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In Cultural Heritage, hyperspectral images are commonly used since they provide extended information regarding the optical properties of materials. Thus, the processing of such high-dimensional data becomes challenging from the perspective of machine learning techniques to be applied. In this paper, we propose a Rank-R tensor-based learning model to identify and classify material defects on Cultural Heritage monuments. In contrast to conventional deep learning approaches, the proposed high order tensor-based learning demonstrates greater accuracy and robustness against over-fitting. Experimental results on real-world data from UNESCO protected areas indicate the superiority of the proposed scheme compared to conventional deep learning models.