Understanding matrix factorization for recommendation (part 1) - preliminary insights on PCA

Foreword: this is the first part of a 4 parts series. Here are parts 2, 3 and 4. This series is an extended version of a talk I gave at PyParis 17.

About ten years ago, Netflix launched the Netflix Prize: an open contest where the goal was to design state-of-the-art algorithms for predicting movie ratings. During 3 years, research teams developed many different prediction algorithms, among which matrix factorization techniques stood out by their efficiency.

The goal of this series of posts is twofold:

I tried to keep the math level of the article as accessible as possible, but without trying to over-simplify things, and avoiding dull statements. My hope is that this article is accessible to ML beginners, while still being insightful to the more experienced.

This article is divided into 4 parts: in this first part, we will clearly define the problem we plan to address, and provide some insights about PCA. In the second part, we will review SVD and see how it models the ratings. In the third part, we will see how to apply our knowledge of SVD to the rating prediction task, and derive an implementation of a matrix-factorization-based algorithm. In the last part, we will implement a matrix factorization algorithm in Python using the Surprise library.

The problem

The problem we propose to address here is that of rating prediction. The data we have is a rating history: ratings of users for items in the interval $[1, 5]$. We can put all this data into a sparse matrix called $R$:

\[R = \begin{pmatrix} 1 & \color{#e74c3c}{?} & 2 & \color{#e74c3c}{?} & \color{#e74c3c}{?}\\ \color{#e74c3c}{?} & \color{#e74c3c}{?} & \color{#e74c3c}{?} & \color{#e74c3c}{?} & 4\\ 2 & \color{#e74c3c}{?} & 4 & 5 & \color{#e74c3c}{?}\\ \color{#e74c3c}{?} & \color{#e74c3c}{?} & 3 & \color{#e74c3c}{?} & \color{#e74c3c}{?}\\ \color{#e74c3c}{?} & 1 & \color{#e74c3c}{?} & 3 & \color{#e74c3c}{?}\\ 5 & \color{#e74c3c}{?} & \color{#e74c3c}{?} & \color{#e74c3c}{?} & 2\\ \end{pmatrix} \begin{matrix} \text{Alice}\\ \text{Bob}\\ \text{Charlie}\\ \text{Daniel}\\ \text{Eric}\\ \text{Frank}\\ \end{matrix}\]

Each row of the matrix corresponds to a given user, and each column corresponds to a given item. For instance here, Alice has rated the first item with a rating of $1$, and Charlie has rated the third item with a rating of $4$. In our case, we will consider that the items are movies and we will use the terms item and movie interchangeably.

The matrix $R$ is sparse (more than 99% of the entries are missing), and our goal is to predict the missing entries, i.e. predict the $\color{#e74c3c}{?}$.

There are many different techniques for rating prediction, and in our case we will factorize the matrix $R$. This matrix factorization is fundamentally linked to SVD, which stands for Singular Value Decomposition. SVD is one of the highlights of linear algebra. It’s a beautiful result. When people tell you that math sucks, show them what SVD can do.

The aim of this article is to explain how SVD can be used for rating prediction purposes. But before we can dive into SVD in the second part, we need to review what PCA is. PCA is only slightly less awesome than SVD, but it is still really cool.

A little bit of PCA

Let’s forget the recommendation problem for 2 minutes. We’ll play around with the Olivetti dataset. It’s a set of greyscale images of faces from 40 people, making up a total of 400 images. Here are the first 10 people:

Friendly, right? Well, just wait a little bit…

Each image size is 64 x 64 pixels. We will flatten each of these images (we thus get 400 vectors, each with 64 x 64 = 4096 elements). We can represent our dataset in a 400 x 4096 matrix:

\[\newcommand{\horzbar}{\Rule{2.5ex}{0.5pt}{0.1pt}} \newcommand{\vertbar}{\Rule{0.5pt}{1pt}{2.5ex}} X= \begin{pmatrix} \horzbar & \text{Face 1} & \horzbar\\ \horzbar & \text{Face 2} & \horzbar\\ & \vdots &\\ \horzbar & \text{Face 400} & \horzbar\\ \end{pmatrix}\]

PCA, which stands for Principal Component Analysis, is an algorithm that will reveal 400 of these guys:

Creepy, right ;)?

We call these guys the principal components (hence the name of the technique), and when they represent faces such as here we call them the eigenfaces. Some really cool stuff can be done with eigenfaces such as face recognition, or optimizing your tinder matches! The reason why they’re called eigenfaces is because they are in fact the eigenvectors of the covariance matrix of $X$ (but we will not detail this, see the references if you want to dive further into it).

We obtain here 400 principal components because the original matrix $X$ has 400 rows (or more precisely, because the rank of $X$ is 400). As you may have guessed, each of the principal component is in fact a vector that has the same dimension as the original faces, i.e. it has 64 x 64 = 4096 pixels.

As far as we’re concerned, we will call these guys the creepy guys. Now, one amazing thing about them is that they can build back all of the original faces. Take a look at this (these are animated gifs, about 10s long):

Here is what’s going on. Each of the 400 original faces (i.e. each of the 400 original rows of the matrix) can be expressed as a (linear) combination of the creepy guys. That is, we can express the first original face (i.e. its pixel values) as a little bit of the first creepy guy, plus a little bit of the second creepy guy, plus a little bit of third, etc. until the last creepy guy. The same goes for all of the other original faces: they can all be expressed as a little bit of each creepy guy. Mathematically, we write it this way:

\(\begin{align*} \text{Face 1}~=~&\alpha_1 \cdot \color{#048BA8}{\text{Creepy guy #1}}\\ +~ &\alpha_2 \cdot \color{#048BA8}{\text{Creepy guy #2}}\\ +~ &\cdots\\ +~ &\alpha_{400} \cdot \color{#048BA8}{\text{Creepy guy #400}} \end{align*} ~~~\) \(\begin{align*} \text{Face 2}~=~&\beta_1 \cdot \color{#048BA8}{\text{Creepy guy #1}}\\ +~ &\beta_2 \cdot \color{#048BA8}{\text{Creepy guy #2}}\\ +~ &\cdots\\ +~ &\beta_{400} \cdot \color{#048BA8}{\text{Creepy guy #400}} \end{align*}\) \(~~~\cdots\)

The gifs you saw above are the very translation of these math equations: the first frame of a gif is the contribution of the first creepy guy, the second frame is the contribution of the first two creepy guys, etc. until the last creepy guy.

If you want to play around a bit with the creepy guys and reproduce the cool gifs, I made a small notebook for you.

Note: there is nothing really special about the fact that the creepy guys can express all the original faces. We could have randomly chosen 400 independent vectors of 64 x 64 = 4096 pixels and still be able to reconstruct any face. What makes the creepy guys from PCA special is a very important result about dimensionality reduction, which we will discuss in part 3.

Latent factors

We’ve actually been kind of harsh towards the creepy guys. They’re not creepy, they’re typical. The goal of PCA is to reveal typical vectors: each of the creepy/typical guy represents one specific aspect underlying the data. In an ideal world, the first typical guy would represent (e.g.) a typical elder person, the second typical guy would represent a typical glasses wearer, and some other typical guys would represent concepts such as smiley, sad looking, big nose, stuff like that. And with these concepts, we could define a face as more or less elder, more or less glassy, more or less smiling, etc. In practice, the concepts that PCA reveals are really not that clear: there is no clear semantic that we could associate with any of the creepy/typical guys that we obtained here. But the important fact remains: each of the typical guys captures a specific aspect of the data. We call these aspects the latent factors (latent, because they were there all the time, we just needed PCA to reveal them). Using barbaric terms, we say that each principal component (the creepy/typical guys) captures a specific latent factor.

Now, this is all good and fun, but we’re interested in matrix factorization for recommendation purposes, right? So where is our matrix factorization, and what does it have to do with recommendation? PCA is actually a plug-and-play method: it works for any matrix. If your matrix contains images, it will reveal some typical images that can build back all of your initial images, such as here. If your matrix contains potatoes, PCA will reveal some typical potatoes that can build back all of your original potatoes. If your matrix contains ratings, well… Here we come.

PCA on a (dense) rating matrix

Until stated otherwise, we will consider for now that our rating matrix $R$ is completely dense, i.e. there are no missing entries. All the ratings are known. This is of course not the case in real recommendation problems, but bear with me.

PCA on the users

Here is our rating matrix, where rows are users and columns are movies:

\[R = \begin{pmatrix} \horzbar & \text{Alice} & \horzbar\\ \horzbar & \text{Bob} & \horzbar\\ & \vdots &\\ \horzbar & \text{Zoe} & \horzbar\\ \end{pmatrix}\]

Looks familiar? Instead of having faces in the rows represented by pixel values, we now have users represented by their ratings. Just like PCA gave us some typical guys before, it will now give us some typical users, or rather some typical raters.

Here again, in an ideal world, the concepts associated with the typical users would have a clear semantic meaning: we would obtain a typical action movie fan, a typical romance movie fan, a typical comedy fan, etc. In practice, the semantic behind the typical users is not clearly defined, but for the sake of simplicity we will assume that they are (it doesn’t change anything, this is just for intuition/explanation purposes).

So here we are: each of our initial users (Alice, Bob…) can be expressed as a combination of the typical users. For instance, Alice could be defined as a little bit of an action fan, a little bit of a comedy fan, a lot of a romance fan, etc. As for Bob, he could be more keen on action movies:

\[\begin{align*} \text{Alice} &= 10\% \color{#048BA8}{\text{ Action fan}} + 10\% \color{#048BA8}{\text{ Comedy fan}} + 50\% \color{#048BA8}{\text{ Romance fan}} +\cdots\\ \text{Bob} &= 50\% \color{#048BA8}{\text{ Action fan}} + 30\% \color{#048BA8}{\text{ Comedy fan}} + 10\% \color{#048BA8}{\text{ Romance fan}} +\cdots\\ \text{Zoe} &= \cdots \end{align*}\]

And the same goes for all of the users, you get the idea. (In practice the coefficients are not necessarily percentages, but it’s convenient for us to think of it this way).

PCA on the movies

What would happen if we transposed our rating matrix? Instead of having users in the rows, we would now have movies, defined as their ratings:

\[R^T = \begin{pmatrix} \horzbar & \text{Titanic} & \horzbar\\ \horzbar & \text{Toy Story} & \horzbar\\ & \vdots &\\ \horzbar & \text{Fargo} & \horzbar\\ \end{pmatrix}\]

In this case, PCA will not reveal typical faces nor typical users, but of course typical movies. And here again, we will associate a semantic meaning behind each of the typical movies, and these typical movies can build back all of our original movies:

And the same goes for all the other movies.

We are now ready to dive into SVD in the next part.

Nicolas Hug

Nicolas Hug

ML Engineer

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