cooperative compressed sensing for decentralized...

Cooperative Compressed Sensing for

Decentralized Networks

Zhi (Gerry) Tian

Dept. of ECE, Michigan Tech Univ.

[email protected]

February 18, 2011

A presentation at

Sparsity-Aware Sensing and Communications Z. Tian, Michigan Tech

1

Ground-Breaking Recent Advances

Compressive sampling [Chen-Donoho-Saunders’98], [Candès et al’04-06]

Given y and H, unknown s can be found with high probability

Least-absolute shrinkage selection operator (Lasso)

Ex. (scalar case) closed-form solution

Sparse regression [Tibshirani’96], [Tipping’01]

(a1) s is sparse (nonzero entries unknown)

(a2) H can be fat (K N);

satisfies restricted isometry property (RIP)

variable selection + estimation


2

Outline

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

“Sparsity-Aware Sensing in Networked Environments”


3

Sensing Network: Signal Model

signal vector is sparse wrt grid points

source locations = grid locations localization as byproduct

Virtual grid Sensor grid

(densely deployed)

# sources (N) = # grid points (Ns) # sources (N) = # sensors (Nr)

Source locations at grid points w/ known locations

Signal source locations and amplitudes


4

Sensor readings are additive:

Hi is distance-dependent, known (learning)

Objective: to recover sparse s from

Network Data Model

Global Info

Global Awareness Global Awareness Local Awareness

Local Info Local Info

Scalability Robustness Lack of Infrastructure

Centralized

FUSION

CENTER

Decentralized

FUSION

CENTER


5

Decentralized Processing for Global Awareness

Spectrum Sensing in Cooperative Cognitive Radio Networks

Decentralized,

Global Awareness

Global Info Local Info

Centralized,

Global Awareness

one-hop communication range: rC

neighboring sensors of sensor i:

long-range or multi-hop

communication needed

security issues


Spectrum Scarcity Problem

fixed spectrum access policies have

useful radio spectrum pre-assigned

US FCC

inefficient utilization 0 1 2 3 4 5 6GHz

PS

D

“Scarcity vs. Underutilization Dilemma”

Source: Spectrum Sharing Inc. 6


7

Cognitive Radio (CR)

CRs opportunistically use the spectrum under user hierarchy

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

legacy users

frequency

pow

er

cognitive radios

legacy users cognitive radio

Secondary User (SU) Primary User (PU)


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

Efficient Sharing Requires Sensing

Source: Office of Communications (UK)

f multiple (random) paths unlikely to fade simultaneously

Spatial diversity against fading

8


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

Distributed Cooperative CR Sensing

9


10

Modeling

Transmitters

Sensing CRs

Frequency bases

Sensed frequencies

Sparsity present in space and frequency


11

Superimposed Tx spectra measured at CR r

Average path-loss

Frequency bases

Linear model in and

Space-Frequency Basis Expansion


12

Sparse Regression

Seek a space s to capture the spectrum measured at all CRr

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


Consensus-based Distributed Optimization

Decentralized

Scalability Robustness Lack of infrastructure

Centralized Lasso:

Decentralized equivalence

Constraints impose consensus across the network

solvable locally

Exchange of local

estimates

13


14

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:

Decentralized Algorithm

Scalable: one-hop communication, local computation

Globally optimal: guaranteed if the network is connected


15

Power Spectrum Cartography

NNLS Lasso

5 sources Ns = 121 candidate locations, Nr = 50 CRs

Sparsity-unaware NNLS is prone to false alarms

As a byproduct, Lasso localizes all sources via variable selection


16

Cooperative Compressed Sensing P

rob.

of

Dete

ction

Prob. of False Alarm

Decentralized,

majority vote

Decentralized

consensus

2 PUs, 3 CRs, SNR=-5 dB; compression = 50%

Performance gain by decentralized fusion over majority vote


17

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]


18

Spectrum Hole/Edge Detection

Spectrum reconstruction Spectrum hole detection

20%

3

3%

50%

7

5%

90%

100%

Compressive sampling at sub-Nyquist rates [ICASSP’07]

Edge detection using Wavelet basis [CROWNCOM’06]


19

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

Decentralized,

Local Awareness

Local Info


20

Localized Event Detection in Large Networks

Sensor grid: Nr = Ns = N

Localized events of limited influence

influence of event sj on sensor vi: hij si

Sparsity-aware formulations

Prior info: sources are sparse wrt grid points

Quadratic programming: bounded noise energy

Linear programming: bounded measurement errors

( )

[centralized]

H


21

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

Consensus with neighbors

Separable objective for sensor i

Decentralized implementation via Iterative ADMoM

iteratively exchange decision vectors with neighbors

heavy communication load for a large network with L >>


22

Localized Event Detection

Reformulation

each sensor i optimizes one scalar variable si for itself

based on linear programming for simplicity

equivalence to centralized optimality if H is localized

22

( )


Iterative Procedure

update local decision variable at sensor i; linear computation

send decision + multiplier scalars to neighbors; one-hop comm.

23

Solution 1 via ADMoM

23

Slack variables

Lagrange multipliers for

measurement constraints

Lagrange multipliers for

nonnegative constraints

Local decision

si(t) si(t+1)


24

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


25

Comparison of Iteration Steps

Local DLP:

Decision Making

Local computation

Information Exchange

One-hop communication

send

collect

send

collect

send

collect

compute

compute

Global Consensus:

Local ADMoM:

compute

[ICASSP’10]


26

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

si = 1 at (3r, 5r)

sj = 0.5 at (5r, 5r)

Influence is distance-dependent

influence function

limited influence

Parameters

Task: identify locations & severity

of damage

26 emulation hij: model


27

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


28

Sleeping Networks

Scalable complexity

Energy saving

Fast convergence

High resolution

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]


29

Multiple measurement vector (MMV) problem

sensors recover signals of different amplitudes, but common support

no need for channel or location information

Cooperative Support Detection

Unknown environments

Known sampling strategy


30

Row Lasso for the MMV problem

Similar to Group Lasso in centralized form [Yuan-Lin’06]

Coupled variables in mixed-norm

Decentralized Support Detection

Q: What to consent on?

Distributed Implementation


31

Energy-based Consensus

Energy vector

Consensus optimization formulation

Consensus-based Support Detection

Centralized

R-Lasso:

solved locally

exchange

in one-hop


32

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:

Decentralized Algorithm


33

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


Future Research

Wireless Sensor Networking

Infrastructure: centralized, decentralized hierarchical

Awareness: global vs. local

Tasks: long-term monitoring vs. time-critical exploration

Collaborative Information Processing

Iterative Consensus Optimization

benefits: one-hop, optimal, scalable, robust, asynchronous

issues: convergence speed

Assessment and optimization

When to collaborate? How to collaborate?

How to speed up the convergence rate?

cooperative compressed sensing for decentralized...

Documents