Hierarchical Shape Classification Using Bayesian Aggregation

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Hierarchical Shape ClassificationUsing Bayesian
Aggregation
Zafer Barutcuoglu Princeton
University Christopher DeCoro
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Shape Matching

• Given two shapes, quantify the difference between
  them
• Useful for search and retrieval, image
  processing, etc.
• Common approach is that of shape descriptors
• Map arbitrary definition of shape into a
  representative vector
• Define a distance measure (i.e Euclidean) to
  quantify similarity
• Examples include GEDT, SHD, REXT, etc.
• A common application is classification
• Given an example, and a set of classes, which
  class is most appropriate for that example?
• Applicable to a large range of applications

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Hierarchical Classification

• Given a hierarchical set of classes,
• And a set of labeled examples for those classes
• Predict the hierarchically-consistent
  classification of a novel example, using the
  hierarchy to improve performance.

Example courtesy of The Princeton Shape
Benchmark, P. Shilane et. al (2004)
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Motivation

• Given these, how can we predict classes for novel
  shapes?
• Conventional algorithms dont apply directly to
  hierarchies
• Binary classification
• Multi-class (one-of-M) classification
• Using binary classification for each class can
  produce predictions which contradict with the
  hierarchy
• Using multi-class classification over the leaf
  nodes loses information by ignoring the hierarchy

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Other heirarchical classification methods, other
domains

• TO ZAFER I need something here about background
  information, other methods, your method, etc.
• Also, Szymon suggested a slide about conditional
  probabilities and bayes nets in general. Could
  you come up with something very simplified and
  direct that would fit with the rest of the
  presentation?

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Motivation (Example)

• Independent classifiers give an inconsistent
  prediction
• Classified as bird, but not classified as flying
  creature
• Also cause incorrect results
• Not classified as flying bird
• Incorrectly classified as dragon

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Motivation (Example)

• We can correct this using our Bayesian
  Aggregation method
• Remove inconsistency at flying creature
• Also improves results of classification
• Stronger prediction of flying bird
• No longer classifies as dragon

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Naïve Hierarchical Consistency
INDEPENDENT
animal
YES
biped
NO
human
YES
Unfair distribution ofresponsibility and
correction
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Our Method Bayesian Aggregation

• Evaluate individual classifiers for each class
• Inconsistent predictions allowed
• Any classification algorithm can be used (e.g.
  kNN)
• Parallel evaluation
• Bayesian aggregation of predictions
• Inconsistencies resolved globally

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Our Method - Implementation

• Shape descriptor Spherical Harmonic Descriptor
• Converts shape into 512-element vector
• Compared using Euclidean distance
• Binary classifier k-Nearest Neighbors
• Finds the k nearest labeled training examples
• Novel example assigned to most common class
• Simple to implement, yet flexible

Rotation Invariant Spherical Harmonic
Representation of 3D Shape Descriptors M.
Kazhdan, et. al (2003)
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A Bayesian Framework
Given predictions g1...gN from kNN,
find most likely true labels y1...yN
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Classifier Output Likelihoods

• P(y1...yN g1...gN) a P(g1...gN y1...yN)
  P(y1...yN)
• Conditional independence assumption
• Classifiers outputs depend only on their true
  labels
• Given its true label, an output is conditionally
  independent of all other labels and outputs
• P(g1...gN y1...yN) ??i P(gi yi)

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Estimating P(gi yi)
The Confusion Matrix obtained using
cross-validation
Predicted negative
Predicted positive
Negative examples
Positive examples
e.g. P(g0 y0) (g0,y0) / (g0,y0)
(g1,y0)
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Hierarchical Class Priors

• P(y1...yN g1...gN) a P(g1...gN y1...yN)
  P(y1...yN)
• Hierarchical dependency model
• Class prior depends only on children
• P(y1...yN) ??i P(yi
  ychildren(i))
• Enforces hierarchical consistency
• The probability of an inconsistent assignment is
  0
• Bayesian inference will not allow inconsistency

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Conditional Probabilities

• P(yi ychildren(i))
• Inferred from known labeled examples
• P(gi yi)
• Inferred by validation on held-out data

y1
y2
y3
y4

• We can now apply Bayesian inference algorithms
• Particular algorithm independent of our method
• Results in globally consistent predictions
• Uses information present in hierarchy to improve
  predictions

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Applying Bayesian Aggregation

• Training phase produces Bayes Network
• From hierarchy and training set, train
  classifiers
• Use cross-validation to generate conditional
  probabilities
• Use probabilities to create bayes net
• Test phase give probabilities for novel examples
• For a novel example, apply classifiers
• Use classifier outputs and existing bayes net to
  infer probability of membership in each class

Hierarchy
Classifiers
Bayes Net
Cross-validation
Training Set
Classifiers
Bayes Net
Class Probabilities
Test Example
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Experimental Results

• 2-fold cross-validation on each class using kNN
• Area Under the ROC Curve (AUC) for evaluation
• Real-valued predictor can be thresholded
  arbitrarily
• Probability that pos. example is predicted over a
  neg. example
• 169 of 170 classes were improved by our method
• Average ?AUC 0.137 (19 of old AUC)
• Old AUC .7004 (27 had AUC of 0.5, random
  guessing)

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AUC Scatter Plot
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AUC Changes

• 169 of 170 classes were improved by our method
• Average ?AUC 0.137 (19 of old AUC)
• Old AUC .7004 (27 had AUC of 0.5, random
  guessing)

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Questions