/
triplet_loss.py
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/
triplet_loss.py
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import torch
import torch.nn.functional as F
import numpy as np
import sklearn
def pairwise_mahalanobis(S1, S2, Cov_1=None):
"""
S1: C1 x K matrix (torch.FloatTensor)
-> C1 K-dimensional semantic features
S2: C2 x K matrix (torch.FloatTensor)
-> C2 K-dimensional semantic features
Sigma_1: K x K matrix (torch.FloatTensor)
-> inverse of the covariance matrix Sigma; used to compute Mahalanobis distances
by default Sigma is the identity matrix (and so distances are euclidean distances)
returns an C1 x C2 matrix corresponding to the Mahalanobis distance between each element of S1 and S2
(Equation 5)
"""
if S1.dim() != 2 or S2.dim() != 2 or S1.shape[1] != S2.shape[1]:
raise RuntimeError("Bad input dimension")
C1, K = S1.shape
C2, K = S2.shape
if Cov_1 is None:
Cov_1 = torch.eye(K)
if Cov_1.shape != (K, K):
raise RuntimeError("Bad input dimension")
S1S2t = S1.matmul(Cov_1).matmul(S2.t())
S1S1 = S1.matmul(Cov_1).mul(S1).sum(dim=1, keepdim=True).expand(-1, C2)
S2S2 = S2.matmul(Cov_1).mul(S2).sum(dim=1, keepdim=True).t().expand(C1, -1)
return torch.sqrt(torch.abs(S1S1 + S2S2 - 2. * S1S2t) + 1e-32) # to avoid numerical instabilities
def distance_matrix(S, mahalanobis=True, mean=1., std=0.5):
"""
S: C x K matrix (numpy array)
-> K-dimensional semantic features of C classes
mahalanobis: indicates whether to use Mahalanobis distance (uses euclidean distance if False)
mean & std: target mean and standard deviation
returns a C x C matrix corresponding to the Mahalanobis distance between each pair of elements of S
rescaled to have approximately target mean and standard deviation while keeping values positive
(Equation 6)
"""
Cov_1 = None
if mahalanobis:
Cov, _ = sklearn.covariance.ledoit_wolf(S) # robust estimation of covariance matrix
Cov_1 = torch.FloatTensor(np.linalg.inv(Cov))
S = torch.FloatTensor(S)
distances = pairwise_mahalanobis(S, S, Cov_1)
# Rescaling to have approximately target mean and standard deviation while keeping values positive
max_zero_distance = distances.diag().max()
positive_distances = np.array([x for x in distances.view(-1) if x > max_zero_distance])
emp_std = float(positive_distances.std())
emp_mean = float(positive_distances.mean())
distances = F.relu(std * (distances - emp_mean) / emp_std + mean)
emp_std = float(distances.std())
emp_mean = float(distances.mean())
distances = F.relu(std * (distances - emp_mean) / emp_std + mean)
return distances
def _pairwise_distances(feature, squared=False):
"""Compute the 2D matrix of distances between all the embeddings.
Args:
feature: tensor of shape (batch_size, embed_dim)
squared: Boolean. If true, output is the pairwise squared euclidean distance matrix.
If false, output is the pairwise euclidean distance matrix.
Returns:
pairwise_distances: tensor of shape (batch_size, batch_size)
"""
# Get the dot product between all embeddings
# shape (batch_size, batch_size)
dot_product = feature@torch.transpose(feature,1,0)
# Get squared L2 norm for each embedding. We can just take the diagonal of `dot_product`.
# This also provides more numerical stability (the diagonal of the result will be exactly 0).
# shape (batch_size,)
square_norm = torch.diag(dot_product)
# Compute the pairwise distance matrix as we have:
# ||a - b||^2 = ||a||^2 - 2 <a, b> + ||b||^2
# shape (batch_size, batch_size)
distances = square_norm.unsqueeze(1) - 2.0 * dot_product + square_norm.unsqueeze(0)
# Because of computation errors, some distances might be negative so we put everything >= 0.0
distances = torch.relu(distances)
if not squared:
# Because the gradient of sqrt is infinite when distances == 0.0 (ex: on the diagonal)
# we need to add a small epsilon where distances == 0.0
mask = distances.eq(0.0).type(torch.cuda.FloatTensor)
distances = distances + mask * 1e-16
distances = torch.sqrt(distances)
# Correct the epsilon added: set the distances on the mask to be exactly 0.0
distances = distances * (1.0 - mask)
return distances
def _get_triplet_mask(labels):
"""Return a 3D mask where mask[a, p, n] is True iff the triplet (a, p, n) is valid.
A triplet (i, j, k) is valid if:
- i, j, k are distinct
- labels[i] == labels[j] and labels[i] != labels[k]
Args:
labels: tf.int32 `Tensor` with shape [batch_size]
"""
# Check that i, j and k are distinct
indices_equal = torch.eye(labels.shape[0]).type(torch.cuda.ByteTensor)
indices_not_equal = ~indices_equal
i,j = indices_not_equal.shape
i_not_equal_j = indices_not_equal.unsqueeze(2)
i_not_equal_k = indices_not_equal.unsqueeze(1)
j_not_equal_k = indices_not_equal.unsqueeze(0)
distinct_indices = (i_not_equal_j&i_not_equal_k)&j_not_equal_k
# Check if labels[i] == labels[j] and labels[i] != labels[k]
label_equal = labels.unsqueeze(0).eq(labels.unsqueeze(1))
label_equal = label_equal.type(torch.cuda.ByteTensor)
i_equal_j = label_equal.unsqueeze(2)
i_equal_k = label_equal.unsqueeze(1)
valid_labels = i_equal_j&(~i_equal_k)
# Combine the two masks
mask = distinct_indices&valid_labels
return mask
def _get_anchor_positive_triplet_mask(labels):
"""Return a 2D mask where mask[a, p] is True iff a and p are distinct and have same label.
Args:
labels: tf.int32 `Tensor` with shape [batch_size]
Returns:
mask: tf.bool `Tensor` with shape [batch_size, batch_size]
"""
# Check that i and j are distinct
indices_equal = torch.eye(labels.shape[0]).type(torch.cuda.ByteTensor)
indices_not_equal = ~indices_equal
# Check if labels[i] == labels[j]
# Uses broadcasting where the 1st argument has shape (1, batch_size) and the 2nd (batch_size, 1)
labels_equal = labels.unsqueeze(0).eq(labels.unsqueeze(1))
labels_equal = labels_equal.type(torch.cuda.ByteTensor)
# Combine the two masks
mask = indices_not_equal&labels_equal
return mask
def _get_anchor_negative_triplet_mask(labels):
"""Return a 2D mask where mask[a, n] is True iff a and n have distinct labels.
Args:
labels: tf.int32 `Tensor` with shape [batch_size]
Returns:
mask: tf.bool `Tensor` with shape [batch_size, batch_size]
"""
# Check if labels[i] != labels[k]
# Uses broadcasting where the 1st argument has shape (1, batch_size) and the 2nd (batch_size, 1)
labels_equal = labels.unsqueeze(0).eq(labels.unsqueeze(1))
mask = ~labels_equal
return mask
def batch_all_triplet_loss(feature, labels, margin, squared=False):
"""Build the triplet loss over a batch of embeddings.
We generate all the valid triplets and average the loss over the positive ones.
Args:
feature: tensor of shape (batch_size, embed_dim)
labels: labels of the batch, of size (batch_size,)
margin: margin for triplet loss
Returns:
triplet_loss: scalar tensor containing the triplet loss
"""
pairwise_dist = _pairwise_distances(feature, squared=squared)
# shape (batch_size, batch_size, 1)
anchor_positive_dist = pairwise_dist.unsqueeze(2)
# shape (batch_size, 1, batch_size)
anchor_negative_dist = pairwise_dist.unsqueeze(1)
# Compute a 3D tensor of size (batch_size, batch_size, batch_size)
# triplet_loss[i, j, k] will contain the triplet loss of anchor=i, positive=j, negative=k
# Uses broadcasting where the 1st argument has shape (batch_size, batch_size, 1)
# and the 2nd (batch_size, 1, batch_size)
triplet_loss = anchor_positive_dist - anchor_negative_dist + margin
# print(triplet_loss.shape)
# Put to zero the invalid triplets
# (where label(a) != label(p) or label(n) == label(a) or a == p)
mask = _get_triplet_mask(labels)
mask = mask.type(torch.cuda.FloatTensor)
# print(mask.shape)
triplet_loss = mask*triplet_loss
# Remove negative losses (i.e. the easy triplets)
triplet_loss = torch.relu(triplet_loss)
# Count number of positive triplets (where triplet_loss > 0)
valid_triplets = torch.gt(triplet_loss,1e-16).type(torch.cuda.FloatTensor)
num_positive_triplets = torch.sum(valid_triplets)
num_valid_triplets = torch.sum(mask)
fraction_positive_triplets = num_positive_triplets / (num_valid_triplets + 1e-16)
# Get final mean triplet loss over the positive valid triplets
triplet_loss = torch.sum(triplet_loss) / (num_positive_triplets + 1e-16)
return triplet_loss, fraction_positive_triplets
def batch_hard_triplet_loss(feature, labels , margin, squared=False):
"""Build the triplet loss over a batch of embeddings.
For each anchor, we get the hardest positive and hardest negative to form a triplet.
Args:
feature: tensor of shape (batch_size, embed_dim)
labels: labels of the batch, of size (batch_size,)
margin: margin for triplet loss
Returns:
triplet_loss: scalar tensor containing the triplet loss
"""
# Get the pairwise distance matrix
pairwise_dist = _pairwise_distances(feature, squared=squared)
# For each anchor, get the hardest positive
# First, we need to get a mask for every valid positive (they should have same label)
mask_anchor_positive = _get_anchor_positive_triplet_mask(labels)
mask_anchor_positive = mask_anchor_positive.type(torch.cuda.FloatTensor)
# We put to 0 any element where (a, p) is not valid (valid if a != p and label(a) == label(p))
anchor_positive_dist = mask_anchor_positive*pairwise_dist
# shape (batch_size, 1)torch.max(a, dim=1, keepdim=True)
hardest_positive_dist = torch.max(anchor_positive_dist, dim=1, keepdim=True)[0]
# print("hardest_positive_dist", torch.mean(hardest_positive_dist))
# For each anchor, get the hardest negative
# First, we need to get a mask for every valid negative (they should have different labels)
mask_anchor_negative = _get_anchor_negative_triplet_mask(labels)
mask_anchor_negative = mask_anchor_negative.type(torch.cuda.FloatTensor)
# We add the maximum value in each row to the invalid negatives (label(a) == label(n))
max_anchor_negative_dist = torch.max(pairwise_dist, dim=1, keepdim=True)[0]
anchor_negative_dist = pairwise_dist + max_anchor_negative_dist * (1.0 - mask_anchor_negative)
# shape (batch_size,)
hardest_negative_dist = torch.min(anchor_negative_dist, dim=1, keepdim=True)[0]
# print("hardest_negative_dist", torch.mean(hardest_negative_dist))
# Combine biggest d(a, p) and smallest d(a, n) into final triplet loss
triplet_loss = torch.relu(hardest_positive_dist - hardest_negative_dist + margin)
# Get final mean triplet loss
triplet_loss = torch.mean(triplet_loss)
return triplet_loss