MAPSS: Manifold-based Assessment of Perceptual Source Separation

## Abstract

Abstract Objective assessment of source-separation systems still mismatches subjective human perception, especially when leakage and self-distortion interact. We introduce the Perceptual Separation (PS) and Perceptual Match (PM), the first pair of measures that functionally isolate these two factors. Our intrusive approach begins with generating a bank of fundamental distortions for each reference waveform signal in the mixture. Distortions, references, and their respective system outputs from all sources are then independently encoded by a pre-trained self-supervised learning model. These representations are aggregated and projected onto a manifold via a nonlinear dimensionality reduction technique called diffusion maps, which aligns Euclidean distances on the manifold with dissimilarities of the encoded waveform representations.
On this manifold, the PM measures the Mahalanobis distance from each output to its attributed cluster that consists of its reference and distortions embeddings, capturing self-distortion. The PS accounts for the Mahalanobis distance of the output to the attributed and to the closest non-attributed clusters, quantifying leakage. Both measures are differentiable and operate at a resolution as low as 50 frames per second, allowing granular optimization and analysis of self-distortion and leakage for every output channel of the separation system.
We further derive, for both measures, deterministic error radius and non-asymptotic, high-probability confidence intervals (CIs).
Experiments on English, Spanish, and music mixtures show that compared to 14 alternative measures, the PS and PM nearly always achieve the highest linear correlation coefficients with human mean-opinion scores, reaching as high as 86.36% for speech and 87.21% for music. We observe, at worst, an error radius of 1.39 % 1.39\% and a probabilistic 95% CI of 12.21% for these coefficients, which improves reliable and informed evaluation. Using mutual information, we show that the measures strongly complement each other, and more so as their values decrease, suggesting they are jointly more informative as system performance degrades. 1 1 1 Code available at https://github.com/Amir-Ivry/MAPSS-measures

## 1 Introduction

Progress in source-separation heavily depends on reliable perceptual evaluation, but gold-standard human listening tests remain prohibitively costly and time-consuming ITU-T. (1996; 2003; 2018).
Consequently, research commonly relies on objective metrics for model development and comparison. However, the most widely adopted metrics conflate two perceptually distinct failure modes: interference from competing talkers and distortion of the target signal.
Measures that better isolate these two modes could help close the gap between automatic evaluation and what listeners actually perceive, potentially enabling an accelerated and more trustworthy development of source separation.

Existing measures typically fall into several groups. Signal-to-distortion ratio (SDR), signal-to-interferences ratio (SIR), signal-to-artifacts ratio (SAR) Vincent et al. (2006), scale-invariant SDR (SI-SDR) Le Roux et al. (2019) and alike usually compute ratios between source to various disturbances purely with waveform energy, offering low complexity and widespread adoption. However, even jointly, they cannot tell by design whether an error stems from leakage or self-distortion.
Classical intrusive perceptual and intelligibility metrics like the PESQ Rix et al. (2001), STOI Taal et al. (2011) and ESTOI Jensen & Taal (2016) map an entire utterance to a mean-opinion score (MOS) scale using hand-crafted auditory features. Designed preliminary for speech enhancement, they perform well for corrupted noisy-reverberant speech utterances but may not account for leakage, while also lacking to provide access to their inherent granular processing.
Learned black-box metrics such as the DNSMOS family Reddy et al. (2022) that are trained end-to-end to predict crowd-sourced MOS, as well as SpeechBERT Tseng et al. (2021) and Sheet-SSQA Huang et al. (2025), have shown promising results on various speech tasks, but do not offer confidence in their decisions. Spectral-distance metrics are interpretable but tend to mask where degradations occur, e.g., the popular Mel-Cepstral Distortion (MCD) Fukada et al. (1992) collapses the spectral envelope into a global value.
Even when taking into account a broader set of metrics, e.g., as available in recently developed speech quality assessment toolkits Shi et al. (2024), no existing family of measures can simultaneously disentangle leakage from distortion, offer granular analysis, and provide error estimates for their decisions.

In this paper, we introduce the Perceptual Separation (PS) and Perceptual Match (PM), the first measures for source separation that functionally disentangle leakage and self-distortion. For every reference frame, at a 50 and 10 frames-per-second rate for speech and music mixtures, respectively, we synthesize a bank of fundamental distortions motivated by psycho-acoustic and auditory theory Gabrielsson & Sjögren (1979); Jekosch (2004); Wilson & Fazenda (2014); Bannister et al. (2024).
Then, a pretrained self-supervised model, e.g., wav2vec 2.0 Baevski et al. (2020), is used to independently encode the waveform of references, distortions, and system outputs across all sources, producing high-dimensional frame-wise representations. These are aggregated and projected via diffusion maps Coifman & Lafon (2006) onto a low-dimensional manifold, where a key property of the diffusion maps ensures that Euclidean distances on this manifold align with dissimilarities between encoded representations.
Harnessing this property and the geometric structures that distortions form around their references on the manifold, then the Mahalanobis distance from an output embedding to its own cluster is used to calculate the PM, whereas PS also accounts for the distance to the nearest foreign cluster.
Because these manifolds are learned at tens of milliseconds resolution, low PS pinpoints cross-talk frames, while low PM isolates self-distortion, indicating actionable debugging cues.
We derive theoretical deterministic error radius and non-asymptotic, high-probability confidence intervals (CIs) for both measures. These bounds enable frame-level guarantees on the reliability of the measures.

Evaluated on the SEBASS database Kastner & Herre (2022) that includes mixtures of English, Spanish, and music, the PS and PM jointly deliver the top linear correlation coefficients with subjective mean-opinion scores (MOS) in nearly every scenario, compared with $14$ competing widely-used measures. Specifically, coefficients reach as high as $87.21\%$ in music mixtures and $86.36\%$ in English mixtures, with maximal error radius of $1.39\%$ and 95% CI of $12.21\%$.
We also achieve the highest ranking-based correlation coefficients in music mixtures with a maximum of $85.54\%$, but scoring $84.69\%$ and $83.41\%$ in English and Spanish places the measures in the top-3 and top-5 places, respectively, with an error radius of 0.4% and a 95% CI of $30.03\%$ at worst. These error margins are imperative to an informed and reliable development of more progressed source-separation systems.
Empirically, we also show that the normalized mutual information between the measures never exceeds 0.2, and that as their values decrease it approaches zero. This raises the possibility that when the performance of the system degrades, the simultaneous information from both measures becomes more informative for a reliable assessment.

## 2 Problem Formulation

We mark column vectors and matrices with bold symbols and the rest with non-bold symbols.

Consider a trial with index $l$ in which a source separation system with index $q$ performs inference on a single-channel mixture, which includes $N^{l}$ time frames where at least two speakers are active Vincent et al. (2018). Let $\mathcal{F}^{l}$ be the set containing these time frame indices, and assume that in time frame ${f\in\mathcal{F}^{l}}$ there are $N^{l}_{f}\geq 2$ active speakers and their indices are contained in the set $\mathcal{S}^{l}_{f}$. It therefore holds that:

$$ $\displaystyle\left|\mathcal{F}^{l}\right|$ $\displaystyle=N^{l},$ (1) $\displaystyle\left|\mathcal{S}^{l}_{f}\right|$ $\displaystyle=N^{l}_{f}.$ (2) $$

Then, we can model the observed mixture as:

$$ $\mathbf{z}^{l}_{f}=\sum_{i\in\mathcal{S}^{l}_{f}}\mathbf{y}^{l}_{i,f}+\mathbf{v}_{f},$ (3) $$

where $\mathbf{z}^{l}_{f}=\big{[}z^{l}_{f}(n-L+1),\dots,z^{l}_{f}(n)\big{]}^{T}\in\mathbb{R}^{L}$, i.e., frame $f$ has $L$ samples and starts and ends in the $n-L+1$-th and the $n$-th sample of its signal, respectively. All vectors in the section follow the same structure and notation as $\mathbf{z}^{l}_{f}$. Each $\mathbf{y}^{l}_{i,f}\in\mathbb{R}^{L}$ is defined as the reference signal of the ${i\text{-th}}$ source in frame $f$ and holds the exact source instance that is mixed, potentially including reverberation, distortion, noise, or other interference inherent to its original conditions. The term $\mathbf{v}_{f}$ represents bounded system and environmental interference, assuming statistical independence of the sources. The estimation of $\mathbf{y}^{l}_{i,f}$ is denoted $\hat{\mathbf{y}}^{q,l}_{i,f}\in\mathbb{R}^{L}$.

The first contribution of this paper is the introduction of two new intrusive measures to evaluate source-separation systems. For each source index $i$:

- •
The perceptual separation (PS) measure quantifies how well $\hat{\mathbf{y}}^{q,l}_{i,f}$ is perceptually separated from all interfering sources $\{\mathbf{y}^{l}_{j,f}\}_{j\neq i}$.
- •
The perceptual match (PM) measure quantifies how closely the estimated source $\hat{\mathbf{y}}^{q,l}_{i,f}$ aligns perceptually with its reference $\mathbf{y}^{l}_{i,f}$.

The second contribution of this paper is the development of theoretical deterministic error radius and non-asymptotic, high-probability confidence intervals for both measures.

## 3 MAPSS: Manifold-based Assessment of Perceptual Speech Separation

This section discusses the theoretical framework and practical steps we apply to calculate the PM and PS measures. We temporarily assume a fixed speech separation system and time frame and omit the indices $l,q,f$ unless otherwise specified.

### 3.1 Diffusion Maps: Theoretical Foundations

Diffusion maps provide a low-dimensional manifold representation of a set of high-dimensional points, by learning their geometric and structural relationships Coifman & Lafon (2006).
Given a set ${\mathcal{X}=\{\mathbf{x}_{0},\mathbf{x}_{1},\dots,\mathbf{x}_{N-1}\}}$ of $N$ data points of dimension $M$, an affinity matrix ${\mathbf{K}\in\mathbb{R}^{N\times N}}$ is calculated using a Gaussian kernel:

$$ $\mathbf{K}_{i,j}=\exp\!\left(-\frac{\|\mathbf{x}_{i}-\mathbf{x}_{j}\|_{2}^{2}}{\sigma_{\mathbf{K}}^{2}}\right),$ (4) $$

where $i,j\in\{0,\ldots,N-1\}$ and $\forall i,j:0\leq K_{i,j}\leq 1$, and:

$$ $\sigma_{\mathbf{K}}^{2}=\operatorname{median}\left\{\left\|\mathbf{x}_{i}-\mathbf{x}_{j}\right\|_{2}^{2}\;\middle|\;i\neq j\right\}.$ (5) $$

To account for non-uniform sampling density of points, an $\alpha$-normalization replaces $\mathbf{K}$ by $\mathbf{K}^{(\alpha)}$:

$$ $\mathbf{K}^{(\alpha)}_{i,j}=\frac{\mathbf{K}_{i,j}}{\left(v_{i}v_{j}\right)^{\alpha}},$ (6) $$

where $\alpha\in\left[0,1\right]$ and $v_{i}=\sum_{j=0}^{N-1}\mathbf{K}_{i,j}$. Define the diagonal degree-matrix $\mathbf{D}^{(\alpha)}\in\mathbb{R}^{N\times N}$, given by:

$$ $\mathbf{D}^{(\alpha)}=\operatorname{diag}\left(v^{(\alpha)}_{0},\dots,v^{(\alpha)}_{N-1}\right),$ (7) $$

with:

$$ $v^{(\alpha)}_{i}=\sum_{j=0}^{N-1}\mathbf{K}^{(\alpha)}_{i,j},$ (8) $$

where $\forall i:v^{(\alpha)}_{i}>0$ by construction. For readability, we neglect the $\alpha$ notation from now on. The probability transition matrix ${\mathbf{P}\in\mathbb{R}^{N\times N}}$ on $\mathbf{K}$ can be defined as:

$$ $\mathbf{P}=\mathbf{D}^{-1}\mathbf{K}.$ (9) $$

Note $\mathbf{P}$ is row-stochastic, so $\forall i,j:\mathbf{P}_{ij}\geq 0,\text{ }\sum_{j=0}^{N-1}\mathbf{P}_{ij}=1$.
Spectral decomposition on $\mathbf{P}$ reveals a trivial right eigenvector ${\mathbf{u}_{0}=\mathbf{1}\in\mathbb{R}^{N}}$ with eigenvalue ${\lambda_{0}=1}$. Remaining eigenvectors $\{\mathbf{u}_{\ell}\}_{\ell=1}^{N-1}$ are associated with eigenvalues $\{\lambda_{\ell}\}_{\ell=1}^{N-1}$ ordered as ${1>\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N-1}>0}$, so that:

$$ $\mathbf{P}\mathbf{u}_{\ell}=\lambda_{\ell}\mathbf{u}_{\ell}.$ (10) $$

After $t\in\mathbb{Z}^{+}$ time steps on $\mathbf{P}$, the diffusion maps embedding of $\mathbf{x}_{i}$ onto manifold $\mathcal{M}$ is ${\Psi_{t}:\mathcal{X}\rightarrow\mathbb{R}^{N-1}}$:

$$ $\mathbf{\Psi}_{t}(\mathbf{x}_{i})=\left(\lambda_{1}^{t}\mathbf{u}_{1}(i),\,\lambda_{2}^{t}\mathbf{u}_{2}(i),\,\dots,\,\lambda_{N-1}^{t}\mathbf{u}_{N-1}(i)\right)^{T},$ (11) $$

where $\{\mathbf{u}_{\ell}\}_{\ell=1}^{N-1}$ are orthonormal with respect to the stationary measure ${\boldsymbol{\pi}=\left[\pi_{0},\ldots,\pi_{N-1}\right]^{T}\in\mathbb{R}^{N\times 1}}$ that satisfies:

$$ $\pi_{i}=\frac{\mathbf{D}_{ii}}{\sum_{j=0}^{N-1}\mathbf{D}_{jj}},$ (12) $$

where $\pi_{i}\in\left(0,1\right)$. Let $D_{t}(i,j)$ be the diffusion distance at time step $t$ between two points $\mathbf{x}_{i}$ and $\mathbf{x}_{j}$:

$$ $\displaystyle D^{2}_{t}(i,j)=\sum_{m=0}^{N-1}\frac{\left(\mathbf{P}^{t}_{im}-\mathbf{P}^{t}_{jm}\right)^{2}}{\boldsymbol{\pi}_{m}}$ (13) $$

where $\mathbf{P}^{t}_{im}$ denotes the probability of transitioning from node $i$ to node $m$ in $t$ time steps. Intuitively, the diffusion distance measures the similarity between the probability distributions of random walks starting from nodes $i$ and $j$. A key strength of diffusion maps we utilize is the following equivalence:

$$ $D^{2}_{t}(i,j)=\big{\|}\boldsymbol{\Psi}_{t}(\mathbf{x}_{i})-\boldsymbol{\Psi}_{t}(\mathbf{x}_{j})\big{\|}_{2}^{2}.$ (14) $$

This equivalence is fundamental to our approach, since it ensures that Euclidean distances on the manifold align with dissimilarities between their matching high-dimensional points.

In practice, the embedding in ([11](https://arxiv.org/html/2509.09212v1#S3.E11)) is truncated to its first $d$ coordinates, with $d<N-1$. This simple yet effective step reduces noise sensitivity and retains only the most meaningful, dominant geometric structures Nadler et al. (2006). Mapping ${\Psi_{t}:\mathcal{X}\rightarrow\mathbb{R}^{d}}$ onto manifold $\mathcal{M}^{\left(d\right)}$ gives:

$$ $\boldsymbol{\Psi}^{(d)}_{t}(\mathbf{x}_{i})=\left(\lambda_{1}^{t}\mathbf{u}_{1}(i),\,\lambda_{2}^{t}\mathbf{u}_{2}(i),\,\dots,\,\lambda_{d}^{t}\mathbf{u}_{d}(i)\right)^{T}.$ (15) $$

We cut off the spectral content by choosing $d$ as follows, given $\tau\in\left[0,1\right]$:

$$ $d=\min\left\{k\in\{1,\dots,N-1\}:\frac{\sum_{\ell=1}^{k}\lambda_{\ell}}{\sum_{\ell=1}^{N-1}\lambda_{\ell}}\geq\tau\right\},$ (16) $$

where we set $\tau=0.99$. Truncating the spectrum to $d$ dimensions breaks the equality in ([14](https://arxiv.org/html/2509.09212v1#S3.E14)), and leads to a truncation error. In App. ([A](https://arxiv.org/html/2509.09212v1#A1)) we derive the expectation and probabilistic tail bound for this truncation error.

### 3.2 Constructing Perceptual Clusters on the Manifold

Let us recall $N^{l}_{f}$ from Section [2](https://arxiv.org/html/2509.09212v1#S2). Given $i\in\{0,\dots,N^{l}_{f}-1\}$, the waveform reference signal of the $i$-th source, $\mathbf{y}_{i}$, undergoes $N_{p}$ hand-crafted perceptual distortions detailed in Appendix [D](https://arxiv.org/html/2509.09212v1#A4).
We define the $i$-th distortion set $\mathcal{D}_{i}$ as:

$$ $\mathcal{D}_{i}=\big{\{}\mathbf{\hat{y}}_{i},\mathbf{y}_{i},\mathbf{y}_{i,1},\ldots,\mathbf{y}_{i,{N_{p}}}\big{\}},$ (17) $$

where $\mathbf{y}_{i,p}\in\mathbb{R}^{L}$ is the $p$-th perceptual distortion of $\mathbf{y}_{i}$, for ${p\in\{1,\ldots,N_{p}\}}$.
Each waveform in $\mathcal{D}_{i}$ is independently encoded via a pre-trained self-supervised model, e.g., wav2vec 2.0 Baevski et al. (2020). Let $\Phi:\mathbb{R}^{L}\to\mathbb{R}^{M}$ be this encoding operator, then:

$$ $\mathbf{x}_{i,p}=\Phi\left(\mathbf{y}_{i,p}\right),\quad\mathbf{x}_{i}=\Phi\left(\mathbf{y}_{i}\right),\quad\hat{\mathbf{x}}_{i}=\Phi\left(\hat{\mathbf{y}}_{i}\right).$ (18) $$

Applying ([17](https://arxiv.org/html/2509.09212v1#S3.E17)), ([18](https://arxiv.org/html/2509.09212v1#S3.E18)) across all $N^{l}_{f}$ speech sources, we get the complete high-dimensional dataset:

$$ $\mathcal{X}=\Big{\{}\hat{\mathbf{x}}_{i},\mathbf{x}_{i},\mathbf{x}_{i,1},\dots,\mathbf{x}_{i,{N_{p}}}\;\Big{|}\;i=0,\dots,N^{l}_{f}-1\Big{\}},$ (19) $$

with $|\mathcal{X}|=N^{l}_{f}(N_{p}+2):=N$. The diffusion maps embedding of $\mathbf{x}\in\mathcal{X}$ on the manifold is thus:

$$ $\Psi_{t}^{(d)}(\mathbf{x})=\left(\lambda_{1}^{t}\mathbf{u}_{1}(k),\dots,\lambda_{d}^{t}\mathbf{u}_{d}(k)\right)^{T}\in\mathbb{R}^{d},$ (20) $$

where $k\in\big{\{}1,\ldots,N\big{\}}$ is the global index of $\mathbf{x}$. We can now define the $i$-th perceptual cluster on the manifold in the truncated dimension $d$ as:

$$ $c_{i}^{(d)}=\left\{\Psi_{t}^{(d)}(\mathbf{x}_{i}),\Psi_{t}^{(d)}(\mathbf{x}_{i,p})\;\middle|\;p=1,\dots,N_{p}\right\}.$ (21) $$

where we exclude the system output embedding $\Psi_{t}^{(d)}(\mathbf{\hat{x}}_{i})$ from $c_{i}^{(d)}$, since this embedding will be measured against the cluster statistics to yield the PS and PM measures. Including it in the cluster statistics would create circular dependency, where the embedding being evaluated would influence its own evaluation metrics.

By creating a group of waveform distortions from each reference, and then embedding their encoded versions using diffusion maps, perceptual geometric clusters are formed around the references on the manifold, as demonstrated in Figure [1](https://arxiv.org/html/2509.09212v1#S3.F1). Crucially, Euclidean distances between each pair of embedded coordinates align with the diffusion distance between their corresponding self-supervised representations ([14](https://arxiv.org/html/2509.09212v1#S3.E14)). We hypothesize and empirically demonstrate that these geometric distances can also be backtracked to perceptual dissimilarity between the corresponding waveforms. This perceptual-geometric correspondence forms the theoretical foundation for our proposed PS and PM measures.

Figure: Figure 1: Two embedded clusters on the manifold, associated with a two-speaker English mixture, created via diffusion maps. Each cluster consists of the reference of its speaker denoted by ‘x’ and its distortions marked by circles. We use the two prime coordinates $\left(\lambda_{1}^{t}\mathbf{u}_{1},\,\lambda_{2}^{t}\mathbf{u}_{2}\right)$ with $t=1$ ([15](https://arxiv.org/html/2509.09212v1#S3.E15)).
Refer to caption: x1.png

### 3.3 The Perceptual Separation (PS) Measure

We aim to quantify the perceptual separation of $\mathbf{\hat{y}}_{i}$ from its non-attributed references $\{\mathbf{y}_{j}\}_{i\neq j}$ with the Mahalanobis distance Evans et al. (2021). The empirical centroid and unbiased covariance matrix of the cluster $c^{(d)}_{j}$ are:

$$ $\displaystyle\hat{\boldsymbol{\mu}}_{j}^{(d)}$ $\displaystyle=\frac{1}{\Big{|}c^{(d)}_{j}\Big{|}}\sum_{\boldsymbol{\psi}\in c_{j}^{(d)}}\boldsymbol{\psi},$ (22) $\displaystyle\widehat{\boldsymbol{\Sigma}}_{j}^{(d)}$ $\displaystyle=\frac{1}{\Big{|}c^{(d)}_{j}\Big{|}-1}\sum_{\boldsymbol{\psi}\in c_{j}^{(d)}}\left(\boldsymbol{\psi}-\hat{\boldsymbol{\mu}}_{j}^{(d)}\right)\left(\boldsymbol{\psi}-\hat{\boldsymbol{\mu}}_{j}^{(d)}\right)^{T},$ (23) $$

where $\hat{\boldsymbol{\mu}}_{j}^{(d)}\in\mathbb{R}^{d},\text{ }\widehat{\boldsymbol{\Sigma}}_{j}^{(d)}\in\mathbb{R}^{d\times d}$.
The Mahalanobis distance from the embedding of the $i$-th output $\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i})$ to $c_{j}^{(d)}$ is given by:

$$ $\displaystyle d_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i});\hat{\boldsymbol{\mu}}_{j}^{(d)},\widehat{\boldsymbol{\Sigma}}_{j}^{(d)}\right)=$ (24) $\displaystyle\sqrt{\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i})-\hat{\boldsymbol{\mu}}_{j}^{(d)}\right)^{T}\left(\widehat{\boldsymbol{\Sigma}}_{j}^{(d)}+\epsilon I^{(d)}\right)^{-1}\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i})-\hat{\boldsymbol{\mu}}_{j}^{(d)}\right)},$ $$

where we use for regularization $\epsilon=10^{-6}$ with the $d$-dimensional identity matrix $I^{(d)}$.
We define the measured Mahalanobis distance from $\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i})$ to its attributed and closest non-attributed clusters as:

$$ $\displaystyle\hat{A}_{i}^{(d)}$ $\displaystyle=d_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i});\hat{\boldsymbol{\mu}}_{i}^{(d)},\widehat{\boldsymbol{\Sigma}}_{i}^{(d)}\right),$ (25) $\displaystyle\hat{B}_{i}^{(d)}$ $\displaystyle=\min_{j\neq i}d_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i});\hat{\boldsymbol{\mu}}_{j}^{(d)},\widehat{\boldsymbol{\Sigma}}_{j}^{(d)}\right),$ (26) $$

Then, the measured PS score for $\mathbf{\hat{y}}_{i}$ in the truncated dimension $d$ is:

$$ $\widehat{\text{PS}}_{i}^{(d)}=1-\frac{\hat{A}_{i}^{(d)}}{\hat{A}_{i}^{(d)}+\hat{B}_{i}^{(d)}},$ (27) $$

where higher values are desirable and by design $\hat{A}_{i}^{(d)}+\hat{B}_{i}^{(d)}>0$.
The distance to the attributed cluster $\hat{A}_{i}^{(d)}$ and
to the nearest foreign cluster $\hat{B}_{i}^{(d)}$ balance fidelity and separation, as demonstrated in Figure [2](https://arxiv.org/html/2509.09212v1#S3.F2). The PS maintains high values whenever $\hat{A}_{i}^{(d)}\ll\hat{B}_{i}^{(d)}$, and falls as $\hat{A}_{i}^{(d)}$ outgrows $\hat{B}_{i}^{(d)}$, representing growing leakage when the output of the $i$-th source in the separation system may perceptually resemble other references more than its attributed one.

Since within each cluster the distribution of distortions may not be isotropic, i.e., some directions capture subtle perturbations while others represent more variable changes, then using Euclidean distance alone would ignore this anisotropy and will not emphasize distortions depending on direction. The Mahalanobis distance resolves this by normalizing with the empirical covariance of each cluster, weighting deviations more heavily in stable directions and less in noisy ones. In this sense, Mahalanobis does not contradict the property ([14](https://arxiv.org/html/2509.09212v1#S3.E14)), it refines it locally and preserves it by incorporating second-order structure of the manifold.

Figure: Figure 2: Functional behavior of the PS measure with $0.25,0.5,0.75$ contour lines, and of the PM measure in three different setups of $\theta^{(d)}_{i}=1,a^{(d)}_{i}=1,k^{(d)}_{i}=1$.
Refer to caption: x2.png

### 3.4 The Perceptual Match (PM) Measure

The PM measure aims to quantify how perceptually aligned the estimated output $\hat{\mathbf{y}}_{i}$ is with its reference $\mathbf{y}_{i}$.
Let $\tilde{c}_{i}^{(d)}=c_{i}^{(d)}\setminus\Psi_{t}^{(d)}(\mathbf{x}_{i})$ denote the reference-free $i$-th cluster. We compute the unbiased empirical covariance matrix of $\tilde{c}_{i}^{(d)}$, relative to its reference embedding:

$$ $\displaystyle\widehat{\widetilde{\boldsymbol{\Sigma}}}_{i}^{(d)}=\frac{1}{\big{|}\tilde{c}^{(d)}_{i}\big{|}-1}\sum_{\mathbf{\boldsymbol{\psi}}\in\tilde{c}_{i}^{(d)}}\left(\boldsymbol{\psi}-\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i})\right)\left(\boldsymbol{\psi}-\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i})\right)^{T}.$ (28) $$

Then, for $p\in\{1,\ldots,N_{p}\}$ the squared Mahalanobis distance from the $p$-th distortion to its attributed reference in the $i$-th cluster, is given by:

$$ $\displaystyle d^{2}_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i,p});\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i}),\widehat{\widetilde{\boldsymbol{\Sigma}}}_{i}^{(d)}\right)=$ (29) $\displaystyle\hskip-60.00009pt\left(\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i,p})-\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i})\right)^{T}\left(\widehat{\widetilde{\boldsymbol{\Sigma}}}_{i}^{(d)}+\epsilon I^{(d)}\right)^{-1}\left(\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i,p})-\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i})\right),$ $$

where as in the PS case, $\epsilon=10^{-6}$. Let us define the set of distances:

$$ $\displaystyle\hat{\mathcal{G}}^{(d)}_{i}=\left\{d^{2}_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i,p});\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i}),\widehat{\widetilde{\boldsymbol{\Sigma}}}_{i}^{(d)}\right)\;\middle|\;p=1,\dots,N_{p}\right\}.$ (30) $$

Empirically, we observed that nearly always these distances are well-approximated by a Gamma distribution, validated using Kolmogorov-Smirnov goodness-of-fit tests Kolmogorov (1986); Smirnov (1948). The sample mean and unbiased variance of $\hat{\mathcal{G}}^{(d)}_{i}$ are estimated by:

$$ $\displaystyle\hat{\mu}_{\mathcal{G}^{(d)}_{i}}=\frac{1}{\Big{|}\hat{\mathcal{G}}^{(d)}_{i}\Big{|}}\sum_{g\in\hat{\mathcal{G}}^{(d)}_{i}}g\text{ ,}$ (31) $\displaystyle\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}=\frac{1}{\Big{|}\hat{\mathcal{G}}^{(d)}_{i}\Big{|}-1}\sum_{g\in\hat{\mathcal{G}}^{(d)}_{i}}{\left(g-\hat{\mu}_{\mathcal{G}^{(d)}_{i}}\right)^{2}},$ (32) $$

and can be moment-matched with a Gamma distribution, assuming $\hat{\mu}_{\mathcal{G}^{(d)}_{i}},\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}>0$, with parameters:

$$ $\displaystyle\hat{k}^{(d)}_{i}=\frac{\hat{\mu}_{\mathcal{G}^{(d)}_{i}}^{2}}{\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}},$ (33) $\displaystyle\hat{\theta}^{(d)}_{i}=\frac{\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}}{\hat{\mu}_{\mathcal{G}^{(d)}_{i}}}.$ (34) $$

Similarly, the squared Mahalanobis distance from the output embedding to its attributed cluster is:

$$ $\displaystyle d^{2}_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i});\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i}),\widehat{\widetilde{\boldsymbol{\Sigma}}}_{i}^{(d)}\right):=\hat{a}_{i}^{(d)}.$ (35) $$

Consider $Q(k,x)=\Gamma(k,x)/\Gamma(k)$ as the regularized upper incomplete Gamma function. Then, the PM score for $\hat{\mathbf{y}}_{i}$ in dimension $d$ is:

$$ $\widehat{\text{PM}}^{(d)}_{i}=Q\!\left(\hat{k}^{(d)}_{i},\frac{\hat{a}_{i}^{(d)}}{\hat{\theta}^{(d)}_{i}}\right),$ (36) $$

where again a higher score is more desirable, and $\hat{k}^{(d)}_{i}$ and $\hat{\theta}^{(d)}_{i}$ are well-defined by design as long as $N_{p}\geq 1$.

Figure: Figure 3: The distribution of PM and PS values across speech and music scenarios from the SEBASS database.
Refer to caption: x3.png

Differently from the PS, here the focus shifts to whether an output remains perceptually close to its own reference. Since each cluster of distortions around a reference may be anisotropic, the Mahalanobis distance is again the natural choice. If the output lies well within the bulk of its distortion cluster, the Gamma-tail probability is near one, which may indicate a strong perceptual match. As the output drifts away, the score decays smoothly toward zero, reflecting degradation. Observing Figure [2](https://arxiv.org/html/2509.09212v1#S3.F2), in edge cases, when distortions are tightly concentrated, even small mismatches in $a_{i}^{(d)}$ lower PM sharply, while, when distortions are more broadly dispersed, the PM remains tolerant to larger deviations.

The empirical distributions of the frame-level values of the measures are shown in Figure [3](https://arxiv.org/html/2509.09212v1#S3.F3).
The PM and PS metrics exhibit contrasting distribution patterns. PM values cluster predominantly around zero with minimal density near one, while PS concentrate near one with virtually no occurrence near zero. Although frame-level human speech quality ratings are not publicly available for direct comparison, these patterns raise comparisons to how humans might perceive audio disturbances.
The PM distribution aligns intuitively with human perception, as listeners typically penalize speech quality severely when disturbances occur, making ratings near the scale minimum unsurprising. However, real granular human ratings would likely show less extreme clustering around zero due to perceptual and rating scale complexities.
The PS behavior presents a more complex interpretative challenge. Previous research suggests that humans perceive leakage as more quality-degrading than self-distortions, particularly in acoustic echo cancellation contexts Khanagha et al. (2024), yet our findings here do not support this hypothesis. Whether this discrepancy stems from dataset characteristics, limitations of the PS measure itself, or the mismatch between granular PS values and aggregated human ratings remains unclear and warrants future investigation beyond the scope of this study.

## 4 Experimental Setup

Figure: Figure 4: Frame-level SNR estimations for English and Spanish references in the SEBASS database.
Refer to caption: x4.png

### 4.1 Database

To benchmark the proposed measures, we rely on the Subjective Evaluation of Blind Audio Source Separation (SEBASS) database (Kastner & Herre, 2022), a publicly-available collection of expertly curated listening tests that pools $11,000$ individual human ratings for more than $900$ separated signals, originating from five evaluation campaigns. SEBASS focuses on distinct scenarios of speech mixtures of 4 male and 4 female speakers, separately, with a pair of English and pair of Spanish speakers in each, and music with both a drums and a no-drums subset, each containing 3 music sources. The split between music mixtures with and without drums is crucial, as percussion transients introduce masking patterns that are perceptually and algorithmically distinct from purely harmonic content, thereby stressing leakage and self-distortion behaviors that do not appear in speech mixtures.

Every mixture was fed to 32 different speech separation systems, spanning classic methods such as non-negative matrix factorization (NMF), spatial filtering and modern deep-learning models.
Every output utterance of 10 s, sampled at a sampling frequency of 16 kHz, was judged by 10 or more predominantly certified human raters according to the MUSHRA standard Schoeffler et al. (2018). Every rater was asked to provide a grade between 0 and $100$ to the question “grade the basic audio quality of the items under test with respect to the reference signal. Any perceived differences between the reference and the other signal must be interpreted as an impairment”. During grading, the user was exposed to the reference as a baseline and to the hidden reference and mixture anchor as comparative outputs. Figure [4](https://arxiv.org/html/2509.09212v1#S4.F4) shows that the speech reference signals have been recorded in a relatively clean environment with signal-to-noise-ratios (SNRs) between 3.9 dB to 41.7 dB, with an average of 25 dB.

The SEBASS dataset suits this study for several reasons. Multilingual coverage of English and Spanish validates language-agnostic behavior, while music tests robustness to highly transient material. Large algorithmic spread creates rich output clusters that stress-test our methodology, and the dense sampling of raters allows for a more reliable estimation of the true mean-opinion score of subjective human opinion. Importantly, SEBASS represents the only available dataset incorporating human quality ratings for source separation tasks. While creating an additional human-rated dataset falls beyond our scope, we would like to note that expanded perceptual data for source separation would significantly benefit and foster future research in this area.

### 4.2 Pre-processing

We first recognize that English and Spanish speakers rarely participate in the same conversation in real-life scenarios. Thus, we separate each 4-speaker mixture into their English and Spanish speakers, creating for each language two mixtures where the one has a pair of male speakers and the other a pair of female speakers. We acknowledge the bias this contributes, as residuals of English may be present in the output signal of a Spanish speaker, and vice versa. It should be mentioned that listening tests have rendered this cross-language leakage extremely negligible, since as expected, source separation systems are able to leverage languages as a meaningful feature to recognize leakage and remove it.

Every waveform, including references, distortions, and outputs from all sources of the mixture, undergoes independent loudness normalization. We use the EBU Recommendation R-128 European Broadcasting Union (2011) (EBU) and set the target level of each waveform to loudness units relative to full scale (LUFS) of -23. If the peak magnitude of the scaled waveform exceeds one, we attenuate it to avoid digital clipping. This step removes loudness bias, known to wrongly affect both human and algorithmic quality judgments, while preserving inter-speaker level relations across the outputs. Since the PS and PM measures address source separation, we also filter out any frames in which there are not at least two active sources using energy-based thresholding.

When applying diffusion maps, we set $\alpha=1$ in equation ([6](https://arxiv.org/html/2509.09212v1#S3.E6)) to eliminate density-dependent bias from the embedding. This choice ensures that the PS and PM measures reflect the intrinsic geometric structure of the manifold rather than sampling density variations, which would introduce instead artificial distortions into the representation. In the specific case of music with drums, however, we choose to make an exception because density encodes naturally occurring frequency and perceptual salience of transient states, as in drum mixing. Letting $\alpha=0$ retains density bias and makes the PS and PM event-weighted, which we assume aligns with the perceptual experience of the average human listener. We set $t=1$ ([11](https://arxiv.org/html/2509.09212v1#S3.E11)) across the board, to keep the diffusion operator focused on local neighborhoods and not being blurred by multi-step mixing.

Unfortunately, granular subjective human scores are not available to the best of our knowledge, although a protocol to obtain them has been established Hansen & Kollmeier (1999). In fact, for source separation, SEBASS provides these scores at utterance-level resolution of approximately 10 seconds. Since our PM and PS measures operate at finer temporal resolutions of 50 and 10 frames-per-second for speech and music, respectively, then aggregation from frame-level to utterance-level is necessary to compare our measure to human MOS. Based on the characteristic of each measure, we have decided to apply average aggregation for the PM and a more perceptual, PESQ-inspired aggregation for the PS.

We now retrieve the indices used in Section [2](https://arxiv.org/html/2509.09212v1#S2) to formalize these aggregations. At trial $l$, let us denote the PM value of the $i$-th output of source-separation system $q$ in time frame $f$ as $\textrm{PM}^{q,l}_{i,f}$. We denote $\mathcal{F}^{l}_{i}\in\mathcal{F}^{l}$ as the subset of time-frame indices in which the $i$-th source is active. Then, the utterance-level PM measure after average aggregation is given by:

$$ ${\textrm{PM}}^{q,l}_{i,\textrm{utt}}=\frac{1}{\big{|}\mathcal{F}^{l}_{i}\big{|}}\sum_{f\in\mathcal{F}^{l}_{i}}{\textrm{PM}^{q,l}_{i,f}}.$ (37) $$

Although average aggregation assumes that human listeners perceive global audio quality by weighing local events equally, which is evidently not the case Rix et al. (2001), we chose to carry it for the PM since its behavior already exhibits strong and frequent granular penalties where the score drops to around zero. Thus, it is assumed that standard human behavior that weighs negative experience heavily in the utterance-level score is implicitly carried out by the nature of the PM measure itself.

However, this is not the case for the PS measure. Here, the aggregation we applied is inspired by the window-based pooling and logistic mapping used inside PESQ Rix et al. (2001). Again, considering only time frame indices in $\mathcal{F}^{l}_{i}$ and dropping the rest, let us consider a window of size $W$ frames that slides across the PS measure with a hop size of $H$ frames. Using the $p$-norm, we define the following:

$$ $\ell^{q,l}_{i,m}=\left(\frac{1}{W}\sum_{w=1}^{W}{\Big{|}\text{PS}^{q,l}_{i,(m-1)H+w}\Big{|}^{p}}\right)^{1/p},$ (38) $$

where $m\in\{1,\ldots,M^{l}_{i}\}$ and $M^{l}_{i}$ is the number of possible windows:

$$ $M^{l}_{i}=\max\left(1,\left\lfloor\frac{\big{|}\mathcal{F}^{l}_{i}\big{|}-W}{H}\right\rfloor\right).$ (39) $$

We then calculate the following root mean square expression:

$$ $\ell^{q,l}_{i}=\sqrt{\frac{1}{M^{l}_{i}}\sum_{m=1}^{M^{l}_{i}}{\left(\ell^{q,l}_{i,m}\right)^{2}}},$ (40) $$

and eventually the aggregated PS measure is given by:

$$ ${\textrm{PS}}^{q,l}_{i,\textrm{utt}}=0.999+\frac{4}{1+\exp\bigl{(}-1.3669\,\ell^{q,l}_{i}+3.8224\bigr{)}},$ (41) $$

where the constants were chosen according to ITU-T (2007). Here, we penalize lower scores explicitly using the $p$-norm to better match human perceptual aggregation.

The parameters we used for different scenarios and measures are summarized in Table [1](https://arxiv.org/html/2509.09212v1#S4.T1).

**Table 1: The parameters we pre-defined across scenarios for the PS and PM measures.**
| Measure | Scenario | $\alpha$ ([6](https://arxiv.org/html/2509.09212v1#S3.E6)) | $t$ ([11](https://arxiv.org/html/2509.09212v1#S3.E11)) | Aggregation |
| --- | --- | --- | --- | --- |
| PS | English | 1 | 1 | PESQ |
| PM | English | 1 | 1 | Average |
| PS | Spanish | 1 | 1 | PESQ |
| PM | Spanish | 1 | 1 | Average |
| PS | Music (no drums) | 1 | 1 | PESQ |
| PM | Music (no drums) | 1 | 1 | Average |
| PS | Music (drums) | 0 | 1 | PESQ |
| PM | Music (drums) | 0 | 1 | Average |

### 4.3 Metrics

The preliminary approach we use to evaluate the PM and PS measures is the correlation between their aggregated, utterance-level values and the MOS of the human listeners given in the SEBASS database, which are also utterance-based. We report on two types of correlation coefficients.

At trial $l$, let the utterance-level MOS of the $i$-th output from separation system $q$ be $v^{q,l}_{i}$. Given $Q$ independent source separation systems such that $q\in\{1,\ldots,Q\}$, consider the $Q$-dimensional vectors:

$$ $\displaystyle{\textrm{{PS}}}^{l}_{i,\textrm{utt}}=$ $\displaystyle\left({\textrm{PS}}^{1,l}_{i,\textrm{utt}},\ldots,{\textrm{PS}}^{Q,l}_{i,\textrm{utt}}\right)^{T},$ (42) $\displaystyle{\textrm{{PM}}}^{l}_{i,\textrm{utt}}=$ $\displaystyle\left({\textrm{PM}}^{1,l}_{i,\textrm{utt}},\ldots,{\textrm{PM}}^{Q,l}_{i,\textrm{utt}}\right)^{T},$ (43) $\displaystyle\mathbf{v}^{l}_{i}=$ $\displaystyle\left(v^{1,l}_{i},\ldots,v^{Q,l}_{i}\right)^{T}.$ (44) $$

Pearson product-moment correlation coefficient (PCC) assumes a linear relationship and therefore captures both rank and spacing between paired observations. High PCC indicates that the metric not only orders systems correctly but also preserves the perceptual distance listeners perceive Benesty et al. (2009). The PCC is measured twice, for the PS and the PM, as follows:

$$ $\displaystyle r^{\textrm{pcc},l}_{i}\left({\textrm{{PS}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right)=$ $\displaystyle\frac{\left({\overline{\textrm{{PS}}}}^{l}_{i,\textrm{utt}}\right)^{T}\mathbf{\overline{v}}^{l}_{i}}{\Big{\|}{\overline{\textrm{{PS}}}}^{l}_{i,\textrm{utt}}\Big{\|}_{2}\Big{\|}\mathbf{\overline{v}}^{l}_{i}\Big{\|}_{2}},$ (45) $\displaystyle r^{\textrm{pcc},l}_{i}\left({\textrm{{PM}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right)=$ $\displaystyle\frac{\left({\overline{\textrm{{PM}}}}^{l}_{i,\textrm{utt}}\right)^{T}\mathbf{\overline{v}}^{l}_{i}}{\Big{\|}{\overline{\textrm{{PM}}}}^{l}_{i,\textrm{utt}}\Big{\|}_{2}\Big{\|}\mathbf{\overline{v}}^{l}_{i}\Big{\|}_{2}},$ (46) $$

where ${\overline{\textrm{{PS}}}}^{l}_{i,\textrm{utt}},{\overline{\textrm{{PM}}}}^{l}_{i,\textrm{utt}}$ and $\mathbf{\overline{v}}^{l}_{i}$ are the centered versions of ${\textrm{{PS}}}^{l}_{i,\textrm{utt}},{\textrm{{PM}}}^{l}_{i,\textrm{utt}}$ and $\mathbf{v}^{l}_{i}$, respectively.

The Spearman rank-order correlation coefficient (SRCC) measures the strength of any monotonic relationship between two variables. Because it depends only on rank, SRCC is insensitive to strictly increasing, possibly non-linear re-scalings of either variable, making it ideal for answering the question “do the PS and PM measures rank systems exactly as listeners do ?” Sedgwick (2014). Let $\mathcal{R}:\mathbb{R}^{Q}\rightarrow R^{Q}$ be the ranking operator, which in the presence of ties assigns the average ranks. Again, the SRCC is measured for the PS and the PM:

$$ $\displaystyle\rho^{\textrm{srcc},l}_{i}\left({\textrm{{PS}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right)=$ $\displaystyle r^{\textrm{pcc},l}_{i}\left(\mathcal{R}\left({\textrm{{PS}}}^{l}_{i,\textrm{utt}}\right),\mathcal{R}\left(\mathbf{v}^{l}_{i}\right)\right),$ (47) $\displaystyle\rho^{\textrm{srcc},l}_{i}\left({\textrm{{PM}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right)=$ $\displaystyle r^{\textrm{pcc},l}_{i}\left(\mathcal{R}\left({\textrm{{PM}}}^{l}_{i,\textrm{utt}}\right),\mathcal{R}\left(\mathbf{v}^{l}_{i}\right)\right).$ (48) $$

We report these correlation coefficients per English, Spanish, and music mixtures scenarios separately. Given a scenario, we assume that it contains $\mathcal{L}$ independent trials such that ${l\in\{1,\ldots,\mathcal{L}\}}$ with $N^{l}_{\textrm{max}}$ speakers in each, where:

$$ $N^{l}_{\textrm{max}}=\max_{f\in\mathcal{F}^{l}}{N^{l}_{f}}.$ (49) $$

Then, for the PS and PM measures, the PCC and SRCC we report per scenario are given by:

$$ $\displaystyle\textrm{PS}^{\textrm{pcc}}=$ $\displaystyle\frac{1}{\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}}\sum_{l=1}^{\mathcal{L}}\sum_{i=1}^{N^{l}_{\textrm{max}}}r^{\textrm{pcc},l}_{i}\left({\textrm{{PS}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right),$ (50) $\displaystyle\textrm{PM}^{\textrm{pcc}}=$ $\displaystyle\frac{1}{\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}}\sum_{l=1}^{\mathcal{L}}\sum_{i=1}^{N^{l}_{\textrm{max}}}r^{\textrm{pcc},l}_{i}\left({\textrm{{PM}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right),$ (51) $\displaystyle\textrm{PS}^{\textrm{srcc}}=$ $\displaystyle\frac{1}{\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}}\sum_{l=1}^{\mathcal{L}}\sum_{i=1}^{N^{l}_{\textrm{max}}}\rho^{\textrm{srcc},l}_{i}\left({\textrm{{PS}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right),$ (52) $\displaystyle\textrm{PM}^{\textrm{srcc}}=$ $\displaystyle\frac{1}{\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}}\sum_{l=1}^{\mathcal{L}}\sum_{i=1}^{N^{l}_{\textrm{max}}}\rho^{\textrm{srcc},l}_{i}\left({\textrm{{PM}}}^{l}_{i,\textrm{utt}},\mathbf{v}^{l}_{i}\right).$ (53) $$

**Table 2: Self-supervised architectures, their pre-trained checkpoints, scenarios, and number of transformer layers.**
| Architecture | Checkpoint | Scenario | Trans. Layers |
| --- | --- | --- | --- |
| WavLM Large | microsoft/wavlm-large | English | 24 |
| WavLM Base | microsoft/wavlm-base | English | 12 |
| Wav2Vec 2.0 Large | facebook/wav2vec2-large-lv60 | English | 24 |
| Wav2Vec 2.0 Base | facebook/wav2vec2-base | English | 12 |
| HuBERT Large | facebook/hubert-large-ll60k | English | 24 |
| HuBERT Base | facebook/hubert-base-ls960 | English | 12 |
| Wav2Vec 2.0 Large | facebook/wav2vec2-large-xlsr-53 | Spanish | 24 |
| AST | MIT/ast-finetuned-audioset-10-10-0.4593 | Music | 12 |

## 5 Experimental Results

Figure: Figure 5: For English mixtures, the effect of transformer layers in different pretrained self-supervised models on the PCC and SRCC values for the PS and PM measures.
Refer to caption: x5.png

All results in this section are derived with zero-shot inference on the entire SEBASS database as described in Section [4](https://arxiv.org/html/2509.09212v1#S4), i.e., no training was involved and no data-driven parameter selection was applied.

We begin by analyzing how performance depends on the choice of the pre-trained self-supervised model, the purpose of which is encoding waveforms into perceptual representations before they are fed into the diffusion maps. Table [2](https://arxiv.org/html/2509.09212v1#S4.T2) lists the models we examine in this study. We consider six different models for English mixtures, based on the Wav2Vec 2.0 Baevski et al. (2020), WavLM Chen et al. (2022), and HuBERT Hsu et al. (2021) backbones, with Figure [5](https://arxiv.org/html/2509.09212v1#S5.F5) demonstrating their layer-wise performance.
When using “Large” versions of the models, for both PCC and SRCC values, earlier layers frequently produce representations that allow superior results that gradually decline toward deeper layers, showing approximately 10% average absolute degradation between extremes.
Existing layer-wise analysis already reported that acoustic and phonetic content is richly represented in intermediate layers, while deeper layers shift toward semantic abstraction Pasad et al. (2022); Vaidya & Kell (2022).
Additional work confirms that distortion sensitivity peaks in the lower or middle layers and diminishes in deeper ones Tamm et al. (2023); Hung et al. (2022), and that pretrained models tend to lose low-level signal fidelity in their deepest layers Moussa & Toneva (2025).
A notable data point appears in the final layers of Wav2Vec 2.0 with a sharp drop in performance, especially for PS. This is likely due to its contrastive learning pretraining objective, which drives later layers to specialize in predicting quantized latent codes rather than preserving acoustic detail.
For the “Base” versions of the models, we observe a somewhat different behavior. At low and middle layers, their performance is often quite competitive with the “Large” variants, and in several cases the former even outperforms the latter in deeper layers. However, for WavLM, the gap widens toward the final layers, with the “Large” version consistently outperforming. Interestingly, Wav2Vec 2.0 Base does not exhibit the sharp degradation observed in its counterpart and instead its deeper layers remain stable and even show improvements for PS, suggesting that the absence of over-specialization to quantized prediction in the “Base” model preserves sensitivity to perceptual distortions.

**Table 3: For English mixtures, comparing PCC and SRCC values between best-layer performance of “Large” and “Base” models. A raw waveform option, i.e. no encoding, is also reported. The highest SRCC and PCC are in bold per PS and PM.**
| Measure | Representation | Transformer Layer | SRCC | PCC |
| --- | --- | --- | --- | --- |
| PS | Wav2Vec 2.0 (Large) | 2 | 84.12% | 83.74% |
| PS | Wav2Vec 2.0 (Base) | 2 | 84.25% | 83.23% |
| PM | Wav2Vec 2.0 (Large) | 2 | 84.69% | 86.36% |
| PM | Wav2Vec 2.0 (Base) | 2 | 82.79% | 80.07% |
| PS | WavLM (Large) | 3 | 84.80% | 84.16% |
| PS | WavLM (Base) | 2 | 84.84% | 84.19% |
| PM | WavLM (Large) | 3 | 85.71% | 81.44% |
| PM | WavLM (Base) | 2 | 82.82% | 77.51% |
| PS | HuBERT (Large) | 3 | 84.48% | 83.09% |
| PS | HuBERT (Base) | 2 | 84.83% | 82.73% |
| PM | HuBERT (Large) | 3 | 84.12% | 82.24% |
| PM | HuBERT (Base) | 2 | 81.37% | 79.47% |
| PS | Waveform (raw) | - | 73.42% | 71.04% |
| PM | Waveform (raw) | - | 69.30% | 66.62% |

Table [3](https://arxiv.org/html/2509.09212v1#S5.T3) narrows these models down to their top performing layer, chosen by the max-min criteria of the PCC and SRCC values, across all layers. A no-encoding option is also reported, where waveforms are skip-connected directly into the diffusion maps, which under-performs compared to encoded modes and emphasizes the effectiveness of the waveform encoding in the proposed pipeline. These results reaffirm that shallow layers achieve optimal performance.
Although point-by-point comparisons show that “Base” models perform comparably to, or occasionally slightly exceed “Large” models, applying the max-min criteria across the models in the table reveals that ‘Large” models are preferable when jointly optimizing for PS and PM. For once, Wav2Vec 2.0 Large achieves for the PM a PCC and SRCC differences from its “Base” counterpart of absolute 6% and 2%, respectively, even when the PS case shows a negligible gap. Among the “Large” model variants, Wav2Vec 2.0 Large with transformer layer 2 emerges as the ideal configuration and we carry it forward as a case study we investigate. It should be noted that among “Large” models, the PS very slightly changes with roughly 1% and 0.5% gaps between extremes for the PCC and SRCC, respectively, while the PM gaps are more meaningful. This suggests that the choice of model will mainly affect the PM scores.

Figure: Figure 6: For all scenarios, the effect of transformer layers in their respective pretrained self-supervised architectures on the PCC and SRCC values for the PS and PM measures.
Refer to caption: x6.png

In Appendix [E](https://arxiv.org/html/2509.09212v1#A5), Figure [11](https://arxiv.org/html/2509.09212v1#A5.F11) shows reference and output spectrograms from an English mixture, time-aligned to corresponding PM and PS values over a 10-second utterance using the “Large” models, with layers specified in Table [3](https://arxiv.org/html/2509.09212v1#S5.T3). While a single example cannot be over-interpreted, the latest observation about the PS gaps across layers is visually supported here, with very similar behavior of all models. The PM shows highly correlated behavior, but with noticeable different values by Wav2Vec 2.0, which exhibits the highest PM value for PCC.
An interesting visual example is shown at approximately the 9 seconds mark, when both speakers exhibit visible self-distortion artifacts accompanied by sharp drops in their PM measures. Listening tests confirmed that leakage is indeed more present in “Speaker 2” than in “Speaker 1”, as supported by the PS plot.

Next, we extend the layer-wise analysis to Spanish and music mixtures and compare them with all “Large” English models in Figure [6](https://arxiv.org/html/2509.09212v1#S5.F6). Spanish and English mixtures, both evaluated with Wav2Vec 2.0 backbones, show broadly similar trends across layers, but Spanish exhibits a sharper decline in deeper layers. This can be explained by the XLSR pretraining data being relatively scarcer in Spanish than in English, leading later layers to emphasize cross-lingual abstractions over fine acoustic detail Conneau et al. (2020).
Music mixtures with drums exhibit the lowest performance among scenarios, which we attribute to the dominance of strong percussive transients. Self-supervised models have demonstrated less stability in these highly non-stationary regions, reducing the ability of PS and PM to capture perceptual degradations Zeghidour et al. (2021). In contrast, music mixtures without drums demonstrate consistently high performance, in some layers even surpassing speech mixtures. This likely stems from the AST backbone being particularly suited in capturing harmonic and timbral structure, allowing the measures to remain faithful to perceptual cues such as instrument texture and vocal clarity Gong et al. (2021).
Interestingly, mixtures with drums exhibit reduced performance in the middle layers, while both shallow and deep layers achieve relatively higher coefficients.
No-drum mixtures, however, show a relatively flat trend across layers, with a modest peak around the mid-layer region. This suggests that the AST representations of percussive onsets are more vulnerable to degradation in intermediate processing stages.
The max-min criteria across all layers, per scenario, shows that the ideal layers for Spanish, drums, and no-drums music mixtures are layers 2, 2, and 6, respectively. As for English, we carry these ideal configurations forward.

Table [4](https://arxiv.org/html/2509.09212v1#S5.T4) benchmarks the proposed PS and PM measures against 14 established metrics for speech and audio quality and also versus its waveform-only version. Per scenario, we report our measures with the best-performing configuration.
For English mixtures, both PS and PM consistently lead the table, except for PESQ taking the top with SRCC higher by an absolute of roughly 1% than our measures.
The advantage of PESQ in rank correlation can be attributed to its long-standing perceptual model, which explicitly encodes aspects of loudness perception, asymmetry, and time-alignment penalties, features that directly penalize separation artifacts.
In Spanish mixtures, PS and PM maintain top performance in terms of PCC, but fall behind PESQ and SDR-based metrics in SRCC. One possible explanation is that the syllable-timed rhythm and steady vowels of Spanish make fidelity-driven metrics such as SI-SDR, CI-SDR, and SDR more predictive of listener rankings, as these metrics emphasize reconstruction accuracy at the waveform level.
For music mixtures, PS and PM achieve the strongest overall correlations across both drums and no-drums conditions. Even though SpeechBERTscore has shown impressive results and is also based on a self-supervised backbone, it is mostly not competitive with our measures, and notably even performing worse than our raw waveform version at times, which projects on the importance of the diffusion maps in the pipeline.
We emphasize that unlike English, we only inspected one backbone model for Spanish or music mixtures. Also, the average and PESQ-like aggregation strategies were not a data-driven optimization but rather an heuristic and reasoning-based choice. Consequently, while the proposed measures already demonstrate strong alignment with human perception, these low-hanging fruits may potentially boost performance.
Quite surprisingly, the first group of STOI, PESQ, and SDR-based measures is consistently preferable to the second group consisting of DNSMOS, speechBERTscore, UTMOS, and others, which rarely achieve more than 70% in performance. One crucial conclusion this table suggests is that measures originally developed for a certain audio application, should not be zero-shot adapted into other applications, and in that case into source separation evaluation. Otherwise, there is a realistic risk of reporting values that highly drift from human opinion, and that may spiral the development of audio technologies instead of accelerating it.
Ultimately, the PM and PS results suggest a strong validation of them as the superior choices to capture leakage and self-distortion in a manner that resonates with human judgments for source separation systems.

**Table 4: SRCC and PCC of the PS and PM measures (underlined), their waveform counterparts, and 14 comparative measures, across scenarios. The top-3 results in every column are in bold.**
|  | English | Spanish | Music (Drums) | Music (No Drums) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Measure | SRCC | PCC | SRCC | PCC | SRCC | PCC | SRCC | PCC |
| PS | 84.12% | 83.74% | 82.33% | 85.01% | 70.63% | 79.14% | 84.35% | 87.21% |
| PM | 84.69% | 86.36% | 83.41% | 85.30% | 67.35% | 73.33% | 85.54% | 83.47% |
| PS (waveform) | 73.42% | 71.04% | 74.69% | 75.05% | 51.75% | 61.83% | 78.88% | 78.95% |
| PM (waveform) | 69.30% | 66.62% | 68.27% | 67.35% | 49.52% | 51.77% | 74.37% | 75.51% |
| STOI | 80.85% | 78.40% | 78.79% | 82.56% | 67.29% | 71.27% | 75.64% | 78.13% |
| eSTOI | 82.14% | 82.28% | 79.20% | 82.68% | 54.68% | 57.35% | 70.06% | 74.45% |
| PESQ | 85.56% | 84.05% | 86.06% | 84.98% | 61.60% | 53.87% | 61.26% | 60.24% |
| SI-SDR | 78.11% | 76.96% | 84.07% | 81.38% | 42.08% | 56.98% | 70.42% | 71.96% |
| SDR | 77.72% | 73.13% | 84.29% | 76.07% | 44.78% | 54.33% | 74.51% | 75.35% |
| SIR | 51.28% | 56.20% | 45.67% | 55.19% | 18.64% | 35.76% | 51.00% | 55.12% |
| SAR | 75.54% | 72.98% | 78.21% | 73.29% | 36.98% | 40.81% | 66.15% | 68.96% |
| CI-SDR | 78.66% | 77.41% | 84.32% | 81.48% | 45.02% | 55.42% | 74.25% | 75.11% |
| DNSMOS-OVRL | 63.70% | 67.77% | 35.34% | 43.57% | 21.79% | 34.27% | 13.81% | 19.47% |
| MCD | 43.05% | 33.86% | 45.90% | 37.97% | 30.27% | 42.23% | 33.49% | 32.19% |
| SpeechBERT | 68.58% | 67.44% | 69.55% | 70.48% | 52.33% | 59.71% | 75.60% | 81.13% |
| Sheet-SSQA | 41.17% | 51.38% | 61.06% | 73.01% | 39.40% | 29.03% | 14.19% | 5.17% |
| UTMOS | 55.53% | 55.43% | 52.22% | 55.75% | -9.24% | -8.25% | 12.59% | 7.72% |
| NISQA | 60.78% | 67.62% | 63.37% | 66.58% | 27.27% | 41.73% | 42.33% | 48.07% |

An additional stress test for our measures concerns their robustness to temporal misalignment between the input and output streams of the separator, a phenomenon commonly introduced by modern communication systems or, e.g., when dealing with references obtained from different, per-speaker microphones, such as in meeting datasets Carletta et al. (2005); Vinnikov et al. (2024).
Figure [7](https://arxiv.org/html/2509.09212v1#S5.F7) illustrates the effect of artificial delays ranging from 0 ms to 100 ms across English, Spanish, and music scenarios. While performance gradually degrades for speech scenarios as misalignment grows, as expected, a 20 ms delay or less still preserves coefficients higher than 80%. Surpassing this threshold, however, often causes a pronounced drop that underscores this weakness in our measures, since human ratings are insensitive to these short latencies. Music mixtures exhibit a different pattern, as performance remains largely stable across delays, with the presence of drums introducing more variability than its counterpart. This robustness can be partly attributed to the Audio Spectrogram Transformer (AST), which processes music with a temporal resolution of 100 ms and a receptive field of 160 ms, making it inherently less sensitive to short misalignments at the cost of reduced granularity.

Figure: Figure 7: The effect of temporal misalignment between references and outputs of the separation system on the PS and PM measures.
Refer to caption: x7.png

Figure: Figure 8: Normalized mutual information between the PS and PM measures across their thresholded values.
Refer to caption: x8.png

We examine the complementary relationship between PS and PM using normalized mutual information (NMI) Danon et al. (2005), which captures instantaneous statistical dependence beyond linear effects, where a lower NMI indicates less shared information. Each measure is first normalized per utterance to the range $\left[0,1\right]$. Then, for a threshold in the set $\{0.1,0.2,\ldots,1\}$, we retain frames with PS values lower than the threshold, and compute the NMI between the aligned pairs of the PS and PM, aggregated across utterances and conditions. We repeat the procedure
with roles reversed, i.e., thresholding on the PM. As shown in Figure [8](https://arxiv.org/html/2509.09212v1#S5.F8), NMI is generally modest and tends to decrease as the threshold tightens, approaching near-zero at the extreme. This pattern suggests that PS and PM convey increasingly complementary information when source separation quality is poor. Conversely, at looser thresholds, NMI rises up to 0.2. Since NMI can reach 1 under full redundancy, however, even this upper bound still indicates a considerable degree of complementarity between the measures. These results support reporting both PS and PM, particularly in challenging conditions, since each measure captures failure modes that the other may miss.
For transparency, we also report the number of samples contributing to each threshold that was used in the NMI estimates in Table [7](https://arxiv.org/html/2509.09212v1#A6.T7) in Appendix [F](https://arxiv.org/html/2509.09212v1#A6).

In the next phase, we investigate the deterministic error radius and probabilistic CIs derived for the PS and PM measures in Appendix [B](https://arxiv.org/html/2509.09212v1#A2). Figure [9](https://arxiv.org/html/2509.09212v1#S5.F9) shows histograms of the frame-level error distributions for speech and music mixtures. As expected, the radius caused by the spectral truncation in the diffusion maps process is typically an order of magnitude smaller than the 95% probabilistic width, originated from finite-sample clusters on the manifold. The error radius is also concentrated mostly near zero, which further confirms its negligibility. CIs typically span 10-50% of the dynamic range of the measures at the frame level, but surprisingly in the PM, between 10-15% of frame-level instances have probabilistic tails that approach zero across scenarios. The immediate contribution of these results are by making development of source separation systems more reliable and informed at the frame-level.

Now, we investigate how these frame-level errors propagate into the reported PCC and SRCC values at the scenario level, following the methodology detailed in Appendix [C](https://arxiv.org/html/2509.09212v1#A3). The resulting deterministic and probabilistic error ingredients are summarized in Table [5](https://arxiv.org/html/2509.09212v1#S5.T5). The former never exceeding $1.39\%$, a bias that even in the worst case barely changes the performance ranking in Table [4](https://arxiv.org/html/2509.09212v1#S5.T4). A more critical finding concerns the latter. To reach 95% confidence in the reported coefficients, PM requires at most an additional $8.98\%$ margin, while PS requires as much as $30.03\%$. This difference shows a trade-off where PS provides valuable sensitivity to source leakage but carries higher statistical uncertainty when propagated to scenario-level correlation coefficients. The practical consequences are twofold. First, while both measures align well with human perception, PM should be viewed as statistically more robust, particularly when system differences are small. Second, the larger confidence requirements for PS suggest that it should be used primarily as a complementary diagnostic, capturing perceptual aspects not reflected in PM but at the cost of greater variability. This complementarity underscores the value of considering PS and PM jointly, i.e., PS reveals perceptual distortions that may otherwise be missed, while PM ensures the stability and reliability of the reported correlations.

Finally, to give the reader an intuitive grasp of how the two error terms evolve in
a time-aligned manner with the PS and PM measures, Figure [12](https://arxiv.org/html/2509.09212v1#A5.F12) in Appendix [E](https://arxiv.org/html/2509.09212v1#A5) illustrates a representative example.

Figure: Figure 9: An histogram view of the frame-level deterministic error radius and the 95% probabilistic tail in the PS and PM measures across scenarios.
Refer to caption: x9.png

## 6 Limitations

Several factors currently limit the broad applicability of the PS and PM measures.

Our validation depends exclusively on the SEBASS database, the only public corpus that provides human ratings for source separation systems.
This corpus does not support variation of languages, accents, recording conditions, or speaker configurations that enables statistically significant conclusions across these verticals.
Moreover, the listening tests used to collect human ratings ask a generic basic quality question, rather than questions that isolate leakage versus self-distortion.
This design choice may attenuate the ground-truth sensitivity to the specific error modes that PS and PM are intended to disentangle, and can introduce a systematic bias that even multi-rater averaging cannot fully cancel. Another noticeable limitation of this research concerns the aggregation techniques we employ to convert frame-level to utterance-level scores. Since neither granular human ratings exist nor is there any documented data-driven mapping from granular to global human ratings, we limit the capability of the PS and PM measures by merely approximating aggregation functions.
In future work we aim to collect and release the first source separation benchmark that incorporates diverse realistic scenarios and granular human subjective ratings with targeted questions for leakage and self-distortions.

On a single NVIDIA A6000 GPU paired with 32 CPU cores with 64 GB of memory, our implementation achieves a real-time factor of 1.2, e.g., when analyzing a 25 ms frame in 30 ms on average.
While this enables offline evaluation and hyper-parameter sweeps, it falls short of strict real-time monitoring and may limit large-scale neural-architecture searches and limit using the PS and PM measures inside loss function during training sessions. Profiling reveals that the dominant bottlenecks are diffusion-map eigendecompositions and repeated Mahalanobis distance computations with per-frame covariance estimation for all distortions points in every cluster. We plan to introduce more efficient implementations as we maintain our code repository.

We also point out that in music mixtures, 0.5% of frames exhibit for the PM measure an error radius that exceeds 1, rendering these observations irrelevant. These cases should be ignored completely, and future work that focuses on the separation of music sources will further investigate this phenomenon.

## 7 Conclusions

To evaluate source separation systems more reliably and consistently, we introduce the PS and PM, a pair of frame-level differentiable measures that assess leakage and self-distortion on a diffusion map manifold using Mahalanobis geometry. Beyond proposing the measures, we established deterministic error envelopes and non-asymptotic confidence intervals, making the scores not only informative but also actionable under quantified uncertainty. Validations across English, Spanish, and music mixtures show that PM and PS correlate more strongly with human mean-opinion scores than existing alternative measures in the majority of tested scenarios. Moreover, it is suggested that as system quality degrades, the more mutual information increases between the measures, indicating that their joint use provides the most diagnostic value.

Looking ahead, we expect this two-fold view to streamline development in three concrete ways. First, as a diagnostic lens, PS and PM can pinpoint whether errors arise from target distortion or cross-talk, accelerating ablation, data curation, and model selection. Second, as training-time guidance, their differentiability enables loss shaping or curriculum triggers that explicitly trade fidelity against separation while monitoring confidence. Third, as a benchmarking layer, the attached uncertainty bounds support fairer hyper-parameter sweeps and reporting practices.

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## Appendix A Expectation and Probabilistic Confidence Bound of the Truncation Error

Assume a point $\mathbf{x}_{i}\in\mathcal{X}$ is drawn from the stationary distribution $\boldsymbol{\pi}$ ([12](https://arxiv.org/html/2509.09212v1#S3.E12)) of the diffusion process, where $i\in\{1,\ldots,N\}$. This assumption is supported by (Hein et al., 2005, Lem. 1), and by showing empirically on $5,000$ graphs that the corresponding eigenvector matches the theoretical stationary distribution up to statistical fluctuations.
Given that the $N-1$-dimensional embedding of $\mathbf{x}_{i}$ is truncated to dimension $d$, then the truncation error is expressed as ([15](https://arxiv.org/html/2509.09212v1#S3.E15)):

$$ $E(\mathbf{x}_{i})=\left(\sum_{\ell=d+1}^{N-1}\lambda_{\ell}^{2t}\mathbf{u}^{2}_{\ell}(i)\right)^{1/2}.$ (54) $$

We define the squared truncation error and analyze it:

$$ $T(\mathbf{x}_{i})=E^{2}(\mathbf{x}_{i})=\sum_{\ell=d+1}^{N-1}\lambda_{\ell}^{2t}\mathbf{u}^{2}_{\ell}(i).$ (55) $$

Since the eigenvectors $\{\mathbf{u_{\ell}}\}_{\ell=0}^{N-1}$ are orthonormal under $\boldsymbol{\pi}$, then:

$$ $\mathbb{E}_{\boldsymbol{\pi}}\left[\mathbf{u}^{2}_{\ell}(i)\right]=\sum_{i=0}^{N-1}\boldsymbol{\pi}_{i}\mathbf{u}_{\ell}^{2}(i)=1,$ (56) $$

from which we derive the expectation of $T(\mathbf{x}_{i})$ under $\boldsymbol{\pi}$:

$$ $\mathbb{E}_{\boldsymbol{\pi}}\left[T(\mathbf{x}_{i})\right]=\mathbb{E}_{\boldsymbol{\pi}}\left(\sum_{\ell=d+1}^{N-1}\lambda_{\ell}^{2t}u^{2}_{\ell}(i)\right)=\sum_{\ell=d+1}^{N-1}\lambda_{\ell}^{2t}.$ (57) $$

Thus, the expectation of the truncation error is given directly by:

$$ $\mathbb{E}_{\boldsymbol{\pi}}\left[E(\mathbf{x}_{i})\right]=\left(\sum_{\ell=d+1}^{N-1}\lambda_{\ell}^{2t}\right)^{1/2}.$ (58) $$

This term decays monotonically as $d$ grows and is typically lower than $10^{-3}$.
To obtain a non-asymptotic and high-probability confidence bound on the truncation error, we derive $\forall\ell,i$ ([56](https://arxiv.org/html/2509.09212v1#A1.E56)):

$$ $\big{|}\mathbf{u}_{\ell}(i)\big{|}\leq\pi_{\min}^{-1/2},\quad\pi_{\min}=\min_{i\in\{1,\ldots,N\}}{\boldsymbol{\pi}}.$ (59) $$

Any bounded variable is sub-Gaussian, and its
$\psi_{2}$-norm is at most the bound divided by $\sqrt{\ln 2}$
(Vershynin, 2024, Example 2.6.5):

$$ $\|\mathbf{u}_{\ell}(i)\|_{\psi_{2},\boldsymbol{\pi}}\leq\frac{\pi_{\min}^{-1/2}}{\sqrt{\ln 2}}:=K.$ (60) $$

Let $m=N-1-d$, so we define $\mathbf{z}_{i}\in\mathbb{R}^{m}$ as:

$$ $\mathbf{z}_{i}=\bigl{(}\mathbf{u}_{d+1}(i),\dots,\mathbf{u}_{N-1}(i)\bigr{)}^{T},$ (61) $$

and the diagonal matrix of weights $\mathbf{D}\in\mathbb{R}^{m\times m}$ as:

$$ $\mathbf{D}=\operatorname{diag}\bigl{(}\lambda_{d+1}^{t},\dots,\lambda_{N-1}^{t}\bigr{)}.$ (62) $$

Then $E(\mathbf{x}_{i})$ and $T(\mathbf{x}_{i})$ can be rewritten as:

$$ $\displaystyle T(\mathbf{x}_{i})=\bigl{\|}\mathbf{D}\mathbf{z}_{i}\bigr{\|}_{2}^{2},$ (63) $\displaystyle E(\mathbf{x}_{i})=\bigl{\|}\mathbf{D}\mathbf{z}_{i}\bigr{\|}_{2}.$ (64) $$

For $\ell>0$, $\mathbf{u}_{\ell}(i)$ is zero-mean under $\boldsymbol{\pi}$. Consequently, the vector $\mathbf{z}_{i}$ is zero-mean and by definition satisfies $\|\mathbf{z}_{i}\|_{\psi_{2},\boldsymbol{\pi}}\leq K\sqrt{m}$. We also notice that multiplication by a fixed matrix scales the
sub-Gaussian norm linearly, and since $\mathbf{D}$ is symmetric and positive:

$$ $\|\mathbf{D}\mathbf{z}(i)\|_{\psi_{2},\boldsymbol{\pi}}\leq K\sqrt{m}\|\mathbf{D}\|_{2}=K\sqrt{m}\max_{\ell>d}\lambda_{\ell}^{t}=K\sqrt{m}\lambda_{d+1}^{t}.$ (65) $$

According to (Vershynin, 2024, Prop. 6.2.1), for an
$m$-dimensional, zero-mean and sub-Gaussian vector $\mathbf{Y}$ with
$\|\mathbf{Y}\|_{\psi_{2},\boldsymbol{\pi}}\leq\kappa$, it holds:

$$ $\mathbb{P}_{\boldsymbol{\pi}}\bigl{\{}\|\mathbf{Y}\|_{2}\geq C\kappa(\sqrt{m}+t)\bigr{\}}\leq e^{-t^{2}},$ (66) $$

where $t\geq 0$ and $C>0$ is a constant.
Setting ${\mathbf{Y}=\mathbf{D}\mathbf{z}_{i}}$ and $\kappa=K\sqrt{m}\lambda_{d+1}^{t}$ gives:

$$ $\mathbb{P}_{\boldsymbol{\pi}}\Bigl{\{}T(\mathbf{x}_{i})>C^{2}\lambda_{d+1}^{2t}K^{2}m\bigl{(}\sqrt{m}+t\bigr{)}^{2}\Bigr{\}}\leq e^{-t^{2}}.$ (67) $$

Let $\delta\in\left(0,1\right)$ and set $t=\sqrt{\ln(1/\delta)}$. We can rewrite ([67](https://arxiv.org/html/2509.09212v1#A1.E67)) as:

$$ $\mathbb{P}_{\boldsymbol{\pi}}\Bigg{\{}T(\mathbf{x}_{i})\leq C^{2}\lambda_{d+1}^{2t}K^{2}m\left(\sqrt{m}+\sqrt{\ln\frac{1}{\delta}}\right)^{2}\Bigg{\}}\geq 1-\delta.$ (68) $$

Thus, the desired confidence bound on the truncation error is:

$$ $\mathbb{P}_{\boldsymbol{\pi}}\Bigg{\{}E(\mathbf{x}_{i})\leq C\lambda_{d+1}^{t}K\left(m+\sqrt{m\ln\frac{1}{\delta}}\right)\Bigg{\}}\geq 1-\delta.$ (69) $$

The choice of $d$ dimensions affects both $m$ that shrinks linearly with $d$ and $\lambda_{d+1}^{t}$ that falls monotonically with $d$. $K$ is affected by the minimal stationary probability $\pi_{\textrm{min}}$, so if the graph contains rare points then $\pi_{\textrm{min}}$ may be tiny, while a well-balanced graph derives $K\sim\sqrt{N}$ and tightens the bound.

## Appendix B Deterministic Error Radius and Probabilistic Tail Bound of the Measures

We derive a deterministic error radius and a high-probability confidence bound on the frame-level PS and PM measures by considering: (i) spectral truncation error due to retaining $d$ diffusion coordinates; (ii) finite-sample uncertainty in estimating the cluster centroid and covariance. We then combine these via union bounds. In this section, we consider a trial $l$, system $q$, and time frame $f$.

### B.1 The PS Measure

Considering source $i,j\in\{0,\ldots,N^{l}_{f}-1\}$, we begin by analyzing the effect of the truncation error, assuming access to cluster statistics. The difference between the embedding of $\hat{\mathbf{x}}_{i}$ and the centroid of cluster $j$ can be expressed in the truncated subspace $\mathbb{R}^{d}$ and in its complement subspace $\mathbb{R}^{m}$, respectively denoted $\boldsymbol{\Delta}^{(d)}_{i,j}$ and $\boldsymbol{\Delta}^{(\perp)}_{i,j}$. Using ([15](https://arxiv.org/html/2509.09212v1#S3.E15)), ([22](https://arxiv.org/html/2509.09212v1#S3.E22)):

$$ $\displaystyle\boldsymbol{\Delta}^{(d)}_{i,j}$ $\displaystyle=\boldsymbol{\Psi}^{(d)}_{t}(\hat{\mathbf{x}}_{i})-\boldsymbol{\mu}^{(d)}_{j}\in\mathbb{R}^{d},$ (70) $\displaystyle\boldsymbol{\Delta}^{(\perp)}_{i,j}$ $\displaystyle=\boldsymbol{\Psi}^{(\perp)}_{t}(\hat{\mathbf{x}}_{i})-\boldsymbol{\mu}^{(\perp)}_{j}\in\mathbb{R}^{m},$ (71) $$

where $m=N-d-1$. For completion, for every $\mathbf{x}\in\mathcal{X}$ and its global index $k\in\big{\{}1,\ldots,N\big{\}}$:

$$ $\displaystyle\boldsymbol{\mu}_{j}^{(\perp)}=$ $\displaystyle\frac{1}{\Big{|}c^{(\perp)}_{j}\Big{|}}\sum_{\boldsymbol{\psi}\in c_{j}^{(\perp)}}\boldsymbol{\psi}$ (72) $\displaystyle c_{j}^{(\perp)}=$ $\displaystyle\left\{\boldsymbol{\Psi}_{t}^{(\perp)}(\mathbf{x}_{j}),\boldsymbol{\Psi}_{t}^{(\perp)}(\mathbf{x}_{j,p})\;\middle|\;p=1,\dots,N_{p}\right\}$ (73) $\displaystyle\boldsymbol{\Psi}^{(\perp)}_{t}(\mathbf{x})=$ $\displaystyle\left(\lambda_{d+1}^{t}\mathbf{u}_{d+1}(k),\,\dots,\,\lambda_{N-1}^{t}\mathbf{u}_{N-1}(k)\right).$ (74) $$

In the full, $N-1$-dimensional space, the cluster $c_{j}$ is given by:

$$ $\displaystyle c_{j}=\left\{\boldsymbol{\Psi}_{t}(\mathbf{x}_{j}),\boldsymbol{\Psi}_{t}(\mathbf{x}_{j,p})\;\middle|\;p=1,\dots,N_{p}\right\},$ (75) $$

with mean $\boldsymbol{\mu}\in\mathbb{R}^{N-1}$, difference ${\boldsymbol{\Delta}_{i,j}\in\mathbb{R}^{N-1}}$ and covariance ${\boldsymbol{\Sigma}_{j}\in\mathbb{R}^{(N-1)\times(N-1)}}$ that hold:

$$ $\boldsymbol{\mu}_{j}=\begin{bmatrix}\boldsymbol{\mu}_{j}^{(d)}\\ \boldsymbol{\mu}_{j}^{(\perp)}\end{bmatrix},\quad\boldsymbol{\Delta}_{i,j}=\begin{bmatrix}\boldsymbol{\Delta}_{i,j}^{(d)}\\ \boldsymbol{\Delta}_{i,j}^{(\perp)}\end{bmatrix},\quad\boldsymbol{\Sigma}_{j}=\begin{bmatrix}\boldsymbol{\Sigma}_{j}^{(d)}&\boldsymbol{C}_{j}\\ \boldsymbol{C}_{j}^{T}&\boldsymbol{\Sigma}_{j}^{(\perp)}\end{bmatrix},$ (76) $$

where $\boldsymbol{\Sigma}_{j}^{(\perp)}\in\mathbb{R}^{m\times m}$ and $\boldsymbol{C}_{j}\in\mathbb{R}^{d\times m}$ are:

$$ $\displaystyle\boldsymbol{\Sigma}_{j}^{(\perp)}=\frac{1}{\Big{|}c^{(\perp)}_{j}\Big{|}-1}\sum_{\boldsymbol{\psi}\in c_{j}^{(\perp)}}\left(\boldsymbol{\psi}-\boldsymbol{{\mu}}^{(\perp)}_{j}\right)\left(\boldsymbol{\psi}-\boldsymbol{{\mu}}^{(\perp)}_{j}\right)^{T},$ (77) $\displaystyle\boldsymbol{C}_{j}=\frac{1}{\Big{|}c^{(\perp)}_{j}\Big{|}-1}\sum_{p=0}^{N_{p}}\left(\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{j,p})-{\boldsymbol{\mu}}_{j}^{(d)}\right)\left(\boldsymbol{\Psi}_{t}^{(\perp)}(\mathbf{x}_{j,p})-{\boldsymbol{\mu}}_{j}^{(\perp)}\right)^{T}.$ (78) $$

According to ([24](https://arxiv.org/html/2509.09212v1#S3.E24)), the squared Mahalanobis distance from $\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i})$ to $c_{j}$ is:

$$ $d_{M}^{2}\left(\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\mu}_{j},\boldsymbol{\Sigma}_{j}\right)=\boldsymbol{\Delta}_{i,j}^{T}\left(\boldsymbol{\Sigma}_{j}+\epsilon I^{(N-1)}\right)^{-1}\boldsymbol{\Delta}_{i,j},$ (79) $$

where inversion is empirically obtained by taking $\epsilon=10^{-6}$ with $I^{(N-1)}$ being the $N-1$-dimensional identity matrix. To evaluate the truncation effect, we perform blockwise inversion on ([79](https://arxiv.org/html/2509.09212v1#A2.E79)) via the Schur complement Horn & Johnson (2013):

$$ $\displaystyle d_{M}^{2}\left(\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\mu}_{j},\boldsymbol{\Sigma}_{j}\right)=\left(\boldsymbol{\Delta}_{i,j}^{(d)}\right)^{T}\left(\boldsymbol{\Sigma}_{j}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{\Delta}_{i,j}^{(d)}+\boldsymbol{r}_{i,j}^{T}\boldsymbol{S}_{j}^{-1}\boldsymbol{r}_{i,j},$ (80) $$

where $\boldsymbol{r}_{i,j}\in\mathbb{R}^{m}$ and the Schur complement $\boldsymbol{S}_{j}\in\mathbb{R}^{m\times m}$ hold:

$$ $\displaystyle\boldsymbol{r}_{i,j}$ $\displaystyle=\boldsymbol{\Delta}^{(\perp)}_{i,j}-\boldsymbol{C}_{j}^{T}\left(\boldsymbol{\Sigma}_{j}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{\Delta}^{(d)}_{i,j},$ (81) $\displaystyle\boldsymbol{S}_{j}$ $\displaystyle=\boldsymbol{\Sigma}_{j}^{(\perp)}-\boldsymbol{C}_{j}^{T}\left(\boldsymbol{\Sigma}_{j}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{C}_{j}.$ (82) $$

We now utilize the inequality:

$$ $\forall a,b\geq 0:\quad\Big{|}\sqrt{a+b}-\sqrt{a}\Big{|}\leq\sqrt{b},$ (83) $$

obtained by the mean-value theorem for $f(\cdot)=\sqrt{\cdot}$ (Rudin, 1976, Ch. 5). Let us set:

$$ $\displaystyle a=\left(\boldsymbol{\Delta}_{i,j}^{(d)}\right)^{T}\left(\boldsymbol{\Sigma}_{j}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{\Delta}_{i,j}^{(d)},$ (84) $\displaystyle b=\boldsymbol{r}_{i,j}^{T}\boldsymbol{S}_{j}^{-1}\boldsymbol{r}_{i,j},$ (85) $$

to obtain:

$$ $\displaystyle|\delta_{i,j}|=\left|d_{M}\left(\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\mu}_{j},\boldsymbol{\Sigma}_{j}\right)-d_{M}\left(\boldsymbol{\Psi}^{(d)}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\mu}^{(d)}_{j},\boldsymbol{\Sigma}^{(d)}_{j}\right)\right|\leq\sqrt{\boldsymbol{r}_{i,j}^{T}\boldsymbol{S}_{j}^{-1}\boldsymbol{r}_{i,j}}.$ (86) $$

Namely, $|\delta_{i,j}|$ is the truncation error of this Mahalanobis distance. From ([25](https://arxiv.org/html/2509.09212v1#S3.E25))-([26](https://arxiv.org/html/2509.09212v1#S3.E26)), it holds that:

$$ $\delta_{i,i}=A_{i}-A^{(d)}_{i},\quad\delta_{i,j^{*}}=B_{i}-B^{(d)}_{i},$ (87) $$

where:

$$ $\displaystyle j^{*}=\operatorname*{arg\,min}_{j\in\{0,\ldots,N^{l}_{f}-1\},\text{ }j\neq i}d_{M}\left(\boldsymbol{\Psi}^{\left(d\right)}_{t}\left(\hat{\mathbf{x}}_{i}\right);\boldsymbol{\mu}^{(d)}_{j},\boldsymbol{\Sigma}^{(d)}_{j}\right)$ (88) $\displaystyle A_{i}=d_{M}\left(\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\mu}_{i},\boldsymbol{\Sigma}_{i}\right),$ (89) $\displaystyle B_{i}=d_{M}\left(\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\mu}_{j^{*}},\boldsymbol{\Sigma}_{j^{*}}\right).$ (90) $$

Consider the $N-1$-dimensional representation of $\text{PS}^{(d)}_{i}$, i.e. $\text{PS}_{i}$ ([27](https://arxiv.org/html/2509.09212v1#S3.E27)):

$$ $\text{PS}_{i}=1-\frac{A_{i}}{A_{i}+B_{i}},$ (91) $$

which is smooth and differentiable in $A_{i}$ and $B_{i}$, since by definition $A_{i}+B_{i}>0$.
We assume and have empirically validated that truncation introduces a small relative change, i.e.:

$$ $\displaystyle|\delta_{i,i}|\ll A^{(d)}_{i}+B^{(d)}_{i},$ (92) $\displaystyle|\delta_{i,j^{*}}|\ll A^{(d)}_{i}+B^{(d)}_{i},$ (93) $$

making the first-order Taylor expansion of $\text{PS}_{i}$ around $\left(A_{i}^{(d)},B_{i}^{(d)}\right)$ valid. We can therefore write:

$$ $\displaystyle\text{PS}_{i}-\text{PS}_{i}^{(d)}$ $\displaystyle\simeq\frac{\partial\text{PS}_{i}}{\partial A_{i}}\left(A_{i}^{(d)},B_{i}^{(d)}\right)\delta_{i,i}+\frac{\partial\text{PS}_{i}}{\partial B_{i}}\left(A_{i}^{(d)},B_{i}^{(d)}\right)\delta_{i,j^{*}}$ (94) $\displaystyle=-\frac{B_{i}^{(d)}}{\left(A_{i}^{(d)}+B_{i}^{(d)}\right)^{2}}\,\delta_{i,i}+\frac{A_{i}^{(d)}}{\left(A_{i}^{(d)}+B_{i}^{(d)}\right)^{2}}\,\delta_{i,j^{*}},$ $$

where the quadratic remainder in the expansion is empirically one order smaller than the first-order term and can be safely dropped. Applying the triangle inequality and ([86](https://arxiv.org/html/2509.09212v1#A2.E86)) yields the deterministic error radius in the PS measure:

$$ $\displaystyle\Big{|}\text{PS}_{i}-\text{PS}_{i}^{(d)}\Big{|}\leq\frac{B_{i}^{(d)}|\delta_{i,i}|+A_{i}^{(d)}|\delta_{i,j^{*}}|}{\left(A_{i}^{(d)}+B_{i}^{(d)}\right)^{2}}=\frac{B_{i}^{(d)}\sqrt{\boldsymbol{r}_{i,i}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,i}}+A_{i}^{(d)}\sqrt{\boldsymbol{r}_{i,j^{*}}^{T}\boldsymbol{S}_{j^{*}}^{-1}\boldsymbol{r}_{i,j^{*}}}}{\left(A_{i}^{(d)}+B_{i}^{(d)}\right)^{2}}.$ (95) $$

We now quantify the uncertainty in $\widehat{\text{PS}}_{i}^{(d)}$ due to finite-sample cluster statistics.
Empirically, we observe that cluster coordinates exhibit weak dependence between one another and derive from Bartlett (1946) the following cut-off rule for the
effective sample size of cluster $c^{(d)}_{j}$:

$$ $n_{j,\mathrm{eff}}=\frac{n_{j}}{1+2\sum_{\ell=1}^{L_{j}}\hat{\rho}_{j,\ell}},\quad n_{j}=\Big{|}c^{(d)}_{j}\Big{|}$ (96) $$

where $\hat{\rho}_{j,\ell}$ is the empirical average Pearson auto-correlation of coordinates at lag $\ell$, and:

$$ $L_{j}=\operatorname*{arg\,min}_{\ell}\Bigg{\{}|\hat{\rho}_{j,\ell}|<\frac{z_{0.975}}{\sqrt{n_{j}-\ell}}\Bigg{\}}.$ (97) $$

Empirical evidence across $5,000$ graphs suggest that on average $\sum_{\ell}\hat{\rho}_{j,\ell}\simeq 0.2$, and so we set $n_{j,\mathrm{eff}}=0.7\,n_{j}$ for all clusters.

To bound the deviation between the estimated and true cluster mean and covariance, we employ the vector and matrix Bernstein (Vershynin, 2024, Props. 2.8.1, 4.7.1) and the dependent Hanson-Wright inequalities (Adamczak, 2015, Thm. 2.5). For every ${\delta^{\textrm{PS}}_{j,\boldsymbol{\mu}},\delta^{\textrm{PS}}_{j,\boldsymbol{\Sigma}}\in\left(0,1/2\right)}$, with respective least probabilities ${1-\delta^{\textrm{PS}}_{j,\boldsymbol{\mu}}}$ and ${1-\delta^{\textrm{PS}}_{j,\boldsymbol{\Sigma}}}$:

$$ $\displaystyle\big{|}{\boldsymbol{\mu}}_{j}-\hat{{\boldsymbol{\mu}}}_{j}\big{|}\leq\sqrt{\frac{2\lambda_{\max}\left(\widehat{\mathbf{\Sigma}}_{j}^{(d)}\right)\,\ln\left(2/\delta^{\textrm{PS}}_{j,\boldsymbol{\mu}}\right)}{n_{j,\mathrm{eff}}}}:=\Delta_{j,\boldsymbol{\mu}}$ (98) $\displaystyle\Big{\|}\boldsymbol{\Sigma}^{(d)}_{j}-\widehat{\boldsymbol{\Sigma}}^{(d)}_{j}\Big{\|}_{2}\leq C\lambda_{\textrm{max}}\left(\widehat{\mathbf{\Sigma}}_{j}^{(d)}\right)\left(\frac{r_{j}}{n_{j,\mathrm{eff}}}+\frac{r_{j}+\ln\left(2/\delta^{\textrm{PS}}_{j,\boldsymbol{\Sigma}}\right)}{n_{j,\mathrm{eff}}}\right):=\Delta_{j,\boldsymbol{\Sigma}},$ (99) $$

with an absolute constant $C>0$ and the ratio:

$$ $r_{j}=\frac{\operatorname{tr\left(\widehat{\mathbf{\Sigma}}_{j}^{(d)}\right)}}{\lambda_{\textrm{max}}\left(\widehat{\mathbf{\Sigma}}_{j}^{(d)}\right)}.$ (100) $$

Let us integrate the definitions of $\widehat{A}_{i}^{(d)}$ ([25](https://arxiv.org/html/2509.09212v1#S3.E25)) and $\widehat{B}_{i}^{(d)}$ ([26](https://arxiv.org/html/2509.09212v1#S3.E26)) with ([98](https://arxiv.org/html/2509.09212v1#A2.E98))-([99](https://arxiv.org/html/2509.09212v1#A2.E99)). Then, with probability of at least $1-\delta^{\textrm{PS}}_{i,\boldsymbol{\mu}}-\delta^{\textrm{PS}}_{i,\mathbf{\Sigma}}$, $\widehat{A}_{i}^{(d)}$ and $\widehat{B}_{i}^{(d)}$ deviate from their true versions by $\varepsilon^{\mathrm{PS}}\left(\widehat{A}_{i}^{(d)}\right)$ and $\varepsilon^{\mathrm{PS}}\left(\widehat{B}_{i}^{(d)}\right)$, bounded by:

$$ $\displaystyle\varepsilon^{\mathrm{PS}}\left(\widehat{A}_{i}^{(d)}\right)\leq 2\sqrt{\widehat{A}_{i}^{(d)}}\Delta_{i,\boldsymbol{\mu}}\sqrt{\frac{\lambda_{\textrm{max}}\left(\widehat{\mathbf{\Sigma}}_{i}^{(d)}\right)}{\tilde{\lambda}_{\textrm{min}}\left(\widehat{\mathbf{\Sigma}}_{i}^{(d)}\right)}}+\widehat{A}_{i}^{(d)}\frac{\Delta_{i,\mathbf{\Sigma}}}{\lambda_{\textrm{max}}\left(\widehat{\mathbf{\Sigma}}_{i}^{(d)}\right)},$ (101) $\displaystyle\varepsilon^{\mathrm{PS}}\left(\widehat{B}_{i}^{(d)}\right)\leq 2\sqrt{\widehat{B}_{i}^{(d)}}\Delta_{j^{\ast},\boldsymbol{\mu}}\sqrt{\frac{\lambda_{\textrm{max}}\left(\widehat{\mathbf{\Sigma}}_{j^{\ast}}^{(d)}\right)}{\tilde{\lambda}_{\textrm{min}}\left(\widehat{\mathbf{\Sigma}}_{j^{\ast}}^{(d)}\right)}}+\widehat{B}_{i}^{(d)}\frac{\Delta_{j^{\ast},\mathbf{\Sigma}}}{\lambda_{\textrm{max}}\left(\widehat{\mathbf{\Sigma}}_{j^{\ast}}^{(d)}\right)},$ (102) $$

where $j^{\ast}$ is defined in ([88](https://arxiv.org/html/2509.09212v1#A2.E88)). We avoid extremely loose bounds by replacing tiny, rarely observable eigenvalues, by a robust floor eigenvalue. Given a matrix $\mathbf{A}\in\mathbb{R}^{d\times d}$, we define $\tilde{\lambda}_{\textrm{min}}\left(\mathbf{A}\right)$ as (Horn & Johnson, 2013, Thm. 4.3.1):

$$ $\tilde{\lambda}_{\textrm{min}}\left(\mathbf{A}\right)=\lambda_{\textrm{min}}\left(\mathbf{A}+\epsilon_{r}\lambda_{\textrm{max}}\left(\mathbf{A}\right)I^{(d)}\right),$ (103) $$

where $\epsilon_{r}=0.05$ is typically taken and $I^{(d)}$ is the identity matrix.
Ultimately, let us define the Euclidean Lipschitz constant $L_{i,\textrm{lip}}$ as:

$$ $L_{i}^{\textrm{PS}}=\sqrt{\left(\frac{\partial\text{PS}_{i}}{\partial A_{i}}\left(A_{i}^{(d)},B_{i}^{(d)}\right)\right)^{2}+\left(\frac{\partial\text{PS}_{i}}{\partial B_{i}}\left(A_{i}^{(d)},B_{i}^{(d)}\right)\right)^{2}}=\frac{\sqrt{\left(\widehat{A}_{i}^{(d)}\right)^{2}+\left(\widehat{B}_{i}^{(d)}\right)^{2}}}{\left(\widehat{A}_{i}^{(d)}+\widehat{B}_{i}^{(d)}\right)^{2}},$ (104) $$

which enables us to bound the finite-sample deviation of $\widehat{\textrm{PS}}_{i}^{(d)}$ with:

$$ $\bigg{|}\widehat{\operatorname{PS}}_{i}^{(d)}-\operatorname{PS}_{i}^{(d)}\bigg{|}\leq L_{i}^{\textrm{PS}}\sqrt{\varepsilon^{\mathrm{PS}}\left(\widehat{A}_{i}^{(d)}\right)+\varepsilon^{\mathrm{PS}}\left(\widehat{B}_{i}^{(d)}\right)}.$ (105) $$

Finally, we employ the triangle inequality on both error sources ([95](https://arxiv.org/html/2509.09212v1#A2.E95)) and ([105](https://arxiv.org/html/2509.09212v1#A2.E105)) and obtain for $\delta_{i}^{\textrm{PS}}=\delta^{\textrm{PS}}_{i,\boldsymbol{\mu}}+\delta^{\textrm{PS}}_{i,\mathbf{\Sigma}}$, with $\delta_{i}^{\textrm{PS}}\in\left(0,1\right)$:

$$ $\displaystyle\mathbb{P}_{\boldsymbol{\pi}}\left\{\vphantom{\frac{\hat{B}_{i}^{(d)}\sqrt{\boldsymbol{r}_{i,i}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,i}}+\hat{A}_{i}^{(d)}\sqrt{\boldsymbol{r}_{i,j^{*}}^{T}\boldsymbol{S}_{j^{*}}^{-1}\boldsymbol{r}_{i,j^{*}}}}{(\hat{A}_{i}^{(d)}+\hat{B}_{i}^{(d)})^{2}}+\frac{\varepsilon_{i}+\varepsilon_{j^{\ast}}}{\hat{A}_{i}^{(d)}+\hat{B}_{i}^{(d)}-(\varepsilon_{i}+\varepsilon_{j^{\ast}})}}\left|\widehat{\text{PS}}_{i}^{(d)}-\text{PS}_{i}\right|\leq\right.$ (106) $\displaystyle\left.\frac{\hat{B}_{i}^{(d)}\sqrt{\boldsymbol{r}_{i,i}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,i}}+\hat{A}_{i}^{(d)}\sqrt{\boldsymbol{r}_{i,j^{*}}^{T}\boldsymbol{S}_{j^{*}}^{-1}\boldsymbol{r}_{i,j^{*}}}}{\left(\hat{A}_{i}^{(d)}+\hat{B}_{i}^{(d)}\right)^{2}}+\right.\left.L_{i}^{\textrm{PS}}\sqrt{\varepsilon^{\mathrm{PS}}\left(\widehat{A}_{i}^{(d)}\right)+\varepsilon^{\mathrm{PS}}\left(\widehat{B}_{i}^{(d)}\right)}\right\}\geq 1-\delta_{i}^{\textrm{PS}}.$ $$

We now analyze the obtained expression separately for the deterministic and probabilistic terms.
In the former term, the two square-root terms are energies that leak into the truncated complement after regressing out the retained $d$ diffusion coordinates.
Intuitively, $\boldsymbol{\Sigma}^{(d)}_{j}$ encodes the local anisotropy of cluster $j$ in the kept coordinates, $\boldsymbol{C}_{j}$ represents coupling of residual energy in the truncated block, and $\boldsymbol{\Sigma}^{(\perp)}_{j}$ is the spread that remains in the truncated block. Therefore, larger $\boldsymbol{S}_{j}$ down-weights complement deviations, reducing the bias, while $\boldsymbol{r}_{i,j}$ and $\boldsymbol{S}_{j}$ co-vary through the cross-covariance $\boldsymbol{C}_{j}$. Namely, increasing $\boldsymbol{C}_{j}$ shrinks both $\boldsymbol{r}_{i,j}$ and $\boldsymbol{S}_{j}$, while decreasing $\boldsymbol{C}_{j}$ does the opposite. The practical rule is to prevent tiny $\lambda_{\min}\left(\boldsymbol{S}_{j}\right)$, e.g., by promoting such directions into the kept set via a local choice of $d$, and shape distortions so the complement is
predictable from the kept coordinates, keeping $\boldsymbol{r}_{i,j}$ small.

For the probabilistic part, its width reflects uncertainty in the empirical centroid and covariance of the attributed and nearest foreign clusters, where $\lambda_{\max}\!\bigl{(}\widehat{\boldsymbol{\Sigma}}^{(d)}_{j}\bigr{)}$ and $n_{j,\mathrm{eff}}$ determine the width primarily. Practically, correlated distortions shrink $n_{j,\mathrm{eff}}$ and widen the bound, and a smaller spectral flatness ratio $r_{j}$ yields tighter matrix concentration. As expected, the probabilistic piece dominates the deterministic as shown in Figure [9](https://arxiv.org/html/2509.09212v1#S5.F9), which underlines the importance of cluster construction and dependence control.

### B.2 The PM Measure

As in the PS case, we start with the truncation error and assume access to cluster statistics. Let us reconsider Eqs. ([70](https://arxiv.org/html/2509.09212v1#A2.E70))–([82](https://arxiv.org/html/2509.09212v1#A2.E82)), but with two adjustments. First, the cluster coordinates are centered around the cluster reference embedding and not the cluster mean. Given $m=N-d-1$, we define:

$$ $\displaystyle\boldsymbol{\Delta}^{(d)}_{i,p}$ $\displaystyle=\boldsymbol{\Psi}^{(d)}_{t}(\mathbf{x}_{i,p})-\boldsymbol{\Psi}^{(d)}_{t}(\mathbf{x}_{i,0})\in\mathbb{R}^{d},$ (107) $\displaystyle\boldsymbol{\Delta}^{(\perp)}_{i,p}$ $\displaystyle=\boldsymbol{\Psi}^{(\perp)}_{t}(\mathbf{x}_{i,p})-\boldsymbol{\Psi}^{(\perp)}_{t}(\mathbf{x}_{i,0})\in\mathbb{R}^{m}.$ (108) $$

Second, the cluster is now absent the reference embedding. Namely, the full $N-1$-dimensional cluster is ([75](https://arxiv.org/html/2509.09212v1#A2.E75)):

$$ $\displaystyle\tilde{c}_{i}=c_{i}\setminus\Psi_{t}(\mathbf{x}_{i}).$ (109) $$

The cluster $\tilde{c}_{i}$ has difference ${\boldsymbol{\Delta}_{i,p}\in\mathbb{R}^{N-1}}$ for every ${p\in\{1,\ldots,N_{p}\}}$ and covariance ${\boldsymbol{\tilde{\Sigma}}_{i}\in\mathbb{R}^{(N-1)\times(N-1)}}$ that hold ([28](https://arxiv.org/html/2509.09212v1#S3.E28)):

$$ $\boldsymbol{\Delta}_{i,p}=\begin{bmatrix}\boldsymbol{\Delta}_{i,p}^{(d)}\\ \boldsymbol{\Delta}_{i,p}^{(\perp)}\end{bmatrix},\quad\boldsymbol{\tilde{\Sigma}}_{i}=\begin{bmatrix}\boldsymbol{\tilde{\Sigma}}_{i}^{(d)}&\boldsymbol{\tilde{C}}_{i}\\ \boldsymbol{\tilde{C}}_{i}^{T}&\boldsymbol{\tilde{\Sigma}}_{i}^{(\perp)}\end{bmatrix},$ (110) $$

with $\boldsymbol{\tilde{\Sigma}}_{i}^{(\perp)}\in\mathbb{R}^{m\times m}$ and $\boldsymbol{\tilde{C}}_{i}\in\mathbb{R}^{d\times m}$ being:

$$ $\displaystyle\boldsymbol{\tilde{\Sigma}}_{i}^{(\perp)}=\frac{1}{\Big{|}\tilde{c}^{(\perp)}_{i}\Big{|}-1}\sum_{\boldsymbol{\psi}\in\tilde{c}_{i}^{(\perp)}}\left(\boldsymbol{\psi}-\Psi_{t}^{(\perp)}(\mathbf{x}_{i})\right)\left(\boldsymbol{\psi}-\Psi_{t}^{(\perp)}(\mathbf{x}_{i})\right)^{T},$ (111) $\displaystyle\boldsymbol{\tilde{C}}_{i}=\frac{1}{\Big{|}\tilde{c}^{(\perp)}_{i}\Big{|}-1}\sum_{p=1}^{N_{p}}\left(\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i,p})-\Psi_{t}^{(d)}(\mathbf{x}_{i})\right)\left(\boldsymbol{\Psi}_{t}^{(\perp)}(\mathbf{x}_{i,p})-\Psi_{t}^{(\perp)}(\mathbf{x}_{i})\right)^{T}.$ (112) $$

In $N-1$ dimensions, the squared Mahalanobis distance from $\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,p})$ to $\tilde{c}_{i}$ is given by ([24](https://arxiv.org/html/2509.09212v1#S3.E24)):

$$ $\displaystyle d_{M}^{2}\left(\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,p});\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,0}),\boldsymbol{\tilde{\Sigma}}_{i}\right)=\boldsymbol{\Delta}_{i,p}^{T}\left(\boldsymbol{\tilde{\Sigma}}_{i}+\epsilon I^{(N-1)}\right)^{-1}\boldsymbol{\Delta}_{i,p},$ (113) $$

where as in ([79](https://arxiv.org/html/2509.09212v1#A2.E79)), inversion has been empirically obtained with $\epsilon=10^{-6}$ and the $N-1$-dimensional identity matrix $I^{(N-1)}$. We again turn to the Schur complement Horn & Johnson (2013) and decompose ([113](https://arxiv.org/html/2509.09212v1#A2.E113)):

$$ $\displaystyle d_{M}^{2}\left(\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,p});\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,0}),\boldsymbol{\tilde{\Sigma}}_{i}\right)=\left(\boldsymbol{\Delta}_{i,p}^{(d)}\right)^{T}\left(\boldsymbol{\tilde{\Sigma}}_{i}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{\Delta}_{i,p}^{(d)}+\boldsymbol{r}_{i,p}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,p},$ (114) $$

with $\boldsymbol{r}_{i,p}\in\mathbb{R}^{m}$ and the Schur complement $\boldsymbol{S}_{i}\in\mathbb{R}^{m\times m}$ being:

$$ $\displaystyle\boldsymbol{r}_{i,p}$ $\displaystyle=\boldsymbol{\Delta}^{(\perp)}_{i,p}-\boldsymbol{\tilde{C}}_{i}^{T}\left(\boldsymbol{\tilde{\Sigma}}_{i}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{\Delta}^{(d)}_{i,p},$ (115) $\displaystyle\boldsymbol{S}_{i}$ $\displaystyle=\boldsymbol{\tilde{\Sigma}}_{i}^{(\perp)}-\boldsymbol{\tilde{C}}_{i}^{T}\left(\boldsymbol{\tilde{\Sigma}}_{i}^{(d)}+\epsilon I^{(d)}\right)^{-1}\boldsymbol{\tilde{C}}_{i}.$ (116) $$

Let us define the set of squared Mahalanobis distances of cluster $\tilde{c}_{i}$ in dimension $N-1$ as:

$$ $\mathcal{G}_{i}=\left\{d^{2}_{M}\left(\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,p});\boldsymbol{\Psi}_{t}(\mathbf{x}_{i}),\widetilde{\boldsymbol{\Sigma}}_{i}\right)\;\middle|\;p=1,\dots,N_{p}\right\},$ (117) $$

in accordance to the truncated version of $\mathcal{G}^{(d)}_{i}$ in ([30](https://arxiv.org/html/2509.09212v1#S3.E30)).
By employing ([114](https://arxiv.org/html/2509.09212v1#A2.E114)), for every ${p\in\{1,\ldots,N_{p}\}}$, we can bound the truncation error of the squared Mahalanobis distance as follows:

$$ $\displaystyle d_{M}^{2}\left(\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,p});\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,0}),\boldsymbol{\tilde{\Sigma}}_{i}\right)-d_{M}^{2}\left(\boldsymbol{\Psi}^{(d)}_{t}(\mathbf{x}_{i,p});\boldsymbol{\Psi}^{(d)}_{t}(\mathbf{x}_{i,0}),\boldsymbol{\tilde{\Sigma}}^{(d)}_{i}\right)=$ (118) $\displaystyle\boldsymbol{r}_{i,p}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,p}:=\delta_{\mathcal{G}_{i},p},$ $$

and the difference between the mean of the elements in $\mathcal{G}_{i}$ and $\mathcal{G}^{(d)}_{i}$ can be expressed as:

$$ $\displaystyle\mu_{\mathcal{G}_{i}}-\mu_{\mathcal{G}^{(d)}_{i}}=\frac{1}{|\mathcal{G}_{i}|}\sum_{g\in\mathcal{G}_{i}}g-\frac{1}{\Big{|}\mathcal{G}^{(d)}_{i}\Big{|}}\sum_{g\in\mathcal{G}^{(d)}_{i}}g=\frac{1}{N_{p}}\sum_{p=1}^{N_{p}}\boldsymbol{r}_{i,p}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,p}=$ (119) $\displaystyle\frac{1}{N_{p}}\sum_{p=1}^{N_{p}}\delta_{\mathcal{G}_{i},p}:=\delta_{\mathcal{G}_{i},\mu}.$ $$

Similarly, we can express the deviation of the variance:

$$ $\displaystyle\sigma^{2}_{\mathcal{G}_{i}}-\sigma^{2}_{\mathcal{G}^{(d)}_{i}}=\frac{1}{|\mathcal{G}_{i}|-1}\sum_{g\in\mathcal{G}_{i}}\left(g-\mu_{\mathcal{G}_{i}}\right)^{2}-\frac{1}{\Big{|}\mathcal{G}^{(d)}_{i}\Big{|}-1}\sum_{g\in\mathcal{G}^{(d)}_{i}}\left(g-\mu_{\mathcal{G}^{(d)}_{i}}\right)^{2},$ (120) $$

and with ([119](https://arxiv.org/html/2509.09212v1#A2.E119)) and the Cauchy-Schwartz inequality, we can obtain:

$$ $\displaystyle\Big{|}\sigma^{2}_{\mathcal{G}_{i}}-\sigma^{2}_{\mathcal{G}^{(d)}_{i}}\Big{|}\leq\frac{N_{p}}{N_{p}-1}\left(2\delta^{\textrm{max}}_{\mathcal{G}_{i},p}\left(\sigma_{\mathcal{G}_{i}}+\sigma_{\mathcal{G}^{(d)}_{i}}\right)+\left(\delta^{\textrm{max}}_{\mathcal{G}_{i},p}\right)^{2}\right),$ (121) $$

where $\delta^{\textrm{max}}_{\mathcal{G}_{i},p}=\max_{p}{\delta_{\mathcal{G}_{i},p}}$. The Gamma-matching parameters in the truncated and full dimensions are ([33](https://arxiv.org/html/2509.09212v1#S3.E33)):

$$ $\displaystyle k^{(d)}_{i}=\frac{\mu^{2}_{\mathcal{G}^{(d)}_{i}}}{\sigma^{2}_{\mathcal{G}^{(d)}_{i}}},\quad k_{i}=\frac{\mu_{\mathcal{G}_{i}}^{2}}{\sigma^{2}_{\mathcal{G}_{i}}},$ (122) $\displaystyle\theta^{(d)}_{i}=\frac{\sigma^{2}_{\mathcal{G}^{(d)}_{i}}}{\mu_{\mathcal{G}^{(d)}_{i}}},\quad\theta_{i}=\frac{\sigma^{2}_{\mathcal{G}_{i}}}{\mu_{\mathcal{G}_{i}}},$ (123) $$

and their deviations can be bounded by considering ([119](https://arxiv.org/html/2509.09212v1#A2.E119)), ([121](https://arxiv.org/html/2509.09212v1#A2.E121)):

$$ $\displaystyle\Big{|}k_{i}-k_{i}^{(d)}\Big{|}$ $\displaystyle\leq C_{1}\delta^{\textrm{max}}_{\mathcal{G}_{i},p}\frac{N_{p}}{N_{p}-1}\frac{\mu_{\mathcal{G}_{i}}+\mu_{\mathcal{G}^{(d)}_{i}}}{\sigma^{2}_{\mathcal{G}^{(d)}_{i}}}:=\delta_{\mathcal{G}_{i},k},$ (124) $\displaystyle\Big{|}\theta_{i}-\theta_{i}^{(d)}\Big{|}$ $\displaystyle\leq C_{2}\delta^{\textrm{max}}_{\mathcal{G}_{i},p}\frac{N_{p}}{N_{p}-1}\frac{\sigma^{2}_{\mathcal{G}_{i}}+\sigma^{2}_{\mathcal{G}^{(d)}_{i}}}{\mu^{2}_{\mathcal{G}^{(d)}_{i}}}:=\delta_{\mathcal{G}_{i},\theta},$ (125) $$

with universal constants $C_{1},C_{2}>0$. Recalling ([35](https://arxiv.org/html/2509.09212v1#S3.E35)), let the squared Mahalanobis distance from the output embedding to the cluster be:

$$ $d^{2}_{M}\left(\boldsymbol{\Psi}_{t}^{(d)}(\hat{\mathbf{x}}_{i});\boldsymbol{\Psi}_{t}^{(d)}(\mathbf{x}_{i}),\boldsymbol{\tilde{\Sigma}}_{i}^{(d)}\right):=a_{i},$ (126) $$

and employing ([118](https://arxiv.org/html/2509.09212v1#A2.E118)) for the output embedding yields:

$$ $\displaystyle d_{M}^{2}\left(\boldsymbol{\Psi}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\Psi}_{t}(\mathbf{x}_{i,0}),\boldsymbol{\tilde{\Sigma}}_{i}\right)-d_{M}^{2}\left(\boldsymbol{\Psi}^{(d)}_{t}(\hat{\mathbf{x}}_{i});\boldsymbol{\Psi}^{(d)}_{t}(\mathbf{x}_{i,0}),\boldsymbol{\tilde{\Sigma}}^{(d)}_{i}\right)=\boldsymbol{r}_{i,a}^{T}\boldsymbol{S}_{i}^{-1}\boldsymbol{r}_{i,a}:=\delta_{\mathcal{G}_{i},a}.$ (127) $$

As in ([36](https://arxiv.org/html/2509.09212v1#S3.E36)), the PM definition in dimension ${N-1}$ can be expressed using the regularized upper incomplete gamma function $Q(k,x)=\Gamma(k,x)/\Gamma(k)$:

$$ $\text{PM}_{i}=Q\left(k_{i},\frac{a_{i}}{\theta_{i}}\right).$ (128) $$

Consider the truncation-induced ellipsoid:

$$ $\displaystyle\mathcal{B}_{i}=\Big{\{}(k^{\prime}_{i},\theta^{\prime}_{i},a^{\prime}_{i}):\Big{|}k^{\prime}_{i}-k_{i}^{(d)}\Big{|}\leq\delta_{\mathcal{G}_{i},k},\Big{|}\theta^{\prime}_{i}-\theta_{i}^{(d)}\Big{|}\leq\delta_{\mathcal{G}_{i},\theta},\Big{|}a^{\prime}_{i}-a_{i}^{(d)}\Big{|}\leq\delta_{\mathcal{G}_{i},a}\Big{\}},$ (129) $$

For $F(k,\theta,a)=Q\!\bigl{(}k,a/\theta\bigr{)}$, the gradient with the partial derivatives with respect to $k$, $\theta$ and $a$ is:

$$ $\displaystyle\nabla F(k,\theta,a)=\begin{pmatrix}\displaystyle\frac{1}{\Gamma(k)}\int_{x}^{\infty}t^{\,k-1}e^{-t}\,\ln t\,\mathrm{d}t-\psi(k)\,Q(k,x)\[14.0pt] \displaystyle\phantom{-}\frac{a}{\theta^{2}}\,\frac{x^{\,k-1}e^{-x}}{\Gamma(k)}\[14.0pt] \displaystyle-\frac{1}{\theta}\,\frac{x^{\,k-1}e^{-x}}{\Gamma(k)}\end{pmatrix},$ (130) $$

where $x=a/\theta$ and $\psi(\cdot)$ is the digamma function. Since $\nabla F(k,\theta,a)$ is continuous and bounded on the compact set $\mathcal{B}_{i}$, we set:

$$ $L_{\mathcal{B}_{i}}=\sup_{(k,\theta,a)\in\mathcal{B}_{i}}\bigl{\|}\nabla F(k,\theta,a)\bigr{\|}_{2}\;<\;\infty.$ (131) $$

To yield the bound on the PM measure due to truncation, we notice that both
$\left(k_{i},\theta_{i},a_{i}\right)$ and $\left(k_{i}^{(d)},\theta_{i}^{(d)},a_{i}^{(d)}\right)$
lie in $\mathcal{B}_{i}$, and apply the multivariate mean-value theorem to yield the following:

$$ $\Bigl{|}\mathrm{PM}_{i}-\mathrm{PM}_{i}^{(d)}\Bigr{|}\;\leq\;L_{\mathcal{B}_{i}}\,\Bigl{(}\delta_{\mathcal{G}_{i},k}^{2}+\delta_{\mathcal{G}_{i},\theta}^{2}+\delta_{\mathcal{G}_{i},a}^{2}\Bigr{)}^{1/2}.$ (132) $$

However, this bound can be tightened. We notice that
$Q(k,x)$ ([36](https://arxiv.org/html/2509.09212v1#S3.E36)) is monotonically increasing in $k$ and decreasing in $x$, for $k,\,x>0$ ([130](https://arxiv.org/html/2509.09212v1#A2.E130)). We assume that on $\mathcal{B}_{i}$, and for all $(\theta,a)\in\mathcal{B}_{i}$, $\partial F/\partial k$ does not change signs, or otherwise we fallback to ([132](https://arxiv.org/html/2509.09212v1#A2.E132)).
Consequently, the maximal change of $Q(k,x)$ inside
$\mathcal{B}_{i}$ is attained at one of its eight corners, and ([132](https://arxiv.org/html/2509.09212v1#A2.E132)) can be tightened to this PM error radius:

$$ $\Bigl{|}\mathrm{PM}_{i}-\mathrm{PM}_{i}^{(d)}\Bigr{|}\;\leq\;\max_{(k_{c},\theta_{c},a_{c})\in\partial\mathcal{B}_{i}}\Big{|}\,Q\left(k_{c},a_{c}/\theta_{c}\right)-Q\left(\hat{k}^{(d)}_{i},\hat{a}^{(d)}_{i}/\hat{\theta}^{(d)}_{i}\right)\Big{|}.$ (133) $$

As in the PS case, we now analyze how the finite number of coordinates in a cluster leads to uncertainty in the PM evaluation. Let $R_{i}$ be the maximal squared Mahalanobis distance in $\mathcal{G}_{i}$, namely:

$$ $R_{i}=\max_{g\in\mathcal{G}_{i}}\,g.$ (134) $$

Again, similarly to the PS case, we utilize the vector and matrix Bernstein (Vershynin, 2024, Props. 2.8.1, 4.7.1) and the dependent Hanson-Wright inequalities (Adamczak, 2015, Thm. 2.5). Let us consider the confidence parameters $\delta^{\textrm{PM}}_{i,\mu},\,\delta^{\textrm{PM}}_{i,\sigma},\,\delta^{\textrm{PM}}_{i,a}\in\left(0,1/3\right)$, so with respective least probabilities of $1-\delta^{\textrm{PM}}_{i,\mu},\,1-\delta^{\textrm{PM}}_{i,\sigma},\,1-\delta^{\textrm{PM}}_{i,a}$:

$$ $\displaystyle\Big{|}\mu_{\mathcal{G}^{(d)}_{i}}-\hat{\mu}_{\mathcal{G}^{(d)}_{i}}\Big{|}\leq\sqrt{\frac{2\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}\ln\left(2/\delta^{\textrm{PM}}_{i,\mu}\right)}{N_{p}}}+\frac{3R_{i}\ln\left(2/\delta^{\textrm{PM}}_{i,\mu}\right)}{N_{p}}:=\Delta_{i,\mu},$ (135) $\displaystyle\Big{|}\sigma_{\mathcal{G}^{(d)}_{i}}-\hat{\sigma}_{\mathcal{G}^{(d)}_{i}}\Big{|}\leq\sqrt{\frac{2R^{2}_{i}\ln\left(2/\delta^{\textrm{PM}}_{i,\sigma}\right)}{N_{p}}}+\frac{3R^{2}_{i}\ln\left(2/\delta^{\textrm{PM}}_{i,\sigma}\right)}{N_{p}}:=\Delta_{i,\sigma},$ (136) $\displaystyle\Big{|}a_{i}-\hat{a}_{i}\Big{|}\leq R_{i}\sqrt{\frac{\ln\left(2/\delta^{\textrm{PM}}_{i,a}\right)}{N_{p}}}:=\Delta_{i,a}.$ (137) $$

Recalling the definition of $k^{(d)}_{i},\,\theta^{(d)}_{i}$ from ([122](https://arxiv.org/html/2509.09212v1#A2.E122)), ([123](https://arxiv.org/html/2509.09212v1#A2.E123)), since by design $\hat{\mu}_{\mathcal{G}^{(d)}_{i}},\,\hat{\sigma}_{\mathcal{G}^{(d)}_{i}}>0$, and since we empirically validate that $\Delta_{i,\mu}\ll\hat{\mu}_{\mathcal{G}^{(d)}_{i}},\,\Delta_{i,\sigma}\ll\hat{\sigma}_{\mathcal{G}^{(d)}_{i}}$, we can apply the first-order Taylor expansions to $k^{(d)}_{i},\,\theta^{(d)}_{i}$ around $\hat{k}^{(d)}_{i},\,\hat{\theta}^{(d)}_{i}$, respectively. Apply the triangle inequality to it gives:

$$ $\displaystyle\Big{|}k^{(d)}_{i}-\hat{k}^{(d)}_{i}\Big{|}\leq\Bigg{|}\frac{\partial k^{(d)}_{i}}{\partial\mu}\Bigg{|}\Delta_{i,\mu}+\Bigg{|}\frac{\partial k^{(d)}_{i}}{\partial\sigma}\Bigg{|}\Delta_{i,\sigma}=\Bigg{|}\frac{2\hat{\mu}_{\mathcal{G}^{(d)}_{i}}}{\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}}\Bigg{|}\Delta_{i,\mu}+\Bigg{|}\frac{-2\hat{\mu}^{2}_{\mathcal{G}^{(d)}_{i}}}{\hat{\sigma}^{3}_{\mathcal{G}^{(d)}_{i}}}\Bigg{|}\Delta_{i,\sigma}:=\Delta_{i,k},$ (138) $\displaystyle\Big{|}\theta^{(d)}_{i}-\hat{\theta}^{(d)}_{i}\Big{|}\leq\Bigg{|}\frac{\partial\theta^{(d)}_{i}}{\partial\mu}\Bigg{|}\Delta_{i,\mu}+\Bigg{|}\frac{\partial\theta^{(d)}_{i}}{\partial\sigma}\Bigg{|}\Delta_{i,\sigma}=\Bigg{|}\frac{-\hat{\sigma}^{2}_{\mathcal{G}^{(d)}_{i}}}{\hat{\mu}^{2}_{\mathcal{G}^{(d)}_{i}}}\Bigg{|}\Delta_{i,\mu}+\Bigg{|}\frac{2\hat{\sigma}_{\mathcal{G}^{(d)}_{i}}}{\hat{\mu}_{\mathcal{G}^{(d)}_{i}}}\Bigg{|}\Delta_{i,\sigma}:=\Delta_{i,\theta}.$ (139) $$

Empirically, rarely $\Delta_{i,k}$, $\Delta_{i,\theta}$ or $\Delta_{i,a}$ become extremely loose. To avoid this behavior, we practically regularize the box by setting:

$$ $\displaystyle\Delta_{i,k}\rightarrow\min\left(\Delta_{i,k},0.5k^{(d)}_{i}\right),$ (140) $\displaystyle\Delta_{i,\theta}\rightarrow\min\left(\Delta_{i,\theta},0.5\theta^{(d)}_{i}\right),$ (141) $\displaystyle\Delta_{i,a}\rightarrow\min\left(\Delta_{i,a},0.5a^{(d)}_{i}\right).$ (142) $$

Let us consider the local box of values:

$$ $\mathcal{B}_{i}^{\mathrm{loc}}=\Big{\{}(k^{\prime}_{i},\theta^{\prime}_{i},a^{\prime}_{i}):k^{\prime}_{i}\in\left[k^{(d)}_{i}\pm\Delta_{i,k}\right],\theta^{\prime}_{i}\in\left[\theta^{(d)}_{i}\pm\Delta_{i,\theta}\right],a^{\prime}_{i}\in\left[a^{(d)}_{i}\pm\Delta_{i,a}\right]\Big{\}}.$ (143) $$

As discussed earlier, $Q(k,x)$ is monotonically increasing in $k$ and decreasing in $x$, for $k,\,x>0$ ([130](https://arxiv.org/html/2509.09212v1#A2.E130)). Consequently, the maximal change of $Q(k,x)$ inside $\mathcal{B}_{i}^{\mathrm{loc}}$ is attained at one of its eight corners. Thus, the finite-sample error of the PM measure in dimension $d$ is bounded by:

$$ $\bigg{|}\mathrm{PM}^{(d)}_{i}-\widehat{\mathrm{PM}}^{(d)}_{i}\bigg{|}\leq\max_{(k_{c},\theta_{c},a_{c})\in\partial\mathcal{B}_{i}^{\mathrm{loc}}}\Big{|}\,Q\left(k_{c},a_{c}/\theta_{c}\right)-Q\left(\hat{k}^{(d)}_{i},\hat{a}^{(d)}_{i}/\hat{\theta}^{(d)}_{i}\right)\Big{|}.$ (144) $$

Ultimately, we combine the deterministic error radius with the probabilistic width. Let ${\delta^{\textrm{PM}}_{i}=\delta^{\textrm{PM}}_{i,\mu}+\delta^{\textrm{PM}}_{i,\sigma}+\delta^{\textrm{PM}}_{i,a}}$, which yields for $\delta^{\textrm{PM}}_{i}\in\left(0,1\right)$:

$$ $\displaystyle\mathbb{P}_{\boldsymbol{\pi}}\left\{\vphantom{L_{Q,i}\,\Bigl{(}\delta_{\mathcal{G}_{i},k}^{2}+\delta_{\mathcal{G}_{i},\theta}^{2}+\delta_{\mathcal{G}_{i},a}^{2}\Bigr{)}^{1/2}}\left|\widehat{\text{PM}}_{i}^{(d)}-\text{PM}_{i}\right|\leq\right.$ (145) $\displaystyle\left.\max_{(k_{c},\theta_{c},a_{c})\in\partial\mathcal{B}_{i}}\Big{|}\,Q\left(k_{c},a_{c}/\theta_{c}\right)-Q\left(\hat{k}^{(d)}_{i},\hat{a}^{(d)}_{i}/\hat{\theta}^{(d)}_{i}\right)\Big{|}+\right.$ $\displaystyle\left.\max_{(k_{c},\theta_{c},a_{c})\in\partial\mathcal{B}_{i}^{\mathrm{loc}}}\Big{|}\,Q\left(k_{c},a_{c}/\theta_{c}\right)-Q\left(\hat{k}^{(d)}_{i},\hat{a}^{(d)}_{i}/\hat{\theta}^{(d)}_{i}\right)\Big{|}\right\}\geq 1-\delta^{\textrm{PM}}_{i}.$ $$

In the deterministic term, large cross-block coupling $\widetilde{\boldsymbol{C}}_{i}$ or residual spread $\widetilde{\boldsymbol{\Sigma}}^{(\perp)}_{i}$ again directly inflate the error radius via the Schur complement.

In the probabilistic part, the local box $\mathcal{B}^{\mathrm{loc}}_{i}$ aggregates two finite-sample pieces. The first is the uncertainty of the moment, with ${\Delta}_{i,\mu}$ and $\Delta_{i,\sigma}$ scale as $1/N_{p}$ but are amplified by the maximal radius of Mahalanobis within the cluster $R_{i}$. Heavy outliers increase $R_{i}$ and widen both bounds. The second is the uncertainty of the distance of the output, contributed by $\Delta_{i,a}$ which is also proportional to $R_{i}$ but scales by $1/\sqrt{N_{p}}$. Again, this emphasizes the importance of the design of distortions.

## Appendix C Error Radius and Probabilistic Confidence Bound of the PCC and SRCC

In this Appendix, we propagate the frame-level error radius and probabilistic widths developed in Appendix [B.1](https://arxiv.org/html/2509.09212v1#A2.SS1) and  [B.2](https://arxiv.org/html/2509.09212v1#A2.SS2) to the reported PCC and SRCC values. e.g., as reported in Table [4](https://arxiv.org/html/2509.09212v1#S5.T4).

We start by fixing a trial $l$, a source separation system $q$, and a time frame $f$. Let the indices of the active sources in frame $f$ be $\mathcal{S}^{l}_{f}$ and consider a source $i\in\mathcal{S}^{l}_{f}$.
The observation of measure $\mathcal{P}\in\{\mathrm{PS},\mathrm{PM}\}$, denoted $\widehat{v}^{q,l,\mathcal{P}}_{i,f}$, can be decomposed as:

$$ $\widehat{v}^{q,l,\mathcal{P}}_{i,f}\;=\;v^{q,l,\mathcal{P}}_{i,f}\;+\;\widetilde{\beta}^{q,l,\mathcal{P}}_{i,f}\;+\;\zeta^{q,l,\mathcal{P}}_{i,f},$ (146) $$

where:

$$ $\widetilde{\beta}^{q,l,\mathcal{P}}_{i,f}=\beta^{q,l,\mathcal{P}}_{i,f}+\mu^{q,l,\mathcal{P}}_{i,f},$ (147) $$

and $\beta^{q,l,\mathcal{P}}_{i,f}$ is an unknown deterministic bias with a provided radius $b^{q,l,\mathcal{P}}_{i,f}$, such that:

$$ $\big{|}\beta^{q,l,\mathcal{P}}_{i,f}\big{|}\leq b^{q,l,\mathcal{P}}_{i,f},$ (148) $$

with $b^{q,l,\mathcal{P}}_{i,f}$ given by either ([95](https://arxiv.org/html/2509.09212v1#A2.E95)) or ([133](https://arxiv.org/html/2509.09212v1#A2.E133)). Regarding the probabilistic side, we define:

$$ $\zeta^{q,l,\mathcal{P}}_{i,f}=\varepsilon^{q,l,\mathcal{P}}_{i,f}-\mu^{q,l,\mathcal{P}}_{i,f},$ (149) $$

where:

$$ $\varepsilon^{q,l,\mathcal{P}}_{i,f}=\widehat{v}^{q,l,\mathcal{P}}_{i,f}-v^{q,l,\mathcal{P}}_{i,f},$ (150) $$

and $\mathbb{E}_{\boldsymbol{\pi}}\left(\zeta^{q,l,\mathcal{P}}_{i,f}\right)=0$. Thus, the two-sided probabilistic half-width $p^{q,l,\mathcal{P}}_{i,f}\geq 0$ can be interpreted as:

$$ $\mathbb{P}_{\boldsymbol{\pi}}\left(\Big{|}\varepsilon^{q,l,\mathcal{P}}_{i,f}-\mu^{q,l,\mathcal{P}}_{i,f}\Big{|}\leq p^{q,l,\mathcal{P}}_{i,f}\right)\geq 1-\delta^{\mathcal{P}}:=c^{\mathcal{P}},$ (151) $$

with $\delta^{\mathcal{P}}$ and the probabilistic bounds defined in ([105](https://arxiv.org/html/2509.09212v1#A2.E105)) and ([144](https://arxiv.org/html/2509.09212v1#A2.E144)). We abbreviate $c^{\mathcal{P}}$ as $c$ from now on.
Consider $z_{c^{\ast}}$ the normal quantile at level $c^{\ast}=(1+c)/2$, so we calibrate the half-widths scale to be:

$$ $\sigma^{q,l,\mathcal{P}}_{i,f}=\frac{c}{z_{c^{\ast}}},$ (152) $$

with tails still reported back as half-widths at the original confidence $c$.

We now propagate these errors from frame to utterance level, based on the aggregations we introduced in ([37](https://arxiv.org/html/2509.09212v1#S4.E37)) and ([41](https://arxiv.org/html/2509.09212v1#S4.E41)). On average, experiments showed that frames more than $g=4$ apart are effectively independent both for speech and music mixtures. Given the set $\mathcal{F}^{l}$ of time frames with two or more active sources, the standard Bartlett block-decimation Bartlett (1946) yields the conservative inflation:

$$ $\mathrm{std}\!\left(\frac{1}{\mathcal{F}^{l}}\sum_{f=1}^{\mathcal{F}^{l}}\zeta^{q,l,\mathcal{P}}_{i,f}\right)\;\leq\;\frac{\sqrt{g+1}}{\sqrt{\mathcal{F}^{l}}}\left(\frac{1}{\mathcal{F}^{l}}\sum_{f=1}^{\mathcal{F}^{l}}\left(\sigma^{q,l,\mathcal{P}}_{i,f}\right)^{2}\right)^{1/2}.$ (153) $$

Let the radius error and the $p$-level probabilistic half-width obtained at the utterance-level using average pooling equal, respectively:

$$ $\displaystyle b^{q,l,\mathcal{P}}_{i,\textrm{average}}$ $\displaystyle=\frac{1}{\mathcal{F}^{l}}\sum_{f=1}^{\mathcal{F}^{l}}b^{q,l,\mathcal{P}}_{i,f},$ (154) $\displaystyle h^{q,l,\mathcal{P}}_{i,\textrm{average}}$ $\displaystyle=z_{c^{\ast}}\,\frac{\sqrt{g+1}}{\sqrt{\mathcal{F}^{l}}}\left(\frac{1}{\mathcal{F}^{l}}\sum_{f=1}^{\mathcal{F}^{l}}\left(\sigma^{q,l,\mathcal{P}}_{i,f}\right)^{2}\right)^{1/2}.$ (155) $$

For the PESQ-like aggregation, let us denote its aggregation function from ([41](https://arxiv.org/html/2509.09212v1#S4.E41)) as:

$$ $s(u)=0.999+4\bigl{(}1+\exp(-1.3669\,u+3.8224)\bigr{)}^{-1}.$ (156) $$

Let $W$ be the window and $H$ the hop of frame used for aggregation, then $M^{l}$ is the maximal number of possible windows. By norm submultiplicativity and the mean-value theorem (Horn & Johnson, 2013, Sec. 5.6):

$$ $\displaystyle b^{q,l,\mathcal{P}}_{i,\textrm{pesq}}$ $\displaystyle=\frac{C_{\mathrm{OL}}}{\sqrt{M^{l}}}\left(\frac{1}{\mathcal{F}^{l}}\sum_{f=1}^{\mathcal{F}^{l}}\left(b^{q,l,\mathcal{P}}_{f}\right)^{2}\right)^{\!1/2}\frac{\partial s}{\partial u},$ (157) $\displaystyle h^{q,l,\mathcal{P}}_{i,\textrm{pesq}}$ $\displaystyle=z_{c^{\ast}}\,\frac{C_{\mathrm{OL}}}{\sqrt{M^{l}}}\left(\frac{1}{\mathcal{F}^{l}}\sum_{f=1}^{\mathcal{F}^{l}}\left(\sigma^{q,l,\mathcal{P}}_{f}\right)^{2}\right)^{\!1/2}\frac{\partial s}{\partial u},$ (158) $$

where $C_{\mathrm{OL}}=\lceil W/H\rceil$ and by construction $\partial s/\partial u\leq 1.3669$ when evaluated at point $u$.

To translate utterance-level errors to source-based PCC and SRCC values, let the integration of utterance-level MOS ratings from every system $q\in\{1,\ldots,Q\}$ be:

$$ $\mathbf{v}^{l}_{i,\mathrm{MOS}}=\left(v^{1,l}_{i,\mathrm{MOS}},\ldots,v^{Q,l}_{i,\mathrm{MOS}}\right),$ (159) $$

and similarly, denoting $\hat{v}^{q,l,\mathcal{P}}_{i,\mathcal{A}}$ as the estimated aggregated measure across systems, where $\mathcal{A}$ is either average or PESQ-like aggregation ([4.2](https://arxiv.org/html/2509.09212v1#S4.SS2)), then its integration is given by:

$$ $\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}=\left(\hat{v}^{1,l,\mathcal{P}}_{i,\mathcal{A}},\ldots,\hat{v}^{Q,l,\mathcal{P}}_{i,\mathcal{A}}\right).$ (160) $$

For every vector $\mathbf{v}$, we denote its centered version by $\tilde{\mathbf{v}}$. Let us denote the PCC value between an observation vector $\mathbf{v}$ and a MOS vector $\mathbf{m}$ as $r^{\mathrm{PCC}}\left(\mathbf{v},\mathbf{m}\right)$, according to ([45](https://arxiv.org/html/2509.09212v1#S4.E45)) and ([46](https://arxiv.org/html/2509.09212v1#S4.E46)). Its gradient with respect to $\mathbf{v}$ at point $\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}$ is Benesty et al. (2009):

$$ $\frac{\partial r^{\textrm{PCC}}}{\partial\mathbf{v}}\Bigg{|}_{\mathbf{v}=\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}}=\frac{\mathbf{v}^{l}_{i,\mathrm{MOS}}}{\Big{\|}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}\Big{\|}_{2}\,\Big{\|}\mathbf{v}^{l}_{i,\mathrm{MOS}}\Big{\|}_{2}}-\frac{r^{\mathrm{PCC}}\left(\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}},\mathbf{v}^{l}_{i,\mathrm{MOS}}\right)}{\Big{\|}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}\Big{\|}_{2}^{2}}\,\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}.$ (161) $$

Consider $\mathbf{b}^{l,\mathcal{P}}_{i,\mathcal{A}}$ the utterance-level bias radii from ([154](https://arxiv.org/html/2509.09212v1#A3.E154)) or ([157](https://arxiv.org/html/2509.09212v1#A3.E157)) across all systems:

$$ $\mathbf{b}^{l,\mathcal{P}}_{i,\mathcal{A}}=\left(b^{1,l,\mathcal{P}}_{i,\mathcal{A}},\ldots,b^{Q,l,\mathcal{P}}_{i,\mathcal{A}}\right).$ (162) $$

Then, the induced PCC bias can be bounded by:

$$ $b^{l,\mathcal{P},\textrm{PCC}}_{i,\mathcal{A}}\;\leq\;\left\|\frac{\partial r^{\textrm{PCC}}}{\partial\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}}\right\|_{2}\,\left\|\tilde{\mathbf{b}}^{l,\mathcal{P}}_{i,\mathcal{A}}\right\|_{2}.$ (163) $$

For the probabilistic half-width, we model independent Gaussian jitters across systems with scales fixed by the utterance half-widths. Consider the $Q$-dimensional Gaussian vector:

$$ $\boldsymbol{\eta}\sim\mathcal{N}\!\left(\mathbf{0},\mathrm{diag}\left(\left(\frac{h^{1,l,\mathcal{P}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)^{2},\ldots,\left(\frac{h^{Q,l,\mathcal{P}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)^{2}\right)\right),$ (164) $$

with $\mathbf{0}\in\mathbb{R}^{Q}$. Using the delta method, first-order error propagation gives:

$$ $h^{l,\mathcal{P},\textrm{PCC}}_{i,\mathcal{A}}=z_{c^{\ast}}\,\sqrt{\left(\frac{\partial r^{\textrm{PCC}}}{\partial\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}}\right)^{T}\mathrm{diag}\left(\left(\frac{h^{1,l,\mathcal{P}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)^{2},\ldots,\left(\frac{h^{Q,l,\mathcal{P}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)^{2}\right)\left(\frac{\partial r^{\textrm{PCC}}}{\partial\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}}\right)}.$ (165) $$

Turning to the SRCC, let $\rho^{\mathrm{SRCC}}(\cdot,\cdot)$ denote Spearman’s rank correlation between two vectors Kendall & Gibbons (1990), as defined in ([47](https://arxiv.org/html/2509.09212v1#S4.E47)) and ([48](https://arxiv.org/html/2509.09212v1#S4.E48)). Because ranks are piecewise-constant, a safe deterministic error radius is obtained by checking the two extreme bias orientations:

$$ $\displaystyle b^{l,\mathcal{P},\textrm{SRCC}}_{i,\mathcal{A}}=\max\!\Big{(}\left|\rho^{\mathrm{SRCC}}\big{(}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}+\mathbf{b}^{l,\mathcal{P}}_{i,\mathcal{A}},\mathbf{v}^{l}_{i,\mathrm{MOS}}\big{)}-\rho^{\mathrm{SRCC}}\big{(}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}},\mathbf{v}^{l}_{i,\mathrm{MOS}}\big{)}\right|,\;$ (166) $\displaystyle\left|\rho^{\mathrm{SRCC}}\big{(}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}-\mathbf{b}^{l,\mathcal{P}}_{i,\mathcal{A}},\mathbf{v}^{l}_{i,\mathrm{MOS}}\big{)}-\rho^{\mathrm{SRCC}}\big{(}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}},\mathbf{v}^{l}_{i,\mathrm{MOS}}\big{)}\right|\Big{)}.$ $$

For the probabilistic half-width we jitter $\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}$ with the same independent Gaussian model in ([164](https://arxiv.org/html/2509.09212v1#A3.E164)) and report the empirical $c^{\ast}$ quantile from Monte Carlo of the following:

$$ $h^{l,\mathcal{P},\textrm{SRCC}}_{i,\mathcal{A}}=\;\mathrm{Quantile}_{c^{\ast}}\!\left(\left|\rho^{\mathrm{SRCC}}\big{(}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}}+\boldsymbol{\eta},\mathbf{v}^{l}_{i,\mathrm{MOS}}\big{)}-\rho^{\mathrm{SRCC}}\big{(}\hat{\mathbf{v}}^{l,\mathcal{P}}_{i,\mathcal{A}},\mathbf{v}^{l}_{i,\mathrm{MOS}}\big{)}\right|\right),$ (167) $$

where we used $10^{4}$ draws for estimation, in the spirit of quantile bootstrap Efron & Tibshirani (1994).

Lastly, we consider the error propagation across all trials and their sources in a given scenario, e.g., English mixtures.
Let $\mathcal{L}$ denote the number of trials in a scenario, and for each trial $l\in\{1,\ldots,\mathcal{L}\}$, assume the number of total speakers in the trial is $N^{l}_{\textrm{max}}$ ([49](https://arxiv.org/html/2509.09212v1#S4.E49)). The values we report average across all $\mathcal{L}$ trials and $N^{l}_{\textrm{max}}$ speakers, following ([50](https://arxiv.org/html/2509.09212v1#S4.E50))-([53](https://arxiv.org/html/2509.09212v1#S4.E53)).

The deterministic error radius of the PCC and SRCC per scenario are respectively given by:

$$ $\displaystyle b^{\textrm{PCC}}$ $\displaystyle=\ \frac{1}{\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}}\sum_{l=1}^{\mathcal{L}}\sum_{i=1}^{N^{l}_{\textrm{max}}}b^{l,\mathcal{P},\textrm{PCC}}_{i,\mathcal{A}},$ (168) $\displaystyle b^{\textrm{SRCC}}$ $\displaystyle=\ \frac{1}{\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}}\sum_{l=1}^{\mathcal{L}}\sum_{i=1}^{N^{l}_{\textrm{max}}}b^{l,\mathcal{P},\textrm{SRCC}}_{i,\mathcal{A}}.$ (169) $$

To yield the probabilistic term, we assume that within any fixed trial $l$, the pairwise correlation between the source jitters has been empirically estimated and is denoted by $\rho_{l}$, while jitters from different trials are independent. This assumption holds by the construction of our trials in every scenario [4.1](https://arxiv.org/html/2509.09212v1#S4.SS1).
Consequently, the $c$-level probabilistic half-width on the scenario mean equals:

$$ $\displaystyle h^{\textrm{PCC}}\;=\;$ (170) $\displaystyle z_{c^{\ast}}\,\sqrt{\frac{1}{\left(\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}\right)^{2}}\sum_{l=1}^{\mathcal{L}}\left(\sum_{i=1}^{N^{l}_{\textrm{max}}}\left(\frac{h^{l,\mathcal{P},\textrm{PCC}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)^{\!2}\;+\;2\,\rho_{l}\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{N^{l}_{\textrm{max}}}\left(\frac{h^{l,\mathcal{P},\textrm{PCC}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)\left(\frac{h^{l,\mathcal{P},\textrm{PCC}}_{j,\mathcal{A}}}{z_{c^{\ast}}}\right)\right)}.$ $\displaystyle h^{\textrm{SRCC}}\;=\;$ (171) $\displaystyle z_{c^{\ast}}\,\sqrt{\frac{1}{\left(\sum_{l=1}^{\mathcal{L}}{N^{l}_{\textrm{max}}}\right)^{2}}\sum_{l=1}^{\mathcal{L}}\left(\sum_{i=1}^{N^{l}_{\textrm{max}}}\left(\frac{h^{l,\mathcal{P},\textrm{SRCC}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)^{\!2}\;+\;2\,\rho_{l}\sum_{\begin{subarray}{c}i,j=1\\ i<j\end{subarray}}^{N^{l}_{\textrm{max}}}\left(\frac{h^{l,\mathcal{P},\textrm{SRCC}}_{i,\mathcal{A}}}{z_{c^{\ast}}}\right)\left(\frac{h^{l,\mathcal{P},\textrm{SRCC}}_{j,\mathcal{A}}}{z_{c^{\ast}}}\right)\right)}.$ $$

Ultimately, for each scenario and each measure $\mathcal{P}$ that uses aggregation technique $\mathcal{A}$, we report the deterministic envelope and probabilistic half-width $b^{\textrm{PCC}}$ and $h^{\textrm{PCC}}$ for PCC values and $b^{\textrm{SRCC}}$ and $h^{\textrm{SRCC}}$ for SRCC values.

## Appendix D Perceptual Distortions and Their Parameters Applied to the PS and PM

**Table 6: Distortion bank used for the PM and PS measures. $f_{\textrm{s}}$ is the sampling frequency, ${A}_{95}$ marks the 95th-percentile absolute amplitude, and $A_{\mathrm{RMS}}$ the RMS amplitude of their respective signals.**
| Distortion | PS | PM |
| --- | --- | --- |
| Notch Filter | Center frequencies:<br>500, 1000, 2000, 4000, 8000 Hz | Number of notches:<br>$\leq$ 20<br>Operating band:<br>80 Hz - 0.45$f_{\textrm{s}}$<br>Notch spacing:<br>$\geq$ 300 Hz<br>Bandwidth:<br>$\pm$60 Hz |
| Comb Filter | Delay:<br>2.5-15 ms<br>Feedback gain:<br>0.4-0.9 | Delay-gain pairs:<br>(2.5 ms, 0.4), (5 ms, 0.5),<br>(7.5 ms, 0.6), (10 ms, 0.7),<br>(12.5 ms, 0.9) |
| Tremolo | Rate:<br>1, 2, 4, 6 Hz<br>Depth:<br>0.3-1.0 | Rate:<br>1, 2, 4, 6 Hz<br>Depth:<br>1 |
| Additive Noise | SNR:<br>-15, -10, -5, 0, 5, 10, 15 dB<br>Noise color:<br>white, pink, brown | SNR:<br>-15, -10, -5, 0, 5, 10, 15 dB<br>Noise color:<br>white, pink, brown |
| Additive Harmonic Tone | Tone frequency:<br>100, 500, 1000, 4000 Hz<br>Amplitude:<br>0.02-0.08 (absolute) | Tone frequency:<br>100, 500, 1000, 4000 Hz<br>Amplitude:<br>$\{0.4,0.6,0.8,1\}\times A_{\mathrm{RMS}}$ |
| Reverberation | $\textrm{RT}_{60}$:<br>0.3-1.1 s<br>Early tail length:<br>5, 10, 15, 20 ms | Exponential tail length:<br>50, 100, 200, 400 ms<br>Decay scaling:<br>0.3, 0.5, 0.7, 0.9 |
| Noise Gate | Threshold:<br>0.005, 0.01, 0.02, 0.04 (absolute) | Threshold:<br>$\{0.05,0.1,0.2,0.4\}\times A_{95}$ |
| Pitch Shift | Offsets:<br>-4, -2, +2, +4 semitones | Offsets:<br>-4, -2, +2, +4 semitones |
| Low-Pass Filter | Cutoff:<br>2000, 3000, 4000, 6000 Hz | Cutoff rule:<br>spectral-energy quintiles:<br>50, 70, 85, 95%<br>Rounding:<br>nearest 100 Hz |
| High-Pass Filter | Cutoff:<br>100, 300, 500, 800 Hz | Cutoff rule:<br>spectral-energy quintiles:<br>5, 15, 30, 50%<br>Rounding:<br>nearest 100 Hz |
| Echo | Delay:<br>5-20 ms<br>Gain:<br>0.3-0.7 | Delay:<br>50, 100, 150 ms<br>Gain:<br>0.4, 0.5, 0.7 |
| Hard Clipping | Threshold:<br>0.3, 0.5, 0.7 (absolute) | Threshold:<br>$\{0.3,0.5,0.7\}\times A_{95}$ |
| Vibrato | Rate:<br>3, 5, 7 Hz<br>Depth:<br>0.001-0.003 (fractional stretch) | Rate:<br>3, 5, 7 Hz<br>Depth:<br>adaptive, clipped to 0.01-0.05 |

## Appendix E Temporal and Spectral Examples of the Measures

Figure: Figure 11: For an English mixture with two speakers, a spectral view of the system signals and aligned with a time-series view of the PS and PM measures of each speaker across different self-supervised architectures. Blank time intervals remain whenever speech does not overlap.
Refer to caption: x10.png

Figure: Figure 12: Time-aligned view of the PM and PS measures and their deterministic error radius and probabilistic tail with 95%, of two English speakers. Time indices where speech does not overlap remain blank.
Refer to caption: x11.png

## Appendix F Frame Counts used for Normalized Mutual Information Calculations

**Table 7: Frame counts used for NMI computation at each threshold, denoted ‘th’ in the table. Columns show counts of frames per scenario, split by PS and PM subsets.**
| Threshold | English | Spanish | Music (Drums) | Music (No Drums) |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- |
|  | PS$\leq$th | PM$\leq$th | PS$\leq$th | PM$\leq$th | PS$\leq$th | PM$\leq$th | PS$\leq$th | PM$\leq$th |
| 0.1 | 583 | 7426 | 622 | 10721 | 734 | 14768 | 1891 | 46183 |
| 0.2 | 1546 | 12086 | 1591 | 15756 | 2095 | 18697 | 6131 | 59124 |
| 0.3 | 3191 | 16350 | 3350 | 19714 | 4392 | 21496 | 12748 | 68927 |
| 0.4 | 5753 | 20118 | 6091 | 23054 | 8042 | 23550 | 22317 | 76990 |
| 0.5 | 9115 | 23697 | 9725 | 25964 | 12318 | 25285 | 34864 | 83607 |
| 0.6 | 13364 | 27232 | 13904 | 28627 | 16135 | 26892 | 47411 | 89667 |
| 0.7 | 18465 | 30758 | 18592 | 30885 | 19345 | 28443 | 59288 | 95242 |
| 0.8 | 24477 | 34076 | 23871 | 32703 | 22436 | 30148 | 70918 | 99536 |
| 0.9 | 31572 | 36507 | 29589 | 33902 | 26080 | 31397 | 85541 | 102589 |
| 1 | 37888 | 37888 | 34496 | 34496 | 31965 | 31965 | 104058 | 104058 |

## Peer Reviews

confidence: 2
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. Can the authors empirically or theoretically justify the assumption that diffusion map Euclidean distances on the perceptual manifold directly correspond to perceptual similarity, especially for out-of-domain distortions?

2. How robust are the PS and PM measures to errors in the choice of self-supervised model?
soundness: 3
strengths: 1. The PS and PM measures distinctly address leakage and self-distortion, providing interpretable, complementary diagnostics as established in both the main text and Figure 1. This is a clear improvement over single-score evaluation methods, which frequently conflate distinct error sources.

2. The derivations for both measures are grounded in Mahalanobis distance computation on a perceptual manifold, with clear steps and full error analysis.

3. The paper goes beyond point estimates, deriving deterministic and probabilistic error bounds.
summary: This paper introduces two frame-level, intrusive metrics: Perceptual Separation and Perceptual Match, for the assessment of audio source separation systems. The approach decomposes perceptual errors into leakage and self-distortion by encoding distorted signal variants through a pre-trained self-supervised model and embedding the resulting features onto a manifold using diffusion maps. Experimental results on the SEBASS benchmark demonstrate that both metrics offer robust correlation with human MOS compared to 14 existing measures, and the paper derives error and confidence bounds for both metrics.
weaknesses: 1. Several directly related and highly contemporary works on perceptual audio metrics, unsupervised perceptual embeddings, and manifold learning for audio are not discussed, despite their clear relevance.

[1] DeePAQ: A Perceptual Audio Quality Metric Based On Foundational Models and Weakly Supervised Learning
[2] NOMAD: Unsupervised Learning of Perceptual Embeddings for Speech Enhancement and Non-matching Reference Audio Quality Assessment

2. The paper states the hypothesis that diffusion distances on the perceptual manifold align with perceived similarity, but aside from empirical validation, this foundational assumption is not theoretically justified.

3. Despite benchmarking against many baselines, the evaluation gives limited direct insight into when/why PS or PM fails relative to credible alternatives (e.g., PESQ, STOI). The qualitative and error-case analysis is not sufficiently granular.

---

confidence: 4
contribution: 3
initial_rating: 2
presentation: 1
questions: - How does the diffusion map step with $t=1$ and $\alpha=1$ relate to Laplacian eigenmaps or a related simple spectral embedding approach ?
- Do you have evidence that the proposed measures "disentangle" leakage from "self-distortion" ?
- What is the "waveform-only" variant of PS/PM ?
- What is the aim and significance of the "error analysis" ?
- Did you conduct new listening test with 15 raters ?
- How were the parameters tuned ? Was there trial and error ?
soundness: 2
strengths: Performance measures reflecting human ratings are very helpful to help design better algorithms. In the specific case of audio source separation, 
Table 1 illustrates well that existing measures can have limited correlation with human scores from the SEBASS database, a situation that is significantly improved by the proposed measures. 
The approach seems sensible, as it combines a first level of "generic" audio features from a pre-trained embedding, a second level of embedding that somehow learns the impact of "standard" distortions in a fine-grained way (source-specific and time-frame specific). Its soundness is confirmed by the experimental evaluation, which can in fact seem surprisingly good for an approach where no human rating is explicitly used to train the designed measures (although given the number of parameters of the method, their selection with "cherry picking" can be interpreted as supervised training). 
For these reasons I rate the contribution to 3.
summary: The paper proposes two new measures of audio source separation quality, called Perceptual Separation (PS) and Perceptual Match (PM), that respectively aim to quantify the amount of
- "leakage": to what extent an estimated source is closer to (distorted versions of) the corresponding clean source than to (distorted versions of) the other clean sources
- "self-distortion": how close is the estimated source from the clean one up to "standard" distortions
The approach is primarily frame-based and relies on the knowledge of reference clean sources, which are data-augmented using various "standard" distortions. 

For a given frame index, the time frame of each clean, augmented, and estimated source is first mapped from the waveform domain to an embedding domain using a pre-trained embedding (wav2vec 2.0 based, or MERT based, using hand-picked intermediate layers). An additional spectral-like embedding (diffusion map -based) is then computed for each source, and used to define two Mahalanobis metrics and two representatives (a centroid, and the spectral embedding of the reference clean time frame) for each source. The distance between the spectrally embedded *estimated* time-frame of a target source and these representatives, measured with these Mahalanobis metrics, then yields quantities that are aggregated to provide the final frame-based measures for each source. 
Eventually, frame-based measures are aggregated over a whole signal either by averaging (for PS) or using a "perceptually weighted scheme inspired by PESQ" not described in the main text (for PM).

After conducting an error analysis, the authors evaluate the proposed measures using the SEBASS database, which provides MUSHRA human ratings over a collection of signals separated using various algorithms, gathered from pre-existing source separation evaluation campaigns. The proposed measures are compared to existing quality measures using two indices : the Pearson product-moment correlation coefficient (PCC), and Spearman's rank-order correlation coefficient (SRCC). In most of the considered settings, the proposed measures achieve the best PCC/SRCC scores (or scores comparable to the best ones).
weaknesses: The paper seems to have been written very hastily, with e.g. L075 "*Two lines about appendices and disclaimers on LLM usage*"

As a result, the abstract is hard to grasp due to the choice of wording. For example,  the notion of "intrusive approach" is certainly not familiar to the average ICLR reader, and it would be better to write more explicitly that the original sources used to generate the mixture need to be available for the evaluation. The notions of "system output" and "non-attributed references" were also not immediately clear, I only guessed its meaning after reading the rest of the paper. The notion of "granular optimization" is also quite cryptic (as well as further usage of the word "granular" in the main text).
 
Even more importantly, the paper misses a clear overview of the overall computing pipeline corresponding to the proposed measure, and I had to make my own drawings to provide the (hopefully correct) summary above. Combined with the fact that the treatment of the different sections is significantly unbalanced,  this makes the paper quite hard to read (hence my score 1 on the presentation).

Section 2 is very short, while one would expect to have there an overview of the overall context and pipeline from a high-level perspective. This would seem to be the right place to highlight that the approach is frame-based, and that the frame index will be dropped later on. I missed it at first read.
Section 3 is essentially a reminder on diffusion maps, in my opinion this would have been better fit to an appendix : isn't it enough to know that it replaces a collection of vectors by a spectral-like embedding of the said vectors ? By the way I believe that it is an abuse of notation to consider $\Psi_t$ (or $\Psi_t^{(d)}$) as a function defined on $\mathbb{R}^D$: is only defined at the considered $x_i$'s, there is no natural out-of-sample extension for an arbitrary $x \in \mathbb{R}^D$. 
Suggestion: since the time-frames are denoted y_*, their (e.g. word2vec) embedding $x_*$, it would lighten the notations and ease the reading to use another letter, e.g. $v_*$ to denote what is currently written $\Psi^{(d)}_t(x_*)$. It would be sufficient to say that the definition of $v_*$ depends on some parameters ($t$, $d$ or $\tau$, but also $\alpha$)
Given the choice of $t=1$ in all experiments, the approach seems to fall back on a simpler method without diffusion, which seems to be very close to a plain Laplacian eigenmaps.

Section 4 is clear, but in my opinion would gain much by reworking the notations as suggested above. 

The claim about the limits of existing measures L042 ("defined with an intrinsic ambiguity) does not seem supported by evidence, and seems in contradiction for example with the very definition of SIR/SAR/SDR.
Table 1 does provide some evidence that the new measures "better correlate with human global quality assessment", yet this does not per se support a better 

The notion of intrusiveness seems to mean in the introduction that clean reference sources are required. I could not relate it to the wording "mildly-intrusive distortions". 

The notations are somewhat unclear, balancing between being too allusive or too explicit. For example, the problem formulation is frame-based, involving an index $f$ (which by the way in a signal-processing context often indicates frequency, I found this confusing at first read), but the indices are then "repurposed" and $f$ is dropped, this is also confusing.

The definition of some acronyms (MOS, NMI) should be recalled when they are first used several pages after being introduced.


There is no explanation on the definition of the "waveform-only" variant of PS/PM.

The aim and significance of the whole Section 5 on "error analysis" is totally unclear (hence my score 2 on soundness), and this reads as a series of tedious but relatively straightforward computations.

Although the results shown in Table 1 seem to provide evidence of the interest of the proposed measures, underlining the scores on the first two lines seems superfluous.
The database description of Section 6.1 is ambiguous : at first I understood the SEBASS precisely provides human ratings. Then L414 seems to suggest that 15 new certified raters were involved. Were they the 15 certified raters from SEBASS ? Can this be clarified, traced and referenced ?
If this paper involved new certified raters, this should be reflected in the Ethics part.

My understanding is that MUSHRA and MOS are different approaches to obtain statistically significant human ratings. The authors however seem to use both acronyms as synonyms.

The values of many parameters ($L$, $\alpha$, $t$, $d/\tau$, $M$) and some choices (e.g., which layers of the wav2vec / MERT models are used to generate the first embedding)
are provided without saying much on how they were selected, and whether this involved trial and error. Can you explain ?
If trial and error was conducted, some mild level of learning to match human ratings has been conducted, and this should be made explicit.

---

confidence: 4
contribution: 3
initial_rating: 6
presentation: 3
questions: Q1. You claim PM and PS are the first metrics to disentangle leakage and self-distortion, but classical BSS Eval already uses SIR (interference from other sources) and SAR (artifacts). Can you give a concrete example (qualitative or quantitative) where SIR says “good” but PS clearly identifies audible leakage, or where SAR says “good” but PM clearly identifies audible self-distortion? Without that, it’s hard to judge whether PM/PS are fundamentally new or essentially SIR/SAR in a learned perceptual space.

Q2. For SEBASS speech mixtures, you evaluate only English-only and Spanish-only overlaps and justify this by saying “realistic conversations are monolingual.” In many deployment settings (code-switching, bilingual overlap), that assumption does not hold. How would PS behave on overlapping multilingual speech, where two simultaneously active speakers may differ by identity but not cleanly by language? Is PS partly exploiting language identity to discriminate sources?

Q3. PM is averaged over frames, but PS is pooled using a “PESQ-like perceptual weighting,” which is not explicitly defined in the paper. Can you provide the exact temporal pooling rule you use for PS? Right now PS is not reproducible from the description, and your correlation claims with MOS depend on that pooling.

Q4. PM and PS depend on a hand-crafted distortion bank that defines the “perceptual neighborhood” of each source. How sensitive are your metrics to the choice of those distortions? For example, if we remove one distortion type or add an unrealistic one, do PM/PS scores or MOS correlations change significantly? This matters for whether others can safely reuse your metric or will overfit to your particular bank.

Q5. The perceptual manifold uses SSL embeddings + diffusion maps with α = 1, diffusion time t = 1, and dimensionality chosen by a 99% spectral-mass rule (≈20–40 dims). Were any of these hyperparameters, or the PS pooling weights, tuned using SEBASS MOS labels (even indirectly)? If yes, how did you prevent test-set leakage when reporting final Pearson/Spearman correlations with MOS?
soundness: 3
strengths: S1. Clear problem framing.
The paper separates two real failure modes in source separation: "you damaged my target" vs "you left the other source in."
This is actually how practitioners think about failure, but most metrics don't expose it.

S3. Human-grounded evaluation.
They compare directly to listening scores (SEBASS), not just SDR.
They include English speech, Spanish speech, and music (with and without drums), and 32 different separators.

S4. Uncertainty / confidence intervals.
Section 5  gives an error band on scores and pushes that through to MOS correlation.


S5. Actionability.
PM and PS are frame-level before pooling. This helps to point where leakage happened" or "this is where I over-gated the vocal."
That's a lot more usable than one global SDR number.
summary: The paper proposes two new perceptual evaluation metrics for blind audio source separation, called Perceptual Match (PM) and Perceptual Separation (PS).

PM is meant to measure self-distortion: how much the separated output of a given source has been degraded (i.e. does it still "sound like me," or did the separator damage me?).

PS is meant to measure leakage: how much other sources are still audible in the separated output for the target source (i.e. how much bleed from others remains?).

How it works:

For each ground-truth source, the authors generate a bank of controlled distortions (filtering, gating, etc.) that are supposed to mimic plausible separator artifacts.

All short frames (tens of ms) from the clean source, its distorted variants, and the separator output are embedded into a perceptual manifold. This manifold is built with self-supervised audio embeddings (wav2vec2-like) followed by diffusion maps.

Each source becomes a cluster in that manifold, with a mean and covariance.

PM: does the separated output fall inside its OWN source's cluster? (High PM = low self-distortion.)

PS: is the separated output closer to ITS OWN cluster than to ANY OTHER source's cluster? (High PS = low leakage.)

PM/PS are computed framewise and then aggregated to clip level.

The authors evaluate on SEBASS (>11k human ratings over ~900 clips, 32 systems, English speech, Spanish speech, and music). They report that PM and PS correlate well with MOS (Pearson and Spearman), often outperforming SDR/SIR/SAR, PESQ/STOI-style metrics, DNSMOS, etc.

The paper also gives per-frame uncertainty for PM/PS: (i) a deterministic bias radius due to manifold truncation and (ii) a high-probability CI due to limited distortion samples. They propagate this to the final MOS correlation numbers.
weaknesses: W1. Novelty vs SDR/SIR/SAR is under-explained.
The paper says PM/PS are the first metrics to "functionally disentangle" leakage and self-distortion.
But classical BSS Eval already has SIR (interference from other sources) and SAR (artifacts).
The paper mentions SDR/SIR/SAR but does not clearly argue why PS is NOT just SIR in a fancier embedding,

why PM is NOT just SAR in a fancier embedding.
Without that, some readers will see this as repackaged SIR/SAR.

W2. Assumptions in the distortion bank.
The method assumes that the authors' hand-designed distortion bank actually spans "plausible" separator artifacts.
If the bank misses common artifacts (phasiness, musical noise, transient smearing), PM could wrongly accept bad audio as "still me," or wrongly reject good audio that falls outside their curated distortion styles. There is no ablation/sensitivity analysis on which distortions are used, so robustness is not proven.

W3. Monolingual-only evaluation in speech.
For speech, they say they split SEBASS mixtures into English-only and Spanish-only pairs "because realistic conversations are monolingual." That is not generally true (code-switching and bilingual overlap are routine in real deployments).
This matters because PS is partly "do you sound like the other source?"
If the other source is always a different language, that discrimination problem is easier.
So PS may look stronger than it would in truly multilingual overlap.
This should be acknowledged as a limitation.

W4. PS pooling over time is not fully specified.
PM is averaged over frames.
PS is said to use a PESQ-like perceptual weighting so that short, audible bleed is penalized more.
But the exact weighting rule is not written in the main text.
Without that, PS is not actually reproducible from the description.



W5. External validity.
All clips are ~10 seconds, 16 kHz, fairly clean, from curated mixtures.
There is no far-field mic, noisy phone audio, long meetings, reverberation, etc.
That is exactly the setting where applied systems struggle.
So the claim is really "PM/PS correlate with MOS on SEBASS-like curated data," not "PM/PS solve perceptual eval in the wild."
The paper should say that clearly.


Decision: Accept (Poster)