Localized Supervised Metric Learning on ... - Researcher - IBM

epresented by a N-dimensional feature vector x. Examples
of features are the mean and variance of the sensor
measures, or Wavelet coefficients. The prior belief
of physicians is captured as labels on some of the patients.
With this formulation, our goal is to learn a generalized
Mahalanobis distance between patient x i and
patient x j defined as:
√
d m (x i , x j ) = (x i − x j ) T P(x i − x j ) (1)
where P ∈ R N×N is called the precision matrix. Matrix
P is positive semi-definite and is used to incorporate
the correlations between different feature dimensions.
The key is to learn the optimal P such that the
resulting distance metric has the following properties:
Figure 1. Retrieving patients based on
their clinical similarity to a query patient
and using the retrieved patients to project
the evolution of patient’s clinical characteristics.
comparison. In this paper, we leverage both statistical
methods and Wavelet methods to extract features over
the temporal data.
<strong>Supervised</strong> metric learning has been studied in the
past [11, 12, 5]. The goal has been to learn a distance
metric such that samples in the same class are close and
those in different classes are far away. The common
treatment is to add constraints and regularization terms
into the objective function and then to solve it using optimization
methods. To avoid a large number of constraints,
in this paper we model this problem as trace
ratio problem which can be solved effectively (similar
to Wang et al. [9]).
3 <strong>Localized</strong> <strong>Supervised</strong> <strong>Metric</strong> <strong>Learning</strong>
In this section we present the supervised metric
learning problem in the context of patient similarity
measure. When a physician looks for similar patients
in a database, the similarity is often based not only
on quantitative measurements such as lab results, sensor
measurements, age and sex, but also on the physician’s
assessment of the disease type and stage. The
assessment would potentially influence the relative importance
a physician places on different measurements
or groups of measurements. To compute this specific
notion of similarity, we propose to learn a distance metric
that can automatically adjust the importance of each
numeric feature by leveraging the physician’s belief.
Formally, quantitative measurements of a patient are
• Within-class compactness: patients of the same label
are close together;
• Between-class scatterness: patients of different labels
are far away from each other.
To formally measure these properties, we use two kinds
of neighborhoods as defined in [10]: The homogeneous
neighborhood of x i , denoted as Ni o , is the k-nearest
patients of x i with the same label. The heterogeneous
neighborhood of x i , denoted as Ni e , is the k-nearest patients
of x i with different labels.
Based on these two neighborhoods, we define the local
compactness of point x i as
C i =
∑
d 2 m(x i , x j ) (2)
x j∈N o i
and the local scatter ness of point x i as
S i =
∑
x k ∈N e i
d 2 m(x i , x k ) (3)
The discriminability of the distance metric d m is defined
as
∑
J = ∑ i C ∑ ∑
i i x j∈N
(x
i
i S =
o i − x j ) T P(x i − x j )
∑ ∑
i i x k ∈N
(x
i
e i − x k ) T P(x i − x k )
(4)
The goal is to find a P that minimizes J , which is
equivalent to minimizing the local compactness and
maximizing the local scatterness simultaneously. In
contrast with linear discriminant analysis [4] , which
seeks for a discriminant subspace in a global sense,
the localized supervised metric aims to learn a distance
metric with enhanced local discriminability. To minimize
J , we formulate the problem as a trace ratio minimization
problem [9] and use the decomposed Newtown’s
method to find the solution [6].