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Hierarchical Shape Classification Using Bayesian Aggregation. Zafer Barutcuoglu Princeton University Christopher DeCoro. Shape Matching. Given two shapes, quantify the difference between them Useful for search and retrieval, image processing, etc.
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Hierarchical Shape ClassificationUsing Bayesian Aggregation Zafer Barutcuoglu Princeton University Christopher DeCoro
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
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)
Motivation • Given these, how can we predict classes for novel shapes? • Conventional algorithms don’t 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
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?
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
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
TOP-DOWN BOTTOM-UP animal animal YES YES biped biped NO NO human human YES YES Naïve Hierarchical Consistency INDEPENDENT animal YES biped NO human YES Unfair distribution ofresponsibility and correction
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
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)
g1 animal y1 g2 biped y2 g3 flying creature y3 g4 superman y4 A Bayesian Framework Given predictions g1...gN from kNN, find most likely true labels y1...yN
Classifier Output Likelihoods P(y1...yN | g1...gN) = αP(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)
Estimating P(gi | yi) The Confusion Matrix obtained using cross-validation Predicted negative Predicted positive Negative examples Positive examples e.g. P(g=0 | y=0) ≈ #(g=0,y=0) / [ #(g=0,y=0) + #(g=1,y=0) ]
Hierarchical Class Priors P(y1...yN | g1...gN) = αP(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
g1 g2 g3 g4 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
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 Cross-validation Bayes Net Training Set Test Example Classifiers Bayes Net Class Probabilities
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)
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)