Kernel Audio Distance (KAD)

• Kernel Audio Distance (KAD) is a distribution-free, unbiased metric that employs the Maximum Mean Discrepancy framework to quantify differences between real and generated audio.
• It leverages a variety of neural audio embeddings and characteristic kernels, such as the Gaussian RBF, to ensure rapid convergence and maintain perceptual fidelity.
• KAD outperforms traditional metrics like FAD by offering faster computation and reduced bias, validated through empirical benchmarks and perceptual evaluations.

Kernel Audio Distance (KAD) is a distribution-free, unbiased, and computationally efficient metric for evaluating generative audio models, grounded in the Maximum Mean Discrepancy (MMD) framework. KAD leverages characteristic kernels and advanced audio embeddings to robustly quantify differences between distributions of real and generated audio, overcoming limitations of traditional Fréchet Audio Distance (FAD), such as Gaussian assumptions, sample size sensitivity, and computational overhead (Chung et al., 21 Feb 2025). In perceptual modeling contexts, KAD can also be constructed from biologically-inspired or data-driven embeddings with positive-definite kernels, reflecting the geometry of perceptual auditory space (Oh et al., 2020).

1. Mathematical Foundation: Maximum Mean Discrepancy

KAD is formulated as a squared MMD statistic between distributions of audio embeddings. Let $X=\{x_i\}_{i=1}^m$ denote real (reference) audio embeddings drawn from $P$ and $Y=\{y_j\}_{j=1}^n$ those from generated samples $Q$. For a positive-definite, characteristic kernel $k(\cdot,\cdot)$ (such as Gaussian RBF), the unbiased estimator is:

$$\mathrm{KAD}(P, Q) = \mathrm{MMD}^2_k(P, Q) = \frac{1}{m(m-1)}\sum_{i\neq j} k(x_i, x_j) + \frac{1}{n(n-1)}\sum_{i\neq j} k(y_i, y_j) - \frac{2}{mn} \sum_{i=1}^m \sum_{j=1}^n k(x_i, y_j)$$

A scaling factor $\alpha=1000$ is commonly applied for interpretability. This MMD-based construction ensures KAD is nonparametric and unbiased for finite sample sizes, with error decaying at $\mathcal{O}(1/\sqrt{N})$ where $N=\min(m,n)$ (Chung et al., 21 Feb 2025).

2. Embedding Selection and Kernel Specification

KAD operates in embedding spaces produced by pretrained neural encoders spanning environmental audio, music, and speech:

• VGGish (128-dim)
• PANNs-WGLM (2048-dim)
• CLAP (512-dim)
• PaSST (1024-dim)
• OpenL3 (512-dim)
• Music-specific: MERT, CLAP-laion-music

Kernel choice is central; the Gaussian RBF kernel is

$$k(x, y) = \exp \left(-\frac{\|x-y\|^2}{2\sigma^2}\right)$$

Bandwidth $\sigma$ is defined using the median-pairwise-distance heuristic: compute all $\|x_i - x_j\|$ within the reference set, using the median as $\sigma$. This yields stable monotonicity across degradations (Chung et al., 21 Feb 2025). In perceptual metric settings, kernels arise as inner products in mapped perceptual space, e.g., $$k(x, x') = \langle f(x), f(x') \rangle_{\mathbb{R}^{d}}$$ where $f$ is a physiologically or data-driven mapping (Oh et al., 2020).

3. Theoretical Properties and Computational Complexity

KAD's key properties include:

• Distribution-free: It avoids assumptions on the latent distribution (versus FAD's Gaussian model).
• Unbiasedness: MMD estimators are unbiased for finite $m,n$.
• Convergence: Faster finite-sample convergence ($\mathcal{O}(1/\sqrt{N})$ error decay) than FAD ($\mathcal{O}(1/N)$ bias).
• Computationally efficient: Complexity is $\mathcal{O}(d N^2)$ for $d$-dimensional embeddings and $N$ samples, dominated by pairwise kernel evaluations. All steps batch efficiently to GPUs, achieving 5–20× speedup over CPU; no expensive matrix square-root as in FAD’s $\mathcal{O}(d^3)$ cost (Chung et al., 21 Feb 2025).

4. Empirical Validation and Perceptual Alignment

KAD demonstrates robust empirical performance across key benchmarks:

• Human-perceptual alignment: On DCASE 2023 Task 7 (9 models, crowd-sourced ratings), KAD using PANNs-WGLM embeddings achieves Spearman $\rho\approx-0.93$ (p < 0.001), outperforming FAD ($\rho\approx-0.80$, p < 0.01). Across embedding families, KAD consistently achieves stronger correlation with perceptual scores.
• Convergence analysis: Evaluations against AudioLDM-generated samples on the Clotho2 dataset show KAD stabilizes rapidly ($N\approx1000$), while FAD remains positively biased for small $N$.
• Compute benchmarks: For $d=2048$ and $N=100$, CPU runtime is $\approx$7 ms (KAD) versus $1,776$ ms (FAD). GPU acceleration yields up to 20× speedup for KAD (Chung et al., 21 Feb 2025).
• Generative family generalizability: Although validated primarily on diffusion and DCASE models, the approach extends to GANs, VAEs, and general autoencoders by swapping embeddings, paralleling the effectiveness of KID/KMMD in computer vision.

5. Perceptual Modeling and Kernel Audio Distance Construction

Auditory perception motivates KAD beyond statistical divergences, encoding nonlinearities such as basilar membrane compression and cross-band interactions (Oh et al., 2020). In this paradigm:

• Audio is reframed as frames of 96-band sound-pressure vectors, mapped via $f:X \to N$ to perceptual space.
• Four mapping strategies exist: BM-only (physiological), data-driven linear, BM with data-tuning, and hybrid BM + downstream linear.
• The pull-back Riemannian metric $g=J_f^\top J_f$ (with $J_f$ the Jacobian of $f$) captures perceptual geometry; the induced Mercer kernel $k(x,x') = \langle f(x), f(x') \rangle$ defines a positive-definite kernel.
• Distances in this space are approximated by $\|f(x)-f(x')\|_2$ or (for frame-averaged clip comparisons) $$D_P(\mathrm{ref,test}) = \frac{1}{64}\sum_i d_P(x_i^{\mathrm{ref}}, x_i^{\mathrm{test}})$$.
• Multiple-kernel constructions (MKL) are possible: $$k_{\mathrm{KAD}}(x,x') = \sum_m \beta_m \langle f_m(x), f_m(x') \rangle$$, $\beta_m \geq 0$.

Performance in subjective MUSHRA-style experiments shows Pearson $r$ of 0.80–0.84 (BM/data-driven variants), comparable or exceeding established objective measures (Oh et al., 2020).

6. Practical Implementation and Usage Guidance

KAD is available in the open-source kadtk toolkit (MIT license) with PyTorch and TensorFlow support. Key usage attributes:

• Sample size recommendation: $N=500$–$2000$ for $<$5% error bars; $N\approx200$ still produces unbiased estimates with higher variance.
• Kernel bandwidth: The median heuristic is default; $0.1\times$ to $10\times$ median are viable, but extreme values reduce monotonicity of scoring.
• Integration: Extract embeddings from dozens of models (PANNs, VGGish, CLAP, PaSST, OpenL3, MERT, HuBERT, Wav2Vec2.0, etc.) and perform direct FAD comparisons.

Example Python invocation:

from kadtk.metrics import compute_kad kad_score = compute_kad( real_embeddings=ref_feats, fake_embeddings=gen_feats, kernel='rbf', bandwidth='median', scale=1000 ) print(f"KAD score: {kad_score:.3f}")

Internally, compute_kad evaluates pairwise squared distances, applies RBF kernel, computes the three MMD sums, then scales the score (Chung et al., 21 Feb 2025).

7. Significance and Application Scope

KAD’s distribution-free, unbiased framework and strong perceptual alignment establish it as a robust and efficient metric for benchmarking generative audio models, supporting rapid, scalable, and accurate evaluation. Its theoretical flexibility accommodates physiologically motivated and data-driven perceptual kernels, enabling broad applicability across audio domains and modeling paradigms (Chung et al., 21 Feb 2025, Oh et al., 2020). The ability to leverage multiple embedding families and kernels facilitates deployment in quantitative studies and subjective perceptual analyses, advancing reliable assessment standards in generative audio research.