Energy Preservation and Stability of Random Filterbanks

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.