What makes waveform-based deep learning so hard? Despite numerous attempts at training convolutional neural networks (convnets) for filterbank design, they often fail to outperform hand-crafted baselines. This is all the more surprising because these baselines are linear time-invariant systems: as such, their transfer functions could be accurately represented by a convnet with a large receptive field. In this article, we elaborate on the statistical properties of simple convnets from the mathematical perspective of random convolutional operators. We find that FIR filterbanks with random Gaussian weights are ill-conditioned for large filters and locally periodic input signals, which both are typical in audio signal processing applications. Furthermore, we observe that expected energy preservation of a random filterbank is not sufficient for numerical stability and derive theoretical bounds for its expected frame bounds.

波形为基础的深度学习为何如此困难？尽管有很多尝试去训练卷积神经网络(convnets)进行滤波器设计，但它们通常无法胜过手工制作的基准线。本文从随机卷积算子的数学角度详细论述了简单卷积神经网络的统计性质。我们发现，具有随机高斯权重的有限脉冲响应(FIR)滤波器对于大滤波器和局部周期性输入信号是病态的，而这两者在音频信号处理应用中很常见。此外，我们观察到随机滤波器组的期望能量保持并不足以确保数值稳定性，并为其期望帧界导出了理论上的限制。

随机滤波器组的能量保持和稳定性