1. Transformation-invariant clustering using the EM algorithm
Brendan Frey and Nebojsa Jojic
IEEE Trans on PAMI, 25(1) 2003
yan karklin. cns presentation 08/10/2004

2. 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?
yan karklin. cns presentation 08/10/2004

3. 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
yan karklin. cns presentation 08/10/2004

4. This is what it would look like for...
a 2x3 image with pixel-shift translations (wrap-around)
z =
{T1...T6} =
x =
yan karklin. cns presentation 08/10/2004

5. The full statistical model
for one feature (one cluster):
data, given latent repr:
joint of all variables:
Gaussian post-transformation with noise Ψ
Gaussian pre-transformation with noise Φ
for multiple features (clusters), mixture model:
yan karklin. cns presentation 08/10/2004

6. 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
yan karklin. cns presentation 08/10/2004

7. Inference and is Gassian marginals for inferring parameters T, c, z:
yan karklin. cns presentation 08/10/2004

8. 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
yan karklin. cns presentation 08/10/2004

9. Experiments recovering 4 clusters 4 clusters w/o transform.
yan karklin. cns presentation 08/10/2004

10. Pre/post transformation noise
yan karklin. cns presentation 08/10/2004

11. Pre/post transformation noise
mean
variance
single Gaussian model of image μ Φ
transformation-invariant model, no post-t noise μ Φ Ψ
transformation-invariant model, with post-t noise
yan karklin. cns presentation 08/10/2004

12. 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
yan karklin. cns presentation 08/10/2004

13. yan karklin. cns presentation 08/10/2004

14. yan karklin. cns presentation 08/10/2004