Automatically Gating Multi-Frequency Patterns through Rectified Continuous Bernoulli Units with Theoretical Principles

Different nonlinearities are only suitable for responding to different frequency signals. The locally-responding ReLU is incapable of modeling high-frequency features due to the spectral bias, whereas the globally-responding sinusoidal function is intractable to represent low-frequency concepts cheaply owing to the optimization dilemma. Moreover, nearly all the practical tasks are composed of complex multi-frequency patterns, whereas there is little prospect of designing or searching a heterogeneous network containing various types of neurons matching the frequencies, because of their exponentially-increasing combinatorial states. In this paper, our contributions are three-fold: 1) we propose a general Rectified Continuous Bernoulli (ReCB) unit paired with an efficient variational Bayesian learning paradigm, to automatically detect/gate/represent different frequency responses; 2) our numerically-tight theoretical framework proves that ReCB-based networks can achieve the optimal representation ability, which is O(m^{η/(d^2)}) times better than that of popular neural networks, for a hidden dimension of m, an input dimension of d, and a Lipschitz constant of η; 3) we provide comprehensive empirical evidence showing that ReCB-based networks can keenly learn multi-frequency patterns and push the state-of-the-art performance.