Feature Dimensions
Overview
Feature dimensions refers to the number of degrees of freedom that a sample datapoint has. In the
standard case, a data point is represented by a vector
{% \vec{v} \in \mathbb{R}^n %}
That is, the vector would look like a
column vector
{%
\begin{bmatrix}
a \\
c \\
e \\
\end{bmatrix}
%}
Projection
An example of a dataset that spans 3 dimensions.
The dataset can be reduced to two dimensions by just ignoring one dimension. The following shows the same dataset but
graphed in only 2 dimensions
The result is effectively to remove one feature, or alternatively, setting it to zero. In this case, this is just projecting
each point down to its shadow on the x-y plane.
This type of transformation is often called a projection.
{% f(\begin{bmatrix}
x \\
y \\
z \\
\end{bmatrix}) \rightarrow \begin{bmatrix}
x \\
y \\
0 \\
\end{bmatrix} %}
Alternative Algorithms
Just removing dimensions is often not the best algorithm, as it tends to discard valuable information. Other algorithms exist
which are designed to retain as much information as possible when pruning dimensions.
Sparsity
