% =========================================================
%  Chapter 9: Recommender Systems
% =========================================================
\chapter{Recommender Systems}

Recommender systems are among the most commercially successful applications of
machine learning. They power suggestions on streaming platforms such as Netflix
and Spotify, product recommendations on Amazon, and content feeds of social
media platforms. At their core, they learn from past interactions between users
and items to predict which items a user is most likely to enjoy next.

Formally: $n$ users, $m$ items, rating matrix $\mathbf{R}\in\R^{n\times m}$
where $r_{ui}$ is user $u$'s rating of item $i$. Because most users rate only
a tiny fraction of items, this matrix is extremely \emph{sparse}. The task is
to infer the missing entries.

%----------------------------------------------------------
\section{Collaborative Filtering}
\label{sec:cf}

\subsection{Core Idea}

Collaborative filtering's fundamental assumption: \emph{users who agreed in the
past tend to agree in the future}. No item content information is required.

\subsection{Memory-Based CF}

\subsubsection{User-Based CF}

Find users similar to the target user; use their ratings to predict:
\begin{equation}
\hat{r}_{ui}=\bar{r}_u+\frac{\sum_{v\in\mathcal{N}(u)}\mathrm{sim}(u,v)(r_{vi}-\bar{r}_v)}
{\sum_{v\in\mathcal{N}(u)}|\mathrm{sim}(u,v)|}.
\end{equation}

\subsubsection{Similarity Measures}

\paragraph{Cosine similarity.}
\[\mathrm{sim}_{\cos}(\mathbf{a},\mathbf{b})=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|_2\|\mathbf{b}\|_2}.\]

\paragraph{Pearson correlation.}
\[\mathrm{sim}_{\text{P}}(u,v)=\frac{\sum_{i\in\mathcal{I}_{uv}}(r_{ui}-\bar{r}_u)(r_{vi}-\bar{r}_v)}
{\sqrt{\sum_i(r_{ui}-\bar{r}_u)^2}\,\sqrt{\sum_i(r_{vi}-\bar{r}_v)^2}}.\]

\subsection{Model-Based CF: Matrix Factorisation}

Approximate the rating matrix as a product of two low-rank factor matrices:
\begin{equation}
\mathbf{R}\approx\mathbf{U}\mathbf{V}^{\!\top},
\label{eq:mf}
\end{equation}
$\mathbf{U}\in\R^{n\times k}$ (user factors), $\mathbf{V}\in\R^{m\times k}$
(item factors), $k\ll\min(n,m)$. Predicted rating:
\[\hat{r}_{ui}=\mathbf{u}_u\cdot\mathbf{v}_i.\]

\paragraph{Learning.} Minimise regularised MSE over observed ratings:
\begin{equation}
J(\mathbf{U},\mathbf{V})=\frac{1}{|\Omega|}\sum_{(u,i)\in\Omega}
(r_{ui}-\mathbf{u}_u\cdot\mathbf{v}_i)^2
+\lambda\!\Bigl(\sum_u\|\mathbf{u}_u\|^2+\sum_i\|\mathbf{v}_i\|^2\Bigr).
\end{equation}

SGD updates for observation $(u,i)$:
\begin{align}
e_{ui}&=r_{ui}-\mathbf{u}_u\cdot\mathbf{v}_i,\\
\mathbf{u}_u&\leftarrow\mathbf{u}_u+\alpha(e_{ui}\mathbf{v}_i-\lambda\mathbf{u}_u),\\
\mathbf{v}_i&\leftarrow\mathbf{v}_i+\alpha(e_{ui}\mathbf{u}_u-\lambda\mathbf{v}_i).
\end{align}

\paragraph{Bias terms.} An augmented model:
\[\hat{r}_{ui}=\mu+b_u+b_i+\mathbf{u}_u\cdot\mathbf{v}_i,\]
where $\mu$ is the global mean, $b_u$ the user bias, $b_i$ the item bias.

\subsection{Limitations of Collaborative Filtering}

\begin{itemize}
\item \textbf{Cold-start problem}: new users or items with no ratings.
\item \textbf{Popularity bias}: over-recommends popular items.
\item \textbf{Filter bubbles}: reinforces existing preferences.
\end{itemize}

%----------------------------------------------------------
\section{Implementation Details}

\subsection{Cost Function}

Let $n_u$ users, $n_m$ movies. Indicator $r^{(i,j)}=1$ if user $j$ rated
movie $i$.

\begin{equation}
J=\frac{1}{2}\sum_{(i,j):\,r^{(i,j)}=1}
\!\!\bigl(\vec{w}^{(j)}\cdot\vec{x}^{(i)}+b^{(j)}-y^{(i,j)}\bigr)^2
+\frac{\lambda}{2}\sum_j\sum_k(w_k^{(j)})^2
+\frac{\lambda}{2}\sum_i\sum_k(x_k^{(i)})^2.
\label{eq:rec-cost}
\end{equation}

Unlike supervised learning, both user parameters $\{(\vec{w}^{(j)},b^{(j)})\}$
and item features $\{\vec{x}^{(i)}\}$ are simultaneously optimised.

\subsection{Mean Normalisation}

A new user with no ratings yields $\vec{w}^{(j)}=\mathbf{0}$,
$\hat{y}^{(i,j)}=0$ --- unhelpful. Mean normalisation:
\[\mu_i=\frac{\sum_{j:\,r^{(i,j)}=1}y^{(i,j)}}{\sum_j r^{(i,j)}},\qquad
y_{\text{norm}}^{(i,j)}=y^{(i,j)}-\mu_i.\]
Prediction: $\hat{y}^{(i,j)}=\vec{w}^{(j)}\cdot\vec{x}^{(i)}+b^{(j)}+\mu_i$.
A new user now gets $\hat{y}=\mu_i$ (average rating) --- a sensible default.

\subsection{Finding Related Items}

Learned item features $\vec{x}^{(i)}$ can be used to find similar items:
\[\mathrm{sim}(i,i')=\|\vec{x}^{(i)}-\vec{x}^{(i')}\|_2^2.\]

%----------------------------------------------------------
\section{Content-Based Filtering}
\label{sec:cbf}

\subsection{Overview}

Content-based filtering recommends items by matching item features to a model
of the user's preferences. It does not need other users' data.

\subsection{User Profile and Prediction}

Let $\vec{x}^{(i)}\in\R^d$ be the feature vector for item $i$. A linear user
profile $\vec{\theta}^{(j)}$ is estimated from rating history:
\[\hat{y}^{(i,j)}=\vec{\theta}^{(j)}\cdot\vec{x}^{(i)}+b^{(j)}.\]

\paragraph{Advantages.}
\begin{itemize}
\item Handles cold-start for new \emph{items} (as long as features are available).
\item Explainable: ``recommended because you liked action movies.''
\item No popularity bias.
\end{itemize}

\paragraph{Limitations.}
\begin{itemize}
\item Requires rich item features.
\item Over-specialisation: keeps recommending similar content.
\item User cold-start remains.
\end{itemize}

\subsection{Hybrid Systems}

Production systems typically combine collaborative and content-based filtering.
Google's YouTube recommender uses a two-tower neural network: one tower
processes user features (including watch history), the other processes video
content features; the final score is the dot product of the two embeddings.

%----------------------------------------------------------
\section{Principal Component Analysis}
\label{sec:pca}

\subsection{Motivation}

Real-world data often contain far fewer independent sources of variation than
the raw number of features suggests. PCA finds the directions of maximum
variance and projects data onto a lower-dimensional subspace.

\subsection{Mathematical Formulation}

Let $\mathbf{X}\in\R^{n\times d}$ be mean-centred. The covariance matrix:
\[\boldsymbol{\Sigma}=\frac{1}{n-1}\mathbf{X}^{\top}\mathbf{X}.\]

The $l$-th principal component is the eigenvector of $\boldsymbol{\Sigma}$
with the $l$-th largest eigenvalue $\lambda_l$. The projection:
\[\mathbf{Z}=\mathbf{X}\mathbf{U}_k,\]
where $\mathbf{U}_k\in\R^{d\times k}$ contains the top-$k$ eigenvectors.

Equivalently via SVD: $\mathbf{X}=\mathbf{U}\mathbf{S}\mathbf{V}^{\top}$;
principal components are the columns of $\mathbf{V}$.

\subsection{Choosing the Number of Components}

Fraction of variance explained:
\[\mathrm{EVR}(k)=\frac{\sum_{l=1}^k\lambda_l}{\sum_{l=1}^d\lambda_l}.\]
Common heuristic: choose $k$ such that $\mathrm{EVR}(k)\ge 0.95$.

\begin{figure}[h]
\centering
\begin{tikzpicture}
\begin{axis}[
  width=0.6\textwidth, height=5cm,
  xlabel={Principal component}, ylabel={Variance explained (\%)},
  xmin=0, xmax=8, ymin=0, ymax=58,
  xtick={1,2,3,4,5,6,7},
  grid=both, grid style={gray!15, thin},
  tick label style={font=\small}, label style={font=\small},
  axis lines=left,
  every axis plot/.append style={line width=1.4pt}]
\addplot[pBlue, mark=*, mark size=2.5pt]
  coordinates{(1,50)(2,20)(3,12)(4,7)(5,4)(6,3)(7,2)};
\addplot[pOrange, dashed, mark=*, mark size=2.5pt]
  coordinates{(1,50)(2,70)(3,82)(4,89)(5,93)(6,96)(7,98)};
\draw[pRed, dashed, thin] (axis cs:0,50) -- (axis cs:8,50);
\node[right, font=\small, pRed] at (axis cs:0.1,52)
  {PC1: 50\% of variance};
\end{axis}
\end{tikzpicture}
\caption{Scree plot (blue, left axis) and cumulative variance explained (orange,
right axis). Most variance is captured by the first few components.}
\label{fig:scree}
\end{figure}

\subsection{PCA in Recommender Systems}

\begin{enumerate}
\item \textbf{Dimensionality reduction}: compress high-dimensional content
  features before feeding into a model.
\item \textbf{Latent Semantic Analysis}: truncated SVD directly on the
  user--item matrix fills in missing ratings:
  $\hat{\mathbf{R}}=\mathbf{U}_k\mathbf{S}_k\mathbf{V}_k^{\top}$.
\item \textbf{Visualisation}: project learned embeddings to 2D/3D.
\item \textbf{Noise reduction}: discard small singular values.
\end{enumerate}

\paragraph{Connection to matrix factorisation.}
For a fully observed matrix with no regularisation, the rank-$k$ matrix
factorisation recovers exactly the truncated SVD:
\[\underset{\mathbf{U},\mathbf{V}}{\argmin}
\|\mathbf{R}-\mathbf{U}\mathbf{V}^{\top}\|_F^2
=\mathbf{U}_k\mathbf{S}_k\mathbf{V}_k^{\top}.\]

%----------------------------------------------------------
\section*{Chapter Summary}
\addcontentsline{toc}{section}{Chapter Summary}

\begin{enumerate}
\item \textbf{Collaborative filtering} exploits shared preferences. Matrix
  factorisation learns compact latent representations optimised by SGD or ALS.
\item \textbf{Mean normalisation} gives sensible default predictions for new
  users.
\item \textbf{Content-based filtering} builds a per-user preference model from
  item features; handles item cold-start but can over-specialise.
\item \textbf{PCA} extracts the principal axes of variation; it underpins matrix
  factorisation and is used for dimensionality reduction, noise filtering, and
  latent semantic analysis.
\item Production systems \textbf{combine} both paradigms in large-scale deep
  learning architectures.
\end{enumerate}