire regularization path can be computed much more efficiently than by solving a family of problems independently.
Power control by geometric programming
 IEEE Trans. on Wireless Commun
, 2005
"... Abstract — In wireless cellular or ad hoc networks where Quality of Service (QoS) is interferencelimited, a variety of power control problems can be formulated as nonlinear optimization with a systemwide objective, e.g., maximizing the total system throughput or the worst user throughput, subject ..."
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Cited by 128 (16 self)
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Abstract — In wireless cellular or ad hoc networks where Quality of Service (QoS) is interferencelimited, a variety of power control problems can be formulated as nonlinear optimization with a systemwide objective, e.g., maximizing the total system throughput or the worst user throughput, subject to QoS constraints from individual users, e.g., on data rate, delay, and outage probability. We show that in the high SignaltoInterference Ratios (SIR) regime, these nonlinear and apparently difficult, nonconvex optimization problems can be transformed into convex optimization problems in the form of geometric programming; hence they can be very efficiently solved for global optimality even with a large number of users. In the medium to low SIR regime, some of these constrained nonlinear optimization of power control cannot be turned into tractable convex formulations, but a heuristic can be used to compute in most cases the optimal solution by solving a series of geometric programs through the approach of successive convex approximation. While efficient and robust algorithms have been extensively studied for centralized solutions of geometric programs, distributed algorithms have not been explored before. We present a systematic method of distributed algorithms for power control that is geometricprogrammingbased. These techniques for power control, together with their implications to admission control and pricing in wireless networks, are illustrated through several numerical examples. Index Terms — Convex optimization, CDMA power control, Distributed algorithms. I.
Adaptation, Coordination and Distributed Resource Allocation in InterferenceLimited Wireless Networks
"... A sensible design of wireless networks involves striking a good balance between an aggressive reuse of the spectral resource throughout the network and managing the resulting cochannel interference. Traditionally this problem has been tackled using a “divide and conquer” approach. The latter consis ..."
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Cited by 35 (3 self)
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A sensible design of wireless networks involves striking a good balance between an aggressive reuse of the spectral resource throughout the network and managing the resulting cochannel interference. Traditionally this problem has been tackled using a “divide and conquer” approach. The latter consists in deploying the network with a static or semidynamic pattern of resource reutilization. The chosen reuse factor, while sacrificing a substantial amount of efficiency, brings the interference to a tolerable level. The resource can then be managed in each cell so as to optimize the per cell capacity using an advanced air interface design. In this paper we focus our attention on the overall network capacity as a measure of system performance. We consider the problem of resource allocation and adaptive transmission in multicell scenarios. As a key instance, the problem of joint scheduling and power control simultaneously in multiple transmitreceive links, which employ capacityachieving adaptive codes, is studied. In principle, the solution of such an optimization hinges on tough issues such as the computational complexity and the requirement for heavy receivertotransmitter feedback and, for cellular networks, celltocell channel state information (CSI) signaling. We give asymptotic properties pertaining to ratemaximizing power control and scheduling in multicell networks. We then present some promising leads for substantial complexity and signaling reduction via the use of newly developed distributed and game theoretic techniques.
Cooperative jamming for secure communications in MIMO relay networks
 IEEE Trans. Signal Process
, 2011
"... Abstract—Secure communications can be impeded by eavesdroppers in conventional relay systems. This paper proposes cooperative jamming strategies for twohop relay networks where the eavesdropper can wiretap the relay channels in both hops. In these approaches, the normally inactive nodes in the rel ..."
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Cited by 29 (12 self)
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Abstract—Secure communications can be impeded by eavesdroppers in conventional relay systems. This paper proposes cooperative jamming strategies for twohop relay networks where the eavesdropper can wiretap the relay channels in both hops. In these approaches, the normally inactive nodes in the relay network can be used as cooperative jamming sources to confuse the eavesdropper. Linear precoding schemes are investigated for two scenarios where single or multiple data streams are transmitted via a decodeandforward (DF) relay, under the assumption that global channel state information (CSI) is available. For the case of single data stream transmission, we derive closedform jamming beamformers and the corresponding optimal power allocation. Generalized singular value decomposition (GSVD)based secure relaying schemes are proposed for the transmission of multiple data streams. The optimal power allocation is found for the GSVD relaying scheme via geometric programming. Based on this result, a GSVDbased cooperative jamming scheme is proposed that shows significant improvement in terms of secrecy rate compared to the approach without jamming. Furthermore, the case involving an eavesdropper with unknown CSI is also investigated in this paper. Simulation results show that the secrecy rate is dramatically increased when inactive nodes in the relay network participate in cooperative jamming. Index Terms—Interference, jamming, physical layer security, relay networks, secrecy, wiretap channel. I.
Robust Secure Transmission in MISO Channels Based on WorstCase Optimization
 IEEE Trans. Signal Process
, 2012
"... Abstract—This paper studies robust transmission schemes for MISO wiretap channels with imperfect channel state information (CSI) for the eavesdropper link. Both the cases of direct transmission and cooperative jamming with a helper are investigated. The error in the eavesdropper’s CSI is assumed to ..."
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Cited by 25 (5 self)
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Abstract—This paper studies robust transmission schemes for MISO wiretap channels with imperfect channel state information (CSI) for the eavesdropper link. Both the cases of direct transmission and cooperative jamming with a helper are investigated. The error in the eavesdropper’s CSI is assumed to be normbounded, and robust transmit covariance matrices are obtained based on worstcase secrecy rate maximization, under both individual and global power constraints. Numerical results show the advantage of the proposed robust design. In particular, under a global power constraint, although cooperative jamming is not necessary for optimal transmission with perfect eavesdropper’s CSI, we show that robust jamming support can increase the secrecy rate in the presence of channel mismatch. I.
A new method for design of robust digital circuits
 Proceedings International Symposium on Quality Electronic Design (ISQED
, 2005
"... As technology continues to scale beyond 100nm, there is a significant increase in performance uncertainty of CMOS logic due to process and environmental variations. Traditional circuit optimization methods assuming deterministic gate delays produce a flat “wall ” of equally critical paths, resulting ..."
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Cited by 17 (1 self)
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As technology continues to scale beyond 100nm, there is a significant increase in performance uncertainty of CMOS logic due to process and environmental variations. Traditional circuit optimization methods assuming deterministic gate delays produce a flat “wall ” of equally critical paths, resulting in variationsensitive designs. This paper describes a new method for sizing of digital circuits, with uncertain gate delays, to minimize their performance variation leading to a higher parametric yield. The method is based on adding margins on each gate delay to account for variations and using a new “soft maximum ” function to combine path delays at converging nodes. PSfrag Using replacements analytic models to predict the means and standard deterministic deviations method of gate delays as posynomial functions of the device sizes, PDF we create a simple, computationally efficient heuristic for uncertaintyaware sizing of digital circuits via Geometric Programming. MonteCarlo simulations on custom 32bit adders and ISCAS’85 benchmarks show that about 10 % to 20 % delay reduction over deterministic sizing methods can be achieved, without any additional cost in area. 1.
Geodesic convexity and covariance estimation
 IEEE Trans. Signal Process
, 2012
"... Abstract—Geodesic convexity is a generalization of classical convexity which guarantees that all local minima of gconvex functions are globally optimal. We consider gconvex functions with positive definite matrix variables, and prove that Kronecker products, and logarithms of determinants are g ..."
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Cited by 15 (6 self)
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Abstract—Geodesic convexity is a generalization of classical convexity which guarantees that all local minima of gconvex functions are globally optimal. We consider gconvex functions with positive definite matrix variables, and prove that Kronecker products, and logarithms of determinants are gconvex. We apply these results to two modern covariance estimation problems: robust estimation in scaledGaussian distributions, andKronecker structured models. Maximum likelihood estimation in these settings involves nonconvexminimizations.We show that these problems are in fact gconvex. This leads to straight forward analysis, allows the use of standard optimization methods and paves the road to various extensions via additional gconvex regularization. Index Terms—Elliptical distributions, geodesic convexity, Kronecker models, logsumexp, martix variate models, robust covariance estimation. I.
Power control for cognitive radio networks under channel uncertainty
 IEEE Trans. Wireless Commun
, 2011
"... Abstract—Cognitive radio (CR) networks can reuse the RF spectrum licensed to a primary user (PU) network, provided that the interference inflicted to the PUs is carefully controlled. However, due to lack of explicit cooperation between CR and PU systems, it is often difficult for CRs to acquire CR ..."
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Cited by 15 (6 self)
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Abstract—Cognitive radio (CR) networks can reuse the RF spectrum licensed to a primary user (PU) network, provided that the interference inflicted to the PUs is carefully controlled. However, due to lack of explicit cooperation between CR and PU systems, it is often difficult for CRs to acquire CRtoPU channels accurately. In fact, if the PU receivers are off, the sensing algorithms cannot obtain the channels for the PU receivers, although they have to be protected nevertheless. In order to achieve aggressive spectrum reuse even in such challenging scenarios, power control algorithms that take channel uncertainty into account are developed. Both lognormal shadowing and smallscale fading effects are considered through suitable approximations. Accounting for the latter, centralized network utility maximization (NUM) problems are formulated, and their KarushKuhnTucker points are obtained via sequential geometric programming. For the case where CRtoCR channels are also uncertain, a novel outage probabilitybased NUM formulation is proposed, and its solution method developed in a unified fashion. Numerical tests verify the performance merits of the novel design. Index Terms—Cognitive radio, power control, network utility maximization, channel uncertainty, interference modeling, geometric programming. I.
Leakage minimization of nanoscale circuits in the presence of systematic and random variations
 Proc. DAC, 2005
"... This paper presents a novel gate sizing methodology to minimize the leakage power in the presence of process variations. The leakage and delay are modeled as posynomials functions to formulate a geometric programming problem. The existing statistical leakage model of [18] is extended to include th ..."
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Cited by 9 (0 self)
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This paper presents a novel gate sizing methodology to minimize the leakage power in the presence of process variations. The leakage and delay are modeled as posynomials functions to formulate a geometric programming problem. The existing statistical leakage model of [18] is extended to include the variations in gate sizes as well as systematic variations. We propose techniques to efficiently evaluate constraints on the αpercentile of the path delays without enumerating the paths in the circuit. The complexity of evaluating the objective function is O(N 2) and that of evaluating the delay constraints is O(N  + E) for a circuit with N  gates and E  wires. The optimization problem is then solved using a convex optimization algorithm that gives an exact solution.