Spectral Algorithms by Ravindran Kannan, Santosh Vempala

By Ravindran Kannan, Santosh Vempala

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Let C be the m × n matrix with C(i) = µr for all i ∈ Tr . We will now state the assumptions under which we will prove that spectral clustering works. 2 Clustering Based on Deterministic Assumptions 195 holds, then the assertions/algorithms work as claimed. ] We first assume Assumption 0 : A − C = ∆ ≤ O(σk (C)/ log n). ] We note that A − C 2 can be viewed as the maximum total distance squared in any direction of the points from their respective centers. So ∆ being small is the same as saying the displacements of A(i) from their respective centers are not “biased” toward any direction, but sort of spread out.

In both cases, this transformation makes the variance of the mixture 1 in every direction, so the principal components give us no insight into the intermean direction. Consider next the effect of the reweighting on the mean of the mixture. For the case of equal mixing weights, symmetry assures that the mean does not shift at all. For imbalanced weights, however, the heavier component, which lies closer to the origin will become heavier still. Thus, the reweighted mean shifts toward the mean of the heavier component, allowing us to detect the intermean direction.

Rescale) Use samples to compute an affine transformation W that makes the distribution nearly isotropic (mean zero, identity covariance matrix). 2. (Reweight) For each of m1 samples, compute a 2 weight e− x /α . 3. (Find Separating Direction) Find the mean √ of the reweighted data µ ˆ. If µ ˆ > w/(32α) (where α > n/w), let h = µ ˆ. Otherwise, find the ˆ covariance matrix M of the reweighted points and let h be its top principal component. 4. (Classify) Project m2 sample points to h and classify the projection based on distances.

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