\[ \begin{align}\begin{aligned}\newcommand{\bs}{\boldsymbol}
\newcommand{\dp}{\displaystyle}
\newcommand{\rm}{\mathrm}
\newcommand{\cl}{\mathcal}
\newcommand{\pd}{\partial}\\\newcommand{\cd}{\cdot}
\newcommand{\cds}{\cdots}
\newcommand{\dds}{\ddots}
\newcommand{\lag}{\langle}
\newcommand{\lv}{\lVert}
\newcommand{\ol}{\overline}
\newcommand{\od}{\odot}
\newcommand{\ra}{\rightarrow}
\newcommand{\rag}{\rangle}
\newcommand{\rv}{\rVert}
\newcommand{\seq}{\subseteq}
\newcommand{\td}{\tilde}
\newcommand{\vds}{\vdots}
\newcommand{\wh}{\widehat}\\\newcommand{\0}{\boldsymbol{0}}
\newcommand{\1}{\boldsymbol{1}}
\newcommand{\a}{\boldsymbol{\mathrm{a}}}
\newcommand{\b}{\boldsymbol{\mathrm{b}}}
\newcommand{\c}{\boldsymbol{\mathrm{c}}}
\newcommand{\d}{\boldsymbol{\mathrm{d}}}
\newcommand{\e}{\boldsymbol{\mathrm{e}}}
\newcommand{\f}{\boldsymbol{\mathrm{f}}}
\newcommand{\g}{\boldsymbol{\mathrm{g}}}
\newcommand{\h}{\boldsymbol{\mathrm{h}}}
\newcommand{\i}{\boldsymbol{\mathrm{i}}}
\newcommand{\j}{\boldsymbol{j}}
\newcommand{\m}{\boldsymbol{\mathrm{m}}}
\newcommand{\n}{\boldsymbol{\mathrm{n}}}
\newcommand{\o}{\boldsymbol{\mathrm{o}}}
\newcommand{\p}{\boldsymbol{\mathrm{p}}}
\newcommand{\q}{\boldsymbol{\mathrm{q}}}
\newcommand{\r}{\boldsymbol{\mathrm{r}}}
\newcommand{\u}{\boldsymbol{\mathrm{u}}}
\newcommand{\v}{\boldsymbol{\mathrm{v}}}
\newcommand{\w}{\boldsymbol{\mathrm{w}}}
\newcommand{\x}{\boldsymbol{\mathrm{x}}}
\newcommand{\y}{\boldsymbol{\mathrm{y}}}
\newcommand{\z}{\boldsymbol{\mathrm{z}}}\\\newcommand{\A}{\boldsymbol{\mathrm{A}}}
\newcommand{\B}{\boldsymbol{\mathrm{B}}}
\newcommand{\C}{\boldsymbol{\mathrm{C}}}
\newcommand{\D}{\boldsymbol{\mathrm{D}}}
\newcommand{\H}{\boldsymbol{\mathrm{H}}}
\newcommand{\I}{\boldsymbol{\mathrm{I}}}
\newcommand{\K}{\boldsymbol{\mathrm{K}}}
\newcommand{\M}{\boldsymbol{\mathrm{M}}}
\newcommand{\N}{\boldsymbol{\mathrm{N}}}
\newcommand{\P}{\boldsymbol{\mathrm{P}}}
\newcommand{\Q}{\boldsymbol{\mathrm{Q}}}
\newcommand{\S}{\boldsymbol{\mathrm{S}}}
\newcommand{\U}{\boldsymbol{\mathrm{U}}}
\newcommand{\W}{\boldsymbol{\mathrm{W}}}
\newcommand{\X}{\boldsymbol{\mathrm{X}}}
\newcommand{\Y}{\boldsymbol{\mathrm{Y}}}
\newcommand{\Z}{\boldsymbol{\mathrm{Z}}}\\\newcommand{\R}{\mathbb{R}}\\\newcommand{\cE}{\mathcal{E}}
\newcommand{\cX}{\mathcal{X}}
\newcommand{\cY}{\mathcal{Y}}\\\newcommand{\ld}{\lambda}
\newcommand{\Ld}{\boldsymbol{\mathrm{\Lambda}}}
\newcommand{\sg}{\sigma}
\newcommand{\Sg}{\boldsymbol{\mathrm{\Sigma}}}
\newcommand{\th}{\theta}
\newcommand{\ve}{\varepsilon}\\\newcommand{\mmu}{\boldsymbol{\mu}}
\newcommand{\ppi}{\boldsymbol{\pi}}
\newcommand{\CC}{\mathcal{C}}
\newcommand{\TT}{\mathcal{T}}\\
\newcommand{\bb}{\begin{bmatrix}}
\newcommand{\eb}{\end{bmatrix}}
\newcommand{\bp}{\begin{pmatrix}}
\newcommand{\ep}{\end{pmatrix}}
\newcommand{\bv}{\begin{vmatrix}}
\newcommand{\ev}{\end{vmatrix}}\\\newcommand{\im}{^{-1}}
\newcommand{\pr}{^{\prime}}
\newcommand{\ppr}{^{\prime\prime}}\end{aligned}\end{align} \]
Part 3 Clustering
Clustering is the task of partitioning the data points into natural groups called clusters, such that points within a group are very similar, whereas points between different groups are as dissimilar as possible. Depending on the data and desired cluster characteristics, there are different types of clustering paradigms such as representative-based, hierarchical, density-based, graph-based, and spectral clustering. Clustering is an unsupervised learning approach since it does not require a separate training dataset to learn the model parameters.