Distributed Subspace Projection in Wireless Sensor Networks using

Distributed Subspace Projection in Wireless Sensor Networks using Computational Codes Xabier Insausti1 1

Pedro M. Crespo1

Baltasar Beferull2

CEIT and TECNUN (University of Navarra). 20018 Donostia-San Sebasti´ an {xinsausti,pcrespo}@ceit.es 2 University of Valencia. 46980, Paterna (Valencia) [email protected]

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Xabier Insausti et al (CEIT, UVEG)

Distributed Projection for Wireless Sensor Networks

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Introduction

Objective: Compute the projection of the observed signal in a distributed fashion and achieve time and energy savings with respect to previous schemes.

Strategy: Use Computation Coding in order to exploit local interference.



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Outline 1

Distributed computation in Wireless Sensor Networks: Gossip Algorithms

2

Projection in Wireless Sensor Networks General Projection problem Distributed Projection problem

3

Neighborhood Gossip Algorithm for computing Projection

4

Computation Coding

5

Fundamental Limits

6

Conclusions



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Distributed computation in Wireless Sensor Networks: Gossip Algorithms Random Sensor Network Consider a network composed of N sensors distributed randomly (uniformly). Sensors are assigned a limited transmission power PT per source symbol. The set of nodes within the coverage area of node i form its neighborhood Nd (i). Nodes only have knowledge of their local neighborhood.

d



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Distributed computation in Wireless Sensor Networks: Gossip Algorithms Random Gossip Algorithm They are mainly used to compute the average. They are fast, efficient, and have low computational cost.



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Distributed computation in Wireless Sensor Networks: Gossip Algorithms Neighborhood Gossip Algorithm It is a modification of the general Random Gossip Algorithm. It uses computation coding to exploit local interference and save energy and time. It includes a final broadcasting stage to compute the average faster.



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General Projection problem Now consider the very common case in which the signal observed by the sensors is smooth (spatially correlated).

Why do we need to project? Due to correlation, the signal belongs to a subspace of dimension smaller than N . Projection reduces the measurement noise. Projection will improve the reliability of the observation.

z

ξ

ξb

Signal Subspace

For a known projection matrix P: b ξ = (ξb1 , . . . , ξbN )> = Pz.



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Distributed Projection problem z

ξ

ξb

Signal Subspace

We are looking for a distributed way to compute the projection in an iterative way, so that b ξ[t + 1] = Wb ξ[t],

t = 0, 1, . . .

W ∈ RN ×N

and we impose the following conditions:

Conditions for matrix W Topology constraint: wij = [W]ij = 0 for j 6∈ Nd (i). Convergence condition: limt→∞ b ξ[t] = Pz = b ξ −→ limt→∞ Wt = P. Minimize convergence time: min ρ(W − P).



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Distributed Projection problem

The previous conditions can be intuitively understood as follows: There should be a communication path between any two nodes with at most N − 1 hops. The number of neighbors of each node should be large enough. This means that the transmission power must be high enough to guarantee convergence.



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Neighborhood Gossip Algorithm for computing Projection: Problem

Computing b ξ[t + 1] = Wb ξ[t] in a distributed fashion requires that all the estimated quantities ξbj are updated the same number of times. This requires a substantial change in the Gossip Algorithms introduced so far, since they are only suitable for computing the average. Here is the problem:

ξb3[0]

ξb6[0] ξb1[0]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[0]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[0]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1]

ξb2[0] ξb4[0]

ξb5[0]



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ξb3[0]

ξb6[0] ξb1[1] ξb5[0]


ξb1 [0]

ξb2[1]


ξb4[0]

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Neighborhood Gossip Algorithm for computing Projection: Solution

Define a master-slave relationship between any pair of nodes. Define two operation modes, gossip-waiting, in which each node can be. gossip

gossip b

ξb3[0]

ξ6[0]

gossip b

gossip

gossip

ξb1[0]

ξb2[0]

ξ4[0]

ξ5[0]


gossip b


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Define a master-slave relationship between any pair of nodes. Define two operation modes, gossip-waiting, in which each node can be. gossip slave b

gossip

ξ6[0] waiting b

ξ1[1]

gossip slave b

gossip slave b

ξ2[0]

gossip

ξb4[0]

ξ5[0]


ξb3[0]


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Define a master-slave relationship between any pair of nodes. Define two operation modes, gossip-waiting, in which each node can be. gossip b

gossip slave b

ξ3[0]

ξ6[0] waiting master b

ξ1[1] gossip b

ξ5[0]


ξb1 [0]

waiting b

ξ2[1]


gossip slave b

ξ4[0]

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Neighborhood Gossip Algorithm for computing Projection: Example Example In the previous consider a subspace whose basis is given by the columns of √  topology,   0 √3/3 0.1667 0.5 0 0 0.3333 0.3333  0  3/3  0.5 0.3 0.4 0 0 0   √   0 3/6  0 0.4 0.2 0 0 0    √ U=  =⇒ W =  0 0 0 1 0 0  1 3/6    0.3333  0 √3/6  0 0 0 0.1826 0.1507 √ 0.3333 0 0 0 0.1507 0.1826 0 3/6 which yields ρ(W − P) = 0.5164 and convergence time τ (W) = 1.5131.



      

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Computation Coding The most energy consuming stage is when all nodes in the local neighborhood transmit their corresponding observations to the central node. The central node only needs the weighted sum of the observations. This can be done very efficiently by using computation coding.

Multiple Access Channel The Multiple Access Channel for the neighborhood Nd (i) is given by X yi [t] = hij [t]xj [t] + ni [t], j∈Nd (i) −α/2 jθij [t]

hij [t] = dij

e

If we assume the sensors know the channel in their local neighborhood, the channel simplifies to a simple adder X yi [t] = xj [t] + ni [t] j∈Nd (i)



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Computation Coding

Computation codes can be designed to compute sums very efficiently and reliably. Here is a sketch of how computation codes behave.

Computing the sum with Computation Codes 1

All nodes in the neighborhood of i encode and send their values using identical linear codebooks.

2

The transmitted codewords will be added on the channel and node i will receive the sum of the codewords.

3

Since the codebook is linear, the sum of the codewords is also a codeword and it is actually the codeword corresponding to the desired sum.

4

Node i has to simply add the received message to its observation weighted by the coefficient wii to get its desired new observation.



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Computation Coding

Computation Coding basics 1

All sensors use the same lattice code.

2

The lattice code is chosen to be simultaneously a good channel and source code.

3

Sensors quantize their weighted observation vectors to the lattice and they transmit simultaneously.

4

The receiver decodes the sum and makes an estimate of it.

5

Transmitters send their quantization errors using the same lattice code.

6

The process continues until the number of channel uses is exhausted and desired precision reached.

Theorem: Existence Choose > 0. Assume each node in a local neighborhood has a length L bounded real-valued weighted observation vector, that is, ksij [t]k2 ≤ LPT ∀i, j, t. For L large enough, there exists a coding scheme such that the receiving node i can bi [ti + 1] of the sum ξi [ti + 1] that make an estimate ξ satisfies:

bi [ti + 1]k2 ≥ kξi [ti + 1] − ξ

P

PT

! −2B

2

<

L

∀i, ti

so long as:

1 T log

+ |Nd (i)|

PT

! > B

(maxj∈Nd (i) dij )−α/2 σ 2

for some choice of T channel uses per observation symbol and precision B bits.



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Fundamental Limits Average Energy Gain We compare our strategy with the best possible separation-based scheme and evaluate the √ energy gain when computing the projection for a given distortion (so long as PT ≥ σ 2 , d < 1/ π) ! 2 σS log m N −m D πd2 1 − πd2 1 + PT log m σ2 ! g = D m log mσ 2 PN N S ! πd2 m 1 − πd2 N −m m=1 m 2 σ log σ 2 +mPT PN N m=1 m

Variables:

Remarks:

N : Density of sensors. σ 2 : Variance of the noise. 2 : Variance of the source. σS D: Maximum distortion. PT : Transmission power. d: Effective range.


The gain is very little dependent on the distortion. The gain grows exponentially as the density of sensors increases.


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Fundamental Limits Example Energy gain of the projection algorithm with computation codes with respect to the best possible separation-based scheme. d = 0.2, σ 2 = 1, σS2 = 1, PT = 3, D = 10−5 . 16

15

14

13

Gain (dB)

12

11

10

9

8

7

6 100

200

300

400

500

600

700

800

900

1000

Density of sensors (nodes/unit square)



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Conclusions

1

The problem of finding the projection of the observed signal into a subspace can be solved in a distributed fashion by solving an SDP problem.

2

A general gossip algorithm has been modified to allow the use of computation coding.

3

Combining computation coding with a gossip algorithm to perform the projection leads to exponentially increasing power savings when the number of nodes increases.



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Distributed Subspace Projection in Wireless Sensor Networks using Computational Codes Xabier Insausti1 1

Pedro M. Crespo1

Baltasar Beferull2


16th of May, 2012



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In-network computation of the Transition Matrix for Distributed Subspace Projection Xabier Insausti1 1

Pedro M. Crespo1

Baltasar Beferull2


16th of May, 2012


In-network computation of the Transition Matrix

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Introduction

Objective: Compute the transition matrix for the projection problem in a distributed fashion.

Strategy: Use gossip algorithms and a distributed genetic algorithm.



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Outline 1

Projection in Wireless Sensor Networks

2

Problem constraints

3

Algorithm for Reduction of Degrees of Freedom

4

Computation of the spectral radius

5

Distributed Genetic Algorithm

6

Simulation results



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Projection in Wireless Sensor Networks Random Sensor Network Consider a network composed of N sensors distributed randomly (uniformly). Sensors are assigned a limited transmission power PT per source symbol. The set of nodes within the coverage area of node i form its neighborhood Nd (i). Nodes only have knowledge of their local neighborhood.

d



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General Projection problem Now consider the very common case in which the signal observed by the sensors is smooth (spatially correlated).

Why do we need to project? Due to correlation, the signal belongs to a subspace of dimension smaller than N . Projection reduces the measurement noise. Projection will improve the reliability of the observation.

z

ξ

ξb

Signal Subspace

For a known projection matrix P: ξb = (ξb1 , . . . , ξbN )> = Pz.



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Distributed Projection problem z

ξ

ξb

Signal Subspace

We are looking for a matrix W, so that the following recursive dynamics b + 1] = Wξ[t], b ξ[t

t = 0, 1, . . .

W ∈ RN ×N

converge to the sought projection ξb as t increases:

Conditions for matrix W Topology constraint: wij = [W]ij = 0 for j 6∈ Nd (i). b = Pz = ξb −→ limt→∞ Wt = P. Convergence condition: limt→∞ ξ[t] Minimize convergence time: min ρ(W − P).



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Problem constraints

1

2

The topology constraint: Each node needs to generate a column of W. This is a vector ωi of length |Nd (i)| + 1. limt→∞ Wt = P ⇐⇒ PW = WP = P: We require W to be symmetric and PW = P. This can be solved locally as b i ωi = coli (P), P

i = 1 . . . N.

ωi = Θi ψi + υi ,

i = 1 . . . N.

and therefore:

3

min ρ(W − P): Iterate over the solution set to minimize the spectral radius by using a Genetic Algorithm.



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Algorithm for Reduction of Degrees of Freedom Each node i will have its own natural equation set, given by ωi = Θi ψi + υi and a set of given equations, in the form Ai ωi = bi where each row of Ai refers to the neighboring node that imposes the equation.

Problem of Degrees of Freedom The overall strategy is based on adding extra equations to the node, i.e. adding rows to c has been Ai , until it runs out of degrees of freedom. At that point a candidate matrix W found.

Symmetrization Every time an equation is fixed at a node, it has to be fixed at the corresponding neighbor as well, in order to keep the solution symmetric.



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Algorithm for Reduction of Degrees of Freedom Example

3 5

6

1

4 2



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3 df=2 df=0

5

6

df=1

1

4

df=1

df=1

df=3


2


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3 df=2 df=0

5

6

df=1

1

4

df=1

df=1

df=3


2


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3 df=2 df=0

5

6

df=1

1

4

df=0

df=1

df=3


2


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3 df=2 df=0

5

6

df=1

1

4

df=0

df=1 ID: 001 df=3


2


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3 df=2 df=0

5

6

df=1

1

4

df=0

df=1 ID: 001 df=2


2


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3 df=2 df=0

5

6

df=1

1

4

df=0

df=1

df=2


2


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3 df=1 df=0

5

6

df=1

1

4

df=0

df=1

df=2


2


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3 df=1 df=0

6

5

ID: 002

df=1

1

4

df=0

df=1

df=2


2


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3 df=1 df=0

6

5

ID: 002

df=1

1

4

df=0

df=0

df=2


2


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3 df=1 df=0

6

5

ID: 002

df=1

4

ID: 002

1

df=0

df=0

df=2


2


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3 df=1 df=0

6

ID: 002

5

Error: 002

df=1

4

ID: 002

1

df=0

df=0

df=2


2


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3 df=1 df=0

6

ID: 002

5

Error: 002

df=1

1

4

df=0

df=1

df=2


2


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3 df=1 Error: 002 df=0

6

5

ID: 002

df=1

1

4

df=0

df=1

df=2


2


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3 df=2 Error: 002 df=0

6

5

ID: 002

df=1

1

4

df=0

df=1

df=2


2


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3 df=2 df=0

5

6

df=1

1

4

df=0

df=1

df=2


2


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3 df=2 df=0

5

6

df=1

1

4

df=0

df=1

df=2


2


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3 df=2 df=0

6

5

ID: 003

df=1

1

4

df=0

df=0

df=2


2


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3 df=2

... and continue until all nodes df=0

6

run out of

4 degrees df=0

df=2


5

ID: 003

df=1

1

of freedom ... df=0

2


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c found with the algorithm will satisfy, by construction, the The tentative solution W c=W c P = P. topology constraint and P W c − P). Now we want to evaluate ρ(W

Power Iteration Method For a square n × n matrix A, satisfying certain conditions, the iteration given by x(k) = Ax(k−1) for any nonzero initial vector, verifies: (k)

xi

lim

k→∞

(k−1)

xi

= λ1 (A)

for any non-zero component i = 1 . . . n.



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c − P verifies the conditions requires by the Power Iteration Method. The matrix W We choose a particular initial vector x(0) , which is orthogonal to the subspace of interest. Then: c − P)x(k−1) x(k) = (W c − P)k x(0) (W k

(0)

c − P)x (W

= = = =

ck x(0) W

c This is the general dynamics of the projection problem given the transition matrix W. c All nodes will be able to know ρ(W − P) in an distributed fashion, although any c completely. node knows W



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Distributed Genetic Algorithm The last step is to use a distributed Genetic Algorithm for computing the optimal W in a distributed way.

Distributed Genetic Algorithm 1

Each node solves PW = P.

2

Nodes perform a symmetrization stage.

3

Nodes evaluate the degrees of freedom, they will need this many variables, and will create a random population of npop individuals for each variable. c and evaluates ρ(W c − P). The network generates npop tentative matrices W

4 5

Elitism: each node keeps the best nbest individuals for the next round.

6

If the best individual has fulfilled our stopping criteria, end here.

7

Crossover: generate the next generation by crossing individuals.

8

Mutation: flip some bits in the new population.

9

Go to step 4 with the new generation.



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Simulation results Example In the previous topology, consider a subspace whose basis is given by the columns of   1/2   0 1 0 0 0 0 0 0 0    0 1/50 0 0 0 0   1/2 √3/3    0 0 0 1/2 1/2     0 U= 0 ⇒ W =    ⇒ ρ(W − P) = 0.5 0 0 0 0 1/50 0 0      1/2 √3/3   0  0 1/2 0 1/2 0 √ 0 0 1/2 0 0 1/2 1/2 3/3



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In-network computation of the Transition Matrix for Distributed Subspace Projection Xabier Insausti1 1

Pedro M. Crespo1

Baltasar Beferull2


16th of May, 2012



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