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.

Different from popular neural networks using quasiconvex activations, non-monotonic networks activated by periodic nonlinearities have emerged as a more competitive paradigm, offering revolutionary benefits: 1) compactly characterizing high-frequency patterns; 2) precisely representing high-order derivatives. Nevertheless, they are also well-known for being hard to train, due to easily over-fitting dissonant noise and only allowing for tiny architectures (shallower than 5 layers). The fundamental bottleneck is that the periodicity leads to many poor and dense local minima in solution space. The direction and norm of gradient oscillate continually during error backpropagation. Thus non-monotonic networks are prematurely stuck in these local minima, and leave out effective error feedback. To alleviate the optimization dilemma, in this paper, we propose a non-trivial soft transfer approach. It smooths their solution space close to that of monotonic ones in the beginning, and then improve their representational properties by transferring the solutions from the neural space of monotonic neurons to the Fourier space of non-monotonic neurons as the training continues. The soft transfer consists of two core components: 1) a rectified concrete gate is constructed to characterize the state of each neuron; 2) a variational Bayesian learning framework is proposed to dynamically balance the empirical risk and the intensity of transfer. We provide comprehensive empirical evidence showing that the soft transfer not only reduces the risk of non-monotonic networks on over-fitting noise, but also helps them scale to much deeper architectures (more than 100 layers) achieving the new state-of-the-art performance.

Recently, deep spectral kernel networks (DSKNs) have attracted wide attention. They consist of periodic computational elements that can be activated across the whole feature spaces. In theory, DSKNs have the potential to reveal input-dependent and long-range characteristics, and thus are expected to perform more competitive than prevailing networks. But in practice, they are still unable to achieve the desired effects. The structural superiority of DSKNs comes at the cost of the difficult optimization. The periodicity of computational elements leads to many poor and dense local minima in loss landscapes. DSKNs are more likely stuck in these local minima, and perform worse than expected. Hence, in this paper, we propose the novel Bayesian random Kernel mapping Networks (BaKer-Nets) with preferable learning processes by escaping randomly from most local minima. Specifically, BaKer-Nets consist of two core components: 1) a prior-posterior bridge is derived to enable the uncertainty of computational elements reasonably; 2) a Bayesian learning paradigm is presented to optimize the prior-posterior bridge efficiently. With the well-tuned uncertainty, BaKer-Nets can not only explore more potential solutions to avoid local minima, but also exploit these ensemble solutions to strengthen their robustness. Systematical experiments demonstrate the significance of BaKer-Nets in improving learning processes on the premise of preserving the structural superiority.

Recently, spectral kernels have attracted wide attention in complex dynamic environments. These advanced kernels mainly focus on breaking through the crucial limitation on locality, that is, the stationarity and the monotonicity. But actually, owing to the inefficiency of shallow models in computational elements, they are more likely unable to accurately reveal dynamic and potential variations. In this paper, we propose a novel deep spectral kernel network (DSKN) to naturally integrate non-stationary and non-monotonic spectral kernels into elegant deep architectures in an interpretable way, which can be further generalized to cover most kernels. Concretely, we firstly deal with the general form of spectral kernels by the inverse Fourier transform. Secondly, DSKN is constructed by embedding the preeminent spectral kernels into each layer to boost the efficiency in computational elements, which can effectively reveal the dynamic input-dependent characteristics and potential long-range correlations by compactly representing complex advanced concepts. Thirdly, detailed analyses of DSKN are presented. Owing to its universality, we propose a unified spectral transform technique to flexibly extend and reasonably initialize domain-related DSKN. Furthermore, the representer theorem of DSKN is given. Systematical experiments demonstrate the superiority of DSKN compared to state-of-the-art relevant algorithms on varieties of standard real-world tasks.