Graph of Audio Processors

• Graph of Audio Processors is a directed acyclic graph that models audio effect modules and signal routing to ensure causality and clarity in signal flow.
• The framework employs differentiable processor implementations, enabling gradient descent optimization and parallel batch computation for robust audio mixing.
• Iterative optimization and pruning strategies effectively reduce computational cost while maintaining audio fidelity, paving the way for practical DSP applications.

A graph of audio processors is a structured representation—typically a directed acyclic graph (DAG)—that models how individual audio effect modules (e.g., equalizer, compressor, reverb, delay) and their connections transform dry source audio into a processed output or mixed signal. This formalism enables precise modeling, optimization, and reverse engineering of realistic audio processing workflows, reflecting the compositional nature of music mixing and production.

1. Formal Definitions and Graph Topology

Audio processing graphs are defined as DAGs, $G = (V, E)$, where nodes $v_i \in V$ are individual processors or auxiliary modules (inputs, summing mixes, outputs), and edges $E$ encode signal routing (cables connecting processor outputs to inputs). In a typical mixing console graph, each input track is routed through a fixed chain of processors before submixing and final mixing. The graph is constructed to ensure causality (directed edges, no cycles), with additional constraints to guarantee signal flow correctness, input/output integrity, and absence of routing ambiguities (Lee et al., 19 Sep 2025).

Mathematically:

• Each processor node $v_i$ receives input signal $u_i[n] = \sum_{j \in \mathcal{N}(i)} y_j[n]$, with $\mathcal{N}(i)$ the set of upstream nodes.
• Processing is split into main function $f_i$ (parameterized by $\overline{p}_i$), followed by a dry/wet mix using weight $w_i$:
  • $\tilde{y}_i[n] = f_i(u_i[n], \overline{p}_i)$
  • $y_i[n] = w_i \tilde{y}_i[n] + (1-w_i) u_i[n]$

This structure facilitates differentiable optimization and pruning (Lee et al., 3 Jun 2024), supports parallel computation, and is amenable to efficient batch scheduling (Lee et al., 6 Aug 2024).

2. Processor Implementation and Differentiability

The framework mandates differentiable processor implementations within platforms such as PyTorch, ensuring that all signal transformations admit gradients for end-to-end optimization. Modules include FIR equalizers (via parameterized log-magnitude spectra and IFFT), compressors (with ballistics/one-pole envelope followers), noisegates, stereo imagers, multitap delays (with continuous delay parameterization), and reverbs (via STFT-based masking). The "dry/wet" mix after each processor is a continuous, learnable blending factor $w_i \in [0,1]$ (Lee et al., 19 Sep 2025).

Parallelization is achieved by partitioning the graph into homogeneous subsets (nodes of the same processor type) and scheduling batch computation so that each subset can be processed in parallel, reducing computational cost and memory access overhead (Lee et al., 6 Aug 2024). Optimal schedules (oracle, greedy, beam search) and reordering strategies further maximize GPU throughput and minimize storage fragmentation.

3. Iterative Optimization and Pruning Strategies

Optimization proceeds in two stages:

• Parameter Tuning: All processor parameters $\{p_i\}$ are optimized via gradient descent to minimize an audio-domain loss, typically a multi-resolution STFT loss between target mix $y$ and synthesized mix $\hat{y}$ (Lee et al., 3 Jun 2024, Lee et al., 19 Sep 2025). Regularization terms are included for gain staging and sparsity.
• Iterative Pruning: Processors with negligible contribution (as measured by $w_i$ values or brute-force loss evaluation) are candidates for removal. Pruning is accepted if the audio loss does not increase beyond a preset tolerance $\tau$, with updates followed by further fine-tuning.

The objective is: $$G^*, P^* = \arg\min_{G, P} \big[ L_a(\hat{y}, y) + L_r(G, P)\big]$$ Subject to: $\min L_a(G_p) \leq \min L_a(G_c) + \tau$ where $G_p$ is the pruned graph, $G_c$ the full console, and $L_r$ incorporates graph/parameter regularization.

Sampling methods for candidates include brute-force, dry/wet weight-based (nodes with lowest $w_i$), and hybrids (Lee et al., 3 Jun 2024, Lee et al., 19 Sep 2025). Empirically, approximately two-thirds of processors can be pruned with negligible perceptual or objective degradation (Lee et al., 19 Sep 2025).

4. Computational Efficiency and Batch Processing

Batch processing leverages graph partitioning into causally ordered, homogeneous subsets $V_0, V_1, ..., V_N$, enabling simultaneous evaluation of multiple nodes. For each time sample $n$, node batch outputs are computed as: $$\mathbf{y}_i^{(1)}[n],..., \mathbf{y}_i^{(N)}[n] = f_i(\mathbf{u}_i^{(1)}[n],..., \mathbf{u}_i^{(M)}[n], p_i)$$ where $$\mathbf{u}_i^{(l)}[n] = \sum_{(j,k) \in \mathbb{N}^+(i,l)} \mathbf{y}_j^{(k)}[n]$$.

Memory access is optimized by contiguous slicing and node index reordering, with scheduling strategies implemented to minimize compute iterations, as detailed in (Lee et al., 6 Aug 2024). The approach accommodates dynamic graph topologies (modifications during optimization or by neural network prediction), separating CPU-based setup from GPU-bound tensor computation.

5. Empirical Validation and Practical Applications

Objective assessment uses metrics including multi-resolution STFT loss, Fréchet Audio Distance, scale-invariant SDR, and various MIR feature distances. Subjective validation via MUSHRA tests confirms that pruned graphs reach effectively the same perceptual quality as full consoles, given appropriate tolerance. The dry/wet parameter shows a weak positive correlation with pruning impact (Lee et al., 19 Sep 2025).

Graph extraction enables large-scale pseudo-supervised dataset creation for training deep learning models in automatic mixing, style analysis, and mixing graph estimation (Lee et al., 3 Jun 2024, Lee et al., 19 Sep 2025). The interpretable, sparse graphs reflect real-world mixing practice and provide efficient structure for both analysis and neural training.

6. Limitations and Open Problems

The described methods operate within a restricted search space, primarily pruning from a preset chain; changing processor order or types beyond the initial full console is not explored in the base algorithms (Lee et al., 3 Jun 2024, Lee et al., 19 Sep 2025). Binary pruning decisions are approximated by relaxed dry/wet weights and may be sensitive to the choice of tolerance parameter $\tau$. Computational cost remains nontrivial for large datasets despite batch efficiency.

7. Future Directions

Advancements include extending the search to reordering and dynamic topology modification, incorporating non-differentiable plugins (integrated via stochastic gradient approximation or black-box wrapping), and refining dataset creation with fully realistic processor inventories and signal types. Applications span beyond mixing, including style transfer, reverse engineering of professional DSP chains, and real-time automated mixing systems. Open-source libraries such as GRAFX provide standardized implementations and facilitate further research (Lee et al., 6 Aug 2024). The emergence of DAW-based pipelines integrating commercial plugin formats further bridges experimental methods with real-world DSP workflows (Yang et al., 14 Jul 2025).

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In summary, the graph of audio processors constitutes a foundational, compositional paradigm for modeling, optimizing, and reverse engineering complex audio signal processing workflows. It unites modular signal processing, differentiable programming, efficient computation, and interpretable modeling—forming the basis for both computational research and practical audio engineering applications.