In projection-based clustering, we note that, if the data is well separated into clusters with means...

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In projection-based clustering, we note that, if the data is well separated into clusters with means μ₁,...,K, then the top K eigenvectors of the data covariance matrix, say (v₁,...,UK), tend to align with the Span(μ₁,...,K). It follows that PCA will approximately preserve the distance between cluster means. This intuition (and choice of projection) implicitly assumes that the Euclidean metric is the right way of measuring distance for our particular data. Where is that assumption coming in? What would you do otherwise, i.e., if it turns out that a different notion of distance dist(Xį, X₁) were more appropriate, would you prefer some other transformation over PCA? Following the notation from the previous problem, let V = [v₁,..., vk] € RdxK, the matrix of top principal directions ||v₁|| = 1. Consider two possible transformations of the data: (i) Xį ← VTX₁, and (ii) Xį ← VVTX₁ Would the clusters learnt from the data Ã; differ from those learnt from the data ¿? If we were to run Lloyd's on the transformed data (i) and separately on the transformed data (ii), would one be quicker to execute than the other, i.e., is there a difference in runtime? In projection-based clustering, we note that, if the data is well separated into clusters with means μ₁,...,K, then the top K eigenvectors of the data covariance matrix, say (v₁,...,UK), tend to align with the Span(μ₁,...,K). It follows that PCA will approximately preserve the distance between cluster means. This intuition (and choice of projection) implicitly assumes that the Euclidean metric is the right way of measuring distance for our particular data. Where is that assumption coming in? What would you do otherwise, i.e., if it turns out that a different notion of distance dist(Xį, X₁) were more appropriate, would you prefer some other transformation over PCA? Following the notation from the previous problem, let V = [v₁,..., vk] € RdxK, the matrix of top principal directions ||v₁|| = 1. Consider two possible transformations of the data: (i) Xį ← VTX₁, and (ii) Xį ← VVTX₁ Would the clusters learnt from the data Ã; differ from those learnt from the data ¿? If we were to run Lloyd's on the transformed data (i) and separately on the transformed data (ii), would one be quicker to execute than the other, i.e., is there a difference in runtime?

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Youre right that PCA makes the implicit assumption that the Euclidean distance is the appropriate me... View the full answer