Pattern Decomposition with Scaling - Cornell Universitylebras/publications/pattern... · 2012. 1....
Transcript of Pattern Decomposition with Scaling - Cornell Universitylebras/publications/pattern... · 2012. 1....
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Constraint Reasoning and Kernel Clustering
for Pattern Decomposition With Scaling
Ronan LeBras Computer Science
Theodoros Damoulas Computer Science
Ashish Sabharwal Computer Science
Carla P. Gomes Computer Science
John M. Gregoire Materials Science / Physics
Bruce van Dover Materials Science / Physics
Sept 15, 2011 CP’11
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Motivation
Ronan Le Bras - CP’11, Sept 15, 2011 2
Cornell Fuel Cell Institute
Mission: develop new materials for fuel cells.
An Electrocatalyst must:
1) Be electronically conducting
2) Facilitate both reactions
Platinum is the best known metal to
fulfill that role, but:
1) The reaction rate is still considered
slow (causing energy loss)
2) Platinum is fairly costly, intolerant
to fuel contaminants, and has a short
lifetime.
Goal: Find an intermetallic compound that is a better catalyst than Pt.
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Motivation
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Recipe for finding alternatives to Platinum
1) In a vacuum chamber, place a silicon wafer.
2) Add three metals.
3) Mix until smooth, using three sputter guns.
4) Bake for 2 hours at 650ºC
Ta
Rh
Pd
(38% Ta, 45% Rh, 17% Pd)
• Deliberately inhomogeneous
composition on Si wafer
• Atoms are intimately mixed
[Source: Pyrotope, Sebastien Merkel]
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Motivation
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Identifying crystal structure using X-Ray Diffraction at CHESS
• XRD pattern characterizes the underlying crystal fairly well
• Expensive experimentations: Bruce van Dover’s research
team has access to the facility one week every year.
Ta
Rh
Pd
(38% Ta, 45% Rh, 17% Pd)
[Source: Pyrotope, Sebastien Merkel]
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Motivation
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Ta
Rh
Pd
α
β
δ
γ
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Motivation
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Ta
Rh
Pd
α
β
α+β
δ
γ
γ+δ
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Motivation
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Ta
Rh
Pd
α
β
δ
γ
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Motivation
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15 20 25 30 35 40 45 50 55 600
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Ta
Rh
Pd
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α
β
δ
γ
α+β
Ronan Le Bras - CP’11, Sept 15, 2011
Pi
Pj
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Motivation
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15 20 25 30 35 40 45 50 55 600
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Ta
Rh
Pd
α
β
α+β
δ
γ
Ronan Le Bras - CP’11, Sept 15, 2011
Pi
Pk
Pj
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Motivation
Fe
Al
Si
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INPUT: pure phase
region
Fe
Al
Si
m phase regions
k pure regions
m-k mixed regions
XRD pattern
characterizing
pure phases
Mixed
phase
region
OUTPUT:
Additional Physical characteristics:
Peaks shift by 15% within a region
Phase Connectivity
Mixtures of 3 phases
Small peaks might be discriminative
Peak locations matter, more than peak
intensities
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Motivation
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Ta
Rh
Pd
Figure 1: Phase regions of Ta-Rh-Pd Figure 2: Fluorescence activity of Ta-Rh-Pd
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Outline
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• Motivation
• Problem Definition
Abstraction
Hardness
• CP Model
• Kernel-based Clustering
• Bridging CP and Machine learning
• Conclusion and Future work
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Problem Abstraction: Pattern Decomposition with Scaling
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Input
Output
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Problem Abstraction: Pattern Decomposition with Scaling
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Input
Output
v1
v2
v3
v4
v5
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Problem Abstraction: Pattern Decomposition with Scaling
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Input
Output
v1
v2
v3
v4
v5
P1
P2
P3
P4
P5
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Problem Abstraction: Pattern Decomposition with Scaling
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Input
Output
v1
v2
v3
v4
v5
P1
P2
P3
P4
P5
M= K = 2, δ = 1.5
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Problem Abstraction: Pattern Decomposition with Scaling
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Input
Output
v1
v2
v3
v4
v5
P1
P2
P3
P4
P5
M= K = 2, δ = 1.5 B2
B1
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Problem Abstraction: Pattern Decomposition with Scaling
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Input
Output
v1
v2
v3
v4
v5
P1
P2
P3
P4
P5
M= K = 2, δ = 1.5 B2
B1
s11=1.00
s12=0.95
s13=0.88
s14=0.84
s15=0
s21=0.68
s22=0.78
s23=0.85
s24=0.96
s25=1.00
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Problem Hardness
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Assumptions: Each Bk appears by itself in some vi / No experimental
noise
The problem can be solved* in polynomial time.
Assumption: No experimental noise
The problem becomes NP-hard (reduction from the “Set Basis” problem)
Assumption used in this work: Experimental noise in the form of
missing elements in Pi
v1
v2
v3
v4
v5
P1
P2
P3
P4
P5
M= K = 2, δ = 1.5 B2
B1
s11=1.00
s12=0.95
s13=0.88
s14=0.84
s15=0
s21=0.68
s22=0.78
s23=0.85
s24=0.96
s25=1.00
Ronan Le Bras - CP’11, Sept 15, 2011
P0=
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Outline
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• Motivation
• Problem Definition
• CP Model
Model
Experimental Results
• Kernel-based Clustering
• Bridging CP and Machine learning
• Conclusion and Future work
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CP Model
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CP Model (continued)
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Advantage: Captures physical properties and relies on peak location rather than height.
Drawback: Does not scale to realistic instances; poor propagation if experimental noise.
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CP Model – Experimental Results
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0
200
400
600
800
1000
1200
1400
0 1 2 3 4 5 6
Ru
nn
ing
tim
e (i
n s
)
Number of unknown phases (K’)
Number of unknown phases vs. running times
(AlLiFe with |P| = 24)
N=10
N=15
N=28
N=218
For realistic instances, K’ = 6 and N ≈ 218...
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Outline
24
• Motivation
• Problem Definition
• CP Model
• Kernel-based Clustering
• Bridging CP and Machine learning
• Conclusion and Future work
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Kernel-based Clustering
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Set of features: X =
Similarity matrices:[X.XT]
Method: K-means on Dynamic Time-Warping kernel
Goal: Select groups of samples that belong to the same phase region to feed the CP
model, in order to extract the underlying phases of these sub-problems.
Better approach: take
shifts into account
Red: similar
Blue: dissimilar
+
+
+
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Bridging CP and ML
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Goal: a robust, physically meaningful, scalable,
automated solution method that combines:
Similarity “Kernels”
& Clustering
Machine Learningfor a “global
data-driven view”
Underlying
Physics
A language for
Constraintsenforcing
“local details”
Constraint Programming model
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Bridging Constraint Reasoning and Machine
Learning: Overview of the Methodology
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Fe
Al
Si
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INPUT:
Peak
detection
Machine Learning:
Kernel methods,
Dynamic Time Wrapping
Machine Learning:
Partial “Clustering”
Fe
Al
SiCP Model & Solver
on sub-problems
, +,
only only
Full CP Model
guided by
partial solutions
OUTPUT
Fix errors
in data
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Experimental Validation
28
Al-Li-Fe instance
with 6 phases:
Ground truth
(known)
Our CP + ML hybrid
approach is much
more robust
Previous work (NMF)
violates many
physical requirements
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Conclusion
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Hybrid approach to clustering under constraints
More robust than data-driven “global” ML approaches
More scalable than a pure CP model “locally” enforcing constraints
An exciting application in close collaboration with physicists
Best inference out of expensive experiments
Towards the design of better fuel cell technology
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Future work
Ongoing work:
Spatial Clustering, to further enhance cluster quality
Bayesian Approach, to better exploit prior knowledge
about local smoothness and available inorganic libraries
Active learning:
Where to sample next, assuming we can interfere with
the sampling process?
When to stop sampling if sufficient information has
been obtained?
Correlating catalytic properties across many thin-films:
In order to understand the underlying physical
mechanism of catalysis and to find promising
intermettalic compounds
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The end
Thank you!
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Extra slides
32Ronan Le Bras - CP’11, Sept 15, 2011
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Experimental Sample
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Example on Al-Li-Fe diagram:
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Applications with similar structure
34
Flight Calls / Bird conservation
Identifying bird population from sound recordings at
night.
Analogy: basis pattern = species
samples = recordings
physical constraints = spatial constraints,
species and season specificities…
Fire Detection
Detecting/Locating fires.
Analogy: basis pattern = warmth sources
samples = temperature recordings
physical constraints = gradient of
temperatures, material properties…
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Previous Work 1: Cluster Analysis (Long et al., 2007)
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xi =
(Feature vector) (Pearson correlation coefficients) (Distance matrix)
(PCA – 3 dimensional approx) (Hierarchical Agglomerative Clustering)
Drawback: Requires sampling of pure phases, detects phase regions (not phases),
overlooks peak shifts, may violate physical constraints (phase continuity, etc.).
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Previous Work 2: NMF (Long et al., 2009)
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xi =
(Feature vector) (Linear positive combination (A) of
basis patterns (S))
(Minimizing squared
Frobenius norm
X = A.S + E Min ║E║
Drawback: Overlooks peak shifts (linear combination only), may violate physical
constraints (phase continuity, etc.).
Ronan Le Bras - CP’11, Sept 15, 2011