*Triplet loss* is a loss function that come from the paper FaceNet: A Unified Embedding for Face Recognition and Clustering. The loss function is designed to optimize a neural network that produces embeddings used for comparison.

The loss function operates on triplets, which are three examples from the dataset:

- \(x_i^a\) – an
*anchor*example. In the context of FaceNet, \(x_i^a\) is a photograph of a person’s face. - \(x_i^p\) – a
*positive*example that has the same identity as the anchor. In FaceNet, this is a second picture of the same person as the picture from the anchor example. - \(x_i^n\) – a
*negative*example that represents a different entity. For FaceNet, this would be an image of a second person–someone different than the person represented by the anchor and positive examples.

The triplet loss function is designed to train the model to produce embeddings such that the positive example \(x_i^p\) is closer to the anchor than the negative example \(x_i^n\).

## Math Details

More formally, for an embedding function \(f(x) \in \mathbb R^d\) that embeds input data \(x\) into a \(d\)-dimensional vector, we want

\[ ||f(x_i^a) - f(x_i^p)||_2^2 + \alpha \leq ||f(x_i^a) - f(x_i^n)||_2^2 \]

for all \(N\) possible triplets of \(x_i^a\), \(x_i^p\), and \(x_i^n\). The \(||x||_2^2\) operator is the square of the Eucledian norm. The \(\alpha\) symbol stands for a margin to ensure that the model doesn’t make the embeddings \(f(x_i^a)\), \(f(x_i^p)\), and \(f(x_i^n)\) equal each other to trivially satisfy the above inequality.

This leads to the following loss function over the \(N\) possible triplets.

\[ \sum_i^N \left [ ||f(x_i^a) - f(x_i^p)||_2^2 - ||f(x_i^a) - f(x_i^n)||_2^2 + \alpha \right ]_+ \]

the \([x]_+\) operator stands for \(\max(0,x)\).

## Triplet selection

In a typical dataset, many triplets of \(x_i^a\), \(x_i^p\), and \(x_i^n\) will satisfy the inequality in the previous section without the algorithm learning a useful embedding. This slows down the training of a machine learning algorithm that uses the triplet loss function.

To speed training back up, it makes sense to train the algorithm on examples where \(f(x_i^a)\) is closer to \(f(x_i^n)\) than \(f(x_i^p)\) in the embedding space (ignoring the term \(\alpha\)).