Vector Quantile Regression on Manifolds

Quantile regression (QR) is a statistical tool for distribution-free estimation of conditional quantiles of a target variable given explanatory features. QR is limited by the assumption that the target distribution is univariate and defined on an Euclidean domain. Although the notion of quantiles was recently extended to multi-variate distributions, QR for multi-variate distributions on manifolds remains underexplored, even though many important applications inherently involve data distributed on, e.g., spheres (climate measurements), tori (dihedral angles in proteins), or Lie groups (attitude in navigation). By leveraging optimal transport theory and the notion of $c$-concave functions, we meaningfully define conditional vector quantile functions of high-dimensional variables on manifolds (M-CVQFs). Our approach allows for quantile estimation, regression, and computation of conditional confidence sets. We demonstrate the approach's efficacy and provide insights regarding the meaning of non-Euclidean quantiles through preliminary synthetic data experiments.

Multi-Source Diffusion Models for Simultaneous Music Generation and Separation

In this work, we define a diffusion-based generative model capable of both music synthesis and source separation by learning the score of the joint probability density of sources sharing a context. Alongside the classic total inference tasks (i.e. generating a mixture, separating the sources), we also introduce and experiment on the partial inference task of source imputation, where we generate a subset of the sources given the others (e.g., play a piano track that goes well with the drums). Additionally, we introduce a novel inference method for the separation task. We train our model on Slakh2100, a standard dataset for musical source separation, provide qualitative results in the generation settings, and showcase competitive quantitative results in the separation setting. Our method is the first example of a single model that can handle both generation and separation tasks, thus representing a step toward general audio models.