## 4 Discussion  
In this study, we introduced a sparse deep convolutional autoencoder called SPAER to generate deep thermocirnce features. It embedded three predominant thermal bases, obtained by low-rank matrix approximation in the form of a multichannel input matrix. We designed the SPAER model to be applicable to any imaging modality, but we tested it for dynamic thermography. The results indicated that SPAER could identify the potential symptomatic patients and aid in the early diagnosis of breast abnormality as a noninvasive, efficient, and fast technique to be used with CBE and before mammography. SPAER potentially surpasses the use of pre-trained deep neural networks to extract high-dimensional imaging throughput, which leads to high collinearity among the features and overfitting the system, known as the _curse of dimensionality_. The results indicate that SPAER significantly reduced the noise throughout the process by having sparse regularizations in dense layers, which were tested with additive noise (Figure 6). Autoencoders are often used for noise elimination due to their nonlinearity. SPAER also behaved the same and showed a significant SNR when facing additive noise [64; 65; 66]. Low-rank matrix  
Figure 6: The robustness of the model in the presence of Gaussian additive noise following our signal-to-noise ratio (SNR) definition for all the parts of the model. approximation used for the extraction of high variance bases also enhanced the stability of the system against noise [37; 46]. This had the opposite effect on the accuracy of sparse NMF, and relatively Sparse PCT, as two-folds sparsity with non-negative constraints mitigating the outcome performance of the overall system. This may be due to reducing the domain of the solution having two penalty terms, which led to a slight decrease in the overall accuracy for sparse NMF-SPAER and even for sparse PCT-SPAER (Table 2). The application of infrared thermography to establish early breast abnormality has been suggested previously [30; 49]. This can be expanded for several nonneoplastic diseases and normal homeostatic processes [21; 22]. Despite some discussions about its reliability as a single applied system for finding breast cancer [20], dynamic thermography showed promising outcomes [46; 47; 48], and this makes it a valuable complementary mechanism along with CBE for increasing the possibility of finding abnormalities at an initial stage (not as a mammography substitute [67; 68]). Moreover, the association of heterogeneous thermal patterns with early breast cancer detection has not been discussed in the literature and significantly increases the novelty of the proposed model. Besides that, the SPAER model was tested for dynamic thermography, yet its applications are not limited to this application, and it can be used for any other imaging modality, which makes SPAER a generalized model for a variety of applications. One of the limitations of this study is the number of patients. Despite having a relatively large cohort of patients, having access to an even larger cohort would increase the validation of the statistical analysis and improve the accuracy of the system. The maximum efficiency of the method would be revealed by a higher number of patients. DMR was the only available dataset with such analyses that we could access, and this limited the possibility of an independent validation set to confirm the performance of the system. Another limitation of this study is related to the lack of comprehensive clinical information. Even with the great effort of the DMR team [30], there could be more potential clinical information, such as pathology information. This could help to find the correlation of different factors with thermal heterogeneity. Moreover, there is a limitation regarding the minimum number of thermal images required for the process, which should be more than four images, as there is a matrix factorization method involved that selects three bases, which requires a minimum of four thermal images. The proposed model provides several advantages.