Beyond holography: The entropic quantum gravity foundations of anisotropic diffusion

Recently, thanks to the development of artificial intelligence (AI), there is increasing scientific attention in establishing the connections between theoretical physics and AI. Traditionally, these connections have been focusing mostly on the relation between string theory and image processing and involve important theoretical paradigms such as holography. Recently, G. Bianconi has formulated the Gravity from Entropy (GfE) approach to quantum gravity in which gravity is derived from the geometric quantum relative entropy (GQRE) between two metrics associated with the Lorentzian spacetime. Here, it is demonstrated that the famous Perona-Malik algorithm for image processing is the gradient flow of the GfE action in its simple warm-up scenario. Specifically, this algorithm is the outcome of the maximization of the GfE action between two Euclidean metrics: the one in support of the image and the one induced by the image. As the Perona-Malik algorithm is known to preserve sharp contours, this implies that the GfE action does not, in general, lead to uniform images upon iteration of the gradient flow dynamics as it would be intuitively expected from entropic actions maximizing classical entropies. Rather, the outcome of the maximization of the GfE action is compatible with the preservation of complex structures. These results provide the geometrical and information theory foundations for the Perona-Malik algorithm and might contribute to establish deeper connections between GfE, machine learning, and brain research.