Welcome to this lesson on factorization machines, an important model in the realm of recommendation systems. Factorization machines (FM) excel at capturing interactions between variables, making them a powerful tool for both regression and classification tasks. For instance, they can predict a rating (regression) or calculate the likelihood of a recommendation (classification).

Before we dive into implementing a factorization machine, let's briefly revisit the dataset preparation process from the previous lesson.

Previously, you learned how to load data from JSON files and represent it as arrays of objects in JavaScript. You also created a user-item interaction matrix using dummy variables (one-hot encoding) and enriched the dataset with auxiliary features such as user preferences and genre similarity. These steps are crucial for building a dataset that can be used for accurate predictions in a recommendation system.

For this lesson, assume your data is structured as an array of objects, where each object represents a user-item interaction with features like:

To train the model, you will need to convert this array of objects into two arrays:

• X: an array of arrays, where each sub-array contains the feature values for one interaction (excluding the target).
• y: an array of target values (e.g., ratings).

Factorization machines leverage interactions between variables by decomposing them into simpler, latent factors. Mathematically, the prediction for a factorization machine can be expressed as:

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

Latent vectors are fundamental components in factorization machines used to capture complex pairwise interactions between features. Each feature in the dataset is represented by a latent vector, and the interaction between different features is determined by the dot product of their latent vectors.

For example, if you have features like user1, user2, item1, item2, uf1, if1, etc., each will have an associated latent vector (e.g., [v1, v2] if the latent factor size is 2).

For a row with active features user2, item3, uf1, uf2, if1, and if2, the interaction between any two features (e.g., user2 and item3) is captured by the dot product of their latent vectors.

Example calculation: If ${\mathbf{v}}_u$ and , then

Let's start by defining the class and initializing the required data in JavaScript. We'll use mathjs for vector operations and ml-matrix for matrix handling, as in the practice section.

• nFactors: Number of components in each latent vector (latent space dimensionality).
• nFeatures: Total number of features in the dataset.
• W: Linear coefficients for each feature.
• V: Matrix of latent vectors for each feature, initialized randomly.

Before implementing the training method, let's briefly discuss gradient descent. Gradient descent is an optimization algorithm that minimizes a function by iteratively moving in the direction opposite to the gradient. In our model, we use gradient descent to adjust parameters and minimize the error between predicted and actual outputs.

The update rule for a parameter $\theta$ is:

$\theta := \theta - \alpha \frac{\partial}{\partial \theta} J(\theta)$

Now, let's implement the fit method in JavaScript using ml-matrix and mathjs for correct and efficient vector/matrix operations.

• Linear Terms: Calculated as the sum of the global bias and the dot product of the feature vector and linear coefficients.
• Interaction Terms: For each latent factor, sum the dot products and squared terms to capture feature interactions.
• Parameter Updates: Update the global bias, linear coefficients, and interaction factors using gradient descent and regularization.

Now, let's implement the predict method in JavaScript, again using ml-matrix and mathjs.

• For each data instance, compute the linear and interaction terms, then sum them to get the prediction.

Let's see how to use the factorization machine in practice. We'll split the data into training and testing sets, train the model, make predictions, and evaluate performance using Mean Absolute Error (MAE).

The Mean Absolute Error (MAE) is defined as:

$\text{MAE}=\frac{1}{n}{\sum }_{i=1}^n\mathrm{\mid }{y}_i-{}_{}$

In this lesson, you implemented and evaluated a factorization machine model for recommendation systems in JavaScript. You learned how to initialize model parameters, train the model using gradient descent, make predictions, and evaluate performance. This concludes our exploration of factorization machines and marks the end of this course module.

Congratulations on completing the course! The skills you've acquired here form a strong foundation for building and understanding recommendation systems. Continue exploring other models and refining your expertise in this dynamic field. Well done!