Sound Matching an Analogue Levelling Amplifier Using the Newton-Raphson Method

1.1 Problem Definition

• The task is to choose θ for a controllable model f so its output ŷ best matches target compressor output y under distance D.Source: §1.1 · p. 1

1.2 Related Work

• Neural-network VA models trade interpretability for modeling ability, while grey-box compressor work adds a GRU and trains parameters with gradient descent.Source: §1.2 · p. 2, §1.3 · p. 2

1.3 Motivation

• Newton-Raphson had not, to the authors' knowledge, been used for grey-box analogue-compressor modeling; fewer than 10 parameters make its quadratic storage feasible, with convergence advantages under stated conditions.Source: §1.3 · p. 2

1.4 Contributions

• The paper formulates virtual-analogue modeling as sound matching and presents an optimized feed-forward compressor, an interpretable parameter mapping, and an open-source VST plugin.Source: §1.4 · p. 2

2 Methodology

• With input fixed, the authors use a feed-forward compressor as base model f for efficiency and high modeling capability.Source: §2 · p. 2

2.1 Feed-Forward Compressor (f )

• They use torchcomp's differentiable compressor and omit the RMS detector because smoothing was negligible and removing it slightly lowered D; its extra freedom may trap NR in a local minimum.Source: §2.1 · pp. 2–3
• The model has five controls—threshold, ratio, attack, release, and make-up gain—with attack/release setting recursive gain-smoothing ballistics and sigmoid transforms bounding constrained controls.Source: §2.1 · p. 3

2.2 Newton-Raphson Optimisation

• Newton-Raphson updates parameters using the gradient and Hessian of square-distance loss; it requires local convexity and twice differentiability, with convexity assessed empirically and the objective made twice differentiable.Source: §2.2 · p. 3

3 Hessian Computation

• The Hessian can be computed by passing each identity-matrix column through the gradient’s VJP, yielding a Hessian row.The columns can be passed in parallel.Source: §3 · p. 4
• Forward-over-reverse computes the outer Jacobian with JVPs as an alternative to reverse-over-reverse VJPs.The paper derives the JVP for backpropagation through the ballistics filter because its recursive filters are non-trivial to implement in PyTorch.Source: §3 · p. 4

3.1 Forward-Mode Gradients for Time-Varying One-Pole Filters

• Forward-mode JVP adds one operation beyond the VJP, but the extra addition lies outside the recursion and is often negligible.The attack/release phase indicator ζ[n] selects the filter phase; the derivation treats the recursive filter separately.Source: §3.1 · p. 4
• In forward mode, gradients through a time-varying one-pole filter are obtained by filtering the gradients with the same filter.This result generalises to higher-order all-pole filters.Source: §3.1 · p. 4

4.1 Dataset

• SignalTrain supplies paired 44.1 kHz LA-2A recordings at peak-reduction settings from 0 to 100 in steps of 5, using the same 20-minute stimulus.The selected recordings are labelled 3c and include musical and synthetic sounds; gain is fixed for compressor and limiter modes.Source: §4.1 · p. 5

4.2 Acceleration Techniques

• Twelve-second chunks with one-second overlap enable batch-level parallelism, while parallel associative scan accelerates recursive filters and pre-emphasis on the GPU.Loss is computed over each chunk’s last 11 seconds after the overlap warm-up.Source: §4.2 · p. 5

4.3 Analysis on Computational Cost

• Rev-rev and fwd-rev are equally fast, while rev-fwd is slowest and most memory-consuming; autograd fits each configuration on one 12 GB RTX 3060.One NR update takes about 3.5 seconds, and full-dataset training takes under 20 minutes rather than the hours reported in.Source: §4.3 · p. 5

4.4 Optimisation Strategy

• Damped Newton–Raphson uses backtracking and an equation solver, warm-starting successive peak-reduction settings; no consistent solution was found below 40.Most cases converged in fewer than 10 iterations; at lower settings, compression is less noticeable and other nonlinearities exceed what the feed-forward compressor captures.Source: §4.4 · pp. 5–6

4.5 Results

• Around 4:1 in compressor mode versus the manufacturer’s stated 3:1, learned attack and release times vary exponentially with peak reduction.The single release-time parameter captures an average of the LA-2A’s documented two-stage release; Fig. 2 shows the mapping from peak reduction to compressor parameters.Source: Figure 2 · p. 6, §4.5 · p. 6

5 Evaluation

• The comparison includes 4A-2A, 4A-2A-G, the official UAD LA-2A, CA-2A, CLA-2A, and the grey-box compressor from.The grey-box comparison model uses Model 5, with a GRU replacing make-up gain, trained on data preprocessed by 4A-2A with γ = 0 dB.Source: §5 · pp. 6–7
• ESR and ΔLDR are the evaluation measures; ΔLDR compares predicted and target loudness dynamic range, with positive values indicating less compression in the prediction.LDR uses short- and long-term RMS integration times of 50 ms and 3 s, respectively.Source: §5 · p. 7
• 4A-2A-G has the lowest ESR, while 4A-2A and 4A-2A-G have ΔLDR close to zero, with 4A-2A-G closest; 4A-2A is comparable to the official UAD plugin.The evaluation compares the models across peak reduction and mode.Source: §5 · p. 7, Figure 3 · p. 7
• At around 75 peak reduction, the LA-2A behaves most like the feed-forward compressor, with the lowest ESR; beyond that, 4A-2A’s ESR increases, and errors are greater in limiter mode.4A-2A was evaluated on its training data because its five-parameter model cannot fully represent the LA-2A; no overfitting was observed on that dataset.Source: §5 · pp. 6–7
• At 100 peak reduction, CA-2A’s ESR exceeds 40% and its ΔLDR mismatch reaches 1 dB; UAD and CLA-2A shift from lighter to heavier to lighter compression across peak reductions 40–60, 60–80, and above 80.Commercial-plugin input gains were tuned to make summed ΔLDR across peak reductions close to zero, then output gains were tuned to minimise overall ESR.Source: §5 · p. 7

6 Application: the 4A-2A Plugin

• The plugin provides five compressor-parameter sliders, plus peak-reduction and mode controls; peak reduction maps to parameters through learned mappings, and users can fine-tune parameters manually.The mode switch selects compressor or limiter profiles, and users can observe behaviour in real time.Source: §6 · p. 8, Figure 4 · p. 8
• Linear and cubic-spline interpolation perform worst at peak reductions 65 and 95, respectively; spline interpolation has slightly higher average ESR, so the plugin uses linear interpolation.The comparison omits peak reductions 45, 55, through 95 and interpolates them from the remaining seven sampled points.Source: §6 · p. 8, Figure 5 · p. 8
• The 4A-2A-G combines the feed-forward compressor trained with Newton-Raphson and a real-time Model 5-style GRU emulator, outperforming all commercial plugins in the evaluation.The GRU model has one layer with eight hidden units and is also made available as a standalone plugin.Source: §6 · p. 8
• Adding GRU make-up gain most noticeably changes residuals above 4 kHz, where the GRU captures analogue-circuit nonlinearities and spreads errors across frequencies.The GRU model is also packaged as a standalone plugin using Neutone.Source: §6 · p. 8

• The Newton-Raphson method finds a unique solution matching the LA-2A response under the square-distance criterion, but 4A-2A and LA-2A signal flows differ.Source: §7 · p. 8
• A digital feedback compressor and explicit two-stage release may improve performance, but backpropagating through feedback is challenging and requires custom implementations.Source: §7 · p. 8
• The LA-2A trial is satisfactory, and the grey-box approach can generalize to any dynamic range controller circuit.Source: §7 · p. 9
• Plugin performance is inferior; UAD and CLA-2A show similar ESR/∇LDR patterns, possibly from similar target circuits or modelling approaches.Source: §7 · p. 9
• For complex effects or longer chains, full-Hessian storage can exceed memory; Hessian-vector conjugate gradients avoid explicitly computing it, while more parameters raise local-minimum risk.Source: §7 · p. 9

• A Newton-Raphson-optimized feed-forward digital compressor emulates the LA-2A optical compressor and converts its response into interpretable parameters for creative control.Source: §8 · p. 9
• Efficient ballistics and a parallel associative scan for gradient computation significantly accelerate optimization.Source: §8 · p. 9
• Forward-mode gradient propagation through the compressor is derived to benchmark Hessian-computation strategies.Source: §8 · p. 9
• Newton-Raphson appears promising for grey-box audio-compressor modeling; future work could model feedback and two-stage release, extend to other effects, and reduce Hessian costs in longer chains.Source: §8 · p. 9