Aitchison Geometry on the Simplex for Uncertainty Quantification in Bayesian Hyperspectral Image Unmixing

1. Problem Statement

Hyperspectral image unmixing aims to estimate the spectral signatures of materials (endmembers) and their relative proportions (abundances) within each pixel. While many algorithms exist, they typically produce point estimates without quantifying the uncertainty of these estimates. This is a critical gap because:

• Lack of Ground Truth: Reliable ground truth is rarely available in remote sensing, making it difficult to validate algorithms or trust results.
• Constraints: Abundance vectors are constrained to the unit simplex (non-negative values summing to 1). Standard Euclidean geometry and statistical tools (like Gaussian distributions) do not naturally respect these constraints, leading to:
  • Difficulties in interpreting uncertainty (e.g., confidence ellipsoids may extend outside the valid simplex).
  • Intractable posterior distributions when combining spatial priors with likelihoods.
• Spatial Information: Existing Bayesian methods often ignore spatial correlations or struggle to incorporate them while maintaining the simplex constraints.

The paper addresses the need for a rigorous framework to model spatial and material-wise uncertainty of abundances $p(A|X)$ while strictly adhering to simplex constraints.

2. Methodology

The authors propose leveraging Aitchison geometry, a framework from compositional data analysis, to transform the problem from the constrained simplex space to an unconstrained Euclidean space.

A. Aitchison Geometry and Transformations

The core of the methodology is the use of the Isometric Log Ratio (ilr) transformation, $\psi(a) = \text{ilr}(a)$, which maps the interior of the simplex $\Delta^{P-1}$ to $\mathbb{R}^{P-1}$.

• Properties: This transformation turns the simplex into a flat Riemannian manifold with a pullback Euclidean metric.
• Geodesics: Unlike Euclidean geodesics (straight lines in the simplex), Aitchison geodesics are curves that respect the relative ratios of components. Interpolation in the simplex corresponds to linear interpolation in the ilr space.
• Prior Definition: The authors define an uninformative prior on the simplex by assuming an isotropic Gaussian distribution in the latent ilr space ($z \sim \mathcal{N}(0, \sigma^2 I)$). When mapped back to the simplex, this results in a distribution that naturally approaches the boundaries (due to the log-barrier) without assigning mass to strictly zero components, unlike Dirichlet priors.

B. Simplex-Valued Gaussian Processes (Pushforward GP)

To incorporate spatial information, the authors design Simplex-Valued Gaussian Processes:

1. Latent Space: They define a vector-valued Gaussian Process (GP) in the latent ilr space $\psi(A)$.
2. Separability: The covariance structure is assumed separable: $K_Z = K_P \otimes K_U$, where $K_U$ is a spatial kernel (using an exponential kernel to handle discontinuities in abundance maps) and $K_P$ encodes correlations between material components.
3. Pushforward: The prior on the abundance image $A$ is the "pushforward" of this latent GP through the inverse ilr transformation (softmax). This allows for closed-form interpolation of partial or noisy abundance maps.

C. Sampling the Posterior (Mirror Langevin)

The posterior distribution $p(A|X)$ is intractable because the likelihood (Euclidean Gaussian) and the prior (ilr-Gaussian) break conjugacy.

• Algorithm: The authors employ Mirror Langevin Dynamics, a sampling algorithm adapted for constrained spaces.
• Mirror Map: Instead of using the standard Alternating Log Ratio (alr) or Euclidean projections, they use the ilr transformation as the mirror map (gradient of the entropy potential). This ensures the sampling geometry is consistent with the prior geometry, improving convergence and respecting the simplex constraints naturally.

D. Uncertainty Quantification (UQ) Diagnostics

The paper proposes specific metrics compliant with Aitchison geometry:

• Geodesic Mean: The posterior mean is calculated as the inverse ilr of the Euclidean mean in latent space (the Fréchet mean on the manifold).
• Confidence Regions: Instead of Euclidean ellipsoids, they define Highest Density Regions (HDR) in the simplex. This captures multimodal posteriors and ensures regions remain within the valid simplex.
• Total Variance: They introduce Geodesic Total Variance (trace of the covariance in ilr space) as a scalar measure of uncertainty. This measures the mean squared geodesic distance to the mean, reflecting changes in relative material ratios rather than just absolute differences.

3. Key Contributions

1. Introduction of Aitchison Geometry to Unmixing: The paper is the first to apply compositional data analysis geometry to hyperspectral unmixing, specifically for Bayesian uncertainty quantification.
2. Simplex-Valued Gaussian Processes: It demonstrates how to construct spatial priors for abundance maps by pushing forward Euclidean GPs through the ilr transformation, enabling closed-form interpolation.
3. Constraint-Aware Sampling: It proposes using Mirror Langevin dynamics with the ilr map to sample from the posterior, avoiding the pitfalls of Euclidean projection methods.
4. Novel UQ Metrics: It defines geodesic mean, HDR-based confidence regions, and geodesic total variance, providing tools to visualize and quantify uncertainty that respect the physical constraints of abundances.

4. Results

The methods were validated on both simulated data and real data (Samson dataset):

• Multimodal Posteriors: In simulations with noisy data, the HDR approach successfully captured multimodal distributions and complex shapes that Euclidean confidence ellipsoids failed to represent (often extending outside the simplex).
• Variance Behavior:
  • Euclidean Variance: High in mixed regions, low in pure regions.
  • Aitchison (Geodesic) Variance: High in pure regions (where small absolute noise causes large relative ratio changes) and lower in mixed regions.
  • This highlights that Aitchison variance better captures the relative uncertainty of material proportions, which is physically more meaningful near the simplex boundaries.
• Spatial Smoothing: The spatial GP prior produced smoother abundance maps and, crucially, smoother uncertainty maps compared to non-spatial priors, effectively filtering noise-induced perturbations.

5. Significance

This work provides a theoretically sound and practical framework for uncertainty quantification in hyperspectral unmixing. By moving away from Euclidean assumptions to Aitchison geometry, the authors:

• Enable the creation of spatially aware Bayesian models that strictly respect abundance constraints.
• Offer interpretable uncertainty metrics that reflect the true nature of compositional data (relative proportions).
• Provide a pathway to validate unmixing algorithms in the absence of ground truth by assessing the difficulty of the unmixing problem through the lens of posterior uncertainty.

Future work suggested by the authors includes exploring other UQ diagnostics and using Aitchison geometry to define probabilistic evaluation metrics for validating Bayesian algorithms.