# Short Text Clustering Algorithms, Application and Challenges: A Survey
## 3 Components of Short Text Clustering
### Dimensionality Reduction
#### 3.3.1 Principal Component Analysis (PCA)  
It is the most common technique in data analysis and dimensionality reduction, and almost all scientific disciplines use it. PCA seeks to find the most meaningful basis for re-expressing a given dataset [101]. This entails identifying new uncorrelated variables and maximizing variance to maintain as much variation as possible [128]. This new basis is expected to reveal hidden structures in the dataset and filter out noise [129]. PCA has numerous applications, including dimensionality reduction, feature extraction, data compression and visualization. A dataset with observations on \(p\) numerical variables for each of \(n\) entities or individuals is the standard context for PCA as an exploratory data analysis tool. These data values define \(p\)\(n\)-dimensional vectors \(x_{1},\ldots,\ x_{p}\), or equivalently, an \(x\times p\) data matrix \(X\), with the \(j\)th column containing the vector \(x_{j}\) of observations of the \(j\)th variable. We are looking \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Method** & **Technique** & **Year** & **Dataset** & **ACC** & **NMI** & **Ref.** \\ \hline \multirow{3}{*}{RecNN} & \multirow{3}{*}{K-means} & \multirow{3}{*}{2017} & StackOverflow & 0.4079 & 0.4058 & \multirow{3}{*}{[25]} \\ \cline{3-4}  & & & Biomedical & 0.3705 & 0.3385 & \\ \cline{3-4}  & & & StackOverflow & 0.4493 & 0.4093 & \\ \cline{3-4}  & & & Biomedical & 0.356 & 0.3403 & \\ \hline \multirow{3}{*}{STC\({}^{2}\)} & \multirow{3}{*}{K-means} & \multirow{3}{*}{StackOverflow} & 0.5114 & 0.4908 & \multirow{3}{*}{[25]} \\ \cline{3-4}  & & & Biomedical & 0.43 & 0.3818 & \\ \hline \multirow{3}{*}{SG-DHNMF} & \multirow{3}{*}{/} & \multirow{3}{*}{2020} & Tweets & 0.86 & \multirow{3}{*}{[89]} \\ \cline{3-4}  & & & StackOverflow & 0.65 & \\ \cline{3-4}  & & & Tweets & 0.79 & \\ \cline{3-4}  & & & StackOverflow & 0.5 & \\ \hline \multirow{3}{*}{TE-GSDMM} & \multirow{3}{*}{K-means++} & \multirow{3}{*}{2022} & Web & \multirow{3}{*}{0.514} & \multirow{3}{*}{[123]} \\ \cline{3-4}  & & & Service & & & \\ \hline \multirow{3}{*}{BERT} & \multirow{3}{*}{K-means} & \multirow{3}{*}{2021} & Tweets & 0.8126 & 0.867 & \multirow{3}{*}{[124]} \\ \cline{3-4}  & & & StackOverflow & 0.6253 & 0.5962 & \\ \hline \multirow{3}{*}{STN-GAE} & \multirow{3}{*}{K-means} & \multirow{3}{*}{2021} & Tweets & 0.4049 & 0.3546 & \multirow{3}{*}{[124]} \\ \cline{3-4}  & & & StackOverflow & 0.4049 & 0.4492 & \\ \hline \multirow{3}{*}{SCA-AE} & \multirow{3}{*}{} & Tweets & 0.8485 & 0.8919 & \multirow{3}{*}{[125]} \\ \cline{3-4}  & & & StackOverflow & 0.7655 & 0.6599 & \\ \hline TAE & \multirow{3}{*}{K-means} & \multirow{3}{*}{2022} & StackOverflow & & 62.8 & [19] \\ \cline{3-4}  & & & AG News & 0.6467 & 0.4151 & \\ \cline{3-4}  & & & AG News & 0.8038 & 0.538 & \\ \hline BERT+ Mean & \multirow{3}{*}{DEC} & AG News & 0.8019 & 0.5383 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Analysis of the studies on the deep learning similarity measures. for a linear combination of the columns of matrix \(X\) with the slightest variance. The linear combination can be written as the following equation (see Equation (3)) [128]:  
\[\sum_{j=1}^{p}a_{j}x_{j}=X_{a}, \tag{3}\]  
where \(a\) represents a vector of constants \(a_{1},\ a_{2},\ldots,a_{p}\). This linear combination's variance is given as Equation (4):  
\[Var(x_{a})=a^{T}sa\, \tag{4}\]  
where \(S\) represents the sample covariance matrix. The goal is to find the linear combination with the least amount of variance. This is equivalent to maximizing Equation (5):  
\[a^{T}s_{a}-\lambda\left(a^{T}a-1\right)\, \tag{5}\]  
where \(\lambda\) is a Lagrange multiplier. ICA is a statistical modelling method that expresses observed data as a linear transformation [130]. A statistical 'latent variables' model can be used to rigorously define ICA [131]. Assume we find \(n\) linear mixtures \(x_{1},\ldots,x_{n}\) of \(n\) independent components.