1. Near-Optimal Sensor Placements in Gaussian Processes
Carlos Guestrin
Andreas Krause Ajit Singh
Carnegie Mellon University

2. Sensor placement applications
Monitoring of spatial phenomena
Temperature
Precipitation
Drilling oil wells
...
Active learning, experimental design, ...
Results today not limited to 2-dimensions
Temperature data from sensor network
Precipitation
data from
Pacific NW

3. Deploying sensors Computer science Spatial statistics Considered in:
This deployment:
Evenly distributed sensors
Considered in:
Computer science
(c.f., [Hochbaum & Maass ’85])
Spatial statistics
(c.f., [Cressie ’91])
Chicken-and-Egg problem: 
assumptions
No data or assumptions
about distribution
But, what are the
optimal placements???
i.e., solving combinatorial
(non-myopic) optimization
Don’t know where to
place sensors

4. Strong assumption – Sensing radius
Becomes a covering problem
Problem is NP-complete
But there are good algorithms with (PTAS) -approximation guarantees
[Hochbaum & Maass ’85]
Node predicts
values of positions
with some radius
Unfortunately, approach is usually not useful… Assumption is wrong on real data! 
For example…

5. Spatial correlation Precipitation data from Pacific NW
Non-local, Non-circular
correlations
Complex positive and negative
correlations

6. Complex, noisy correlations
Complex, uneven sensing “region”
Actually, noisy correlations, rather than sensing region

7. Combining multiple sources of information
Temp
here?
Individually, sensors are bad predictors
Combined information is more reliable
How do we combine information?
Focus of spatial statistics

8. Gaussian process (GP) - Intuition
GP – Non-parametric; represents uncertainty;
complex correlation functions (kernels)
Uncertainty
after observations
are made
less sure
here
more sure
here
y - temperature
x - position

9. Prediction after observing
Gaussian processes
Posterior
mean temperature
Posterior
variance
Kernel function:
Prediction after observing
set of sensors A:

10. Gaussian processes for sensor placement
Posterior
mean temperature
Posterior
variance
Goal:
Find sensor placement with least uncertainty after observations
Problem is still NP-complete 
Need approximation

11. Non-myopic placements
Consider myopically selecting
This can be seen as an attempt to non-myopically maximize
H(A1)
+ H(A2 | {A1})
H(Ak | {A1 ... Ak-1})
most uncertain
This is exactly the joint entropy H(A) = H({A1 ... Ak})
most uncertain given A1
most uncertain given A1 ... Ak-1
swap order with previous

12. Entropy criterion (c.f., [Cressie ’91])
A Ã ;
For i = 1 to k
Add location Xi to A, s.t.:
Entropy places
sensors along borders
Temperature data
placements:
Entropy
“Wasted”
information
observed by
[O’Hagan ’78]
Entropy
High uncertainty
given current set A –
X is different
Uncertainty (entropy) plot
Entropy criterion wastes information [O’Hagan ’78],
Indirect, doesn’t consider sensing region –
No formal non-myopic guarantees 

13. Proposed objective function: Mutual information
Locations of interest V
Find locations AµV maximizing mutual information:
Intuitive greedy rule:
Temperature data
placements:
Entropy
Mutual information
Uncertainty of
uninstrumented
locations
before sensing
Uncertainty of
uninstrumented
locations
after sensing
Intuitive criterion – Locations that
are both different and informative
We give formal
non-myopic guarantees 
High uncertainty
given A –
X is different
Low uncertainty
given rest –
X is informative

14. An important observation
Selecting T1 tells sth. about T2 and T5
Selecting T3 tells sth. about T2 and T4
In many cases, new information is worth less if we know more (diminishing returns)! T2 T1 T3 T5 T4
Now adding T2 would not help much

15. Submodular set functions
Submodular set functions are a natural formalism for this idea:
f(A [ {X}) – f(A)
Maximization of SFs is NP-hard 
But…
¸ f(B [ {X}) – f(B) for A µ B B A
{X}

16. How can we leverage submodularity?
Theorem [Nemhauser et al. ’78]: The greedy algorithm guarantees (1-1/e) OPT approximation for monotone SFs, i.e.
~ 63%

17. How can we leverage submodularity?
Theorem [Nemhauser et al. ’78]: The greedy algorithm guarantees (1-1/e) OPT approximation for monotone SFs, i.e.
~ 63%

18. Mutual information and submodularity
Even though MI is submodular,
can’t apply Nemhauser et al.
Or can we… 
Mutual information is submodular 
F(A) = I(A;V\A)
So, we should be able to use Nemhauser et al.
Mutual information is not monotone!!! 
Initially, adding sensor increases MI; later adding sensors decreases MI
F(;) = I(;;V) = 0
F(V) = I(V;;) = 0
F(A) ¸ 0
mutual information
A=;
A=V
num. sensors

19. Approximate monotonicity of mutual information
If H(X|A) – H(X|V\A) ¸ 0, then MI monotonic
Solution: Add grid Z of unobservable locations
If H(X|A) – H(X|ZV\A) ¸ 0, then MI monotonic
H(X|A) << H(X|V\A)
MI not monotonic
For sufficiently fine Z:
H(X|A) > H(X|ZV\A) -  MI approximately monotonic
V\A A
Z –
unobservable X

20. Theorem: Mutual information sensor placement
Greedy MI algorithm provides constant factor approximation: placing k sensors, 8 >0:
Approximate monotonicity
for sufficiently discretization – poly(1/,k,,L,M)
– sensor noise, L – Lipschitz const. of kernels,
M – maxX K(X,X)
Result of
our algorithm
Constant
factor
Optimal
non-myopic
solution

21. Different costs for different placements
Theorem 1: Constant-factor approximation of optimal locations – select k sensors
Theorem 2: (Cost-sensitive placements)
In practice, different locations may have different costs
Corridor versus inside wall
Have a budget B to spend on placing sensors
Constant-factor approximation – same constant (1-1/e)
Slightly more complicated than greedy algorithm [Sviridenko / Krause, Guestrin]

22. Mutual information has 3 times less variance than entropy criterion
Model learned from 54 sensors
Deployment results
“True” temp.
prediction
“True” temp.
variance
Mutual information has 3 times less variance than entropy criterion
Used initial deployment to select 22 new sensors
Learned new GP on test data using just these sensors
Posterior
mean
variance
Entropy criterion
Mutual information criterion

23. Comparing to other heuristics
Greedy
Algorithm we analyze
Random placements
Pairwise exchange (PE)
Start with a some placement
Swap locations while improving solution
Our bound enables a posteriori
analysis for any heuristic
Assume, algorithm TUAFSPGP gives results which are 10% better than the results obtained from the greedy algorithm
Then we immediately know, TUAFSPGP is within 70% of optimum!
Better
mutual information

24. Precipitation data Entropy criterion Mutual information Entropy
Better
Entropy
criterion
Mutual
information
Entropy
Mutual information

25. Computing the greedy rule
At each iteration
For each candidate position i 2{1,…,N}, must compute:
Requires inversion of NxN matrix – about O(N3)
Total running time for k sensors: O(kN4)
Polynomial!
But very slow in practice 
Exploit sparsity in kernel matrix

26. Usually, matrix is only almost sparse
Local kernels
 =
Covariance matrix may have many zeros!
Each sensor location correlated with a small number of other locations
Exploiting locality:
If each location correlated with at most d others
A sparse representation, and a priority queue trick
Reduce complexity from O(kN4) to:
Only about O(N log N)
Usually, matrix is only almost sparse

27. Approximately local kernels
Covariance matrix may have many elements close to zero
E.g., Gaussian kernel
Matrix not sparse
What if we set them to zero?
Sparse matrix
Approximate solution
Theorem:
Truncate small entries ! small effect on solution quality
If |K(x,y)| · , set to 0
Then, quality of placements only O() worse

28. Effect of truncated kernels on solution – Rain data
Improvement
in running time
Effect on
solution quality
Better
Better
About 3 times faster,
minimal effect on solution quality

29. Summary
Mutual information criterion for sensor placement in general GPs
Efficient algorithms with strong approximation guarantees: (1-1/e) OPT-ε
Exploiting local structure improves efficiency
Superior prediction accuracy for several real-world problems
Related ideas in discrete settings presented at UAI and IJCAI this year
Effective algorithm for sensor placement and
experimental design; basis for active learning

30. A note on maximizing entropy
Entropy is submodular [Ko et al. `95], but…
Function F is monotonic iff:
Adding X cannot hurt
F(A[X) ¸ F(A)
Remark:
Entropy in GPs not monotonic (not even approximately)
H(A[X) – H(A) = H(X|A)
As discretization becomes finer H(X|A) ! -1
Nemhauser et al. analysis for submodular
functions not applicable directly to entropy

31. How do we predict temperatures at unsensed locations?
Far away
points?
Interpolation?
Overfits
temperature
position

32. How do we predict temperatures at unsensed locations?
Regression
y = a + bx + cx2 + dx3
Few parameters,
less overfitting 
less sure
here
more sure
here
How sure are we
about the prediction?
y - temperature
x - position
But, regression function has no notion of uncertainty!!! 