# Optimal spectral transportation with application to music transcription

arXiv: https://arxiv.org/abs/1609.09799 · HTML: https://mlsanity.com/paper/1609.09799/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Sep 2016.

## Built on by (most cited first, 12 of 12)

- [Semi-relaxed Gromov-Wasserstein divergence with applications on graphs](https://mlsanity.com/paper/2110.02753/) (2021)
- [SNEkhorn: Dimension Reduction with Symmetric Entropic Affinities](https://mlsanity.com/paper/2305.13797/) (2023)
- [Unsupervised Harmonic Parameter Estimation Using Differentiable DSP and Spectral Optimal Transport](https://mlsanity.com/paper/2312.14507/) (2023)
- [Uncovering Challenges of Solving the Continuous Gromov-Wasserstein Problem](https://mlsanity.com/paper/2303.05978/) (2023)
- [A note on the relations between mixture models, maximum-likelihood and entropic optimal transport](https://mlsanity.com/paper/2501.12005/) (2025)
- [Optimal Transport with Adaptive Regularisation](https://mlsanity.com/paper/2310.02925/) (2023)
- [Data-driven mapping between functional connectomes using optimal transport](https://mlsanity.com/paper/2107.01303/) (2021)
- [Estimation of Stationary Optimal Transport Plans](https://mlsanity.com/paper/2107.11858/) (2021)
- [Wasserstein Nonnegative Tensor Factorization with Manifold Regularization](https://mlsanity.com/paper/2401.01842/) (2024)
- [Scalable Approximate Algorithms for Optimal Transport Linear Models](https://mlsanity.com/paper/2504.04609/) (2025)
- [Translation-Equivariant Self-Supervised Learning for Pitch Estimation with Optimal Transport](https://mlsanity.com/paper/2508.01493/) (2025)
- [Robust blind unmixing: A geometric approach to overcoming basis variation](https://mlsanity.com/paper/2610.04091/) (2026)

## Abstract (arXiv)

Many spectral unmixing methods rely on the non-negative decomposition of spectral data onto a dictionary of spectral templates. In particular, state-of-the-art music transcription systems decompose the spectrogram of the input signal onto a dictionary of representative note spectra. The typical measures of fit used to quantify the adequacy of the decomposition compare the data and template entries frequency-wise. As such, small displacements of energy from a frequency bin to another as well as variations of timber can disproportionally harm the fit. We address these issues by means of optimal transportation and propose a new measure of fit that treats the frequency distributions of energy holistically as opposed to frequency-wise. Building on the harmonic nature of sound, the new measure is invariant to shifts of energy to harmonically-related frequencies, as well as to small and local displacements of energy. Equipped with this new measure of fit, the dictionary of note templates can be considerably simplified to a set of Dirac vectors located at the target fundamental frequencies (musical pitch values). This in turns gives ground to a very fast and simple decomposition algorithm that achieves state-of-the-art performance on real musical data.