Signal Processing - November 2017 - 78

4

4

3

9

7

h(2)

2
1

w(2), b(2)

0
−1

h(1)

−2
−3

w(1), b(1)

−4
−5
−5

0
(a)

x

5

0.02

4

7

FIGURE 2. A simple illustration of a feed-forward neural network architecture

9

used in many DML methods [23]. The input to the network is x, and the output
of the hidden layer and the top layer is h (1) and h (2), respectively, in which
W (m) and b (m) are weights and biases of this neural network, 1 # m # 2.

0.015
0.01
0.005

has been employed by many DML algorithms recently. Generally, CNNs comprise several convolutional layers, subsampling
layers, and fully connected layers. Specifically, the feed-forward
network in Figure 2 is the fully connected part of CNN architecture in Figure 3.

0
−0.005
−0.01
−0.03 −0.02 −0.01

0
(b)

0.01

0.02

0.03

DML via Siamese networks

FIGURE 1. An example on the MNIST data set to illustrate how metric
learning works. For ease of visualization, these samples are embedded
into the two-dimensional feature spaces (a) and (b) by principal component analysis and LDA, respectively.

of layers, r (m) is the number of neural units in the mth layer,
{: R 7 R is a nonlinear activation function (e.g., sigmoid and
tanh). In this way, the output of this neural network at the most
top layer can be represented as:
	

f (x) = h (M ) = { ^W (M ) h (M -1) + b (M ) h ! R r
h

(1)

= { ^W x + b h,
(1)

(1)

r (0)



(2)

	

d f (x i, x j) = d ( f (x i), f (x j)) = f (x i) - f (x i) 2 .(3)

/

h (d f (x i, x j) - x 1)

( i, j ) ! S

+

/

h (x 2 - d f (x i, x j)),

	

(4)

(i, j) ! D

r (M )

The goal of DML is to learn the mapping f under certain constraints, where f is parameterized by the weights and biases of
the neural network.
Figure 3 shows another widely used architecture of neural
network, called a convolutional neural network (CNN), which
78

L ^{W (m), b (m)} mM= 1 h =

(M )

is a parametric nonwhere the mapping f : R " R
linear function which is determined by a set of parameters
{W (m), b (m)} mM= 1 .
Let f be the mapping function of a neural network. For an
input x, f (x) is its output through this neural network. According
to Lemma 1, the distance of data points x i and x j in the deep
metric space is to calculate the Euclidean distance between f (x i)
and f (x j) as:
	

Typically, there are two main types of neural networks used in
DML methods: Siamese networks and triplet networks. Figure 4
shows the diagrams of Siamese networks and triplet networks for
DML. For a pair of data points ^x i, x jh, we say they are a similar pair (or positive pair) if x i and x j are semantically similar,
and they are called a dissimilar pair (or negative pair) if they are
semantically dissimilar. Let S = {(i, j)} be an index set consisting of similar pairs, and D = {(i, j)} be an index set consisting of
dissimilar pairs, respectively. The Siamese networks-based DML
framework is trained by minimizing a contrastive loss function:

where h (x) = max (0, x) is the hinge loss function, and x 1 and
x 2 are two positive thresholds, x 1 1 x 2 . By minimizing this
contrastive loss function, we expect the distance d f (x i, x j) for
a positive pair to be less than a smaller parameter x 1 and that of
a negative pair to be larger than a larger parameter x 2 . Figure 5
shows the key idea of such DML methods.
Dimensionality reduction by learning an invariant -mapping
(DrLIM) [18], [19] is an important work on DML via Siamese
networks for face verification. DrLIM exploited discriminative
information from neighborhood relationships of samples to learn
the mapping function. There are four characteristics in their
method: 1) it only needs -neighborhood relationships between
training samples; 2) it learns distance functions that are robust
to nonlinear transformations of the input signals; 3) the learned
function can handle the unseen classes problem so that the new

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November 2017

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Table of Contents for the Digital Edition of Signal Processing - November 2017
