Principal Component Analysis (PCA)

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Transformation matrix for PCA • PCA transforms the point x (original coordinates) into the point c (new coordinates) by subtracting the mean from x (z = x – m) and multiplying the result by V T ⎡← ⎢ ⎢ ← ⎢ ⎢ ⎣← V v1 v2 M v n T →⎤ → ⎥ ⎥ ⎥ ⎥ →⎦ ⎡↑⎤ ⎢ ⎥ ⎢z ⎥ ⎢ ⎥ ⎣↓⎦ • this is a rotation because V T is an orthonormal matrix • this is a projection of z onto PC axes, because v 1 • z is projection of z onto PC1 axis, v 2 • z is projection onto PC2 axis, etc. z Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 14 = ⎡v ⎢ ⎢ v ⎢ ⎢ ⎣v 1 2 n c M ⋅ ⋅ ⋅ z⎤ z ⎥ ⎥ ⎥ ⎥ z⎦
• PCA expresses the mean-subtracted point, z = x – m, as a weighted sum of the eigenvectors vi . To see this, start with: T ⎡← ⎢ ⎢ ← ⎢ ⎢ ⎣← V v v 1 2 M v D →⎤ → ⎥ ⎥ ⎥ ⎥ →⎦ • multiply both sides of the equation from the left by V: VV T z = Vc | exploit that V is orthonormal z = Vc | i.e. we can easily reconstruct z from c z ⎡↑⎤ ⎢ ⎥ ⎢z ⎥ ⎢ ⎥ ⎣↓⎦ = = ⎡ ↑ ⎢ ⎢v ⎢ ⎣ ↓ 1 ↑ v 2 ↓ V c ⎡c1 ⎤ ↑ ⎤ ⎢ ⎥ ⎥ ⎢ c2 v ⎥ n ⎥ ⎢ M ⎥ ↓ ⎥ ⎦ ⎢ ⎥ ⎣cn ⎦ Jochen Triesch, UC San Diego, http://cogsci.ucsd.edu/~triesch 15 z ⎡↑⎤ ⎢ ⎥ ⎢z ⎥ ⎢ ⎥ ⎣↓⎦ = c = ⎡ ↑ ⎤ ⎢ ⎥ ⎢v1 ⎥ ⎢ ⎥ ⎣ ↓ ⎦ c ⎡ c ⎢ ⎢ c ⎢ M ⎢ ⎣c + 1 2 D ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ c ⎡ ↑ ⎢ ⎢v ⎢ ⎣ ↓ L 1 2 2 ⎤ ⎥ ⎥ ⎥ ⎦ + L + c n ⎡ ↑ ⎢ ⎢v ⎢ ⎣ ↓ n ⎤ ⎥ ⎥ ⎥ ⎦
Page 1 and 2: Principal Component Analysis (PCA)
Page 3 and 4: Example: • consider projecting x
Page 5 and 6: • continuing with some linear alg
Page 7 and 8: Notes: • if v is an eigenvector o
Page 9 and 10: What are the variances of the PCA p
Page 11 and 12: PCA for multinomial Gaussian • We
Page 13: Transformation matrix for PCA • L
Page 17 and 18: • Equivalent view: To compress n-
Page 19 and 20: Application: PCA on face images •
Page 21 and 22: The mean face Jochen Triesch, UC Sa
Page 23 and 24: The transpose trick The eigenvector
Page 25 and 26: Eigenfaces • Eigenfaces (the prin
Page 27 and 28: PCA for low-dimensional low dimensi
Page 29 and 30: The mean face Jochen Triesch, UC Sa
Page 31 and 32: Approximating a face using the firs
Page 33 and 34: Approximating a face using all 97 e
Page 35: Eigenfaces in 3D • Blanz and Vett

Principal Component Analysis (PCA)

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