CPSC 340: Machine Learning and Data Mining. More PCA Fall 2016
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1 CPSC 340: Machine Learning and Data Mining More PCA Fall 2016
2 A2/Midterm: Admin Grades/solutions posted. Midterms can be viewed during office hours. Assignment 4: Due Monday. Extra office hours: Thursdays from 4:30-5:30 in ICICS X836.
3 1. Decision trees 2. Naïve Bayes classification 3. Ordinary least squares regression 4. Logistic regression 5. Support vector machines 6. Ensemble methods 7. Clustering algorithms 8. Principal component analysis 9. Singular value decomposition 10.Independent component analysis
4 Last Week: Principal Component Analysis (PCA) PCA is a linear model for unsupervised learning. Represents features as linear combination of latent factors: But we re learning the latent factors W and latent features z i. Can also be viewed as an approximate matrix factorization:
5 Last Week: Principal Component Analysis (PCA) PCA is a linear model for unsupervised learning. Represents features as linear combination of latent factors: Uses: dimensionality reduction, visualization, factor discovery.
6 Last Week: Principal Component Analysis (PCA) PCA is a linear model for unsupervised learning. Represents features as linear combination of latent factors: Uses: dimensionality reduction, visualization, factor discovery.
7 Maximizing Variance vs. Minimizing Error Our synthesis view that PCA minimizes approximation error. Makes connection to k-means and our tricks for linear regression. Classic analysis view: PCA maximizes variance in z i space. You pick W to explain as much variance in the data as possible.
8 Choosing Number of Latent Factors Common approach to choosing k : Compute error with k=0: Compare to error with non-zero k : Gives a number between 0 and 1, giving how much variance remains. If you want to explain 90% of variance, choose smallest k where ratio is < 0.10.
9 PCA Computation The PCA objective with general d and k : 3 common ways to solve this problem: Singular value decomposition (SVD) classic non-iterative approach. Alternating between updating W and updating Z. Stochastic gradient: gradient descent based on random i and j. (Or just plain gradient descent). Not convex, all these methods work with random initialization.
10 PCA Computation The PCA objective with general d and k : Where we ve subtracted mean µ j from each feature. At test time, to find optimal Z given W for new data:
11 We have the scaling problem: PCA Non-Uniqueness
12 PCA Non-Uniqueness But with multiple PCs, we have new problems: Factors could be non-orthogonal (components interfere with each other): You can still rotate the factors and also have label switching. A fix is to fit the PCs sequentially (can be done with SVD approach):
13 Basis, Orthogonality, Sequential Fitting
14 Basis, Orthogonality, Sequential Fitting
15 Basis, Orthogonality, Sequential Fitting
16 Basis, Orthogonality, Sequential Fitting
17 Do need all this math?
18 Colour Opponency in the Human Eye Classic model of the eye is with 4 photoreceptors: Rods (more sensitive to brightness). L-Cones (most sensitive to red). M-Cones (most sensitive to green). S-Cones (most sensitive to blue). Two problems with this system: Correlation between receptors (not orthogonal). Particularly between red/green. We have 4 receptors for 3 colours.
19 Colour Opponency in the Human Eye Bipolar and ganglion cells seem to code using opponent colors : 3-variable orthogonal basis: This is similar to PCA (d = 4, k = 3)
20 Colour Opponency Representation
21 Application: Face Detection Consider problem of face detection: Classic methods use eigenfaces as basis: PCA applied to images of faces.
22 Eigenfaces Collect a bunch of images of faces under different conditions:
23 Eigenfaces
24 Eigenfaces
25 Eigenfaces
26 Eigenfaces
27 Eigenfaces
28 Eigenfaces
29 Eigenfaces
30 Eigenfaces
31 Eigenfaces
32 Eigenfaces
33 Eigenfaces
34 Eigenfaces
35 Eigenfaces
36 Representing Faces But how should we represent faces? K-means: Grandmother cell : one neuron = one face. Almost certainly not true: too few neurons. PCA: Distributed representation. Coded by pattern of group of neurons. Can represent more concepts. But PCA uses positive/negative cancelling parts. Non-negative matrix factorization (NMF): Latent-factor where W and Z are non-negative. Example of sparse coding : Coded by small number of neurons in group. NMF makes object out of of parts.
37 Why sparse coding? Representing Faces Parts are intuitive, and brains seem to use sparse representation. Energy efficiency if using sparse code. Increase number of concepts you can memorize? Some evidence in fruit fly olfactory system.
38 Summary Analysis view of PCA is that it maximizes variance. We can choose k to explain x% of the variance in the data. Orthogonal basis and sequential fitting of PCs: Leads to non-redundant PCs with unique directions. Biological motivation for orthogonal and/or sparse latent factors. Non-negative matrix factorization leads to sparse LFM. Next time: modifying PCA so it splits faces into eyes, mouths, etc.
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