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

A Deep Neural Network's Loss Surface Contains Every Low-dimensional Pattern