From Mohammed J. Zaki

Dmcourse: Assign2

Assignment 2

Due Date: Fri 28th Sep, Before Midnight

Part I and II have to be done by both CSCI4390 and CSCI6390. There is an extra question for CSCI6390. The bonus question can be attempted by both CSCI4390 and CSCI6390.

(Part I) Diagonals in High Dimensions (50 points total)

Your goal is the compute the probability mass function for the random variable \(X\) that represents the angle (in degrees) between any two diagonals in high dimensions.

Assume that there are \(d\) primary dimensions (the standard axes in cartesian coordinates), with each of them ranging from -1 to 1. There are \(2^{d}\) additional half-diagonals in this space, one for each corner of the \(d\)-dimensional hypercube.

half-diagonals in the d-dimensional hypercube, and computes the angle between them (in degrees).

values of \(d\), as follows \(d=\{10,100,1000\}\). Recall that PMF is simply the plot of the angle versus the probability of observing that angle in the sample of \(n\) points for a given value of \(d\). What is the min, max, value range, mean and variance of \(X\) for each value of \(d\)?

What would you have expected to have happened analytically? In other words, derive formulas for what should happen to angle between half-diagonals as \(d \to \infty\). Does the PMF conform to this trend? Explain why? or why not?

What is the expected number of occurrences of a given angle \(\theta\) between two half-diagonals, as a function of d (the dimensionality) and n (the sample size)?

(Part II) Kernel Principal Components Analysis (50 points total)

compute the projected points along the first two kernel PCs, and create a scatter plot of the projected points.

What to submit

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