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Principal Manifolds and Probabilistic Subspaces for Visual Recognition. Baback Moghaddam TPAMI, June 2002. John Galeotti Advanced Perception February 12, 2004. It’s all about subspaces. Traditional subspaces PCA ICA Kernel PCA (& neural network NLPCA) Probabilistic subspaces.

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principal manifolds and probabilistic subspaces for visual recognition

Principal Manifolds and Probabilistic Subspaces for Visual Recognition

Baback Moghaddam

TPAMI, June 2002.

John Galeotti

Advanced Perception

February 12, 2004

it s all about subspaces
It’s all about subspaces
  • Traditional subspaces
    • PCA
    • ICA
    • Kernel PCA (& neural network NLPCA)
  • Probabilistic subspaces
linear pca
Linear PCA
  • We already know this
  • Main properties
    • Approximate reconstruction

x ≈ y

    • Orthonormality of the basis 


    • Decorrelated principal components

E{yiyj}i≠j = 0

linear ica
Linear ICA
  • 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


    • Near factorization of the joint distribution P(y)

P(y)≈ ∏ p(yi)

nonlinear pca nlpca
Nonlinear PCA (NLPCA)
  • AKA principal curves
  • Essentially nonlinear regression
  • Finds a curved subspace passing “through the middle of the data”
nonlinear pca nlpca1
Nonlinear PCA (NLPCA)
  • Main properties
    • Approximate reconstruction

y = f(x)

    • Nonlinear projection

x ≈ g(y)

    • No prior knowledge regarding joint distribution of the components (typical)

P(y) = ?

  • Two main methods
    • Neural network encoder
    • Kernel PCA (KPCA)
nlpca neural network encoder
NLPCA neural network encoder
  • 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): NL, N

    • Covariance is computed with dot-products
    • For economy, make (x) implicit

k(xi,xj) = ( (xi) (xj) )

  • 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
nearest neighbor recognition
Nearest-neighbor recognition
  • 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

probabilistic similarity
Probabilistic similarity
  • Similarity based on probability that ∆ is characteristic of incidental variations
    • ∆ = image-difference vector (N-dim)
    • ΩI = incidental (intrapersonal) variations
    • ΩE = critical (extrapersonal) variations
probabilistic similarity1
Probabilistic similarity
  • 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
subspace density estimation
Subspace density estimation
  • 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
subspace density estimation1
Subspace density estimation
  • 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
efficient similarity computation
Efficient similarity computation
  • 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
efficient similarity computation1
Efficient similarity computation
  • The whitening transform reduces the marginal Gaussian calculations in the principal subspaces F to simple Euclidean distances
  • The denominators are easy to precompute
efficient similarity computation2
Efficient similarity computation
  • 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
similarity comparison
Similarity Comparison

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

Recognition accuracy (percent)


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 
my results
My results
  • 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 results1
My results
  • My data
  • His data