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.

Fitting Auditory Filterbanks with Multiresolution Neural Networks

Waveform-based deep learning faces a dilemma between nonparametric and parametric approaches. On one hand, convolutional neural networks (convnets) may approximate any linear time-invariant system; yet, in practice, their frequency responses become more irregular as their receptive fields grow. On the other hand, a parametric model such as LEAF is guaranteed to yield Gabor filters, hence an optimal time-frequency localization; yet, this strong inductive bias comes at the detriment of representational capacity. In this paper, we aim to overcome this dilemma by introducing a neural audio model, named multiresolution neural network (MuReNN). The key idea behind MuReNN is to train separate convolutional operators over the octave subbands of a discrete wavelet transform (DWT). Since the scale of DWT atoms grows exponentially between octaves, the receptive fields of the subsequent learnable convolutions in MuReNN are dilated accordingly. For a given real-world dataset, we fit the magnitude response of MuReNN to that of a well-established auditory filterbank: Gammatone for speech, CQT for music, and third-octave for urban sounds, respectively. This is a form of knowledge distillation (KD), in which the filterbank ''teacher'' is engineered by domain knowledge while the neural network ''student'' is optimized from data. We compare MuReNN to the state of the art in terms of goodness of fit after KD on a hold-out set and in terms of Heisenberg time-frequency localization. Compared to convnets and Gabor convolutions, we find that MuReNN reaches state-of-the-art performance on all three optimization problems.

Convex Geometry of ReLU-layers, Injectivity on the Ball and Local Reconstruction

The paper uses a frame-theoretic setting to study the injectivity of a ReLU-layer on the closed ball of $\mathbb{R}^n$ and its non-negative part. In particular, the interplay between the radius of the ball and the bias vector is emphasized. Together with a perspective from convex geometry, this leads to a computationally feasible method of verifying the injectivity of a ReLU-layer under reasonable restrictions in terms of an upper bound of the bias vector. Explicit reconstruction formulas are provided, inspired by the duality concept from frame theory. All this gives rise to the possibility of quantifying the invertibility of a ReLU-layer and a concrete reconstruction algorithm for any input vector on the ball.

Fast Matching Pursuit with Multi-Gabor Dictionaries

Finding the best K-sparse approximation of a signal in a redundant dictionary is an NP-hard problem. Suboptimal greedy matching pursuit (MP) algorithms are generally used for this task. In this work, we present an acceleration technique and an implementation of the matching pursuit algorithm acting on a multi-Gabor dictionary, i.e., a concatenation of several Gabor-type time-frequency dictionaries, each of which consisting of translations and modulations of a possibly different window and time and frequency shift parameters. The technique is based on pre-computing and thresholding inner products between atoms and on updating the residual directly in the coefficient domain, i.e., without the round-trip to the signal domain. Since the proposed acceleration technique involves an approximate update step, we provide theoretical and experimental results illustrating the convergence of the resulting algorithm. The implementation is written in C (compatible with C99 and C++11) and we also provide Matlab and GNU Octave interfaces. For some settings, the implementation is up to 70 times faster than the standard Matching Pursuit Toolkit (MPTK).

Phase-Based Signal Representations for Scattering

The scattering transform is a non-linear signal representation method based on cascaded wavelet transform magnitudes. In this paper we introduce phase scattering, a novel approach where we use phase derivatives in a scattering procedure. We first revisit phase-related concepts for representing time-frequency information of audio signals, in particular, the partial derivatives of the phase in the time-frequency domain. By putting analytical and numerical results in a new light, we set the basis to extend the phase-based representations to higher orders by means of a scattering transform, which leads to well localized signal representations of large-scale structures. All the ideas are introduced in a general way and then applied using the STFT.

Audio Inpainting via $\ell_1$-Minimization and Dictionary Learning

Audio inpainting refers to signal processing techniques that aim at restoring missing or corrupted consecutive samples in audio signals. Prior works have shown that $\ell_1$- minimization with appropriate weighting is capable of solving audio inpainting problems, both for the analysis and the synthesis models. These models assume that audio signals are sparse with respect to some redundant dictionary and exploit that sparsity for inpainting purposes. Remaining within the sparsity framework, we utilize dictionary learning to further increase the sparsity and combine it with weighted $\ell_1$-minimization adapted for audio inpainting to compensate for the loss of energy within the gap after restoration. Our experiments demonstrate that our approach is superior in terms of signal-to-distortion ratio (SDR) and objective difference grade (ODG) compared with its original counterpart.

Inpainting of long audio segments with similarity graphs

We present a novel method for the compensation of long duration data loss in audio signals, in particular music. The concealment of such signal defects is based on a graph that encodes signal structure in terms of time-persistent spectral similarity. A suitable candidate segment for the substitution of the lost content is proposed by an intuitive optimization scheme and smoothly inserted into the gap, i.e. the lost or distorted signal region. Extensive listening tests show that the proposed algorithm provides highly promising results when applied to a variety of real-world music signals.