# Audio synthesizer inversion in symmetric parameter spaces with approximately equivariant flow matching

arXiv: https://arxiv.org/abs/2506.07199 · HTML: https://mlsanity.com/paper/2506.07199/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Audio, speech and music, one of 14 directions in Diffusion models.
- Builds on 37 of its 55 references that are in the map.
- First posted on arXiv 8 Jun 2025.

## Placed in

- Diffusion models / [Audio, speech and music](https://mlsanity.com/diffusion/audio-speech-and-music/)

## Builds on (most cited first, 30 of 37)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Scalable Diffusion Models with Transformers](https://mlsanity.com/paper/2212.09748/) (2022)
- [Classifier-Free Diffusion Guidance](https://mlsanity.com/paper/2207.12598/) (2022)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [On Layer Normalization in the Transformer Architecture](https://mlsanity.com/paper/2002.04745/) (2020)
- [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023)
- [AST: Audio Spectrogram Transformer](https://mlsanity.com/paper/2104.01778/) (2021)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [FSD50K: An Open Dataset of Human-Labeled Sound Events](https://mlsanity.com/paper/2010.00475/) (2020)
- [Multisample Flow Matching: Straightening Flows with Minibatch Couplings](https://mlsanity.com/paper/2304.14772/) (2023)
- [Using the Output Embedding to Improve Language Models](https://mlsanity.com/paper/1608.05859/) (2016)
- [Equivariant Flows: Exact Likelihood Generative Learning for Symmetric Densities](https://mlsanity.com/paper/2006.02425/) (2020)
- [Neural Audio Synthesis of Musical Notes with WaveNet Autoencoders](https://mlsanity.com/paper/1704.01279/) (2017)
- [DDSP: Differentiable Digital Signal Processing](https://mlsanity.com/paper/2001.04643/) (2020)
- [CREPE: A Convolutional Representation for Pitch Estimation](https://mlsanity.com/paper/1802.06182/) (2018)
- [Guided Flows for Generative Modeling and Decision Making](https://mlsanity.com/paper/2311.13443/) (2023)
- [SetVAE: Learning Hierarchical Composition for Generative Modeling of Set-Structured Data](https://mlsanity.com/paper/2103.15619/) (2021)
- [Deep Set Prediction Networks](https://mlsanity.com/paper/1906.06565/) (2019)
- [Equivariant Flow Matching with Hybrid Probability Transport](https://mlsanity.com/paper/2312.07168/) (2023)
- [Fast Point Cloud Generation with Straight Flows](https://mlsanity.com/paper/2212.01747/) (2022)
- [Mixed Continuous and Categorical Flow Matching for 3D De Novo Molecule Generation](https://mlsanity.com/paper/2404.19739/) (2024)
- [FSPool: Learning Set Representations with Featurewise Sort Pooling](https://mlsanity.com/paper/1906.02795/) (2019)
- [ET-Flow: Equivariant Flow-Matching for Molecular Conformer Generation](https://mlsanity.com/paper/2410.22388/) (2024)
- [Conditional Set Generation with Transformers](https://mlsanity.com/paper/2006.16841/) (2020)
- [Scalable Normalizing Flows for Permutation Invariant Densities](https://mlsanity.com/paper/2010.03242/) (2020)
- [A Review of Differentiable Digital Signal Processing for Music & Speech Synthesis](https://mlsanity.com/paper/2308.15422/) (2023)
- [I'm Sorry for Your Loss: Spectrally-Based Audio Distances Are Bad at Pitch](https://mlsanity.com/paper/2012.04572/) (2020)

## Built on by (most cited first, 6 of 6)

- [Modulation Discovery with Differentiable Digital Signal Processing](https://mlsanity.com/paper/2510.06204/) (2025)
- [DDSynth-RL: Audio Synthesizer Inversion via Discrete Diffusion with Reinforcement Learning](https://mlsanity.com/paper/2608.03032/) (2026)
- [Equivariant Music Transformer](https://mlsanity.com/paper/2608.03920/) (2026)
- [Diff2Mix: Controllable Music Mixing via Diffusion Models and Differentiable Audio Effects](https://mlsanity.com/paper/2608.05442/) (2026)
- [Off-manifold robustness in synthesizer inversion with joint distribution flow matching](https://mlsanity.com/paper/2609.29320/) (2026)
- [Synth-JEPA: Joint Embedding Prediction for Renderer-Free Synthesizer Parameter Search](https://mlsanity.com/paper/2609.31024/) (2026)

## Abstract (arXiv)

Many audio synthesizers can produce the same signal given different parameter configurations, meaning the inversion from sound to parameters is an inherently ill-posed problem. We show that this is largely due to intrinsic symmetries of the synthesizer, and focus in particular on permutation invariance. First, we demonstrate on a synthetic task that regressing point estimates under permutation symmetry degrades performance, even when using a permutation-invariant loss function or symmetry-breaking heuristics. Then, viewing equivalent solutions as modes of a probability distribution, we show that a conditional generative model substantially improves performance. Further, acknowledging the invariance of the implicit parameter distribution, we find that performance is further improved by using a permutation equivariant continuous normalizing flow. To accommodate intricate symmetries in real synthesizers, we also propose a relaxed equivariance strategy that adaptively discovers relevant symmetries from data. Applying our method to Surge XT, a full-featured open source synthesizer used in real world audio production, we find our method outperforms regression and generative baselines across audio reconstruction metrics.