## 3 Problem Representation  
A dictionary \(\mathbf{\Phi}\in R^{M\times K}\) containing \(K\) atoms is defined as  
\[\mathbf{A}\approx\mathbf{\Phi}_{\mathbf{\alpha}}, \tag{1}\]  
where \(\mathbf{A}\in R^{M\times N}\) is training data that contain \(C\) classes, i.e., \(\mathbf{A}^{1},\mathbf{A}^{2}\ldots\mathbf{A}^{c}\ldots\mathbf{A}^{C}\), indexed in \(I_{N}\). \(I_{c}\) represents a set containing the indices of the training examples belonging to the \(c\)th class, and equivalently, \(\sum\limits_{c=1}^{C}|I_{c}|=N\), where \(|.|\) is the cardinality of a set. In Equation (1), \(\mathbf{\alpha}\in R^{K\times N}\) is the matrix of sparse codes representing examples in \(\mathbf{A}\). In terms of classes of examples, a class of data can be expressed as  
\[\mathbf{A}^{c}\approx\mathbf{\Phi}\mathbf{\alpha}^{c}, \tag{2}\]  
where \(\mathbf{\alpha}^{c}\in R^{K\times|I_{c}|}\) is the matrix consisting of sparse representation vectors of the examples belonging to \(\mathbf{A}^{c}\). The constrained optimization problem of dictionary and sparse codes learning can be put in the form  
\[<\mathbf{\Phi},\mathbf{\alpha}>=\underset{\mathbf{\Phi},\mathbf{\alpha}}{min}||\mathbf{A}-\bm {\Phi}\mathbf{\alpha}||_{F}^{2}\ \ s.t.\ \forall i,||\mathbf{\alpha}_{i}||_{p}\leq t, \tag{3}\]  
\begin{table}
\begin{tabular}{c l l l l} \hline \hline
**Authors** & **Method(s)** & **Datasets** & \begin{tabular}{c} **Accuracy** \\ **(\%)** \\ \end{tabular}  & \begin{tabular}{c} **Train vs. Test Examples (\%)—Other** \\ **Conditions** \\ \end{tabular}  \\ \hline \multirow{8}{*}{ \begin{tabular}{c} Zhang \\ \end{tabular}  } & Twin-Incoherent Self-Expressive & \begin{tabular}{c} YaleB \\ AR \\ CMU PIE \\ \end{tabular}  & \begin{tabular}{c} 98.2 \\ 98.60 \\ 95.30 \\ 95.30 \\ \end{tabular}  & \begin{tabular}{c} 50/50 \\ 77/23 \\ 24/76 \\ 15 random split train/test—3240 \\ examples (324 images per person) \\ \end{tabular}  \\  & Pair Learning for Classification & \ \begin{tabular}{c} UMIST \\ 15 Scene \\ ETH80 \\ \end{tabular}  &
\begin{tabular}{c} 93.40 \\ 98.80 \\ 98.30 \\ 98.30 \\ 98.30 \\ 98.30 \\ 98.40 \\ 98.50 \\ 98.90 \\ 98.50 \\ 98.50 \\ 98.  
where \(\mathbf{\alpha}_{i}\in R^{K}\) is the \(i\)th column of \(\mathbf{\alpha}\) that represents the sparse codes of the \(i\)th example of data \(\mathbf{A}\). The constant \(t\) controls the sparsity of the columns of the sparse codes matrix. The \(||.||_{F}\) and \(||.||_{p}\) represent the Frobenius norm and \(l_{p}\)-norm of a vector, respectively. The sparse codes learned during dictionary learning can be used for training the parameters of a linear classifier as below:  
\[\mathbf{B}=\underset{\mathbf{B}}{min}\sum_{i=1}^{N}\mathbf{\varepsilon}\{\mathbf{ h}_{i},f(\mathbf{\alpha}_{i},\mathbf{B})\}+\lambda||\mathbf{B}||_{F}^{2}, \tag{4}\]  
where \(\mathbf{B}\in R^{\mathbb{C}\times K}\), \(\mathbf{\varepsilon}\), \(\mathbf{h}_{i}\in\{0,1\}^{C}\), \(\lambda\), and \(f(. )\) represent classifier, loss function, class label for \(\mathbf{a}_{i}\), regularization constant, and predicted label for \(\mathbf{a}_{i}\), respectively. Though the aforementioned approach provides a baseline for using the dictionary learning domain in the classification domain, this does not exhibit joint and discriminative behavior. The joint and discriminatory learning dictionary and classifier obtain the label information of the training data induced into the learning process, and the efficiency of the classifier increases [8; 40; 50]. Among such joint discriminative approaches, the approach followed by [24] became very prominent and knocked out other approaches. This approach used a Bayesian nonparametric framework with beta process [51]. The authors in [52] introduced the beta process for image restoration and compressive sensing. This concept was further exploited by [PERSON] et al. [24; 25] for object and scene classification, as well as face and action recognition. Basically, [PERSON] and [PERSON] [51] developed a beta process for nonparametric factor analysis.