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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
[2/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Data aggregation in WSNs
[3/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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?
[4/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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?
[4/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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 length
I [Cons]: introduces the dense measurement problem if Φ isdense. (i.e., a linear project would involve all the sensorreadings
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[5/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Data aquisiton in WSNsCompressed Sensing (CS)
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
[7/29] 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 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
I [Lee2009] spatially-locailzed sparse projections asmeasurment matrix, not easy to generate
[7/29] 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 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
I [Lee2009] spatially-locailzed sparse projections asmeasurment matrix, not easy to generate
[7/29] 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 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
I [Lee2009] spatially-locailzed sparse projections asmeasurment matrix, not easy to generate
[7/29] 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 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
I [Lee2009] spatially-locailzed sparse projections asmeasurment matrix, not easy to generate
[7/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach
I A sparse binary matrix based on unbalanced expandergraph for two purposes
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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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 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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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 sensors
I query only a necessary number of measurements (i.e.,O(k log(n)) is enough for guaranteed CS data recovery)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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)
[8/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[9/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[9/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[10/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[11/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[11/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[11/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[11/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.)
[12/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[13/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[13/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[13/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[13/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[13/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[13/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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)
[14/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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∗.
[15/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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 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∗.
[15/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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∗.
[15/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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∗.
[15/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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∗.
[15/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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∗.
[15/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach: DCSS algorithm
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
[16/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach: DCSS algorithm
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 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
[16/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach: DCSS algorithm
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 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
[16/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach: DCSS algorithm
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 thedesignated sensor Di , if φij 6= 0
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
[16/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach: DCSS algorithm
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 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
[16/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Our approach: DCSS algorithmAn example
[17/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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))
[18/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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))
[18/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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]
[19/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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
[20/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
ExperimentsExact sparse signal recovery
[21/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
ExperimentsNoisy sparse signals recovery
[22/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
ExperimentsCompressible signals recovery
[23/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
ExperimentsReal signal: Intel lab data (light intensity at node 19)
[24/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
ExperimentsReal signal: Intel lab data (light intensity at node 19)
[25/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Real signal: intel lab data (light intensity at node 19)
[26/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[27/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[27/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[28/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[28/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
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.
[28/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013
Thank you!!
[29/29] Shuangjiang Li, Hairong Qi, AICIP Lab “Distributed Data Aggregation for Sparse Recovery in Wireless Sensor Networks.” DCOSS 2013