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Capsule Networks

This is a PyTorch implementation/tutorial of Dynamic Routing Between Capsules.

Capsule network is a neural network architecture that embeds features as capsules and routes them with a voting mechanism to next layer of capsules.

Unlike in other implementations of models, we’ve included a sample, because it is difficult to understand some concepts with just the modules. This is the annotated code for a model that uses capsules to classify MNIST dataset

This file holds the implementations of the core modules of Capsule Networks.

I used jindongwang/Pytorch-CapsuleNet to clarify some confusions I had with the paper.

Here’s a notebook for training a Capsule Network on MNIST dataset.

33import torch.nn as nn
34import torch.nn.functional as F
35import torch.utils.data
36
37from labml_helpers.module import Module

#

Squash

This is squashing function from paper, given by equation $(1)$.

$\mathbf{v}_j = \frac{{\lVert \mathbf{s}_j \rVert}^2}{1 + {\lVert \mathbf{s}_j \rVert}^2} \frac{\mathbf{s}_j}{\lVert \mathbf{s}_j \rVert}$

$\frac{\mathbf{s}_j}{\lVert \mathbf{s}_j \rVert}$ normalizes the length of all the capsules, whilst $\frac{{\lVert \mathbf{s}_j \rVert}^2}{1 + {\lVert \mathbf{s}_j \rVert}^2}$ shrinks the capsules that have a length smaller than one .

40class Squash(Module):

#

55    def __init__(self, epsilon=1e-8):
56        super().__init__()
57        self.epsilon = epsilon

#

The shape of s is [batch_size, n_capsules, n_features]

59    def forward(self, s: torch.Tensor):

#

${\lVert \mathbf{s}_j \rVert}^2$

65        s2 = (s ** 2).sum(dim=-1, keepdims=True)

#

We add an epsilon when calculating $\lVert \mathbf{s}_j \rVert$ to make sure it doesn’t become zero. If this becomes zero it starts giving out nan values and training fails. $\mathbf{v}_j = \frac{{\lVert \mathbf{s}_j \rVert}^2}{1 + {\lVert \mathbf{s}_j \rVert}^2} \frac{\mathbf{s}_j}{\sqrt{{\lVert \mathbf{s}_j \rVert}^2 + \epsilon}}$

71        return (s2 / (1 + s2)) * (s / torch.sqrt(s2 + self.epsilon))

#

Routing Algorithm

This is the routing mechanism described in the paper. You can use multiple routing layers in your models.

This combines calculating $\mathbf{s}_j$ for this layer and the routing algorithm described in Procedure 1.

74class Router(Module):

#

in_caps is the number of capsules, and in_d is the number of features per capsule from the layer below. out_caps and out_d are the same for this layer.

iterations is the number of routing iterations, symbolized by $r$ in the paper.

85    def __init__(self, in_caps: int, out_caps: int, in_d: int, out_d: int, iterations: int):

#

92        super().__init__()
93        self.in_caps = in_caps
94        self.out_caps = out_caps
95        self.iterations = iterations
96        self.softmax = nn.Softmax(dim=1)
97        self.squash = Squash()

#

This is the weight matrix $\mathbf{W}_{ij}$. It maps each capsule in the lower layer to each capsule in this layer

101        self.weight = nn.Parameter(torch.randn(in_caps, out_caps, in_d, out_d), requires_grad=True)

#

The shape of u is [batch_size, n_capsules, n_features]. These are the capsules from the lower layer.

103    def forward(self, u: torch.Tensor):

#

$\hat{\mathbf{u}}_{j|i} = \mathbf{W}_{ij} \mathbf{u}_i$ Here $j$ is used to index capsules in this layer, whilst $i$ is used to index capsules in the layer below (previous).

112        u_hat = torch.einsum('ijnm,bin->bijm', self.weight, u)

#

Initial logits $b_{ij}$ are the log prior probabilities that capsule $i$ should be coupled with $j$. We initialize these at zero

117        b = u.new_zeros(u.shape[0], self.in_caps, self.out_caps)
118
119        v = None

#

Iterate

122        for i in range(self.iterations):

#

routing softmax $c_{ij} = \frac{\exp({b_{ij}})}{\sum_k\exp({b_{ik}})}$

124            c = self.softmax(b)

#

$\mathbf{s}_j = \sum_i{c_{ij} \hat{\mathbf{u}}_{j|i}}$

126            s = torch.einsum('bij,bijm->bjm', c, u_hat)

#

$\mathbf{v}_j = squash(\mathbf{s}_j)$

128            v = self.squash(s)

#

$a_{ij} = \mathbf{v}_j \cdot \hat{\mathbf{u}}_{j|i}$

130            a = torch.einsum('bjm,bijm->bij', v, u_hat)

#

$b_{ij} \gets b_{ij} + \mathbf{v}_j \cdot \hat{\mathbf{u}}_{j|i}$

132            b = b + a
133
134        return v

#

Margin loss for class existence

A separate margin loss is used for each output capsule and the total loss is the sum of them. The length of each output capsule is the probability that class is present in the input.

Loss for each output capsule or class $k$ is, $\mathcal{L}_k = T_k \max(0, m^{+} - \lVert\mathbf{v}_k\rVert)^2 + \lambda (1 - T_k) \max(0, \lVert\mathbf{v}_k\rVert - m^{-})^2$

$T_k$ is $1$ if the class $k$ is present and $0$ otherwise. The first component of the loss is $0$ when the class is not present, and the second component is $0$ if the class is present. The $\max(0, x)$ is used to avoid predictions going to extremes. $m^{+}$ is set to be $0.9$ and $m^{-}$ to be $0.1$ in the paper.

The $\lambda$ down-weighting is used to stop the length of all capsules from falling during the initial phase of training.

137class MarginLoss(Module):

#

157    def __init__(self, *, n_labels: int, lambda_: float = 0.5, m_positive: float = 0.9, m_negative: float = 0.1):
158        super().__init__()
159
160        self.m_negative = m_negative
161        self.m_positive = m_positive
162        self.lambda_ = lambda_
163        self.n_labels = n_labels

#

v, $\mathbf{v}_j$ are the squashed output capsules. This has shape [batch_size, n_labels, n_features]; that is, there is a capsule for each label.

labels are the labels, and has shape [batch_size].

165    def forward(self, v: torch.Tensor, labels: torch.Tensor):

#

$\lVert \mathbf{v}_j \rVert$

173        v_norm = torch.sqrt((v ** 2).sum(dim=-1))

#

$\mathcal{L}$ labels is one-hot encoded labels of shape [batch_size, n_labels]

177        labels = torch.eye(self.n_labels, device=labels.device)[labels]

#

$\mathcal{L}_k = T_k \max(0, m^{+} - \lVert\mathbf{v}_k\rVert)^2 + \lambda (1 - T_k) \max(0, \lVert\mathbf{v}_k\rVert - m^{-})^2$ loss has shape [batch_size, n_labels]. We have parallelized the computation of $\mathcal{L}_k$ for for all $k$.

183        loss = labels * F.relu(self.m_positive - v_norm) + \
184               self.lambda_ * (1.0 - labels) * F.relu(v_norm - self.m_negative)

#

$\sum_k \mathcal{L}_k$

187        return loss.sum(dim=-1).mean()