Dimensionality reduction
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1 Dimensionality Reduction PCA continued Machine Learning CSE446 Carlos Guestrin University of Washington May 22, 2013 Carlos Guestrin Dimensionality reduction n Input data may have thousands or millions of dimensions! e.g., text data has n Dimensionality reduction: represent data with fewer dimensions easier learning fewer parameters visualization hard to visualize more than 3D or 4D discover intrinsic dimensionality of data n high dimensional data that is truly lower dimensional Carlos Guestrin
2 Lower dimensional projections n Rather than picking a subset of the features, we can new features that are combinations of existing features n Let s see this in the unsupervised setting just X, but no Y Carlos Guestrin Linear projection and reconstruction x 2 project into 1-dimension z 1 x 1 reconstruction: only know z 1, what was (x 1,x 2 ) Carlos Guestrin
3 Principal component analysis basic idea n Project d-dimensional data into k-dimensional space while preserving information: e.g., project space of words into 3-dimensions e.g., project 3-d into 2-d n Choose projection with minimum reconstruction error Carlos Guestrin Linear projections, a review n Project a point into a (lower dimensional) space: point: x = (x 1,,x d ) select a basis set of basis vectors (u 1,,u k ) n we consider orthonormal basis: u i u i =1, and u i u j =0 for i j select a center x, defines offset of space best coordinates in lower dimensional space defined by dot-products: (z 1,,z k ), z i = (x-x) u i n minimum squared error Carlos Guestrin
4 PCA finds projection that minimizes reconstruction error n Given m data points: x i = (x 1i,,x di ), i=1 N n Will represent each point as a projection: where: and N N n PCA: Given k<d, find (u 1,,u k ) minimizing reconstruction error: N x 2 x 1 Carlos Guestrin Understanding the reconstruction error n Note that x i can be represented exactly by d-dimensional projection: d Given k<d, find (u 1,,u k ) minimizing reconstruction error: N n Rewriting error: Carlos Guestrin
5 Reconstruction error and covariance matrix N d N N Carlos Guestrin Minimizing reconstruction error and eigen vectors n Minimizing reconstruction error equivalent to picking (ordered) orthonormal basis (u 1,,u d ) minimizing: n Eigen vector: N d n Minimizing reconstruction error equivalent to picking (u k+1,,u d ) to be eigen vectors with smallest eigen values Carlos Guestrin
6 Basic PCA algoritm n Start from m by n data matrix X n Recenter: subtract mean from each row of X X c X X n Compute covariance matrix: Σ 1/N X c T X c n Find eigen vectors and values of Σ n Principal components: k eigen vectors with highest eigen values Carlos Guestrin PCA example Carlos Guestrin
7 PCA example reconstruction only used first principal component Carlos Guestrin Eigenfaces [Turk, Pentland 91] n Input images: n Principal components: Carlos Guestrin
8 Eigenfaces reconstruction n Each image corresponds to adding 8 principal components: Carlos Guestrin Scaling up n Covariance matrix can be really big! Σ is d by d Say, only features finding eigenvectors is very slow n Use singular value decomposition (SVD) finds to k eigenvectors great implementations available, e.g., R or Matlab svd Carlos Guestrin
9 SVD n Write X = W S V T X data matrix, one row per datapoint W weight matrix, one row per datapoint coordinate of x i in eigenspace S singular value matrix, diagonal matrix n in our setting each entry is eigenvalue λ j V T singular vector matrix n in our setting each row is eigenvector v j Carlos Guestrin PCA using SVD algoritm n Start from m by n data matrix X n Recenter: subtract mean from each row of X X c X X n Call SVD algorithm on X c ask for k singular vectors n Principal components: k singular vectors with highest singular values (rows of V T ) Coefficients become: Carlos Guestrin
10 What you need to know n Dimensionality reduction why and when it s important n Simple feature selection n Principal component analysis minimizing reconstruction error relationship to covariance matrix and eigenvectors using SVD Carlos Guestrin Bayes optimal classifier Naïve Bayes Machine Learning CSE446 Carlos Guestrin University of Washington May 22, 2013 Carlos Guestrin
11 Classification n Learn: h:x Y X features Y target classes n Suppose you know P(Y X) exactly, how should you classify? Bayes optimal classifier: Carlos Guestrin Bayes Rule Which is shorthand for: Carlos Guestrin
12 How hard is it to learn the optimal classifier? n Data = n How do we represent these? How many parameters? Prior, P(Y): n Suppose Y is composed of k classes Likelihood, P(X Y): n Suppose X is composed of d binary features n Complex model! High variance with limited data!!! Carlos Guestrin Conditional Independence n X is conditionally independent of Y given Z, if the probability distribution governing X is independent of the value of Y, given the value of Z n e.g., n Equivalent to: Carlos Guestrin
13 What if features are independent? n Predict Thunder n From two conditionally Independent features Lightening Rain Carlos Guestrin The Naïve Bayes assumption n Naïve Bayes assumption: Features are independent given class: More generally: d n How many parameters now? n Suppose X is composed of d binary features Carlos Guestrin
14 The Naïve Bayes Classifier n Given: Prior P(Y) d conditionally independent features X given the class Y For each X i, we have likelihood P(X i Y) n Decision rule: d n If assumption holds, NB is optimal classifier! Carlos Guestrin MLE for the parameters of NB n Given dataset Count(A=a,B=b) == number of examples where A=a and B=b n MLE for NB, simply: Prior: P(Y=y) = Likelihood: P(X i =x i Y=y) = Carlos Guestrin
15 Subtleties of NB classifier 1 Violating the NB assumption n Usually, features are not conditionally independent: d n Actual probabilities P(Y X) often biased towards 0 or 1 n Nonetheless, NB is the single most used classifier out there NB often performs well, even when assumption is violated [Domingos & Pazzani 96] discuss some conditions for good performance Carlos Guestrin Subtleties of NB classifier 2 Insufficient training data n What if you never see a training instance where X 1 =a when Y=b? e.g., Y={Spam }, X 1 ={ Enlargement } P(X 1 =a Y=b) = 0 n Thus, no matter what the values X 2,,X d take: P(Y=b X 1 =a,x 2,,X d ) = 0 n Solution : smoothing Add fake counts, usually uniformly distributed Equivalent to Bayesian Learning Carlos Guestrin
16 Text classification n Classify s Y = {Spam,NotSpam} n Classify news articles Y = {what is the topic of the article?} n Classify webpages Y = {Student, professor, project, } n What about the features X? The text! Carlos Guestrin Features X are entire document X i for i th word in article Carlos Guestrin
17 NB for Text classification n P(X Y) is huge!!! Article at least 1000 words, X={X 1,,X 1000 } X i represents i th word in document, i.e., the domain of X i is entire vocabulary, e.g., Webster Dictionary (or more), 10,000 words, etc. n NB assumption helps a lot!!! P(X i =x i Y=y) is just the probability of observing word x i in a document on topic y Carlos Guestrin Bag of words model n Typical additional assumption Position in document doesn t matter: P(X i =x i Y=y) = P(X k =x i Y=y) Bag of words model order of words on the page ignored Sounds really silly, but often works very well! When the lecture is over, remember to wake up the person sitting next to you in the lecture room. Carlos Guestrin
18 Bag of words model n Typical additional assumption Position in document doesn t matter: P(X i =x i Y=y) = P(X k =x i Y=y) Bag of words model order of words on the page ignored Sounds really silly, but often works very well! in is lecture lecture next over person remember room sitting the the the to to up wake when you Carlos Guestrin Bag of Words Approach aardvark 0 about 2 all 2 Africa 1 apple 0 anxious 0... gas 1... oil 1 Zaire 0 Carlos Guestrin
19 NB with Bag of Words for text classification n Learning phase: Prior P(Y) n Count how many documents you have from each topic (+ prior) P(X i Y) n For each topic, count how many times you saw word in documents of this topic (+ prior) n Test phase: For each document n Use naïve Bayes decision rule Carlos Guestrin Twenty News Groups results Carlos Guestrin
20 Learning curve for Twenty News Groups Carlos Guestrin
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