Distributed Data Aggregation for Sparse Recovery in ...€¦ · Data aquisiton in WSNs Compressed...

Distributed Data Aggregation for SparseRecovery in Wireless Sensor Networks

Shuangjiang Li, Hairong Qi

Advanced Imaging and Collaborative Information Processing (AICIP) LabUniversity of Tennessee, Knoxville

May 21, 2013

The 9th IEEE International Conference on DistributedComputing in Sensor Systems (DCOSS 2013)

[1/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013

Outline

I Background and motivationI Prior worksI Our approachI ExperimentsI Conclusion


Data aggregation in WSNs


Data aquisiton in WSNs

Traditionally: sample then compress/aggregationI i.e., average, mean of the data or transform the data in

another domain (frequency, wavelet etc)I to save energy and storage by sampling all the data first

and then discarding most of them?

I How to aquire informative data efficiently?


Data aquisiton in WSNs

Traditionally: sample then compress/aggregationI i.e., average, mean of the data or transform the data in

another domain (frequency, wavelet etc)I to save energy and storage by sampling all the data first

and then discarding most of them?I How to aquire informative data efficiently?


Data aquisiton in WSNsCompressed Sensing (CS)

I Direclty through y = Φx , (y ∈ Rm, x ∈ Rn and m� n) wecan recovery x with no/little informance loss.

I Underdetermined system of linear equations which leads toinfinite solutions

I What if we put conditions on x and Φ??x is sparse/compressible, Φ satisfies certain property, (RIP,Null space property)unique solution!!

I [Pros]: reduces the sample lengthI [Cons]: introduces the dense measurement problem if Φ is

dense. (i.e., a linear project would involve all the sensorreadings





I What if we put conditions on x and Φ??

x is sparse/compressible, Φ satisfies certain property, (RIP,Null space property)unique solution!!







I What if we put conditions on x and Φ??x is sparse/compressible, Φ satisfies certain property, (RIP,Null space property)

unique solution!!I [Pros]: reduces the sample lengthI [Cons]: introduces the dense measurement problem if Φ is







I [Pros]: reduces the sample length

I [Cons]: introduces the dense measurement problem if Φ isdense. (i.e., a linear project would involve all the sensorreadings



Figure : Compressed sensing 1

1Image courtsey of Professor Richard Baraniuk at Rice University[6/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013

Prior works

CS based data aggregation and routing in WSNs

I [Bajwa2007, Luo2009] single-hop comm. to sink, denseCS matrix

I [Quer2009, Wang2011] routing based on measurementmatrix, random routing, both require grid networks and notconsider fairness of sensor selection

I [Lee2009] spatially-locailzed sparse projections asmeasurment matrix, not easy to generate


Prior works

CS based data aggregation and routing in WSNsI [Bajwa2007, Luo2009] single-hop comm. to sink, dense

CS matrix

I [Quer2009, Wang2011] routing based on measurementmatrix, random routing, both require grid networks and notconsider fairness of sensor selection



Prior works

CS based data aggregation and routing in WSNsI [Bajwa2007, Luo2009] single-hop comm. to sink, dense

CS matrixI [Quer2009, Wang2011] routing based on measurement

matrix, random routing, both require grid networks and notconsider fairness of sensor selection



Our approach

I A sparse binary matrix based on unbalanced expandergraph for two purposes

I as the CS measurement matrixI works as good as ”dense” CS matrices (Random Gaussian,

Scrambled Fourier matrices)I reduces computational complexity and communication costs

I as the sensor selection matrixI needs to be uniform selection or be fair

I A Distributed Compressive Sparse Sampling (DCSS)algorithm

I randomly choose m designated sensorsI query only a necessary number of measurements (i.e.,O(k log(n)) is enough for guaranteed CS data recovery)


Our approach


I as the CS measurement matrix

I works as good as ”dense” CS matrices (Random Gaussian,Scrambled Fourier matrices)

I reduces computational complexity and communication costsI as the sensor selection matrix

I needs to be uniform selection or be fairI A Distributed Compressive Sparse Sampling (DCSS)

algorithmI randomly choose m designated sensorsI query only a necessary number of measurements (i.e.,O(k log(n)) is enough for guaranteed CS data recovery)


Our approach



Scrambled Fourier matrices)

I reduces computational complexity and communication costsI as the sensor selection matrix

I needs to be uniform selection or be fairI A Distributed Compressive Sparse Sampling (DCSS)

algorithmI randomly choose m designated sensorsI query only a necessary number of measurements (i.e.,O(k log(n)) is enough for guaranteed CS data recovery)


Our approach








Our approach






I randomly choose m designated sensors

I query only a necessary number of measurements (i.e.,O(k log(n)) is enough for guaranteed CS data recovery)


Our approach








Our approach: measurement matrix design

I Instead of using traditional random CS measurementmatrices. We use sparse graph codes (i.e., expandergraphs) as CS measurement matrix

I [Berinde2008, Theorem 1,2,3] studied the relationshipbetween the expander graph and CS measurementmatrices, which serves as the theoretic foundation of ourapproach.


Our approach: measurement matrix designExpander graph

I Let X ⊂ U, N(X ) be set of neighbors of X in VI G(U,V ,E) is called (k , ε)-expander if

∀X ⊂ U, |X | ≤ k ⇒ |N(X )| ≥ (1− ε)d |X |I Each set of nodes on the left expands to N(X ) number of

nodes on right


Our approach: measurement matrix designSparse binary matrix from expander graph

A sparse binary matrix of m rows and n columns is generatedin the following way:

I Step 1: For each column, randomly generate τ integerswhose values are between 1 and m and place 1’s in thoserows indexed by the τ numbers;

I Step 2: If the τ numbers in one column are not distinct,repeat Step 1 until they are (this is not really an issue whenτ � m).

τ is the degree of the expander graph, τ = 8 in our experiment


Our approach: measurement matrix designSparse binary matrix for sensor selection

For each row of the sparse binary matrix Φi ∈ Rn (1 ≤ i ≤ m)

1 0 0 0 1 · · · 0 1 0

can be seen as an n-dimensional row vector (binary indicatorfunction) for sensor subset selection (i.e., select sensor j to beactive for sensing when Φij = 1, and inactive otherwise.)


Our approach: measurement matrix designFairness of sensor selection

I Each sensor will be selected equally τ times for sensingbased on the design of the matrix.

I m designated sensors and routing sensors will causeunbalanced energy consumption, however

I they are chosen each time when you want to sense thewhole environment (i.e., selected very infrequently)

I since m� n, each time only a random small amount ofsensors will be selected.

I In the long run, the energy consumption can still bebalanced


Our approach: DCSS algorithmNetwork model

I Consider a wireless network of n sensors with diameter dhops, each measures a real data value xi (i = 1,2, · · · ,n),which is sparse or compressible under sometransformation domains

I y = Φx (Φ is sparse binary matrix, i.e., φij ∈ {0,1})I Randomly choose m designated sensors from n sensors

(m� n)


Our approach: DCSS algorithm

I Input:I Φ ∈ Rm×n,I sensor neighborhood information,I environmental reading vector x ∈ Rn.

I Output: recovered vector x∗ ∈ Rn.I Steps:

I The Fusion Center(FC) generates the sparse binary matrixΦ and broadcasts the matrix and neighborhood information,

I For each sensor sj , (1 ≤ j <≤ n), sends its reading xj to thedesignated sensor Di through shortest path if Φij 6= 0,

I For the m designated sensors D1, · · · ,Dm, each computesand stores the sum of the reading it receives,

I The m designated sensors send their results to the FC andFC performs CS recovery for x∗.




I Output: recovered vector x∗ ∈ Rn.

I Steps:I The Fusion Center(FC) generates the sparse binary matrix

Φ and broadcasts the matrix and neighborhood information,I For each sensor sj , (1 ≤ j <≤ n), sends its reading xj to the

designated sensor Di through shortest path if Φij 6= 0,I For the m designated sensors D1, · · · ,Dm, each computes

and stores the sum of the reading it receives,I The m designated sensors send their results to the FC and

FC performs CS recovery for x∗.




I Output: recovered vector x∗ ∈ Rn.I Steps:

I The Fusion Center(FC) generates the sparse binary matrixΦ and broadcasts the matrix and neighborhood information,

I For each sensor sj , (1 ≤ j <≤ n), sends its reading xj to thedesignated sensor Di through shortest path if Φij 6= 0,

I For the m designated sensors D1, · · · ,Dm, each computesand stores the sum of the reading it receives,

I The m designated sensors send their results to the FC andFC performs CS recovery for x∗.



φ11 φ12 · · · φ1nφ21 φ22 · · · φ2n· · · · · · · · · · · ·φm1 φm2 · · · φmn

x1x2· · ·xn

=

y1y2· · ·ym

For each sensor:I φij · xj means sensor id sj sends its readings xj to the

designated sensor Di , if φij 6= 0For m designated sensors:

I Designated sensor Di computes and stores the summationof the sensor reading it receives, for all 1 ≤ i ≤ m

I Report y1, · · · , ym as observations y ∈ Rm




x1x2· · ·xn

=

y1y2· · ·ym

For each sensor:

I φij · xj means sensor id sj sends its readings xj to thedesignated sensor Di , if φij 6= 0

For m designated sensors:I Designated sensor Di computes and stores the summation

of the sensor reading it receives, for all 1 ≤ i ≤ mI Report y1, · · · , ym as observations y ∈ Rm




x1x2· · ·xn

=

y1y2· · ·ym

For each sensor:


For m designated sensors:

I Designated sensor Di computes and stores the summationof the sensor reading it receives, for all 1 ≤ i ≤ m

I Report y1, · · · , ym as observations y ∈ Rm




x1x2· · ·xn

=

y1y2· · ·ym

For each sensor:


For m designated sensors:I Designated sensor Di computes and stores the summation

of the sensor reading it receives, for all 1 ≤ i ≤ mI Report y1, · · · , ym as observations y ∈ Rm


Our approach: DCSS algorithmAn example


Our approach: DCSS algorithmCommunication cost

I Based on avergate bit-hop cost per readingI Assume ω to be the average row weight of the sparse

binary measurement matrix, τω is the cost of gathering thesensor readings for each projection

I Assume d as the cost to send the projection to FCI For generation of O(k log(n)) projection for data recovery,

the total communication cost is: O(k(τω + d) log(n))


Our approach: DCSS algorithmCommunication cost

I DS: dense samplingI SRP: sparse random projection in [Wang2007IPSN]I CDS (RW): compressive distributed sensing using random

walk in [Sartipi2011DCC]


ExperimentsSetup

I Data aggregation schemes comparsionI Dense sampling matrices (Random Gaussian, Scrambled

Fourier measusrement matrices)I Sparse random projection matrices in [Wang2007IPSN]

with various sparse level s

I Evaluation metrics

ε =‖x− x∗‖22‖x∗‖22

where x is the value of the original signal, while x∗ is thereconstructed signal.

I Using `1-magic package for CS recovery


ExperimentsExact sparse signal recovery


ExperimentsNoisy sparse signals recovery


ExperimentsCompressible signals recovery


ExperimentsReal signal: Intel lab data (light intensity at node 19)


Real signal: intel lab data (light intensity at node 19)


Conclusions

I A sparse binary measurement matrix was designed basedon expaned graph.

I Can be used for CS measurment matrix and sensor subsetselection.

I The recovery result is as good as traditional random denseCS measurement matrix and worked the best oncompressible data.

I Resolved the dense measurement problem.

I A structure free data aggregation algorithm (DCSS) wasproposed. Results from both synthetic and real dataexperiments demonstrated the usefulness of thealgorithms.


Conclusions

I A sparse binary measurement matrix was designed basedon expaned graph.

I Can be used for CS measurment matrix and sensor subsetselection.

I The recovery result is as good as traditional random denseCS measurement matrix and worked the best oncompressible data.

I Resolved the dense measurement problem.I A structure free data aggregation algorithm (DCSS) was

proposed. Results from both synthetic and real dataexperiments demonstrated the usefulness of thealgorithms.


Key ReferencesI [Berinde2008] R. Berinde, A. Gilbert, P. Indyk, H. Karloff,

and M. Strauss, ”Combining geometry and combinatorics:A unified approach to sparse signal recovery,” inCommunication, Control, and Computing, 2008 46thAnnual Allerton Conference on. IEEE, 2008, pp. 798–805.

I [Wang2007IPSN] W. Wang, M. Garofalakis, and K.Ramchandran, ”Distributed sparse random projections forrefinable approximation,” in Proc. Int. Conf. on Info.Processing in Sensor Networks, Cambridge, MA, Apr.2007.

I [Sartipi2011DCC] M. Sartipi and R. Fletcher,”Energy-efficient data acquisition in wireless sensornetworks using compressed sensing,” in DataCompression Conference (DCC), 2011. IEEE, 2011, pp.223–232.


Key References

I [Luo2009] C. Luo, F. Wu, J. Sun, and C.Chen,”Compressive data gathering for large-scale wirelesssensor networks,” in Proceedings of the 15th annualinternational conference on Mobile computing andnetworking. ACM, 2009, pp. 145-156.

I [Wang2011] X. Wang, Z. Zhao, Y. Xia, and H. Zhang,”Compressed sensing for efficient random routing inmulti-hop wireless sensor networks,” International Journalof Communication Networks and Distributed Systems, vol.7, no. 3, pp. 275–292, 2011.

I [Lee2009] S. Lee, S. Pattem, M. Sathiamoorthy, B.Krishnamachari, and A. Ortega, ”Spatially-localizedcompressed sensing and routing in multi-hop sensornetworks,” Geosensor Networks, pp. 11–20, 2009.


Key References

I [Quer2009] G. Quer, R. Masiero, D. Munaretto, M. Rossi, J.Widmer, and M. Zorzi, ”On the interplay between routingand signal representation for compressive sensing inwireless sensor networks,” in Information Theory andApplications Workshop, 2009. IEEE, 2009, pp. 206–215.

I [Bajwa2007] W. Bajwa, J. Haupt, A. Sayeed, and R.Nowak, ”Joint source–channel communication fordistributed estimation in sensor networks,” InformationTheory, IEEE Transactions on, vol. 53, no. 10, pp.3629–3653, 2007.


Thank you!!


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