Multiple Instance Hidden Markov Model: Application to Landmine Detection in GPR
Data
Jeremy Bolton, Seniha Yuksel, Paul Gader
CSI Laboratory University of Florida
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CSI Laboratory
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Highlights• Hidden Markov Models (HMMs) are
useful tools for landmine detection in GPR imagery
• Explicitly incorporating the Multiple Instance Learning (MIL) paradigm in HMM learning is intuitive and effective
• Classification performance is improved when using the MI-HMM over a standard HMM
• Results further support the idea that explicitly accounting for the MI scenario may lead to improved learning under class label uncertainty
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Outline
I. HMMs for Landmine detection in GPRI. DataII. Feature ExtractionIII.Training
II. MIL Scenario
III.MI-HMM
IV.Classification Results
HMMs for landmine detection
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GPR Data• GPR data
– 3d image cube• Dt, xt, depth
– Subsurface objects are observed as hyperbolas
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GPR Data Feature Extraction• Many features extracted from in GPR data
measure the occurrence of an “edge” – For the typical HMM algorithm (Gader et al.),
• Preprocessing techniques are used to emphasize edges
• Image morphology and structuring elements can be used to extract edges
Image Preprocessed Edge Extraction
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4-d Edge Features
Edge Extraction
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Concept behind the HMM for GPR
• Using the extracted features (an observation sequence when scanning from left to right in an image) we will attempt to estimate some hidden states
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Concept behind the HMM for GPR
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HMM Features• Current AIM viewer by Smock
Image Feature Image
Rising Edge Feature
Falling Edge Feature
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Sampling HMM Summary• Feature Calculation
– Dimensions (Not always relevant whether positive or negative diagonal is observed …. Just simply a diagonal is observed)
• HMMSamp: 2d– Down sampling depth
• HMMSamp: 4
• HMM Models– Number of States
• HMMSamp : 4– Gaussian components per state (Fewer total components
for probability calculation)• HMMSamp : 1 (recent observation)
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Training the HMM• Xuping Zhang proposed a Gibbs Sampling algorithm for
HMM learning– But, given an image(s) how do we choose the training
sequences?– Which sequence(s) do we choose from each image?
• There is an inherent problem in many image analysis settings due to class label uncertainty per sequence
• That is, each image has a class label associated with it, but each image has multiple instances of samples or sequences. Which sample(s) is truly indicative of the target?– Using standard training techniques this translates to
identifying the optimal training set within a set of sequences– If an image has N sequences this translates to a search of 2N
possibilities
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Training Sample Selection Heuristic
• Currently, an MRF approach (Collins et al.) is used to bound the search to a localized area within the image rather than search all sequences within the image.– Reduces search space,
but multiple instance problem still exists
TM46-MB @ 1"
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-4000%Change in LL: 0.0017
H0 Segmentation H1 Segmentation
Original Data
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Data + Bounding Box
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Multiple Instance Learning
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Standard Learning vs. Multiple Instance Learning
• Standard supervised learning– Optimize some model (or learn a target concept) given
training samples and corresponding labels
• MIL– Learn a target concept given multiple sets of samples
and corresponding labels for the sets.– Interpretation: Learning with uncertain labels / noisy
teacher
},...,{},,...,{ 11 nn yyYxxX
?}?,...,{,1},,...,{ 11 ii iniiinii yyYxxX
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Multiple Instance Learning (MIL)
• Given: – Set of I bags
– Labeled + or -
– The ith bag is a set of Ji samples in some feature space
– Interpretation of labels
• Goal: learn concept– What characteristic is common to the positive bags that
is not observed in the negative bags
},...,,,..{ 11
Iii BBBBB
},...,{ 1 iiJii xxB
1)(: iji xlabeljB
0)(, iji xlabeljB
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Standard learning doesn’t always fit: GPR Example
• Standard Learning– Each training sample
(feature vector) must have a label
– But which ones and how many compose the optimal training set?
• Arduous task: many feature vectors per image and multiple images
• Difficult to label given GPR echoes, ground truthing errors, etc …
• Label of each vector may not be known
EHD: Feature Vector
1x?1 y2y3y?4 y
ny
2x
3x
4x
nx
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POSITIVE BAGS(Each bag is an image)
Learning from Bags• In MIL, a label is attached to a set of
samples. • A bag is a set of samples• A sample within a bag is called an
instance. • A bag is labeled as positive if and only if
at least one of its instances is positive.NEGATIVE BAGS(Each bag is an image)
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MI Learning: GPR Example
• Multiple Instance Learning– Each training bag
must have a label
– No need to label all feature vectors, just identify images (bags) where targets are present
– Implicitly accounts for class label uncertainty …
154321 ,...,,,, xxxxxY
EHD: Feature Vector
Multiple Instance Learning HMM: MI-HMM
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MI-HMM• In MI-HMM, instances are
sequences
NEGATIVE BAGS
POSITIVE BAGS
Direction of movement
21
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MI-HMM• Assuming independence between the bags
and assuming the Noisy-OR (Pearl) relationship between the sequences within each bag
• where
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MI-HMM learning• Due to the cumbersome nature of
the noisy-OR, the parameters of the HMM are learned using Metropolis – Hastings sampling.
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Sampling• HMM parameters are sampled from Dirichlet
• A new state is accepted or rejected based on the ratio r at iteration t + 1
• where P is the noisy-or model. 24
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Discrete Observations• Note that since we have chosen a Metropolis
Hastings sampling scheme using Dirichlets, our observations must be discretized.
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MI-HMM Summary• Feature Calculation
– Dimensions• HMMSamp: 2d• MI-HMM: 2d features are descretized into 16 symbols
– Down sampling depth• HMMSamp: 4 • MI-HMM: 4
• HMM Models– Number of States
• HMMSamp : 4• MI-HMM: 4
– Components per state (Fewer total components for probability calculation)
• HMMSamp : 1 Gaussian• MI-HMM: Discrete mixture over 16 symbols
Classification Results
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MI-HMM vs Sampling HMM• Small Millbrook HMM Samp (12,000)
MI-HMM (100)
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What’s the deal with HMM Samp?
Concluding Remarks
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Concluding Remarks• Explicitly incorporating the Multiple
Instance Learning (MIL) paradigm in HMM learning is intuitive and effective
• Classification performance is improved when using the MI-HMM over a standard HMM– More effective and efficient
• Future Work– Construct bags without using MRF heuristic– Apply to EMI data: spatial uncertainty
Back up Slides
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Standard Learning vs. Multiple Instance Learning
• Standard supervised learning– Optimize some model (or learn a target concept) given
training samples and corresponding labels
• MIL– Learn a target concept given multiple sets of samples
and corresponding labels for the sets.– Interpretation: Learning with uncertain labels / noisy
teacher
},...,{},,...,{ 11 nn yyYxxX
?}?,...,{,1},,...,{ 11 ii iniiinii yyYxxX
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Multiple Instance Learning (MIL)
• Given: – Set of I bags
– Labeled + or -
– The ith bag is a set of Ji samples in some feature space
– Interpretation of labels
• Goal: learn concept– What characteristic is common to the positive bags that
is not observed in the negative bags
},...,,,..{ 11
Iii BBBBB
},...,{ 1 iiJii xxB
1)(: iji xlabeljB
0)(, iji xlabeljB
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MIL Application: Example GPR
• Collaboration: Frigui, Collins, Torrione
• Construction of bags– Collect 15 EHD
feature vectors from the 15 depth bins
– Mine images = + bags
– FA images = - bags 154321 ,...,,,, xxxxx
EHD: Feature Vector
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Standard vs. MI Learning: GPR Example
• Standard Learning– Each training sample
(feature vector) must have a label
• Arduous task – many feature vectors
per image and multiple images
– difficult to label given GPR echoes, ground truthing errors, etc …
– label of each vector may not be known
EHD: Feature Vector
1x1y2y3y4y
ny
2x
3x
4x
nx
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Standard vs MI Learning: GPR Example
• Multiple Instance Learning– Each training bag
must have a label
– No need to label all feature vectors, just identify images (bags) where targets are present
– Implicitly accounts for class label uncertainty …
154321 ,...,,,, xxxxxY
EHD: Feature Vector
Random Set Framework for Multiple Instance Learning
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Random Set Brief
• Random Set
)(R)(R, B
))(,( B
)),(,( PB R
)),(,( PB
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How can we use Random Sets for MIL?
• Random set for MIL: Bags are sets
– Idea of finding commonality of positive bags inherent in random set formulation
• Sets have an empty intersection or non-empty intersection relationship
• Find commonality using intersection operator• Random sets governing functional is based on intersection operator
– Capacity functional : T
It is NOT the case that EACH element is NOT the
target concept
Xx
xTXT )(11)(
},...,{ 1 nxxX
A.K.A. : Noisy-OR gate (Pearl 1988)
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Random Set Functionals• Capacity functionals for intersection calculation
• Use germ and grain model to model random set– Multiple (J) Concepts
– Calculate probability of intersection given X and germ and grain pairs:
– Grains are governed by random radii with assumed cumulative:
)()( XTXP
J
jjj
1
)}({
jj
jjTj
jjjj xrrr
rRPrRPxTj
,)exp(1
22)(1)(})({
j Xx
xTXTj
)(11)(
Random Set model parameters
},{ Germ Grain
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RSF-MIL: Germ and Grain Model
• Positive Bags = blue
• Negative Bags = orange
• Distinct shapes = distinct bags
x
x
x
x
x x
x
x
x
TT
T
T
T
Multiple Instance Learning with Multiple Concepts
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Multiple Concepts: Disjunction or Conjunction?
• Disjunction– When you have multiple types of concepts– When each instance can indicate the presence
of a target• Conjunction
– When you have a target type that is composed of multiple (necessary concepts)
– When each instance can indicate a concept, but not necessary the composite target type
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Conjunctive RSF-MIL
• Previously Developed Disjunctive RSF-MIL (RSF-MIL-d)
• Conjunctive RSF-MIL (RSF-MIL-c)
j Xx
xTXTj
)(11)(
j Xx
xTXTj
)(11)(
Standard noisy-OR for one concept j
Noisy-AND combination across concepts
Noisy-OR combination across concepts and samples
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Synthetic Data Experiments
• Extreme Conjunct data set requires that a target bag exhibits two distinct concepts rather than one or none
AUC (AUC when initialized near solution)
Application to Remote Sensing
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Disjunctive Target Concepts
Target Concept Type 1
NoisyOR
…
NoisyOR
Target Concept Type 2
Target Concept Type n
NoisyOR
OR
Target Concept Present?
• Using Large overlapping bins (GROSS Extraction) the target concept can be encapsulated within 1 instance: Therefore a disjunctive relationship exists
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What if we want features with finer granularity
• Fine Extraction– More detail about image and more
shape information, but may loose disjunctive nature between (multiple) instances
…
NoisyOR
NoisyOR
AND
Target Concept Present?
Constituent Concept 1
(top of hyperbola)
Constituent Concept 2(wings of
hyperbola)
Our features have more granularity, therefore our concepts
may be constituents of a target, rather than encapsulating the
target concept
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GPR Experiments• Extensive GPR Data set
– ~800 targets– ~ 5,000 non-targets
• Experimental Design– Run RSF-MIL-d (disjunctive) and RSF-MIL-c
(conjunctive)– Compare both feature extraction methods
• Gross extraction: large enough to encompass target concept
• Fine extraction: Non-overlapping bins
• Hypothesis– RSF-MIL will perform well when using gross extraction
whereas RSF-MIL-c will perform well using Fine extraction
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Experimental Results• Highlights
– RSF-MIL-d using gross extraction performed best – RSF-MIL-c performed better than RSF-MIL-d when
using fine extraction– Other influencing factors: optimization methods for
RSF-MIL-d and RSF-MIL-c are not the same
Gross Extraction
Fine Extraction
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Future Work• Implement a general form that can learn
disjunction or conjunction relationship from the data
• Implement a general form that can learn the number of concepts
• Incorporate spatial information • Develop an improved optimization
scheme for RSF-MIL-C
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HMM Model VisualizationDTXTHMM DTXTHMM
0 0.5 1 1.5 20
1
2
1 2 30
0.5
1Initial Probs Transition Probs
1 2 3
123
FallingDiagonal FallingDiagonal
Rising Diagonal Rising Diagonal
Points =
Gaussian Component meansPoints =
Gaussian Component means
Color =
State IndexColor =
State Index
State index1State index 2State index 3
Initial probabilitiesInitial probabilitiesTransition
probabilities from state to state (red =
high probability)
Transition probabilities from
state to state (red = high probability)
Pattern Characterized
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Backup Slides
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MIL Example (AHI Imagery)• Robust learning tool
– MIL tools can learn target signature with limited or incomplete ground truth
Which spectral signature(s) should we
use to train a target model or classifier?
1. Spectral mixing2. Background signal
3. Ground truth not exact
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CSI Laboratory
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MI-RVM• Addition of set observations and
inference using noisy-OR to an RVM model
• Prior on the weight w
)exp(1
1)(
)(11)|1(1
zz
xwXyPK
jj
T
),0|()( 1 AwNwp
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SVM review• Classifier structure
• Optimization
by )()( T xφwx
,0,1))((:
2
1min
2
,
iiiT
i
ii
bw
btist
C
xφw
w
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MI-SVM Discussion• RVM was altered to fit MIL problem by
changing the form of the target variable’s posterior to model a noisy-OR gate.
• SVM can be altered to fit the MIL problem by changing how the margin is calculated– Boost the margin between the bag (rather
than samples) and decision surface– Look for the MI separating linear discriminant
• There is at least one sample from each bag in the half space
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mi-SVM• Enforce MI scenario using extra
constraints
1:,1
,1:,12
1
Ii
IiI
i
TIt
TIt
}1,1{,0,1))((:
2
1minmin
2
,}{
iiiiT
i
ii
bwt
tbtist
Ci
xφw
w
Mixed integer program: Must find optimal hyperplane and optimal labeling
set
At least one sample in each positive bag must have a label
of 1.All samples in each negative bag must have a label of -1.
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Current Applications
I. Multiple Instance LearningI. MI ProblemII. MI Applications
II.Multiple Instance Learning: Kernel MachinesI. MI-RVMII. MI-SVM
III. Current Applications I. GPR imageryII. HSI imagery
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HSI: Target Spectra Learning• Given labeled areas of interest: learn
target signature• Given test areas of interest: classify
set of samples
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Overview of MI-RVM Optimization
• Two step optimization1. Estimate optimal w, given posterior of
w• There is no closed form solution for the
parameters of the posterior, so a gradient update method is used
• Iterate until convergence. Then proceed to step 2.
2. Update parameter on prior of w• The distribution on the target variable has
no specific parameters.• Until system convergence, continue at step
1.
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1) Optimization of w• Optimize posterior (Bayes’ Rule) of
w
• Update weights using Newton-Raphson method
)(log)|(logmaxargˆ wpwXpww
MAP
gww tt 11 H
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2) Optimization of Prior• Optimization of covariance of prior
• Making a large number of assumptions, diagonal elements of A can be estimated
dwAwpwXpAXpAAA
)|()|(maxarg)|(maxargˆ
12
1
iii
newi Hwa
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Random Sets: Multiple Instance Learning
• Random set framework for multiple instance learning– Bags are sets– Idea of finding commonality of positive bags
inherent in random set formulation• Find commonality using intersection operator• Random sets governing functional is based on
intersection operator
)()( KPKT
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MI issues
• MIL approaches– Some approaches are biased to believe
only one sample in each bag caused the target concept
– Some approaches can only label bags– It is not clear whether anything is
gained over supervised approaches
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RSF-MIL
• MIL-like • Positive
Bags = blue
• Negative Bags = orange
• Distinct shapes = distinct bags
x
x
x
x
x x
x
x
x
TT
T
T
T
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Side Note: Bayesian Networks• Noisy-OR Assumption
– Bayesian Network representation of Noisy-OR
– Polytree: singly connected DAG
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Side Note• Full Bayesian network may be intractable
– Occurrence of causal factors are rare (sparse co-occurrence)
• So assume polytree• So assume result has boolean relationship with causal
factors– Absorb I, X and A into one node, governed by
randomness of I• These assumptions greatly simplify inference calculation• Calculate Z based on probabilities rather than
constructing a distribution using X
j
jXZPXXXXZP )|1(11}),,,{|1( 4321
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Diverse Density (DD)• Probabilistic Approach
– Goal:• Standard statistics approaches identify areas in a feature space
with high density of target samples and low density of non-target samples
• DD: identify areas in a feature space with a high “density” of samples from EACH of the postitive bags (“diverse”), and low density of samples from negative bags.
– Identify attributes or characteristics similar to positive bags, dissimilar with negative bags
– Assume t is a target characterization– Goal:
– Assuming the bags are conditionally independent
tBBBBP mnt
|,...,,,...,maxarg 11
jj
ii
ttBPtBP ||maxarg
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Diverse Density
• Calculation (Noisy-OR Model):
• Optimization
j
iji BtPBtP )|(11)|( },...,{ 1 iiJii xxB
j
iji BtPBtP )|(1)|(
22
expexp)|( txtBBtP ijijij
It is NOT the case that EACH element is NOT the
target concept
jj
ii
ttBPtBP ||maxarg
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Random Set Brief
• Random Set
)(R)(R, B
))(,( B
)),(,( PB R
)),(,( PB
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Random Set Functionals• Capacity and avoidance
functionals
– Given a germ and grain model
– Assumed random radii
)()( KPKT
in
j
ijiji
1
)}({
ijijijij
Tij
ijijij
ij
xrrr
rRPrRP
xTxPij
,)exp(1
2)(1)(
})({)|}({
)()( KPKQ
)(1)( KQKT
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When disjunction makes sense
• Using Large overlapping bins the target concept can be encapsulated within 1 instance: Therefore a disjunctive relationship exists
ORTarget
Concept Present
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Theoretical and Developmental Progress
• Previous Optimization:• Did not necessarily promote
diverse density
• Current optimization• Better for context learning and MIL
• Previously no feature relevance or selection (hypersphere)– Improvement: included learned weights
on each feature dimension
j
jji
ii BQBT )()(maxarg ,,
j
jji
ii BQBT )()(maxarg ,,
• Previous TO DO list• Improve Existing Code
– Develop joint optimization for context learning and MIL
• Apply MIL approaches (broad scale)• Learn similarities between feature sets of
mines• Aid in training existing algos: find “best”
EHD features for training / testing• Construct set-based classifiers?
• Previous TO DO list• Improve Existing Code
– Develop joint optimization for context learning and MIL
• Apply MIL approaches (broad scale)• Learn similarities between feature sets of
mines• Aid in training existing algos: find “best”
EHD features for training / testing• Construct set-based classifiers?
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How do we impose the MI scenario?: Diverse Density (Maron et al.)
• Calculation (Noisy-OR Model):– Inherent in Random Set formulation
• Optimization
– Combo of exhaustive search and gradient ascent
j
iji BtPBtP )|(11)|( },...,{ 1 iiJii xxB
j
iji BtPBtP )|(1)|(
22
expexp)|( txtBBtP ijijij
jj
ii
tBtPBtP ||maxarg
It is NOT the case that EACH element is NOT the
target concept
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How can we use Random Sets for MIL?
• Random set for MIL: Bags are sets– Idea of finding commonality of positive bags inherent in
random set formulation• Sets have an empty intersection or non-empty intersection
relationship• Find commonality using intersection operator• Random sets governing functional is based on intersection operator
• Example:
Bags with target{l,a,e,i,o,p,u,f}{f,b,a,e,i,z,o,u}
{a,b,c,i,o,u,e,p,f}{a,f,t,e,i,u,o,d,v}
Bags without target
{s,r,n,m,p,l}{z,s,w,t,g,n,c}
{f,p,k,r}{q,x,z,c,v}
{p,l,f}
{a,e,i,o,u,f}
intersection
union
{f,s,r,n,m,p,l,z,w,g,n,c,v,q,k}Target concept = \ = {a,e,i,o,u}