Gaussian Process Concept Attribution

• Gaussian Process Concept Attribution (GP-CA) is a principled method that decomposes model predictions into feature and high-level concept attributions with rigorous uncertainty quantification.
• The approach utilizes analytical techniques like Integrated Gradients and closed-form GP evaluations to enable clear, local explanations for regression, classification, and robotics applications.
• GP-CA demonstrates computational efficiency and enhanced robustness by outperforming traditional attribution methods such as SHAP and sensitivity analysis in various structured data domains.

Gaussian Process Concept Attribution (GP-CA) encompasses a principled family of methods that leverage the structure of Gaussian processes (GPs) to explain model predictions in terms of underlying causes or “concepts.” In contemporary applications, GP-CA refers both to analytical decomposition of predictions in regression/classification models as well as human-interpretable uncertainty attribution in structured data domains such as point cloud registration. The essential premise is that, given the GP correspondence for broad classes of neural architectures (Yang, 2019), and the theoretical tractability of GPs under differentiation and integration (Butler et al., 11 Mar 2024, Seitz, 2022), it becomes possible to yield closed-form, uncertainty-calibrated explanations attributing outputs to internal concepts or well-defined sources of error. GP-CA provides both local feature attributions and high-level concept explanations with rigorous uncertainty quantification, and its computational efficiency and robustness have been demonstrated across regression, classification, and robotics domains.

1. Theoretical Foundations: GP Correspondence and Tensor Programs

Gaussian process concept attribution is grounded in the observation that wide neural networks—across architectures including MLPs, CNNs, RNNs (LSTMs, GRUs), attention modules, and normalization layers—converge to GP priors in the infinite-width limit (Yang, 2019). The tensor programs formalism enables rigorous encoding of such network computations. In tensor language, model computations are decomposed into three variable types: G-vars (“Gaussian” variables), H-vars (“hidden” variables), and A-vars (random weights), and manipulated by MatMul, LinComb, and Nonlin operations, with extensions (Nonlin⁺, empirical Moment rules) encompassing normalization and attention.

The centerpiece is the Master Theorem: for any controlled evaluation function $\psi$ applied to G-vars, the empirical average converges (as layer width $n\to\infty$) to the expectation with respect to an explicit Gaussian whose covariance is recursively propagatable via the “V-transform” of network nonlinearities. For example, the recursive kernel updates in a deep MLP are:

$$K_\ell(x, x') = \sigma_w^2\,\mathbb{E}_{(z, z') \sim \mathcal{N}(0, K_{\ell-1})}[\phi(z)\phi(z')] + \sigma_b^2$$

Given this correspondence, for any neural architecture expressible in tensor programs, its wide limit can be described entirely by the GP kernel, which encodes both prediction and uncertainty structure.

2. Attribution Methodologies: Feature and Concept Attribution

GP-CA methods decompose model outputs into contributions of features or abstract concepts. In regression settings, feature attribution is formalized via Integrated Gradients (IG):

$$\mathrm{IG}_k(x) = (x_k - \tilde{x}_k) \int_0^1 \frac{\partial f(\tilde{x} + \alpha(x-\tilde{x}))}{\partial x_k} \,d\alpha$$

where $f$ can be a sample from a GP prior/posterior, and $\tilde{x}$ is a reference baseline. For GPs, since differentiation and integration are linear operators under which the GP prior is closed, IG attributions themselves form GPs (Butler et al., 11 Mar 2024):

• Mean: $$\mu_k(x) = (x_k - \tilde{x}_k) \int_0^1 \partial m(\tilde{x} + t(x-\tilde{x}))/\partial x_k\,dt$$
• Covariance: $$\kappa_k(x, x') = (x_k - \tilde{x}_k)(x'_k - \tilde{x}_k) \int_0^1\int_0^1 \frac{\partial^2 k(\tilde{x} + s(x-\tilde{x}), \tilde{x} + t(x'-\tilde{x}))}{\partial x_k \partial x'_k} ds dt$$

This analysis satisfies the completeness property: $\sum_k \mathrm{IG}_k(x) = f(x) - f(\tilde{x})$, ensuring that attributions sum to the deviation in prediction from the baseline.

For non-Gaussian likelihoods (e.g., classification), analytical solutions are intractable, but the GP structure allows approximate solutions via Taylor expansion or Monte Carlo integration (Seitz, 2022). For binary classification with sigmoid/softmax links, attributions are proportional to expectations of $$\sigma'(\tilde{f}(x)) \cdot \tilde{f}^{(\partial_k)}(x)$$, with recursive formulations involving Stirling numbers and law of total expectation.

In concept attribution, particularly in structured data (e.g., point clouds), GP-CA re-frames the explanation task to attribute model uncertainty not simply to input features, but to semantic “concepts” such as noise, pose error, or occlusion (Gaus et al., 23 Sep 2025). Here, a GP-based classifier is trained on high-level neural embeddings (from DGCNN, for instance) to output probability distributions over concepts, with predictive means interpreted as concept attributions.

3. Uncertainty Quantification in Attribution

Because GP models are Bayesian, every attribution method yields not only a mean value but an uncertainty quantification. For IG attributions with GP regression, both posterior mean and variance can be computed in closed form (Butler et al., 11 Mar 2024). When the model function $F$ has posterior $\mathcal{N}(m, K)$, feature attributions have:

$$\text{attr}_i(x|\text{data}) \sim \mathcal{N}(\mu_i(x), \Sigma_{ii'}(x))$$

with variance growing with $(x_i - \tilde{x}_i)^2$. This principled uncertainty propagation is preserved under the IG operator because it is linear.

In multi-concept settings, GP-CA further expresses uncertainty as the variance of concept probability predictions (epistemic uncertainty), estimated as:

$$v_c = \frac{1}{M} \sum_{m=1}^M (p_c^{(m)}(h) - \hat{s}_c)^2$$

where $p_c^{(m)}(h)$ are Monte Carlo samples from the variational GP posterior.

Uncertainty quantification is essential for robust interpretability, especially where explanations are used to guide real-time decision making or safety-critical recovery actions.

4. Algorithmic and Implementation Details

Implementations of GP-CA span both analytical and neural-network-based pipelines:

• Analytical IG for GP regression/classification: Direct evaluation of mean and covariance expressions for attributions, exploiting the closure under differentiation and integration.
• Approximate solutions for non-Gaussian likelihoods: Taylor series expansion, MC integration, or Riemann sums for IG path integrals (Seitz, 2022), with clear computational tractability and uncertainty estimates.
• Deep feature pipeline for concept attribution (Gaus et al., 23 Sep 2025): Input point cloud $\to$ ICP alignment $\to$ DGCNN embedding $\to$ multi-concept GP classifier, posterior inference via variational Gaussian approximation, Monte Carlo sampling, and softmax likelihood.

Active learning is implemented via the BALD (Bayesian Active Learning by Disagreement) criterion—maximizing mutual information between class labels and model parameters—followed by clustering (k-means) and targeted labeling, enabling efficient vocabulary expansion for uncertainty sources.

Run-time and data efficiency are empirically substantiated: GP-CA achieves per-instance runtimes of 2–4 seconds, with concept attribution accuracies up to 100% on robotic datasets, outperforming SHAP (85+ seconds/sample) and typical sensitivity analysis (Gaus et al., 23 Sep 2025).

5. Comparative Analysis and Applications

The GP-CA framework is compared against ARD kernels (static, non-contextual importance), SHAP (post-hoc, black-box, computationally intensive), and sensitivity analysis (gradient-based, lacks uncertainty quantification) (Seitz, 2022, Gaus et al., 23 Sep 2025). GP-CA supports local, context-sensitive, uncertainty-calibrated attribution and is computationally efficient for GP regression, tractable for classification, and scalable to high-dimensional inputs (images, RGB-D point clouds).

Applications demonstrated include:

• Feature attribution in regression (housing, medical prognosis, wine quality) with uncertainty scaling and accurate decomposition.
• Classification attribution in image domains (MNIST) with heatmap evidence, counterfactual attribution, and per-feature uncertainty.
• Concept attribution in point cloud registration, guiding robotic recovery strategies through interpretable uncertainty diagnosis (e.g., identifying occlusion, pose error, noise) (Gaus et al., 23 Sep 2025).

6. Impact and Future Directions

GP-CA represents a robust and mathematically rigorous paradigm for explainable AI in settings requiring both accounting for uncertainty and semantically informative attribution. Ongoing research targets:

• Expanding multi-class concept attribution beyond independent GPs to correlated GP priors, improving predictive and explanatory fidelity (Seitz, 2022).
• Refining approximation techniques for classification, especially in high-dimensional feature spaces.
• Integrating human priors and domain expertise for regularization and fairness via explicit priors over explanation distributions.

This suggests a convergence of explainability and Bayesian modeling, with plausible implications for improved model debugging, bias mitigation, and enhanced trust in autonomous systems. GP-CA holds particular promise for robotics and perception, where explanations directly inform corrective behaviors, and in scientific domains where uncertainty-aware decompositions are essential.

7. Summary and Significance

GP-CA spans a set of analytical and probabilistic algorithms that decompose GP-based model outputs into local feature attributions or human-interpretable conceptual explanations, with explicit, calibrated uncertainty quantification. Its theoretical basis lies in the broad GP correspondence of modern neural architectures (Yang, 2019), its methodologies are mathematically principled (Butler et al., 11 Mar 2024, Seitz, 2022), and its practical efficiency and effectiveness are demonstrated in challenging domains such as real-time robotics (Gaus et al., 23 Sep 2025). GP-CA offers a principled approach for interpretable machine learning where attribution needs to be both context-sensitive and robust under data/model uncertainty.