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Principal Manifolds and Probabilistic Subspaces for Visual Recognition

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Principal Manifolds and Probabilistic Subspaces for Visual Recognition

Baback Moghaddam

TPAMI, June 2002.

John Galeotti

Advanced Perception

February 12, 2004

- Traditional subspaces
- PCA
- ICA
- Kernel PCA (& neural network NLPCA)

- Probabilistic subspaces

- We already know this
- Main properties
- Approximate reconstruction
x ≈ y

- Orthonormality of the basis
T=I

- Decorrelated principal components
E{yiyj}i≠j = 0

- Approximate reconstruction

- Like PCA, but the components’ distribution is designed to be sub/super Gaussian statistical independence
- Main properties
- Approximate reconstruction
x ≈ Ay

- Nonorthogonality of the basis A
ATA≠I

- Near factorization of the joint distribution P(y)
P(y)≈ ∏ p(yi)

- Approximate reconstruction

- AKA principal curves
- Essentially nonlinear regression
- Finds a curved subspace passing “through the middle of the data”

- Main properties
- Approximate reconstruction
y = f(x)

- Nonlinear projection
x ≈ g(y)

- No prior knowledge regarding joint distribution of the components (typical)
P(y) = ?

- Approximate reconstruction
- Two main methods
- Neural network encoder
- Kernel PCA (KPCA)

- Trained to match the output to the input
- Uses a “bottleneck” layer to force a lower-dimensional representation

- Similar to kernel-based nonlinear SVM
- Maps data to a higher dimensional space in which linear PCA is applied
- Nonlinear input mapping
(x):NL, N<L

- Covariance is computed with dot-products
- For economy, make (x) implicit
k(xi,xj) = ( (xi) (xj) )

- Nonlinear input mapping

- Does not require nonlinear optimization
- Is not subject to overfitting
- Requires no prior knowledge of network architecture or number of dimensions
- Requires the (unprincipled) selection of an “optimal” kernel and its parameters

- Find labeled image most similar to N-dim input vector using a suitable M-dim subspace
- Similarity ex: S(I1,I2) || ∆ ||-1,∆ = I1 - I2
- Observation: Two types of image variation
- Critical:Images of different objects
- Incidental:Images of same object under
different lighting, surroundings, etc.

- Problem:Preceding subspace projections do
not help distinguish variation type

when calculating similarity

- Similarity based on probability that ∆ is characteristic of incidental variations
- ∆ = image-difference vector (N-dim)
- ΩI = incidental (intrapersonal) variations
- ΩE = critical (extrapersonal) variations

- Likelihoods P(∆|Ω) estimated using subspace density estimation
- Priors P(Ω) are set to reflect specific operating conditions (often uniform)
- Two images are of the same object if P(ΩI|∆) > P(ΩE|∆) S(∆) > 0.5

- Necessary for each P(∆|Ω),Ω { ΩI, ΩE }
- Perform PCA on training-sets of ∆ for each Ω
- The covariance matrix (∑) will define a Gaussian

- Two subspaces:
- F = M-dimensional principal subspace of ∑
- F = non-principal subspace orthogonal to F

- yi = ∆ projected onto principal eigenvectors
- i = ranked eigenvalues
- Non-principal eigenvalues are typically unknown and are estimated by fitting a function of the form f -n to the known eigenvalues

- 2(∆) = PCA residual (reconstruction error)
- = density in non-principal subspace
- ≈ average of (estimated) F eigenvalues

- P(∆|Ω) is marginalized into each subspace
- Marginal density is exact in F
- Marginal density is approximate in F

- After doing PCA, use a whitening transform to preprocess the labeled images into single coefficients for each of the principal subspaces:
where and V are matrices of the principal eigenvalues and eigenvectors of either ∑I or ∑E

- At run time, apply the same whitening transform to the input image

- The whitening transform reduces the marginal Gaussian calculations in the principal subspaces F to simple Euclidean distances
- The denominators are easy to precompute

- Further speedup can be gained by using a maximum likelihood (ML) rule instead of a maximum a posteriori (MAP) rule:
- Typically, ML is only a few percent less accurate than MAP, but ML is twice as fast
- In general, ΩE seems less important than ΩI

Probabilistic Similarity

Eigenface (PCA) Similarity

- 21x12 low-res faces, aligned and normalized
- 5-fold cross validation
- ~ 140 unique individuals per subset
- No overlap of individuals between subsets to test generalization performance
- 80% of the data only determines subspace(s)
- 20% of the data is divided into labeled images and query images for nearest-neighbor testing

- Subspace dimensions = d = 20
- Chosen so PCA ~ 80% accurate

- KPCA
- Empirically tweaked Gaussian, polynomial, and sigmoidal kernels
- Gaussian kernel performed the best, so it is used in the comparison

- MAP
- Even split of the 20 subspace dimensions
- ME = MI = d/2 = 10 so that ME + MI = 20

- Even split of the 20 subspace dimensions

Recognition accuracy (percent)

N-Dimensional

Nearest Neighbor

(no subspace)

Recognition accuracy vs subspace dimensionality

Note:data split 50/50 for

training/testing rather

than using CV

- Bayesian matching outperforms all other tested methods and even achieves ≈ 90% accuracy with only 4 projections (2 for each class of variation)
- Bayesian matching is an order of magnitude faster to train than KPCA
- Bayesian superiority with higher resolution images verified in independent US Army FERIT tests
- Wow!
- You should use this

- 50% Accuracy
- Why so bad?
- I implemented all suggested approximations
- Poor data--hand registered
- Too little data

Note:data split 50/50 for

training/testing rather

than using CV

- My data
- His data