# June 2021

# Linear Algebra for Machine Learning

The TensorFlow channel on YouTube recently uploaded a video I made on some elementary ideas from linear algebra and how they're used in machine learning (ML). It's a very nontechnical introduction — more of a bird's-eye view of some basic concepts and standard applications — with the simple goal of whetting the viewer's appetite to learn more.

I've decided to share it here, too, in case it may be of interest to anyone!

I imagine the content here might be helpful for undergraduate students who are in their first exposure to linear algebra and/or to ML, or for anyone else who's new to the topic and wants to get an idea for what it is and some ways it's used.

The video covers three basic concepts — **vectors** and **matrix factorizations** and **eigenvectors/eigenvalues **— and explains a few ways these concepts arise in ML — namely, as **data representations**, to find **vector embeddings**, and for **dimensionality reduction **techniques, respectively.

Enjoy!

# Warming Up to Enriched Category Theory, Part 2

Let's jump right in to where we left off in part 1 of our warm-up to enriched category theory. If you'll recall from last time, we saw that the set of **truth values** $\{0, 1\}$ and the **unit interval** $[0,1]$ and the **nonnegative extended reals** $[0,\infty]$ were not just *sets* but actually preorders* *and hence categories. We also hinted at the idea that a "category enriched over" one of these preorders (whatever *that *means — we hadn't defined it yet!) looks something like a collection of objects $X,Y,\ldots$ where there is at most one arrow between any pair $X$ and $Y$, and where that arrow can further be "decorated with" —or simply replaced by* — *a number from one of those three exemplary preorders.

With that background in mind, my goal in today's article is to say exactly what a **category enriched over** **a preorder **is. The formal definition — and the intuition behind it — will then pave the way for the notion of a category enriched over an *arbitrary* (and sufficiently nice) category, not just a preorder.

En route to this goal, it will help to make a couple of opening remarks.

## Two things to think about.

First, take a closer look at the picture on the right. I've written "$\text{hom}(X,Y)$" in quotation marks because the notation $\text{hom}(-,-)$ is often used for a *set *of morphisms in ordinary category theory. But the point of this discussion is that we're not just interested in sets! So we should use better notation: let's refer to the number associated to a pair of objects $XY$ and $Y$ as $\mathcal{C}(X,Y)$, where the letter "$\mathcal{C}$" reminds us there's an (enriched) $\mathcal{C}$ategory being investigated.

Second, for the theory to work out nicely, it turns out that preorders need a little *more *added to them.

# Warming Up to Enriched Category Theory, Part 1

*It's no secret that I like category theory.* It's a common theme on this blog, and it provides a nice lens through which to view old ideas in new ways — and to view *new *ideas in new ways! Speaking of new ideas, my coauthors and I are planning to upload a new paper on the arXiv soon. I've really enjoyed the work and can't wait to share it with you. But first, you'll have to know a little something about **enriched category theory**. (And before *that*, you'll have to know something about ordinary category theory... here's an intro!) So that's what I'd like to introduce today.

A warm up, if you will.

## What is enriched category theory?

As the name suggests, it's like a "richer" version of category theory, and it all starts with a simple observation. * (Get your category theory hats on, people. We're jumping right in!)*

In a category, you have some objects and some arrows between them, thought of as relationships between those objects. Now in the formal definition of a category, we usually ask for a *set's worth* of morphisms between any two objects, say $X$ and $Y$. You'll typically hear something like, "The hom set $\text{hom}(X,Y)$ bla bla...."

Now here's the thing. Quite often in mathematics, the set $\text{hom}(X,Y)$ may not just be a set. It could, for instance, be a set equipped with extra structure. You already know lots of examples. Let's think about about linear algebra, for a moment.