Shared Gaussian Process Latent Variables Models - Oxford Brookes ...

2.1. INTRODUCTION 15
data can be equally well described by two different models. However, data from
the input domain outside the training data might result in different behavior from
each model. Similarly different assumptions often leads to different models. The
degrees of freedom of a representation is equal to the number of parameters or
dimensions. However, this parameterization does not need to be representing the
data in the correct way. When each dimension in the representation describes or
models a single degree of freedom in the data we say that the data is in its intrinsic
representation.
We will separate the task of modeling into data driven and model driven. Data
driven modeling is, when given a set of training data we try to learn a model from
the data, this is different from model based modeling when we try to fit or match
a specific model to the data. This thesis will focus on data driven models.
2.1.1 Curse of Dimensionality
We spend our lives in a world that is essentially three dimensional. It is in this
world we build our understanding of concepts such as distance and volume. In
Machine Learning we often deal with data compromising many more dimensions.
Many of the concepts we learn to recognize in two and three dimensions cannot
easily be extrapolated to higher dimensions. One example is the relationship be-
tween the volume of a hyper-sphere of diameter 2 and a cube with side length
2.
Vcube(d) = (2) d
Vsphere(d) = 2d Π d
2
d · Γ( d
2 )
(2.1)
(2.2)