Download - Learning Classifiers For Non-IID Data
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Learning Classifiers For Non-IID Data Balaji Krishnapuram, Computer-Aided Diagnosis and TherapySiemens Medical Solutions, Inc.
Collaborators: Volkan Vural, Jennifer Dy [North Eastern], Ya Xue [Duke], Murat Dundar, Glenn Fung, Bharat Rao [Siemens]
Jun 27, 2006
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OutlineImplicit IID assumption in traditional classifier design
Often, not valid in real life. Motivating CAD problems
Convex algorithms for Multiple Instance Learning (MIL)
Bayesian algorithms for Batch-wise classification Faster, approximate algorithms via mathematical programming
Summary / Conclusions
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IID assumption in classifier designTraining data D={(xi,yi)i=1N: xi 2 Rd, yi 2 {+1,-1}}, Testing data T ={(xi,yi)i=1M: xi 2 Rd, yi 2 {+1,-1}},
Assume each training/testing sample drawn independently from identical distribution:(xi,yi) ~ PXY(x,y)
This is why we can classify one test sample at a time, ignoring the features of the other test samplesEg. Logistic Regression: P(yi=1|xi,w)=1/(1+exp(-wT xi))
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Evaluating classifiers: learning-theoryBinomial test set bounds: With high probability over the random draw of M samples in testing set T, if M large and a classifier w is observed to be accurate on T, with high probability its expected accuracy over a random draw of a sample from PXY(x,y) will be high
If the IID assumption fails, all bets are off !Thought experiment: repeat same test sample M times
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Training classifiers: learning theoryWith high probability over the random draw of N samples in training set D, the expected accuracy on a random sample from PXY(x,y) for the learnt classifier w will be high iffaccurate on the training set D; and N largesatisfies intuition before seeing data (prior, large margin etc)
PAC-Bayes, VC-theory etc rely on iid assumptionRelaxation to exchangeability being explored
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CAD: Correlations among candidate ROI
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Hierarchical Correlation Among Samples
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Additive Random Effect ModelsThe classification is treated as iid, but only if given bothFixed effects (unique to sample) Random effects (shared among samples)
Simple additive model to explain the correlationsP(yi|xi,w,ri,v)=1/(1+exp(-wT xi vT ri)) P(yi|xi,w,ri)=s P(yi|xi,w,ri,v) p(v|D) dv Sharing vT ri among many samples correlated prediction
But only small improvements in real-life applications
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CAD detects early stage colon cancer
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Candidate Specific Random Effects Model: PolypsSpecificitySensitivity
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CAD algorithms: domain-specific issuesMultiple (correlated) views: one detection is sufficient
Systemic treatment of diseases: e.g. detecting one PE sufficient
Modeling the data acquisition mechanismErrors in guessing class labels for training set.
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The Multiple Instance Learning ProblemA bag is a collection of many instances (samples)
The class label is provided for bags, not instances
Positive bag has at least one +ve instance in it
Examples of bag definition for CAD applications:Bag=samples from multiple views, for the same regionBag=all candidates referring to same underlying structureBag=all candidates from a patient
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CH-MIL Algorithm: 2-D illustration
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CH-MIL Algorithm for Fishers DiscriminantEasy implementation via Alternating OptimizationScales well to very large datasetsConvex problem with unique optima
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Lung CADDR CADLung Nodules& Pulmonary EmboliComputed TomographyAX*Pending FDA Approval
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CH-MIL: Pulmonary Embolisms
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CH-MIL: Polyps in Colon
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Classifying a Correlated Batch of SamplesLet classification of individual samples xi be based on uiEg. Linear ui = wT xi ; or kernel-predictor ui= j=1N j k(xi,xj)
Instead of basing the classification on ui, we will base it on an unobserved (latent) random variable zi
Prior: Even before observing any features xi (thus before ui), zi are known to be correlated a-priori, p(z)=N(z|0,)
Eg. due to spatial adjacency = exp(-D), Matrix D=pair-wise dist. between samples
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Classifying a Correlated Batch of SamplesPrior: Even before observing any features xi (thus before ui), zi are known to be correlated a-priori, p(z)=N(z|0,)
Likelihood: Let us claim that ui is really a noisy observation of a random variable zi :p(ui|zi)=N(ui|zi, 2)
Posterior: remains correlated, even after observing the features xiP(z|u)=N(z|(-12+I)-1u, (-1+2I)-1)Intuition: E[zi]=j=1N Aij uj ; A=(-12+I)-1
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SVM-like Approximate AlgorithmIntuition: classify using E[zi]=j=1N Aij uj ; A=(-12+I)-1What if we used A=( + I) instead?Reduces computation by avoiding inversion.Not principled, but a heuristic for speed.Yields an SVM-like mathematical programming algorithm:
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Detecting Polyps in Colon
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Detecting Pulmonary Embolisms
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Detecting Nodules in the Lung
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ConclusionsIID assumption is universal in MLOften violated in real life, but ignoredExplicit modeling can substantially improve accuracyDescribed 3 models in this talk, utilizing varying levels of informationAdditive Random Effects Models: weak correlation informationMultiple Instance Learning: stronger correlations enforcedBatch-wise classification models: explicit information Statistically significant improvement in accuracyOnly starting to scratch the surface, lots to improve!
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