Week 3 Check-in Quiz

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Please complete the following quiz by class on Wed. 9/30/2020.

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The maximum cardinality set of mutually orthogonal vectors in d dimensional space has size:

1 point

d-1

2^d

Clear selection

As a function of d, the volume of the unit cube in d dimensions grows more slowly than that volume of the unit ball.

1 point

True

False

Clear selection

Let g be a d dimensional vector containing i.i.d. random Gaussian entries. Let z be the random vector z = g/||g||. z is identically distributed to which of the following (check all that apply):

1 point

A random point on the sphere in d dimensions (surface of the unit ball in d dimensions).

A vector with entries drawn uniformly at random from [-1,1].

A vector drawn uniformly from the d dimensional cube.

x/||x||, where x is a vector drawn uniformly from the d dimensional unit ball.

Let x and y be d dimensional vectors whose entries are all i.i.d. Gaussians with mean 0 and variance 1. What is the value of V = E[<x,y>^2], where <x,y> denotes the inner product and E the expectation.

2 points

Suppose we have n vectors x_1, ..., x_n. We select a properly scale random JL matrix Pi with k rows. How large does k need to be so that, with probability (1-delta), (1-epsilon)||x_i|| <= ||Pi x_i|| <= (1+epsilon) ||x_i||, for all i.

1 point

k = O(log(1/delta)/epsilon^2)

k = O(log(n)/epsilon^2)

k = O(log(n/delta)/epsilon^2)

k = O(log(n)*log(1/delta)/\epsilon^2)

Clear selection

Let Pi be a matrix whose entries are drawn uniformly from the set {0,1}. I.e., they are binary random variables. Do you think that it would be possible to prove that Pi satisfies the JL Lemma? Why or why not?

2 points

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