Unsupervised Kernel Dimension Reduction Supplemental Material
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1 Unsupervised Kernel Dimension Reduction Supplemental Material Meihong Wang Dept. of Computer Science U. of Southern California Los Angeles, CA Fei Sha Dept. of Computer Science U. of Southern California Los Angeles, CA Michael I. Jordan Dept. of Statistics U. of California Berkeley, CA Relationship between ĴYY X and ĴXY and Proof of Proposition We start by noting that conditional independence X Y B X does not necessarily imply the correlation between B X and Y is maximized. To see this, let X be a Gaussian random vairable with zero mean and diagonal covariance matrix. Assume B is an identity matrix and Y = X = (B X) (elementwise square for a vectorial X). The conditional independence is obviously satisfied yet the correlation between B X and Y is zero. This observation is yet another example showing the limitation of Spearson s correlation measures, which detect only linear dependence between random variables. In the following, we show that when measured in the RKHS, the two measures ĴY Y X and ĴXY are equivalent. Assume we use Gaussian RBF kernel for both ĴYY X(B X,Y) and ĴXY(B X,Y): K(x i,x ) = exp ( x i x /σ N) where σn is the bandwidth. It should be chosen to match roughly the scale of the data. While ĴXY(B X,Y) depends only on σ N, Ĵ YY X (B X,Y) depends on σ N and ǫ N, which need to be adusted in empirical studies. Since both parameters control the smoothness of the kernel matrix K B X, it is sensible to adust the relative scale of the two parameters. Particularly, the following statement establishes the link between the two measures: Proposition. Let N + and ǫ N. Additionally, assume the samples are distributed uniformly on the unit sphere. If σ N ǫ N, then up to a constant, Ĵ YY X (B X,Y) c N ǫ NĴXY(B X,Y) () Therefore, It is equivalent to minimizing ĴYY X(B X,Y) and maximizing ĴXY(B X,Y). We sketch the proof in the following. The proof consists of two main steps. In the first step, we bound the largest eigenvalue λ of K B X from above. In the second step, we show that applying the bound allows us to approximate the inverse (G B X +Nǫ N I N ) with K B X. Bound on λ. With Gaussian RBF kernel, the elements of the kernel matrix K B X is strictly positive. By Perron-Frobenius theorem, the spectral radius is λ and is bound by λ max K B i X(i,), () namely, the maximum of the row sums. The (i,)-th element of the kernel matrix is given by ( ) K B X(i,) = exp m B m x i B m x /σ N ()
2 whereb m is them-th columnofthe matrixb. Since eachcolumncontributesnonnegatively to the exponent, it is obvious λ max exp ( B m x i B m x /σ ) N () i for any m. Furthermore, since x i are assumed to be uniformly distributed on the unit sphere, we choose an arbitrary coordinate system such that B m = [ ]. This gives rise to λ exp ( (x i x ) /σn). () Note that we have removed the max i operation based on the symmetry argument. The first element x i of x i can be set as without loss of generality. Moreover, the first elements x of x can be parameterized as cosθ where θ is uniformly distributed between and π. This leads to λ exp ( ( cosθ ) /σn). () When N +, the right-hand-size tends to an integral λ N π e ( cos θ) σ N dθ. () The integral does not have a analytic closed-form. Our intention is to approximate it with a function of σ N. In particular, σ N needs to tend to zero when N +. Therefore, our next step is to identify an asymptotic expansion in σ N. After a few variable substitutions ( cosθ = cos θ/, then cosθ/ = t, the integral of eq. () is transformed (omitting constants) to I(σ N ) = t e t /σ N dt () ApplyingWatsonLemma[]tothe aboveintegralas/σn +, weobtainthe asymptotic expansion up to (and including) the first-order of / t, I(σ N ) / Γ(/) σ N /+O(σ / N ) (σ N ). () This gives us a bound on λ λ c N σ N () where c is a constant. Approximate ĴY Y X(B X,Y). We apply the Woodbury matrix inversion lemma to the independence measure. Let δ N = Nǫ N, we have Ĵ Y Y X (B X,Y) () = Trace [ G Y (HK B XH +δ N I N ) ] () = Trace [ { G Y δ N I N δ N H(K B X +Hδ N H) Hδ }] N () = δ N Trace[ G Y (K B X +δ N H) ] +const () where we have used the identities HH = H and HG Y H = G Y. We have also assumed that K B X is invertible. This is reasonable as we are using Gaussian RBF kernels. If we further assume that B X yields different B x, B x,... and B x N, then the kernel matrix is full ranked and invertible. We consider the limiting case when N +. We have shown in above that the largest eigenvalueofk B X isboundfromabovebyn σ N. Thismeansthatthesmallesteigenvalue /λ ofk B X isbound frombelowby/(n σ N ). Notethatthe largesteigenvalueofδ N H is /(Nǫ N ). Therefore, if /(N σ N ) /(Nǫ N ), ()
3 we can approximate the inversion with (K B X +δ N H) (K B X ) = K B X. () The condition eq. () corresponds to the condition σ N ǫ N in the Proposition, which leads to (after substituting eq. () into eq.()), Ĵ Y Y X (B X,Y) = δ NĴXY(B X,Y)+const () Relation between t-sne and UKDR t-sne aims to preserve local conditional probability structure computed in the original space of X []. The structure is encoded as the random walk probability of p i = P(x i x ) = exp{ x i x /σ i } i exp{ x i x /σ i }. () The low dimensional coordinates Z are estimated such that the structure computed in the low dimensional space q i C(z i,z ) = + z i z () has the smallest KL divergence: z = argminkl( P Q) where Q s elements are q i. P = (P + P )/ is a symmetric version of P. Minimizing the KL divergence is equivalent to maximize the conditional entropy P i logq i = P i logc(z i,z ) log C(z i,z ) = Trace[P logc] log C () i i i where the logarithm of the matrix C is taken element-wisely. There is a strong analogy of this obective function to ĴXY(B X,X) with random walk kernel over X and Cauchy kernel over B X, Ĵ XY (B X,X) = Trace[PP HCH] Trace[PP C] /N C () where N is the number of data points. The approximation is valid if we assume data are well-clustered and the distances between data in the same clusters are roughly the same. We gain further insights about t-sne by comparing the two obective functions of eq. () and eq. (). Apart from using different yet related kernels (tsne s kernel is not positive semidefinite), both obective functions have two terms which function similarly. The trace terms try to match the similarities between low dimensional coordinates (encoded by C and log C respectively) with the similarities in the high dimensional coordinates. The normalization terms, to be minimized, try to push data points far away from each other. Effect of sparsity in RSN UKDR For nonlinear unsupervised kernel dimension reduction, using random and sparse features (RSF) extracted from X has significant computational advantage than using transformed X by radial basis networks. RSF features are sparse and easy to compute. In particular, the dimensionality of RSF features depends on the dimensionality of the data while RBF transformation depends on the number of data points in the training data set. Furthermore, in computing RSF features, the bias constant b can be used to yield sparser feature vectors. In Fig., we investigate the effect of the sparsity level on embeddings. The setup is similar to what is used for generating the embedding in Fig.(i)(in the main paper) where b is and leads to a feature vector with sparsity level of %. We change b to, and, obtaining sparsity levels of %, % and % in Fig.. Through visual inspection, it is clear that a sparsity level of % does not bring detrimental effects, while the higher % starts to show fragmented clustering of data. Therefore, we deem RSF features as a viable option in handling high dimension data for nonlinear UKDR.
4 (a) Sparsity: % (b) Sparsity: % (c) Sparsity: % Figure : D Embedding of USPS- dataset, computed using random sparse features with various degrees of sparsity. The higher the sparsity is, the sparser the features are. Comparison to other dimensionality reduction method We compare our UKDR method to other dimensionality reduction methods, including SNE [], t-sne, MUSHIC (Colored Maximum Variance Unfolding) [] and PCA. The first dataset is USPS, which contains digit,,,, and. The second dataset is UPSPS, which contains images of digits. The third dataset is discussion postings from Newsgroups with posting in each group. It is a subset of the Newsgroups used in [].. USPS Fig. shows the D embedding results on dataset USPS. (a) UKDR (Nonparametric embedding) Figure : D embedding results on USPS by our method UKDR and other methods. UKDR performs much better than other methods. Results on USPS Fig. shows the D embedding results on dataset USPS.. Results on Newsgroup Fig. shows the D embedding results on dataset Newsgroups
5 (a) RBN UKDR Figure : D embedding results on USPS by our method UKDR and other methods. Only UKDR and t-sne separate all classes reasonably well Group Group Group Group Group Group Group Group Group Group Group Group Group Group Group (a) UKDR Group Group Group Group Group Group x Group Group Group Group x Figure : D embedding results on Newsgroupsby our method UKDR and other methods. UKDR, t-sne and MUHSIC works well. References [] E. T. Copson. Asymptotic expansions. Cambridge University Press,. [] L. van der Maaten and G. Hinton. Visualizing data using t-sne. Journal of Machine Learning Research, :,. [] G. Hinton and S. Roweis. Stochastic neighbor embedding. Advances in Neural Information Processing Systems, pages,. [] L. Song, A. Smola, K. Borgwardt, and A. Gretton. Colored maximum variance unfolding. Advances in Neural Information Processing Systems, pages,.
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