Nested Inductive Bias Framework for Curvature-Aligned SPD Manifold Learning

In Geometric Deep Learning, inductive biases serve two primary functions: enforcing manifold constraints and embedding relational priors. Currently, representation learning on SPD manifolds frequently relies on pullback Euclidean metrics to satisfy the former. While computationally efficient in avoiding domain boundary violations, these metrics induce a flat geometry that may fail to capture the intrinsic relational priors of datasets. While metrics such as the Poincaré metric are widely utilized to induce domain-aligned relational priors, generalizing them from standard vector representations to the SPD manifold has remained a challenge. To bridge this gap, we introduce a Nested Inductive Bias framework that utilizes a multistage diffeomorphic composition to formally pull back non-Euclidean target geometries onto the SPD manifold. Moreover, we identify that mapping unbounded Euclidean representations onto manifolds with finite injectivity radii induces a "Norm Mismatch," leading to degraded performance and diffeomorphic failure. We resolve this by introducing the Rational Conformal Metric (RCM), which preserves bijectivity as well as provides a strong regularization against outliers. Empirical evaluations on kinematic and signal processing benchmarks, together with synthetic experiments, demonstrate that deep manifold networks experience degradation in class separability due to curvature/norm mismatch between the metric and the intrinsic data distribution. Finally, our curvature-aligned Riemannian classifiers resolve the computational intractability of classical non-flat SPD metrics (like AIRM), achieving a 95% reduction in computational overhead.