## 1 Introduction
These approaches also introduced the adaptive learning feature of the size of the dictionary and classifier, and the association of labels of the training examples with dictionary and classifier atoms to make them discriminative to enhance classification accuracy. Beta-Bernoulli processes enabled discriminative learning by associating training examples' labels with the dictionary and classifier atoms. The same Bernoulli distributions were learned for the selection of dictionary and classifier atoms to represent training examples and the corresponding labels, respectively. Consequently, the joint discriminative and adaptive learning ability of the model enhanced the classification efficiency. These approaches jointly learn two different sets of sparse weights to represent training examples and their corresponding class labels over the dictionary and the classifier, respectively. Two steps are followed to predict a test sample. In the first step, sparse weights of the test sample are computed over the learned dictionary, using the OMP (orthogonal matching pursuit) algorithm. In the second step, these sparse weights are used as inputs to the learned classifier to predict the label of the test sample. The classifier acts like a dictionary, and its label reconstruction ability determines its efficiency in predictingthe test sample. In the second step, it is expected that the dictionary classifier will precisely construct the corresponding class label of the test sample using these sparse weights computed in the first step. However, during training, different sparse weights were learned to represent the corresponding labels of the training examples. Therefore, the performance of the classifier will be decreased. The use of weights computed over the dictionary will not effectively allow the classifier to reconstruct the corresponding label correctly. These approaches [24, 25] enhanced discriminative behavior while learning a dictionary and linear classifier, along with a feature of adaptively learning the size of the dictionary. However, these approaches are not strongly effective in establishing a strong joint between the dictionary and the classifier, as their strategies are of different sparse weights learning, and choosing the sparse weights computed over the dictionary instead of over the classifier is not intuitively feasible. To mitigate this issue and enhance the coupling between the dictionary and the dictionary classifier, we introduce the same representation weights learning approach. Our approach learns the same sparse weights over the dictionary to represent the training examples and over the classifier to represent the corresponding training labels, respectively. At the prediction stage, the representation weights of a test sample, computed over the dictionary, will also serve as the representation weights of the corresponding label over the classifier. This way, the reconstruction of the corresponding label or mapping to the corresponding label by the dictionary classifier will be more accurate. Therefore, the sparse codes of a signal computed over the dictionary should either be used separately for classification [9] or, if these are to be integrated with the dictionary classifier in joint learning settings [24, 25], the same sparse codes should be further learned/optimized over the dictionary classifier, instead of learning different codes over the dictionary classifier. We point this out as a research gap that needs to be addressed. By learning the same weights, we address this gap in the existing research. We tailor the sparse weights being learned over the dictionary, representing training examples, to further optimize for the representation of corresponding labels over the dictionary classifier, instead of learning another set of weights that are not accessible at the stage of prediction. This way, the sparse weights get truly integrated with the dictionary and the classifier in joint learning settings.