Transformation-invariant clustering using the EM algorithm

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Transformation-invariant clustering using the EM
algorithm
Brendan Frey and Nebojsa Jojic
IEEE Trans on PAMI, 25(1) 2003
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Goal

• unsupervised learning of image structure
  regardless of transformation
• probabilistic description of the data
• clustering as density modeling grouping
  similar images together

Invariance

• manifold in data space
• all points on manifold equivalent
• complex even for basic transformations
• how to approximate?

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Approximating the Invariance Manifold

• discrete set of points
• sparse matrices Ti map cannonical feature z into
  transformed feature x (observed)
• as a Gaussian probability model,
• all possible transformations T enumerated

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This is what it would look like for...

• a 2x3 image with pixel-shift translations
  (wrap-around)

z
T1...T6
x
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The full statistical model

• for one feature (one cluster)
• data, given latent repr
• joint of all variables
• Gaussian post-transformation with noise ?
• Gaussian pre-transformation with noise F
• for multiple features (clusters), mixture model

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The full statistical model

• the generative equation

• for each feature, have a cannonical mean and
  cannonical variance
• image contains one of the cannonical features
  (mixture model) that has undergone one
  transformation

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Inference
and is Gassian

• marginals for inferring parameters T, c, z

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Adapting the rest of parameters

• pre-transformation noise

• post-tranformation noise

• all learned with EM
• E-step assume known params, infer P(z, T, c)
• M-step update parameters

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Experiments
recovering 4 clusters
4 clusters w/o transform.
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Pre/post transformation noise
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Pre/post transformation noise
mean
variance
single Gaussian model of image
µ
F
transformation-invariant model, no post-t noise
µ
F
?
transformation-invariant model, with post-t noise
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Conclusions

• fast (uses sparse matrices, FFT)
• incorporates pre- and post-transformation noise
• works on artificial data, clustering simple image
  sets, cleaning up somewhat contrived examples
• can be extended to make use of time series data,
  account for more transformations
• poor transformation model
• fixed, pre-specified transformations
• must be sparse
• poor feature model
• Gaussian representation of structure

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