Global Separation Rate (GSR)

• Global Separation Rate (GSR) is a quantitative metric that defines the ability to differentiate classes in frozen embedding spaces and segregate species in granular flows.
• It employs a methodology based on comparing intra-class and inter-class distances (or fluxes in particulate systems) with permutation or numerical calibration to ensure robust evaluation.
• GSR’s implications extend to training-free diagnostics in deep learning and precise control in continuum granular models, enhancing transfer learning and process efficiency.

The Global Separation Rate (GSR) is a term that denotes fundamentally different quantities in two distinct scientific domains: (1) a statistical metric measuring geometric class separability in high-dimensional embedding spaces, and (2) a transport parameter in continuum models of granular segregation. In both contexts, GSR serves as a quantitative probe of how distinctly entities can be differentiated—by class in a feature space, or by species in physical flows—yielding metrics of intrinsic structure or process efficiency. Both applications have precise, formal definitions, rigorous computational protocols, and play key roles in the standardization of evaluations and the calibration of predictive models (Basha et al., 10 Dec 2025, Fry et al., 2020).

1. GSR in Frozen Embedding Spaces: Zero-shot Class Separation

In the context of audio representation learning, Global Separation Rate quantifies the intrinsic, point-wise ability of a frozen embedding space to geometrically separate samples from different classes without any post-training adaptation. The primary motivation is to provide a training-free evaluation of class discrimination capacity, distinct from typical supervised transfer benchmarks. This protocol underlies the VocSim benchmark, aggregating large test collections spanning diverse sound types to probe content identity and structural generalization (Basha et al., 10 Dec 2025).

Formal Mathematical Definition

Given a test set $X = \{x_1, ..., x_N\}$ with class labels $y_i \in C$ and a fixed embedding function $f(\cdot)$, the computation proceeds as follows:

• For each point $i$, let $I_i$ denote the set of same-class indices ($j \ne i, y_j = y_i$) and $E_i$ the set of different-class indices ($y_k \ne y_i$).
• Fix a symmetric distance function $d(\cdot, \cdot)$ (cosine, Euclidean, etc.).
• Compute:
  • $$\mathrm{AvgID}_i = (1/|I_i|)\sum_{j \in I_i} d(x_i, x_j)$$
  • $\mathrm{NID}_i = \min_{k \in E_i} d(x_i, x_k)$
  • $$\mathrm{LocalScore}_i = \dfrac{\mathrm{NID}_i - \mathrm{AvgID}_i}{\mathrm{NID}_i + \mathrm{AvgID}_i + \epsilon}$$, with $\epsilon > 0$ for numerical safety.

Aggregating over all $i$:

• Compute the mean, $\mu = (1/N)\sum_{i=1}^N \mathrm{LocalScore}_i$
• Normalize: $\mathrm{GSR}_{\mathrm{norm}} = (\mu + 1)/2$
• Scale: $$\mathrm{GSR} = 100 \times \mathrm{GSR}_{\mathrm{norm}}$$ (range [0, 100]).

A value of $\mathrm{GSR} = 50\%$ denotes a geometry where intra-class and inter-class separations are, on average, equal.

2. Computational Protocol and Calibration

GSR measurement isolates geometry-driven separability by fixing the embedding and prohibiting re-training. The recommended pipeline is:

1. Embedding Extraction: Compute fixed-length feature vectors for all test instances (ex: Whisper encoder, time-frequency pooling, PCA).
2. Pairwise Distance Matrix: Form $D_{i,j} = d(v_i, v_j)$.
3. Local Scores: For each $i$, compute intra-class mean and nearest inter-class distances, apply the local score formula.
4. Global Aggregation: Average local scores, renormalize, and report as a percentage.
5. Permutation Baseline Calibration: Because geometric clustering can inflate GSR even with random labels, VocSim employs an empirical baseline:
   • Permute class labels (typically >1000 times), recompute GSRs, report their mean as $\mathrm{GSR}_{\mathrm{baseline}}$.
   • Lift is $$\mathrm{GSR}_{\mathrm{obs}} - \mathrm{GSR}_{\mathrm{baseline}}$$. Confidence intervals and p-values are derived from the permutation distribution.

A large lift (e.g., >10%) indicates strong class separability significantly exceeding chance expectations. Raw GSR values near 70–80% reflect robust global separability, but lift should always be reported to control for geometry-induced bias (Basha et al., 10 Dec 2025).

3. Interpretation and Application of GSR in Embedding Evaluation

Global Separation Rate quantifies the geometric “strictness” of class boundaries in the embedding. It operates without classifier training, providing a low-variance, label-aware metric of content permanence and transfer potential. In VocSim, GSR reveals generalization gaps (e.g., sharp local retrieval drops in low-resource speech) and allows quantitative comparison across embedding models. GSR values close to 50% after permutation calibration indicate a collapse of geometric structure (no class discrimination), while large positive lifts signal intrinsic content alignment (Basha et al., 10 Dec 2025).

GSR correlates with downstream biometric and classification performance: foundation models with high GSR lift predict avian perceptual similarity, improve bioacoustic classification, and reach state-of-the-art on the HEAR benchmark, supporting its role as a proxy for general-purpose feature utility.

4. GSR as the Segregation Coefficient in Granular Flow Transport

In granular physics, “Global Separation Rate” (also termed segregation coefficient $S$) models the net rate of size-driven segregation in continuum descriptions of particulate flows. The parameter captures the characteristic percolation speed due to shear-driven separation of species, notably in bounded heap flow of size-bidisperse mixtures (Fry et al., 2020).

Continuum Formulation

Consider the local concentration $c(x, z, t)$ of large particles, with velocity fields $u(x, z)$ (streamwise), $w(x, z)$ (normal), and granular diffusion $D(x, z)$.

The transport equation is: $$\frac{\partial c}{\partial t} +\frac{\partial}{\partial x}(u\,c) +\frac{\partial}{\partial z}(w\,c) +\frac{\partial}{\partial z}(w_{s}\,c) = \frac{\partial}{\partial x}\left( D\,\frac{\partial c}{\partial x} \right) +\frac{\partial}{\partial z}\left( D\,\frac{\partial c}{\partial z} \right)$$ where the species-specific segregation (percolation) velocity is: $$w_s(x, z, c) = S \cdot |\dot{\gamma}(x, z)| \cdot [1 - c(x, z)]$$ with $\dot{\gamma} = \partial u/\partial z$ the shear rate.

This structure enforces vanishing segregation at pure-species limits ($c \to 0$ or $c \to 1$) and introduces $S$ (units of length) as the physics-motivated GSR. The segregation flux is then $J_s = S\,|\dot{\gamma}|\,c(1-c)$.

5. Measurement and Calibration of $S$ in Granular Systems

Experimental or simulation-based estimation of $S$ involves:

• Construction of a quasi-2D bounded heap setup: transparent sidewalls, controlled feed of a well-mixed bidisperse granular material.
• Flow properties (surface velocities) measured via PIV; the flowing-layer thickness extracted from mass conservation and surface kinematics.
• Post-deposition: the deposited concentration profile $c_{\mathrm{exp}}(x)$ is measured via image analysis of narrow streamwise bins.
• The continuum transport equation is solved numerically (e.g., MATLAB’s pdepe) for $c_{\mathrm{model}}(x, z;S)$; concentration is sampled at the base.
• The best-fit $S$ minimizes the $L_2$ misfit:

$$J(S) = \sum_j [c_{\exp}(x_j) - c_{\mathrm{model}}(x_j; S)]^2$$

or equivalently, $\arg\min_S J(S)$.

Accuracy depends critically on the velocity profile assumption and sidewall effects; for wall gaps $T/d \lesssim 10$–15, a uniform profile yields $S$ within ≈10% of the true value, while spanwise variation beyond this threshold introduces bias (Fry et al., 2020).

Sensitivity analysis reveals that $S$ is only weakly coupled to the diffusion coefficient, which is typically set by literature priors ($D = 0.1\ \dot{\gamma}\ \bar{d}^2$). Reliable calibration requires minimizing geometric and kinematic uncertainty, especially in the spanwise direction.

6. Comparative Table: GSR in Embedding vs. Granular Flow Contexts

Domain	Definition/Computation	Interpretation
Audio Embedding Spaces	Normalized mean local score: nearest wrong-class vs. same-class distance; permutation-calibrated	Measures point-wise, training-free class separability (0–100%)
Particulate Granular Flow	Best-fit segregation coefficient $S$ from continuum transport; size units via L2 minimization	Quantifies net segregation speed due to percolation; controls flux in continuum models

Both usages formalize separation efficiency but over distinct physical or statistical substrates, calibrated carefully to decouple intrinsic structure from incidental geometric bias.

7. Significance and Broader Impact

GSR is central in standardizing geometric evaluation for zero-shot embedding spaces and robustly parameterizing segregation dynamics in particulate materials. In deep learning, it enables training-free, interpretable diagnostics of transfer potential and exposes structural limits in generalization (e.g., phonotactic collapse in low-resource subsets) (Basha et al., 10 Dec 2025). In granular flow, $S$ (GSR) underpins predictive continuum modeling with minimal empirical inputs, transferable across geometries and flow regimes (Fry et al., 2020). Accurate GSR measurement—using permutation baselines in statistics or kinematic control in granular physics—ensures robust characterization of system-intrinsic differentiability, supporting both fundamental understanding and technological application.