## Derivatives and matrix-vector products (part I)

December 26, 2017

We don’t usually think of linear algebra being compatible with derivatives, but it very useful to be able to know the derivative of certain elements in order to adjust them—basically using gradient-descent.

We might therefore ask ourselves questions like, how do I compute the derivative of a scalar product relative to one of the vector elements? Of a matrix-vector product? Through a matrix inverse? Through a product of matrices? Let’s answer the first two questions for now.

## Taylor Series

December 19, 2017

A Taylor series for a function $f(x)$ around $x_0$ that is $n$ times differentiable is given by

$\displaystyle f(x) \approx f(x_0)+f'(x_0)(x-x_0)+\frac{f''(x_0)}{2}(x-x_0)^2+\frac{f'''(x_0)}{6}(x-x_0)^3+\cdots$

or

$\displaystyle f(x) \approx \sum_{i=0}^{n} \frac{f^{(i)}(x_0)}{i!}(x-x_0)^i$,

where $f^{(i)}(x)$ is the $i$th derivative of $f$ at $x$.

Have you ever wondered where the coefficients in a Taylor series come from? Well, let’s see!

## Building a large text corpus (Part I)

December 12, 2017

Getting good text data for language-model training isn’t as easy as it sounds. First, you have to find a large corpus. Second, you must clean it up!

## Reinventing the Wheel (or not)

December 5, 2017

About a week ago, some dude drops on IRC that he’s beat memcpy “by a lot”. That’d be interesting, except that we couldn’t get neither code nor test methodology out of him. But, how hard can making a better memcpy be? Turns out, harder than you think!

If you think this is a typical case of “reinventing the wheel”, I mostly agree with you. But while reinventing will be hard, can improvements be made?