Homework 2

Due: Wednesday 3/1/2017 11:59pm on CMS
Up to 5 members in each group. Please include NetID and names on front page.

Problem 1: The Curse of Dimensionality

In this problem, we take a more rigorous look at the curse of dimensionality.

We aim to demonstrate two crucial claims, this time for normally distributed data:

  1. As data dimensionality increases, the distance between data points increases.
  2. As data dimensionality increases, this distance growth overwhelms any degree of closeness between points.

To do this, we will treat the squared Euclidean distance between two arbitrary points in a Gaussian point cloud as a single random variable, and prove certain properties of its expectation and variance. More formally, let:

$$D = ||\vec{x}-\vec{z}||^2 = \sum_{\alpha=1}^d (x_\alpha-z_\alpha)^2$$

where $d$ is the dimensionality of the data, $x_\alpha$ and $z_\alpha$ are i.i.d. for all $\alpha$, and $x_\alpha,z_\alpha \sim N(0,1)$ for all $\alpha$.

It will help to keep in mind the definition of variance: $\mathbb{V}[X]=\mathbb{E}[(X-\mathbb{E}[X])^2]=\mathbb{E}[X^2]-\mathbb{E}[X]^2$.

a)

First, calculate the expectation and variance of $(x_\alpha-z_\alpha)$ for arbitrary $\alpha$, given that $x_\alpha,z_\alpha \sim N(0,1)$. Is this distribution a Gaussian normal distribution?

b)

Now, using (a), calculate the expectation of $D$ and simplify it to a function of $d$:

$$\mu_D^{\,} = \mathbb{E}[D] = \mathbb{E}\left[\sum_{\alpha=1}^d (x_\alpha-z_\alpha)^2\right]$$ This supports our first claim, showing that as dimensionality increases, the expected squared Euclidean distance between normally distributed points also increases.
c)

Using (a) and (b), calculate the variance of $D$ and simplify it to a function of $d$:

$$\sigma_D^2 = \mathbb{V}[D] = \mathbb{V}\left[\sum_{\alpha=1}^d (x_\alpha-z_\alpha)^2\right]$$

You may use the fact that $\mathbb{E}[X^4]=3\sigma^4$ if $X \sim N(0,\sigma^2)$. (Hint: recall that $x_\alpha$ and $z_\alpha$ are i.i.d. for all $\alpha$)

d)

We now know the expectation $\mu_D^{\,}$ and variance $\sigma_D^2$ for our squared-distance random variable $D$.

Chebyshev's Inequality states that for any draw $a$ from distribution $D$,

$$P(|\mu_D^{\,} - a| \ge k\sigma_D^{\,}) \le 1/k^2$$ $$P(|\mu_D^{\,} - a| \ge 2\sigma_D^{\,}) \le .25$$

Thus, we can expect at least $75\%$ of point pairs in a Gaussian point cloud to have squared Euclidean distances within $4\sigma_D^{\,}$ of each other. Using (b) and (c), show that $$\frac{4\sigma_D^{\,}}{\mu_D^{\,}}\rightarrow 0$$ as the dimensionality $d\rightarrow \infty$.

This supports our second claim: as the dimensionality increases, the distance growth between normally distributed points overwhelms the degree to which some of those points are closer together than others.

Problem 2: Perceptron

In this problem, we examine the effect that the ordering of training data points has on the perceptron algorithm.
a)
Construct a data set of labeled vectors such that presenting the vectors to the perceptron in one order causes it to converge more slowly than it does for another order. Show that this is the case.
b)
Let there exist a set of vectors $x_1 ... x_n$ such that the perceptron finds a separating hyperplane after no more than a single pass through the data set, regardless of order of presentation. What is the greatest possible difference between the fastest and slowest convergence times, where time is measured in number of updates to the weight vector?

If we assume instead that the perceptron might take more than one pass over the data set to converge, does this maximum difference increase?

c)
The convergence proof given in class guarantees that the perceptron will find a separating hyperplane as long as one exists, but there could exist a wide range of such hyperplanes. Can some separating hyperplanes be considered "better" than others? What would be one way to measure the "goodness" of a hyperplane (only using the training set)?
d)
Describe how you could utilize the effects of data ordering to improve upon the perceptron algorithm as presented in class - i.e. to find a better hyperplane, as defined by your criteria defined in question c).

Problem 3: Maximum Likelihood Estimation

For each of the following distributions, assume we have observed $n$ draws $X_1 ... X_n$. State the log-likelihood function, and estimate the parameters for each distribution using MLE. You will find that many of these answers are highly intuitive.
a)
Estimate $\lambda$ for the Exponential distribution with PDF: $$f(x;\lambda) = \lambda e^{-\lambda x}$$ where $x$ is non-negative.
b)
Estimate $p$ for the Geometric distribution with PMF: $$f(x;p) = p(1-p)^x$$ where $x$ is non-negative.

How does this relate to your answer for (a)? Why do you think this is?

c)
Estimate $\mu$ and $\sigma^2$ for the Gaussian distribution with PDF: $$f(x;\mu,\sigma^2) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}$$