FaceNet

Model structure

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The network consists of a batch input layer and a deep CNN followed by L2 normalization, which results in the face embedding. This is followed by the triplet loss during training.

Namely, the authors strive for an embedding $f(x)$, from an image $x$ into a feature space $\mathbf{R}^d$, such that:

• squared distance between all faces, independent of imaging conditions, of the same identity is small.

• the squared distance between a pair of face images from different identities is large.

Triplet Loss

The embedding is represented by $f (x) \in \mathbf{R}^d$ . It embeds an image x into a d-dimensional Euclidean space. Additionally, we constrain this embedding to live on the d-dimensional hypersphere, i.e. $||f(x)||^2 = 1$.

Here the objective is that we want to make sure that an image $x^a_i$(anchor) of a specific person is closer to all other images $x^p_i$ (positive) of the same person than it is to any image $x^n_i$ (negative) of any other person.

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• $\|f(x^a_i)-f(x^p_i)\|^2 +\alpha<\|f(x^a_i)-f(x^n_i)\|^2$ for $\forall (f(x^a_i),f(x^p_i),f(x^n_i))\in \tau$

• $\alpha$ is a margin that is enforced between positive and negative pairs.

• $\tau$ is the set of all possible triplets in the training set and has cardinality $N$

• The the objective is to minimize the Loss: $L = \sum [\|f(x^a_i)-f(x^p_i) \|^2-\|f(x^a_i)-f(x^n_i)\|^2+\alpha]_+$

Triplet Selection

Given $x^a_i$ , we want to select an $x^p_i$ (hard positive) such that $argmax_{x^p_i} \|f (x^a_i )-f (x^p_i )\|^2$ and similarly $x^n_i$ (hard negative) such that $argmin_{x^n_i}\|f(x^a_i )-f (x^n_i )\|^2$ .

Two obvious choices:

• Generate triplets offline every n steps, using the most recent network checkpoint and computing the argmin and argmax on a subset of the data.

• Generate triplets online. This can be done by select- ing the hard positive/negative exemplars from within a mini-batch.

Online Triplets Generation

• To have a meaningful representation of the anchor- positive distances, it needs to be ensured that a minimal number of exemplars of any one identity is present in each minibatch.

  • around 40 faces are selected per identity per minibatch.

  • randomly sampled negative faces are added to each mini-batch.

• Instead of picking the hardest positive, we use all anchor-positive pairs in a mini-batch while still selecting the hard negatives. For all anchor-positive method was more stable and converged slightly faster at the beginning of training.

• Selecting the hardest negatives can in practice lead to bad local minima early on in training, specifically it can result in a collapsed model (i.e. $f(x) = 0$). In order to mitigate this, it helps to select $x^n_i$ such that

  $\|f(x^a_i)-f(x^p_i)\|^2_2<\|f(x^a_i)-f(x^n_i)\|^2_2$

  • these negative exemplars semi-hard​, as they are further away from the anchor than the positive exemplar, but still hard because the squared distance is close to the anchor-positive distance.

  • Those negatives lie inside the margin $\alpha$

• In most experiments the author used a batch size of around 1,800 exemplars.

Deep Convolutional Networks

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• Use rectified linear units as the non-linear activation function

• Now use Inception+ResNet

Model Evaluation

The author evaluate our method on the face verification task. I.e. given a pair of two face images a squared L2 distance threshold $D(x_i,x_j)$ is used to determine the classification of same and different. All faces pairs $(i, j)$ of the same identity are denoted with $\mathcal{P}_{same}$, whereas all pairs of different identities are denoted with $\mathcal{P}_{diff}$

• the set of all true accepts as:

  $TA(d) = \{ (i,j) \in \mathcal{P}_{same}$, with $\space D(xi,xj) <= d \}$.

  These are the face pairs $(i, j)$ that were correctly classified as same at threshold $d$.

• the set of all pairs that was incorrectly classified as same(false accept) is:

  $TP(d) = \{ (i,j) \in \mathcal{P}_{diff}$, with $\space D(xi,xj) <= d \}$.

• The validation rate $VAL(d)$ and the false accept rate $FAR(d)$ for a given face distance $d$ are then defined as:

  $VAL(d)=\frac{|TA(d)|}{|P_{same}|}$ ,

  $FAR(d)=\frac{|TP(d)|}{|P_{diff}|}$