OK, so this is an old paper, but it was pretty impactful at the time, even if it's less relevant now. Basically, instead of just using a linear regression on basic features, this paper used a linear regression on features plus pairwise interactions between feature variables. This helped deal with the very sparse data common in recommendation algorithm problems, where the number of possible user-item interactions is typically much larger than the number of training samples. And since everything {I believe} is still essentially just a linear regression {with feature special sauce}, it's extremely computationally efficient {linear time}.

Cool stuff, although in modern-day times, this approach is probably very rarely used, at least when you have a lot of data and can throw a neural network at the problem instead.

This paper introduced a new {at the time, 2010!} model called Factorization Machines {FM} designed to effectively handle data sparsity, a common problem in many recommendation systems.

Factorization Machines are a generic approach that allows you to mimic most factorization models by feature engineering. This is a significant advantage over standard factorization models, which are usually bound to specific problem definitions and thus do not allow for the incorporation of additional side information {e.g., user demographics in a movie recommendation system}.

The model architecture of FM is defined by the equation:

[ \hat{y}(\mathbf{x}) = w_0 + \sum_{i=1}^{n} w_i x_i + \sum_{i=1}^{n} \sum_{j=i+1}^{n} \langle \mathbf{v}_i, \mathbf{v}_j \rangle x_i x_j ]

where:

- (\hat{y}(\mathbf{x})) is the predicted target variable.
- (w_0) is the global bias.
- (w_i) are the weights of the model.
- (\mathbf{x}) is the feature vector.
- (\mathbf{v}_i) and (\mathbf{v}_j) are latent vectors that capture interactions between pairs of variables.
- (\langle \mathbf{v}_i, \mathbf{v}_j \rangle) is the dot product of these latent vectors.

The first two terms on the right side of the equation represent a linear regression model. The third term captures pairwise interactions between variables, with each interaction being weighted by the dot product of the corresponding latent vectors. This term is what distinguishes FM from a standard linear model and enables it to effectively handle high-dimensional sparse data.

The model parameters {i.e., the weights and latent vectors} can be learned using any standard optimization algorithm, such as stochastic gradient descent or alternating least squares.

The main advantage of Factorization Machines is their ability to model interactions between features even in problems with very sparse data. This makes them particularly useful in recommendation systems, where the number of possible user-item interactions is typically much larger than the number of observed interactions.

Another advantage of FM is their computational efficiency. The model can be calculated in linear time with respect to the number of non-zero elements in the input data, which makes it feasible to use even on very large datasets.

The paper also demonstrates the effectiveness of FM through a series of experiments on real-world datasets. The results show that FM outperforms other factorization models in terms of prediction accuracy, especially in cases with very sparse data.

This work has significant implications for the field of recommendation systems and more generally for any problem involving high-dimensional sparse data. By providing a flexible and efficient way to model feature interactions, Factorization Machines offer a powerful tool for predictive modeling in these settings.