A Short Note on Automatic Algorithm Optimization via Fast Matrix Exponentiation

Alexander Borzunov has written an interesting article about his Python code that uses fast matrix exponentiation to automatically optimize certain algorithms. It's definitely a recommended read.

In his article, Alexander mentions that it's difficult to directly derive a matrix exponentiation algorithm for recursively-defined sequences such as $F_n = \begin{cases} 0, & n = 0\\ 1, & n = 1\\ 1, & n = 2\\ 7(2F_{n-1} + 3F_{n-2})+4F_{n-3}+5n+6, & n \geq 3 \end{cases}$ While it's true that it's not entirely simple, there is a relatively straightforward way to do this that's worth knowing.

The only difficultly is due to the term $$5n+6$$, but we can eliminate it by setting
$$F_n = G_n + an+b$$, then solving for appropriate values of $$a, b$$.

Substituting and grouping terms we have $G_n + an+b = 7(2G_{n-1} + 3G_{n-2})+4G_{n-3} + 39an-68a+39b+5n+6,$ and equating powers of $$n$$ we need to solve the equations \begin{align*} a &= 39a+5,\\ b &= -68a+39b+6. \end{align*} Setting $$a = -\frac{5}{38}, b = -\frac{142}{361}$$ we end up with the homogeneous system $G_n = \begin{cases} \frac{142}{361}, & n = 0\\ \frac{379}{722}, & n = 1\\ \frac{237}{361}, & n = 2\\ 7(2G_{n-1} + 3G_{n-2})+4G_{n-3}, & n \geq 3 \end{cases}$ This is now easily cast into matrix exponentiation form using the standard method.

A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

Notes on Setting up a Titan V under Ubuntu 17.04

I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under Ubuntu 17.04.

Old install:
Ubuntu 16.04
EVGA GeForce GTX Titan SuperClocked 6GB
2TB Seagate NAS HDD

New install:
Ubuntu 17.04
Titan V 12GB
/ partition on a 250GB Samsung 840 Pro SSD (had an extra around)
/home partition on a new 1TB Crucial MX500 SSD
New WD Blue 4TB HDD