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Principal component analysis

Principal component analysis. Hypothesis: Hebbian synaptic plasticity enables perceptrons to perform principal component analysis. Outline. Variance and covariance Principal components Maximizing parallel variance Minimizing perpendicular variance Neural implementation covariance rule.

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Principal component analysis

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  1. Principal component analysis

  2. Hypothesis: Hebbian synaptic plasticity enables perceptrons to perform principal component analysis

  3. Outline • Variance and covariance • Principal components • Maximizing parallel variance • Minimizing perpendicular variance • Neural implementation • covariance rule

  4. direction of maximum variance in the input space principal eigenvector of the covariance matrix goal: relate these two definitions Principal component

  5. Variance • A random variablefluctuating about its mean value. • Average of the square of the fluctuations.

  6. Covariance • Pair of random variables, each fluctuating about its mean value. • Average of product of fluctuations.

  7. Covariance examples

  8. Covariance matrix • N random variables • NxN symmetric matrix • Diagonal elements are variances

  9. Principal components • eigenvectors with k largest eigenvalues • Now you can calculate them, but what do they mean?

  10. Handwritten digits

  11. Principal component in 2d

  12. One-dimensional projection

  13. Covariance to variance • From the covariance, the variance of any projection can be calculated. • Let w be a unit vector

  14. Maximizing parallel variance • Principal eigenvector of C • the one with the largest eigenvalue.

  15. Orthogonal decomposition

  16. Total variance is conserved • Maximizing parallel variance = Minimizing perpendicular variance

  17. Rubber band computer

  18. Correlation/covariance rule • presynaptic activity x • postsynaptic activity y • Hebbian

  19. Stochastic gradient ascent • Assume data has zero mean • Linear perceptron

  20. Preventing divergence • Bound constraints

  21. Oja’s rule • Converges to principal component • Normalized to unit vector

  22. Multiple principal components • Project out principal component • Find principal component of remaining variance

  23. clustering vs. PCA • Hebb: output x input • binary output • first-order statistics • linear output • second-order statistics

  24. Data vectors • xa means ath data vector • ath column of the matrix X. • Xia means matrix element Xia • ith component of xa • x is a generic data vector • xi means ith component of x

  25. Correlation matrix • Generalization of second moment

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