Implementing Soft Nearest Neighbor Loss in PyTorch | by Abien Fred Agarap | Nov, 2023

Representation finding out is the obligation of finding out most likely probably the most salient choices in a given dataset by a deep neural group. It’s typically an implicit course of completed in a supervised finding out paradigm, and it’s a important concern inside the success of deep finding out (Krizhevsky et al., 2012; He et al., 2016; Simonyan et al., 2014). In numerous phrases, illustration finding out automates the tactic of attribute extraction. With this, we are going to use the realized representations for downstream duties resembling classification, regression, and synthesis.

We’ll moreover have an effect on how the realized representations are common to cater explicit use situations. Inside the case of classification, the representations are primed to have data elements from the an identical class to flock collectively, whereas for know-how (e.g. in GANs), the representations are primed to have elements of precise data flock with the synthesized ones.

Within the an identical sense, we’ve bought beloved the utilization of principal parts analysis (PCA) to encode choices for downstream duties. Nonetheless, we shouldn’t have any class or label knowledge in PCA-encoded representations, subsequently the effectivity on downstream duties may be extra improved. We’ll improve the encoded representations by approximating the class or label knowledge in it by finding out the neighborhood development of the dataset, i.e. which choices are clustered collectively, and such clusters would point out that the choices belong to the an identical class as per the clustering assumption inside the semi-supervised finding out literature (Chapelle et al., 2009).

To mix the neighborhood development inside the representations, manifold finding out strategies have been launched resembling regionally linear embeddings or LLE (Roweis & Saul, 2000), neighborhood parts analysis or NCA (Hinton et al., 2004), and t-stochastic neighbor embedding or t-SNE (Maaten & Hinton, 2008).

Nonetheless, the aforementioned manifold finding out strategies have their very personal drawbacks. For instance, every LLE and NCA encode linear embeddings in its place of nonlinear embeddings. Within the meantime, t-SNE embeddings finish outcome to completely completely different buildings counting on the hyperparameters used.

To steer clear of such drawbacks, we are going to use an improved NCA algorithm which is the tender 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 each hidden layer of a neural group in its place of solely on the ultimate encoding layer. This loss function is used to optimize the entanglement of things in a dataset.

On this context, entanglement is printed as how shut class-similar data elements to at least one one other are compared with class-different data elements. A low entanglement implies that class-similar data elements are rather a lot nearer to at least one one other than class-different data elements (see Decide 1). Having such a set of knowledge elements will render downstream duties rather a lot easier to carry out with a good greater effectivity. Frosst et al. (2019) expanded the SNNL aim by introducing a temperature concern T. Thus giving us the subsequent as the final word loss function,

the place d is a distance metric on each raw enter choices or hidden layer representations of a neural group, and T is the temperature concern that’s immediately proportional to the distances amongst data elements in a hidden layer. For this implementation, we use the cosine distance as our distance metric for further safe computations.

The intention of this textual content is to help readers understand and implement the tender nearest neighbor loss, and so we are going to dissect the loss function to have the ability to understand it greater.

Distance Metric

The very very first thing we must always all the time compute are the distances amongst data elements, which will be each the raw enter choices or hidden layer representations of the group.

For our implementation, we use the cosine distance metric (Decide 3) for further safe computations. On the time being, permit us to disregard the denoted subsets ij and ik inside the decide above, and permit us to easily consider computing the cosine distance amongst our enter data elements. We accomplish this by the use of the subsequent PyTorch code:

normalized_a = torch.nn.helpful.normalize(choices, dim=1, p=2)
normalized_b = torch.nn.helpful.normalize(choices, 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)

Inside the code snippet above, we first normalize the enter choices in traces 1 and a few using Euclidean norm. Then in line 3, we get the conjugate transpose of the second set of the normalized enter choices. We compute the conjugate transpose to account for complex vectors. In traces 4 and 5, we compute the cosine similarity and distance of the enter choices.

Concretely, ponder the subsequent set of choices,

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 hole metric we outlined above, we obtain the subsequent 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 Chance

We’ll now compute the matrix that represents the chance of choosing each attribute given its pairwise distances to all completely different choices. That’s merely the chance of choosing i elements based totally on the distances between i and j or okay elements.

We’ll compute this by the use of the subsequent code:

pairwise_distance_matrix = torch.exp(
-(distance_matrix / temperature)
) - torch.eye(choices.kind[0]).to(model.system)

The code first calculates the exponential of the damaging of the hole matrix divided by the temperature concern, scaling the values to optimistic values. The temperature concern dictates the easiest way to administration the importance given to the distances between pairs of things, as an illustration, at low temperatures, the loss is dominated by small distances whereas exact distances between broadly separated representations flip into a lot much less associated.

Earlier to the subtraction of torch.eye(choices.kind[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 hole matrix to remove all self-similarity phrases (i.e. the hole or similarity of each degree to itself).

Subsequent, we are going to compute the sampling chance for each pair of knowledge elements by the use of the subsequent code:

pick_probability = pairwise_distance_matrix / (
torch.sum(pairwise_distance_matrix, 1).view(-1, 1)
+ stability_epsilon
)

Masked Sampling Chance

Thus far, the sampling chance we’ve bought computed doesn’t comprise any label knowledge. We mix the label knowledge into the sampling chance by masking it with the dataset labels.

First, we’ve bought 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 utilize the label knowledge to isolate the chances for elements that belong to the an identical class:

masked_pick_probability = pick_probability * masking_matrix

Subsequent, we compute the sum chance for sampling a particular attribute by computing the sum of the masked sampling chance per row,

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

Lastly, we are going to compute the logarithm of the sum of the sampling probabilities for choices for computational consolation with an additional computational stability variable, and get the frequent to behave because the closest neighbor loss for the group,

snnl = torch.indicate(
-torch.log(summed_masked_pick_probability + stability_epsilon
)

We’ll now string these parts collectively in a forward transfer function to compute the tender nearest neighbor loss all through all layers of a deep neural group,

def forward(
self,
model: torch.nn.Module,
choices: 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, choices if self.unsupervised else labels
)
activations = self.compute_activations(model=model, choices=choices)
layers_snnl = []
for key, price in activations.devices():
price = price[:, : self.code_units]
distance_matrix = self.pairwise_cosine_distance(choices=price)
pairwise_distance_matrix = self.normalize_distance_matrix(
choices=price, 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.indicate(
-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.concern, snn_loss))
return train_loss, primary_loss, snn_loss

Visualizing Disentangled Representations

We educated an autoencoder with the tender nearest neighbor loss, and visualize its realized disentangled representations. The autoencoder had (x-500–500–2000-d-2000–500–500-x) fashions, and was educated on a small labelled subset of the MNIST, Development-MNIST, and EMNIST-Balanced datasets. That’s to simulate the scarcity of labelled examples since autoencoders are alleged to be unsupervised fashions.

We solely visualized an arbitrarily chosen 10 clusters for easier and cleaner visualization of the EMNIST-Balanced dataset. We’ll see inside the decide above that the latent code illustration turned further clustering-friendly by having a set of well-defined clusters as indicated by cluster dispersion and correct cluster assignments as indicated by cluster colors.

Closing Remarks

On this text, we dissected the tender nearest neighbor loss function as to how we might implement it in PyTorch.

The tender nearest neighbor loss started was first launched by Salakhutdinov & Hinton (2007) the place it was used to compute the loss on the latent code (bottleneck) illustration of an autoencoder, after which the talked about illustration was used for downstream kNN classification course of.

Frosst, Papernot, & Hinton (2019) then expanded the tender nearest neighbor loss by introducing a temperature concern and by computing the loss all through all layers of a neural group.

Lastly, we employed an annealing temperature concern for the tender nearest neighbor loss to extra improve the realized disentangled representations of a group, and as well as tempo up the disentanglement course of (Agarap & Azcarraga, 2020).

The whole code implementation is obtainable in GitLab.

References

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