Efficient Backpropagation for Time-Varying All-Pole Filters in Differentiable Audio Systems

Core Concepts

This paper proposes an efficient backpropagation algorithm and implementation for time-varying all-pole filters that can be used to model various analog audio systems end-to-end using gradient descent.

Abstract

The paper presents a solution to the problem of efficiently training time-varying audio systems that employ infinite impulse response (IIR) filters, which are commonly used in subtractive synthesizers, phaser effects, and dynamic range compression. The authors derive and implement an efficient backpropagation algorithm for a time-varying all-pole filter, which can be used within any audio system containing filters with poles for efficient gradient evaluation.

The key highlights and insights are:

1. The proposed time-domain (TD) implementation significantly increases the forward and backpropagation speed of time-varying recursive filters without introducing any approximation to the filter.
2. Systems trained with the TD implementation can be converted to real-time operation without any generalization issues, unlike previous frequency-sampling (FS) approaches.
3. The TD implementation efficiently and accurately models time-varying analog audio circuits with a recursive structure, outperforming FS methods in terms of accuracy and training efficiency, especially when using the systems at the sample-rate level.
4. The authors demonstrate the effectiveness of the proposed method on three commonly used dynamic audio systems: a phaser, a time-varying subtractive synthesizer, and a feed-forward compressor.
5. The authors make their code available and provide the trained audio effect and synth models in a VST plugin.

Stats

The paper does not contain any explicit numerical data or statistics to support the key logics. The results are presented in the form of qualitative evaluations and comparisons between the proposed time-domain (TD) method and the frequency-sampling (FS) method.

Quotes

"We significantly increase the forward and backpropagation speed of time-varying recursive filters without introducing any approximation to the filter."
"The systems trained with our implementation can be converted to real time without any generalisation issues."
"We show that our implementation efficiently and accurately models time-varying analog audio circuits with a recursive structure."

Key Insights Distilled From

Differentiable All-pole Filters for Time-varying Audio Systems

by Chin... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2404.07970.pdf

Deeper Inquiries

How can the proposed backpropagation algorithm be extended to handle differentiable initial conditions for the all-pole filters

To extend the proposed backpropagation algorithm to handle differentiable initial conditions for the all-pole filters, we can introduce additional parameters or variables that represent the initial conditions of the system. These initial conditions can be incorporated into the computation graph during the backpropagation process, allowing the gradients to flow through them as well. By including the initial conditions as part of the optimization process, the algorithm can learn to adjust them along with the other parameters to minimize the loss function effectively. This extension would involve modifying the existing algorithm to account for the initial conditions and ensure that they are updated appropriately during training.

What are the potential numerical stability issues that may arise when using the forward-mode differentiation of the proposed all-pole filter implementation, and how can they be addressed

When using the forward-mode differentiation of the proposed all-pole filter implementation, potential numerical stability issues may arise due to the accumulation of errors in the calculations. Forward-mode differentiation involves computing the derivatives of the output with respect to the input variables, which can lead to numerical inaccuracies, especially in complex systems with multiple layers of computations. To address these stability issues, it is essential to use high-precision arithmetic, such as double precision, to minimize rounding errors and maintain accuracy in the calculations. Additionally, implementing robust error handling mechanisms and numerical stability techniques, such as gradient clipping or regularization, can help mitigate instability and ensure reliable results during the forward-mode differentiation process.

Could the proposed method be applied to model other types of time-varying audio systems, such as those with feedback loops or nonlinear components, and what would be the key challenges in doing so

The proposed method could be applied to model other types of time-varying audio systems, including those with feedback loops or nonlinear components, by adapting the backpropagation algorithm to accommodate the specific characteristics of these systems. One key challenge in modeling systems with feedback loops is handling the recursive nature of the filters and ensuring stable convergence during training. Incorporating feedback loops into the computational graph and designing appropriate mechanisms to propagate gradients through the feedback paths would be crucial in effectively modeling such systems. Additionally, modeling nonlinear components would require extending the algorithm to capture the nonlinearities in the system and adjust the parameters accordingly. Addressing these challenges would involve developing specialized techniques for handling feedback loops and nonlinearities within the backpropagation framework to accurately model complex time-varying audio systems.