machine learning - Are Gaussian clusters linearly separable? -

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imagine have 2 gaussian probability distributions in two-dimensions first centered @ (0,1) , second @ (0,-1). (for simplicity, assume have same variance.) can 1 consider clusters of data points sampled these 2 gaussians linearly separable?

intuitively, it's clear boundary separating 2 distributions linear, namely abscissa in our case. however, formal requirement linear separability convex hulls of clusters not overlap. cannot case gaussian-generated clusters since underlying probability distributions pervade of r^2 (albeit negligible probabilities far away mean).

so, gaussian-generated clusters linearly separable? how can 1 reconcile requirement of convex hulls fact straight line conceivable "boundary"? or, perhaps, boundary ceases linear once non-equal variances come in pictures?

the gaussian cluster instances might separable or not. depends on outcome, not on process generating it.

linear separability can defined a existence of plane separating 2 sets of points, such 1 set of points entirely on 1 side of plane, , other set of points entirely on other side of plane.

take specific gaussian distributions. possible generated 2 linearly-seperable sets (either @ abscissa or not). however, probability 1, if variance non-zero, , let processes generate enough points, result not linearly separable.

so, again, question of outcome, not of process.


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