This page collects my notes and thoughts on the interesting papers that I read.

Astronomy

The emptiness inside: Finding gaps, valleys, and lacunae with geometric data analysis. (Contardo et al., 2022)
- The paper proposes a smooth approximation of the Gaussian kernel $k_N(s)\propto\exp\left(-\frac{s^2}{2}\right)$ within the 3-sigma region that has a compact support:
  $k(s)=\begin{cases}1 - 6\left(\frac{s}{3} \right)^2+8\left(\frac{s}{3}\right)^3-3\left(\frac{s}{3} \right)^4 & \text{ for } s\leq 3,\\ 0& \text{ for }s>3. \end{cases}$ .
  Note that this kernel is the solution of a spline k(z) with the power of 4:
  $k(s)=\begin{cases}k(0)=1,\, k(1)=0,\\ k'(0)=k'(1)=0,\\ k''(1)=0,\end{cases}$
  where $z=\frac{s}{3}$ .

Statistics

Machine Learning