Sobolev Norms in Neural Embeddings Measure Audio Morphing Regularity

Théo Chasle Cauchy; Modan Tailleur; Barbara Pascal; Fanny Roche; Mathieu Lagrange

cs.SDOct 6, 2026

Sobolev Norms in Neural Embeddings Measure Audio Morphing Regularity

Authors: Théo Chasle Cauchy, Modan Tailleur, Barbara Pascal, Fanny Roche, Mathieu Lagrange

Organizations: Nantes Universit´e, ´Ecole Centrale Nantes, CNRS, LS2N, UMR6004, F-44000 Nantes, France. · Arturia, Montbonnot Saint-Martin, France

Abstract

Morphing has recently gained renewed interest with the emergence of generative models, particularly in audio and image generation. In musical sound synthesis, morphing can generate intermediate sounds between two targets, helping musicians and sound engineers explore new sounds with interesting perceptual properties. As morphing is inherently defined in perceptual terms, evaluating this task is challenging. In this work, we introduce Sobolev Distances to Ideal Morphing (SDIM), a novel objective metric to quantify the regularity of audio morphing trajectories in perceptually relevant audio embedding spaces. Leveraging a physics-based sound synthesizer, we evaluate the discriminative power of SDIM on controlled morphing trajectories with varying degrees of regularity and compare it with that of existing audio morphing metrics. Results show that, contrary to state-of-the-art metrics, the proposed metric reliably discriminates desirable trajectories from adversarial ones.

Figures & tables

Figure 1: Morphing trajectories from two sounds

s

and

t

in a perceptually correlated embedding space. Ideal linear regular trajectory (solid green), good quality morphing trajectory (solid blue), and random

\mathsf{Null}

trajectory (dotted red).

Figure 2: Adversarial morphing trajectories in a perceptually correlated embedding space. (

\mathsf{NUC}

, solid orange) Nonuniform Configuration where the angles are preserved but the spacing regularity is not; (

\mathsf{EQC}

, solid violet) Equidistant Configuration where the distance to the source is the same as in the ideal trajectory but the angle between

\vec{sm}_{i}

and

\vec{st}

is randomly sampled.

Model ( # parameters)	$S_{0,2}^{\text{$ \mathsf{LUM} $}}/S_{0,2}^{\text{$ \mathsf{Null} $}}\downarrow$	$S_{1,2}^{\text{$ \mathsf{LUM} $}}/S_{1,2}^{\text{$ \mathsf{Null} $}}\downarrow$
MFCC [ 7 ] (-)	.18 (.07)	.07 (.03)
VGGish [ 12 ] (72M)	.23 (.08)	.11 (.04)
MSCLAP [ 8 ] (159M)	.15 (.03)	.08 (.01)
CLAP-Music [ 31 ] (158M)	.23 (.05)	.15 (.03)
CLAP-Audio [ 31 ] (158M)	.19 (.06)	.12 (.03)
MERT-v0-public [ 18 ] (95M)	.12 (.05)	.04 (.02)

Table 1: Normalized SDIM metrics for different embeddings. Normalized SDIM metrics means (standard deviations) are estimated from

1,000

\mathsf{LUM}

trajectories and

1,000

\mathsf{Null}

trajectories.

Metric	$\mathsf{LUM}$	$\mathsf{NUC}$	$\mathsf{EQC}$	$\mathsf{Null}$
$\mathsf{Corresp.}$ $\mathsf{SM}$ $\downarrow$	.19 (.02)	.38 (.01)	.14 (.03)	.03 (0.02)
$\mathsf{Smooth.}$ $\mathsf{MF}$ $\uparrow$	.96 (.01)	.75 (.01)	.97 (.03)	.11 (0.17)
$\mathsf{Interm.}$ $\mathsf{SM}$ $\downarrow$	.23 (.04)	.21 (.04)	1.15 (.24)	5.59 (0.73)
$\mathsf{Smooth.}$ $\mathsf{SM}$ $\downarrow$	.02 (.00)	.02 (.00)	.09 (.02)	.52 (0.07)
$10^{-2}\times S_{0,2}$ $\downarrow$	.69 (.09)	1.14 (.20)	1.72 (.32)	8.41 (1.27)
$10^{-3}\times S_{1,2}$ $\downarrow$	.35 (.05)	.69 (.12)	2.59 (.51)	.013 (1.54)

Table 2: Average metrics on the four configurations of trajectories in the MERT embeddings space. Means (standard deviations) estimated from

1000

examples. Only the SDIM metrics (last two rows) are robust to all of the adversarial configurations. Underlined red font indicates failure in measuring the true regularity.

Figure 3: Angle dispersion for the

\mathsf{LUM}

\mathsf{EQC}

, and

\mathsf{NUC}

configuration trajectories. Angles are measured through scalar products between the segment connecting the source and the intermediate point and the source and the target (ideal linear trajectory). Histograms are computed over

11

intermediate points across

1000

trajectories.

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