Visualizing Data by Projection

Description:

• We can visualize the data set that each data point has a vector, by projecting each data point on a one-, two- or three-dimensional space.
  • Each “view” corresponds to a particular projection on a direction
• In the 1D case, the projection on a given line passing through $x^0$ and direction $a$ is related to a scoring function, of the form $f:x\to a^{T}x+b$ where $x\in {\mathbb{R}}^n$ is a data point, and $a\in {\mathbb{R}}^n,b\in R$ are given.
  • WLOG we can set $||a|{|}_2=1$

Centering:

• We can center the scores so that the average score across data points is zero, that is:
  • $0=\frac{1}{m}\sum _{i=1}^{m}f(x_i)=\frac{1}{m}\sum _{i=1}^m(a^T{x}_i+b)=a^T\hat{x}+b$
• This implies a condition on $b:b=-a^T\bar{x}$, where $\hat{x}:=\frac{1}{n}\sum _{i=1}^n{x}_i\in {\mathbb{R}}^n$ is the center of the points for that $a$ projection
• Then we can adjust the scoring function to be $f(x)=a^T(x-\hat{x})$
• Equivalently we project the centered data points on a line passing through $0$