## The category neighborhood of a dataset will be learned using soft nearest neighbor loss

In this text, we discuss learn how to implement the soft nearest neighbor loss which we also talked about here.

Representation learning is the duty of learning essentially the most salient features in a given dataset by a deep neural network. It is frequently an implicit task done in a supervised learning paradigm, and it is a vital think about the success of deep learning (Krizhevsky et al., 2012; He et al., 2016; Simonyan et al., 2014). In other words, representation learning automates the technique of feature extraction. With this, we will use the learned representations for downstream tasks similar to classification, regression, and synthesis.

We may influence how the learned representations are formed to cater specific use cases. Within the case of classification, the representations are primed to have data points from the identical class to flock together, while for generation (e.g. in GANs), the representations are primed to have points of real data flock with the synthesized ones.

In the identical sense, now we have enjoyed using principal components evaluation (PCA) to encode features for downstream tasks. Nevertheless, we shouldn’t have any class or label information in PCA-encoded representations, hence the performance on downstream tasks could also be further improved. We will improve the encoded representations by approximating the category or label information in it by learning the neighborhood structure of the dataset, i.e. which features are clustered together, and such clusters would imply that the features belong to the identical class as per the clustering assumption within the semi-supervised learning literature (Chapelle et al., 2009).

To integrate the neighborhood structure within the representations, manifold learning techniques have been introduced similar to locally linear embeddings or LLE (Roweis & Saul, 2000), neighborhood components evaluation or NCA (Hinton et al., 2004), and t-stochastic neighbor embedding or t-SNE (Maaten & Hinton, 2008).

Nevertheless, the aforementioned manifold learning techniques have their very own drawbacks. As an example, each LLE and NCA encode linear embeddings as an alternative of nonlinear embeddings. Meanwhile, t-SNE embeddings result to different structures depending on the hyperparameters used.

To avoid such drawbacks, we will use an improved NCA algorithm which is the *soft nearest neighbor loss* or SNNL (Salakhutdinov & Hinton, 2007; Frosst et al., 2019). The SNNL improves the NCA algorithm by introducing nonlinearity, and it’s computed for every hidden layer of a neural network as an alternative of solely on the last encoding layer. This loss function is used to optimize the *entanglement* of points in a dataset.

On this context, *entanglement* is defined as how close class-similar data points to one another are in comparison with class-different data points. A low entanglement signifies that class-similar data points are much closer to every apart from class-different data points (see Figure 1). Having such a set of information points will render downstream tasks much easier to perform with a fair higher performance. Frosst et al. (2019) expanded the SNNL objective by introducing a temperature factor *T*. Thus giving us the next as the ultimate loss function,

where *d* is a distance metric on either raw input features or hidden layer representations of a neural network, and *T* is the temperature factor that’s directly proportional to the distances amongst data points in a hidden layer. For this implementation, we use the cosine distance as our distance metric for more stable computations.

The aim of this text is to assist readers understand and implement the soft nearest neighbor loss, and so we will dissect the loss function with a purpose to understand it higher.

## Distance Metric

The very first thing we must always compute are the distances amongst data points, which are either the raw input features or hidden layer representations of the network.

For our implementation, we use the cosine distance metric (Figure 3) for more stable computations. On the time being, allow us to ignore the denoted subsets *ij* and *ik* within the figure above, and allow us to just deal with computing the cosine distance amongst our input data points. We accomplish this through the next PyTorch code:

`normalized_a = torch.nn.functional.normalize(features, dim=1, p=2)`

normalized_b = torch.nn.functional.normalize(features, dim=1, p=2)

normalized_b = torch.conj(normalized_b).T

product = torch.matmul(normalized_a, normalized_b)

distance_matrix = torch.sub(torch.tensor(1.0), product)

Within the code snippet above, we first normalize the input features in lines 1 and a pair of using Euclidean norm. Then in line 3, we get the conjugate transpose of the second set of the normalized input features. We compute the conjugate transpose to account for complex vectors. In lines 4 and 5, we compute the cosine similarity and distance of the input features.

Concretely, consider the next set of features,

`tensor([[ 1.0999, -0.9438, 0.7996, -0.4247],`

[ 1.2150, -0.2953, 0.0417, -1.2913],

[ 1.3218, 0.4214, -0.1541, 0.0961],

[-0.7253, 1.1685, -0.1070, 1.3683]])

Using the gap metric we defined above, we gain the next distance matrix,

`tensor([[ 0.0000e+00, 2.8502e-01, 6.2687e-01, 1.7732e+00],`

[ 2.8502e-01, 0.0000e+00, 4.6293e-01, 1.8581e+00],

[ 6.2687e-01, 4.6293e-01, -1.1921e-07, 1.1171e+00],

[ 1.7732e+00, 1.8581e+00, 1.1171e+00, -1.1921e-07]])

## Sampling Probability

We will now compute the matrix that represents the probability of picking each feature given its pairwise distances to all other features. This is solely the probability of picking *i* points based on the distances between *i* and *j *or* k* points.

We will compute this through the next code:

`pairwise_distance_matrix = torch.exp(`

-(distance_matrix / temperature)

) - torch.eye(features.shape[0]).to(model.device)

The code first calculates the exponential of the negative of the gap matrix divided by the temperature factor, scaling the values to positive values. The temperature factor dictates learn how to control the importance given to the distances between pairs of points, as an example, at low temperatures, the loss is dominated by small distances while actual distances between widely separated representations develop into less relevant.

Prior to the subtraction of `torch.eye(features.shape[0])`

(aka diagonal matrix), the tensor was as follows,

`tensor([[1.0000, 0.7520, 0.5343, 0.1698],`

[0.7520, 1.0000, 0.6294, 0.1560],

[0.5343, 0.6294, 1.0000, 0.3272],

[0.1698, 0.1560, 0.3272, 1.0000]])

We subtract a diagonal matrix from the gap matrix to remove all self-similarity terms (i.e. the gap or similarity of every point to itself).

Next, we will compute the sampling probability for every pair of information points through the next code:

`pick_probability = pairwise_distance_matrix / (`

torch.sum(pairwise_distance_matrix, 1).view(-1, 1)

+ stability_epsilon

)

## Masked Sampling Probability

Thus far, the sampling probability now we have computed doesn’t contain any label information. We integrate the label information into the sampling probability by masking it with the dataset labels.

First, now we have to derive a pairwise matrix out of the label vectors:

`masking_matrix = torch.squeeze(`

torch.eq(labels, labels.unsqueeze(1)).float()

)

We apply the masking matrix to make use of the label information to isolate the possibilities for points that belong to the identical class:

`masked_pick_probability = pick_probability * masking_matrix`

Next, we compute the sum probability for sampling a specific feature by computing the sum of the masked sampling probability per row,

`summed_masked_pick_probability = torch.sum(masked_pick_probability, dim=1)`

Finally, we will compute the logarithm of the sum of the sampling probabilities for features for computational convenience with a further computational stability variable, and get the typical to act as the closest neighbor loss for the network,

`snnl = torch.mean(`

-torch.log(summed_masked_pick_probability + stability_epsilon

)

We will now string these components together in a forward pass function to compute the soft nearest neighbor loss across all layers of a deep neural network,

`def forward(`

self,

model: torch.nn.Module,

features: torch.Tensor,

labels: torch.Tensor,

outputs: torch.Tensor,

epoch: int,

) -> Tuple:

if self.use_annealing:

self.temperature = 1.0 / ((1.0 + epoch) ** 0.55)primary_loss = self.primary_criterion(

outputs, features if self.unsupervised else labels

)

activations = self.compute_activations(model=model, features=features)

layers_snnl = []

for key, value in activations.items():

value = value[:, : self.code_units]

distance_matrix = self.pairwise_cosine_distance(features=value)

pairwise_distance_matrix = self.normalize_distance_matrix(

features=value, distance_matrix=distance_matrix

)

pick_probability = self.compute_sampling_probability(

pairwise_distance_matrix

)

summed_masked_pick_probability = self.mask_sampling_probability(

labels, pick_probability

)

snnl = torch.mean(

-torch.log(self.stability_epsilon + summed_masked_pick_probability)

)

layers_snnl.append(snnl)

snn_loss = torch.stack(layers_snnl).sum()

train_loss = torch.add(primary_loss, torch.mul(self.factor, snn_loss))

return train_loss, primary_loss, snn_loss

## Visualizing Disentangled Representations

We trained an autoencoder with the soft nearest neighbor loss, and visualize its learned disentangled representations. The autoencoder had (x-500–500–2000-d-2000–500–500-x) units, and was trained on a small labelled subset of the MNIST, Fashion-MNIST, and EMNIST-Balanced datasets. That is to simulate the scarcity of labelled examples since autoencoders are presupposed to be unsupervised models.

We only visualized an arbitrarily chosen 10 clusters for easier and cleaner visualization of the EMNIST-Balanced dataset. We will see within the figure above that the latent code representation became more clustering-friendly by having a set of well-defined clusters as indicated by cluster dispersion and proper cluster assignments as indicated by cluster colours.

## Closing Remarks

In this text, we dissected the soft nearest neighbor loss function as to how we could implement it in PyTorch.

The soft nearest neighbor loss was first introduced by Salakhutdinov & Hinton (2007) where it was used to compute the loss on the latent code (bottleneck) representation of an autoencoder, after which the said representation was used for downstream kNN classification task.

Frosst, Papernot, & Hinton (2019) then expanded the soft nearest neighbor loss by introducing a temperature factor and by computing the loss across all layers of a neural network.

Finally, we employed an annealing temperature factor for the soft nearest neighbor loss to further improve the learned disentangled representations of a network, and likewise speed up the disentanglement process (Agarap & Azcarraga, 2020).

The complete code implementation is obtainable in GitLab.

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