Signal Processing - November 2017 - 77
of DML is to explicitly learn a set of hierarchical nonlinear
transformations to map data points into other feature space
for comparing or matching by exploiting the architecture of
neural networks in deep learning, which unifies feature learning and metric learning into a joint learning framework. The
goal of this article is to provide an overview of recent advances
in DML techniques and their various applications in different
visual understanding tasks.
Mathematical background
To have a deep understanding of the concept of metric learning, we briefly introduce some necessary mathematical background. This section simply introduces the basic definitions
of a metric space and how to find a well-defined metric (or
pseudo-metric) over the original inputs by finding a mapping
into a Euclidean space.
Definition 1
A metric over a set X is a mapping d: X # X " R + and this
mapping d satisfies the following properties (axioms) for all
x, y, z ! X:
1) d (x, y) $ 0
2) d (x, y) = d (y, x)
3) d (x, z) # d (x, y) + d (y, z)
4) d (x, x) = 0
5) d (x, y) = 0 , x = y.
In Definition 1, axiom 1) is called the nonnegativity axiom,
axiom 2) is known as the symmetry axiom, axiom 3) is called the
triangle inequality axiom, axiom 4) is referred to as the identity
axiom, and axiom 5) is known as the identity of indiscernibles
axiom. A pair (X, d ), in which X is a set and d is a metric, is
called a metric space.
Definition 2
A pseudo-metric over a set X is a mapping d: X # X " R + satisfying the following properties (axioms) for all x, y, z ! X:
1) d (x, y) $ 0
2) d (x, y) = d (y, x)
3) d (x, z) # d (x, y) + d (y, z)
4) d (x, x) = 0.
A pair (X, d ), in which X is a set and d is a pseudo--metric,
is called a pseudo-metric space. We find that the pseudo--metric
doesn't need to satisfy the identity of indiscernibles axiom of the
metric. In metric learning, we may consider the pseudo-metrics
sometimes instead of metrics and refer to them as metrics.
The Euclidean distance is a widely used metric, which is usually adopted to measure the dissimilarity of data points. Give two
data points x and y, the Euclidean distance between x and y is
defined as
d (x, y) = x - y
2
= (x - y) T (x - y), (1)
in which a large distance means the dissimilarity of x and y, and
a small distance denotes the similarity of x and y.
The main objective of metric learning is to learn a metric over
the input data points. One widely used method to learn a metric
is to first map the input data points of the original space into a
Euclidean metric space and then compute the Euclidean distance
after the mapping. The following lemma declares this method.
Lemma 1
Let X = {x, y, z, g} be a set, f : X " R n be any well-de--
fined mapping, and d: R n # R n " R + be the Euclidean
metric over R n, then d f : X # X " R + defined by d f (x, y) =
d ( f (x), f (y)) = f (x) - f (y) 2 is a well-defined pseudo-metric
over X.
As Definition 2 (pseudo-metric) keeps for all data points
f (x), f (y), f (z) and it is independent of the selection of mapping
f, Lemma 1 is verified.
With Lemma 1, metric learning is the procedure of learning
the mapping function f. In addition, from the perspective of feature representation, the goal of metric learning can be obviously
interpreted as finding a new feature representation h = f (x)
of the data point x to better suit the Euclidean space. Thus, the
objective of metric learning is to find mapping f under various
loss functions and constraints.
An illustration
To simply illustrate how metric learning works, we conducted
an experiment on the MNIST data set [36]. We sampled 150
samples from three classes of handwritten digits: four, seven, and
nine, where each class contains 50 samples. Each digit sample is
a 28 × 28 grayscale image, and we lexicographically converted
it into a 784-dimensional feature vector. We employed the linear
discriminant analysis (LDA) as a metric learning method to project data points from the original space to the transformed space.
Figure 1 shows an example of how metric learning works on this
real-world data set. As seen, samples from different classes are
mixed in the original space, and they are well separated in the
transformed space.
In this article, we focus on DML, which explicitly learns a
nonlinear mapping f to map data points into a new feature space
by exploiting the architecture of deep neural networks, in which
the nonlinear mapping f is parameterized by the weights and
biases of deep neural network.
DML
In this section, we introduce the basic concepts of DML, and discuss
the similarities and differences among the existing DML methods.
Basic concepts
From Lemma 1, DML is to explicitly learn a nonlinear mapping
f to map data points into a new feature space by exploiting the
architecture of deep neural networks, in which the nonlinear
mapping f is parameterized by the weights and biases of deep
neural network.
Given a simple neural network architecture as shown in
(0)
Figure 2, for an input x ! R r , its output of the first layer is
(1)
h (1) = { ^W (1) x + b (1)h ! R r , and its output of the mth layer
(m)
is h (m) = { ^W (m) h (m -1) + b (m)h ! R r , 1 # m # M, h (0) = x,
(m)
(m -1)
(m)
where matrix W (m) ! R r # r
and vector b (m) ! R r are weights
and biases of this neural network, M is the total number
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
|
77
Table of Contents for the Digital Edition of Signal Processing - November 2017
Signal Processing - November 2017 - Cover1
Signal Processing - November 2017 - Cover2
Signal Processing - November 2017 - 1
Signal Processing - November 2017 - 2
Signal Processing - November 2017 - 3
Signal Processing - November 2017 - 4
Signal Processing - November 2017 - 5
Signal Processing - November 2017 - 6
Signal Processing - November 2017 - 7
Signal Processing - November 2017 - 8
Signal Processing - November 2017 - 9
Signal Processing - November 2017 - 10
Signal Processing - November 2017 - 11
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Signal Processing - November 2017 - 13
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Signal Processing - November 2017 - 15
Signal Processing - November 2017 - 16
Signal Processing - November 2017 - 17
Signal Processing - November 2017 - 18
Signal Processing - November 2017 - 19
Signal Processing - November 2017 - 20
Signal Processing - November 2017 - 21
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Signal Processing - November 2017 - 25
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Signal Processing - November 2017 - 28
Signal Processing - November 2017 - 29
Signal Processing - November 2017 - 30
Signal Processing - November 2017 - 31
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Signal Processing - November 2017 - 148
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Signal Processing - November 2017 - 150
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Signal Processing - November 2017 - 157
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Signal Processing - November 2017 - 171
Signal Processing - November 2017 - 172
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Signal Processing - November 2017 - 175
Signal Processing - November 2017 - 176
Signal Processing - November 2017 - Cover3
Signal Processing - November 2017 - Cover4
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