Cooperative Compressed Sensing for Decentralized Networks Zhi (Gerry) Tian Dept. of ECE, Michigan Tech Univ. A presentation at ztian@mtu.edu February 18, 2011
Ground-Breaking Recent Advances (a1) s is sparse (nonzero entries unknown) (a2) H can be fat (K N); satisfies restricted isometry property (RIP) Compressive sampling [Chen-Donoho-Saunders 98], [Candès et al 04-06] Given y and H, unknown s can be found with high probability Sparse regression [Tibshirani 96], [Tipping 01] Least-absolute shrinkage selection operator (Lasso) Ex. (scalar case) closed-form solution variable selection + estimation 1
Outline Sparsity-Aware Sensing in Networked Environments Sparsity-aware sensing for global awareness e.g., spectrum sensing in cognitive radio networks Sparsity-aware sensing for local awareness e.g., localized event detection in wireless sensor networks Decentralized cooperative sensing Summary and future research 2
Sensing Network: Signal Model Source locations at grid points w/ known locations Signal source locations and amplitudes Virtual grid Sensor grid (densely deployed) # sources (N) = # grid points (N s ) # sources (N) = # sensors (N r ) signal vector is sparse wrt grid points source locations = grid locations localization as byproduct 3
Sensor readings are additive: Network Data Model H i is distance-dependent, known (learning) Objective: to recover sparse s from Centralized FUSION CENTER Scalability Robustness Lack of Infrastructure Decentralized FUSION CENTER Global Info Local Info Local Info Global Awareness Global Awareness Local Awareness 4
Decentralized Processing for Global Awareness Spectrum Sensing in Cooperative Cognitive Radio Networks Global Info Local Info Centralized, Global Awareness Decentralized, Global Awareness long-range or multi-hop communication needed security issues one-hop communication range: r C neighboring sensors of sensor i: 5
Spectrum Scarcity Problem Scarcity vs. Underutilization Dilemma US FCC fixed spectrum access policies have useful radio spectrum pre-assigned PSD inefficient utilization 0 1 2 3 4 5 6GHz Source: Spectrum Sharing Inc. 6
Cognitive Radio (CR) CRs opportunistically use the spectrum under user hierarchy legacy users Secondary User (SU) cognitive radio Primary User (PU) legacy users power frequency cognitive radios Cognitive radio network problems Finding holes in the spectrum: wideband spectrum sensing Allocating the open spectrum: dynamic resource allocation Adjusting the transmit waveforms: waveform adaptation 7
Efficient Sharing Requires Sensing Multiple CRs jointly detect the spectrum [Ghasemi-Sousa 05, Ganesan-Li 05, Bazarque-Giannakis 08, Tian 08] Benefits: spatial diversity gain mitigates multipath fading and shadowing reduced sensing time and local processing increase of reliability and ability to detect hidden terminals Tradeoff: cooperation gain vs network overhead Spatial diversity against fading Source: Office of Communications (UK) multiple (random) paths unlikely to fade simultaneously f 8
Distributed Cooperative CR Sensing Idea: CRs collaborate to form a spatial map of the spectrum Goal: Specifications: coarse approx. suffices Approach: basis expansion of Compressive Sampling (CS) possible to form the PSD data 9
Modeling Transmitters Sensing CRs Frequency bases Sensed frequencies Sparsity present in space and frequency 10
Space-Frequency Basis Expansion Superimposed Tx spectra measured at CR r Average path-loss Frequency bases Linear model in and 11
Sparse Regression Seek a space s to capture the spectrum measured at all CR r Lasso: Soft threshold shrinks noisy estimates to zero Similar to Akaike s Information Criterion, it penalizes the number of parameters spectrum selection + estimation via. 1 penalty Power spectrum is non-negative non-negativity constraints 12
Consensus-based Distributed Optimization Centralized Lasso: Decentralized Decentralized equivalence Scalability Robustness Lack of infrastructure solvable locally Exchange of local estimates Constraints impose consensus across the network 13
Decentralized Algorithm Alternating-direction method of multipliers (ADMoM) Augmented Lagrange function Iterative implementation each CR i reconstructs locally: each CR i updates multipliers: broadcasts local decision one-hop: Scalable: one-hop communication, local computation Globally optimal: guaranteed if the network is connected 14
Power Spectrum Cartography 5 sources N s = 121 candidate locations, N r = 50 CRs NNLS Lasso Sparsity-unaware NNLS is prone to false alarms As a byproduct, Lasso localizes all sources via variable selection 15
Cooperative Compressed Sensing 2 PUs, 3 CRs, SNR=-5 db; compression = 50% Prob. of Detection Decentralized consensus Decentralized, majority vote Prob. of False Alarm Performance gain by decentralized fusion over majority vote 16
Other Scenarios of Global Awareness Compressive sampling at sub-nyquist rates [ICASSP 07] Edge detection using Wavelet basis [CROWNCOM 06] Cooperative Sensing at sub-nyquist rates [GLOBECOM 08] Cooperative sensing of common PU s spectrum in the presence of local interference [ICC 10, JSAC 11] Cooperative detection of multiple signals with common support [ICASSP 11] 17
Spectrum Hole/Edge Detection Compressive sampling at sub-nyquist rates [ICASSP 07] Edge detection using Wavelet basis [CROWNCOM 06] Spectrum reconstruction Spectrum hole detection 90% 75% 100% 50% 33% 20% 18
Decentralized Processing for Local Awareness Localized Event Detection in Large Networks Characteristics: events are sparse and local, with limited influence Applications: radioactive sources, targets, structural damages Network considerations: energy efficiency, scalability, robustness Local Info Decentralized, Local Awareness 19
Localized Event Detection in Large Networks Sensor grid: N r = N s = N Localized events of limited influence influence of event s j on sensor v i : h ij s i Sparsity-aware formulations Prior info: sources are sparse wrt grid points Quadratic programming: bounded noise energy H Linear programming: bounded measurement errors ( ) [centralized] 20
Global vs. Local Awareness Global Awareness via Consensus each sensor optimizes one local copy of the decision vector s all local copies are forced to consent via one-hop comm. equivalence to centralized optimality if network is connected Separable objective for sensor i Consensus with neighbors Decentralized implementation via Iterative ADMoM iteratively exchange decision vectors with neighbors heavy communication load for a large network with L >> 21
Localized Event Detection ( ) Reformulation each sensor i optimizes one scalar variable s i for itself based on linear programming for simplicity equivalence to centralized optimality if H is localized 22
Solution 1 via ADMoM Iterative Procedure update local decision variable at sensor i; linear computation Slack variables Lagrange multipliers for measurement constraints Lagrange multipliers for nonnegative constraints Local decision s i (t) s i (t+1) send decision + multiplier scalars to neighbors; one-hop comm. 23
Solution 2 via DLP Parallel computing under diagonal dominance [Tseng 90] Reformulation: Decentralized Linear Programming (DLP) uncoupled objective and constraints solution per sensor: Iterative implementation send one decision scalar to neighboring sensors Global optimality is H is localized and diagonal dominance Simple computation, low-cost comm., fast convergence 24
Comparison of Iteration Steps [ICASSP 10] Decision Making Local computation Information Exchange One-hop communication Global Consensus: compute collect send Local ADMoM: compute collect send Local DLP: compute collect send 25
Simulation Setup Sensors on grid (structural health monitoring) sensor locations: (xr, yr), x,y = 1,,L network size: N = LxL: L=10, N=100 Damages to detect s i = 1 at (3r, 5r) s j = 0.5 at (5r, 5r) Influence is distance-dependent influence function limited influence Parameters Task: identify locations & severity of damage 26 h ij : model emulation 26
Convergence: ADMoM & DLP ADMoM: converges in 20-30 steps DLP: converges in < 4 steps scalable costs in communication and computation global optimality via local cooperation faster convergence than global awareness 27
Sleeping Networks Randomly turns off a fraction of sensors to induce compression Active sensors make decisions for self & neighboring sleeping sensors, but not the entire network [Qing-Tian 2010] Scalable complexity Energy saving Fast convergence High resolution 28
Cooperative Support Detection Multiple measurement vector (MMV) problem sensors recover signals of different amplitudes, but common support no need for channel or location information Unknown environments Known sampling strategy 29
Decentralized Support Detection Row Lasso for the MMV problem Similar to Group Lasso in centralized form [Yuan-Lin 06] Coupled variables in mixed-norm Distributed Implementation Q: What to consent on? 30
Consensus-based Support Detection Centralized R-Lasso: Energy-based Consensus Energy vector Consensus optimization formulation solved locally exchange in one-hop 31
Decentralized Algorithm Alternating-direction method of multipliers (ADMoM) Augmented Lagrange function Iterative implementation each CR i reconstructs locally: each CR i updates multipliers: broadcasts local decision one-hop: 32
Cooperative Support Detection 20 channels, 5 PUs, 6 cooperative CRs, SNR = 5dB, 25% compression 33
Summary Exploiting sparsity in networked environments Global awareness Decentralized cooperation via consensus optimization Flexible problem formulations Local awareness Suitable for large networks that monitor localized phenomena Improved convergence and reduced network overhead 34
Wireless Sensor Networking Future Research Infrastructure: centralized, decentralized hierarchical Awareness: global vs. local Tasks: long-term monitoring vs. time-critical exploration When to collaborate? How to collaborate? Collaborative Information Processing Iterative Consensus Optimization benefits: one-hop, optimal, scalable, robust, asynchronous issues: convergence speed Assessment and optimization How to speed up the convergence rate?