A Deep Neural Networks Loss Surface Contains Every Low-dimensional Pattern

The work Loss Landscape Sightseeing with Multi-Point Optimization (Skorokhodov and Burtsev, 2019) demonstrated that one can empirically find arbitrary 2D binary patterns inside loss surfaces of popular neural networks. In this paper we prove that: (i) this is a general property of deep universal approximators; and (ii) this property holds for arbitrary smooth patterns, for other dimensionalities, for every dataset, and any neural network that is sufficiently deep and wide. Our analysis predicts not only the existence of all such low-dimensional patterns, but also two other properties that were observed empirically: (i) that it is easy to find these patterns; and (ii) that they transfer to other data-sets (e.g. a test-set).