Nested Inductive Bias Framework for SPD Manifold Learning

1 INTRODUCTION

• SPD matrices are used across medical imaging, neural signal decoding, kinematic modeling, and radio-frequency target identification.Riemannian metrics have supported adapting neural-network components such as residual connections and normalization layers to manifold-valued data.Source: §1 · p. 1
• Pullback Euclidean metrics such as LEM preserve the SPD constraint but impose flat geometry that may overlook latent structures such as kinematic hierarchies.The passage states that forcing tree-like distributions into LEM geometry may degrade model performance.Source: §1 · pp. 1–2
• Nested Inductive Biases map SPD matrices through the Log-Euclidean tangent space and then a target manifold, pulling target metrics back onto SPD data.The framework supports hyperbolic and spherical biases and enables native RMLR classification.Source: §1 · p. 2
• The framework also introduces RCM, a bounded geometry designed to regularize against outliers in standard vectorized architectures.The contribution summary also reports empirical validation using Linear Probes, SPDNet, and deep RResNet, including controlled synthetic geometric ablations.Source: §1 · p. 2

2 PRELIMINARIES:THE κ-STEREOGRAPHIC MODEL

• The generalized κ-stereographic model uses κ = K and c = |K| to represent target geometries of varying constant sectional curvature.The model transitions among the Poincaré ball for κ < 0, Euclidean space for κ = 0, and the Projected Hypersphere for κ > 0.Source: §2 · p. 2
• The exponential and logarithmic maps connect the origin-centered tangent space and manifold using generalized trigonometric functions.For κ < 0 these correspond to tanh and arctanh; for κ > 0 they correspond to tan and arctan.Source: §2 · p. 2

3 THE NESTED INDUCTIVE BIASES FRAMEWORK

• The two-stage pullback maps target geometries onto the SPD cone, inducing constant non-zero curvature with computational efficiency akin to pullback Euclidean metrics.The framework also emphasizes using a metric aligned with the dataset.Source: §3 · p. 3

3.1 THE TWO-STAGE DIFFEOMORPHIC COMPOSITION

• The nested map Φ(S) = ϕ₂(ϕ₁(S)), using the principal matrix logarithm and vech followed by the target exponential map, is a diffeomorphism globally for K ≤ 0 and within the injectivity radius π/√c for K > 0.Source: §3.1 · p. 3

3.2 INDUCED CURVATURE VIA ISOMETRIC PULLBACK

• Pulling the target metric back through Φ makes the mapping an isometry, so the SPD manifold inherits the target's constant sectional curvature.The pullback metric is defined on the SPD manifold using the target metric and the pushforward differential of Φ.Source: §3.2 · p. 4

3.3 NESTED RIEMANNIAN MULTINOMIAL LOGISTIC REGRESSION (TRACK A)

• Nested RMLR classifies in the target geometry, using a Riemannian inner product at each class centroid as a linear logit.The tangent-space projection encodes signed orthogonal distance to the decision boundary; its parameter is optimized in the origin tangent space and parallel transported to the centroid.Source: §3.3 · p. 4

3.4 THE RATIONAL CONFORMAL METRIC (RCM)

• RCM uses a radially decreasing conformal factor to shrink lengths toward infinity, compressing extreme deviations without altering the central distribution's angular geometry.The metric is pulled back onto the SPD cone through the nested framework.Source: §3.4 · pp. 4–5

3.5 TANGENT SPACE MAPPING FOR GEODESICALLY INCOMPLETE SPACES (TRACK B)

• Under RCM, TSM maps unbounded vectors uniquely into the open ball B1/α, while the inverse exponential is restricted to that ball.The mapping uses rational polynomials rather than transcendental functions and square roots used by standard constant-curvature manifolds.Source: §3.5 · p. 5

4 EXPERIMENTAL METHODOLOGY

• The nested framework composes Log-Euclidean mapping with κ-stereographic geometry to define NHM and NSM metrics for curvature-aware RMLR boundaries.These boundaries respect SPD constraints and the target geometry's relational curvature.Source: §4 · pp. 5–6

4.1 DATASETS AND PREPROCESSING

• FPHA evaluates 45-class action recognition on 63 × 63 covariance matrices, while Radar uses 20 × 20 covariance matrices with 5-fold cross-validation.FPHA uses the official test split; Radar evaluates generalization across varying signal-to-noise conditions.Source: §4.1 · p. 6
• Linear Probe, dimension-preserving SPDNet, and compressed-latent Deep RResNet test metric separability and robustness across different network capacities.The architectures isolate metric effects, with Deep RResNet using dimensionality-reducing BiMap layers before residual blocks.Source: §4.1 · p. 6
• Track A compares NHM and NSM RMLR against LEM and AIRM; Track B projects RCM outputs to vectors for dense Euclidean classification.Track A uses target-manifold metrics, while Track B applies BatchNorm1d before the classifier.Source: §4.1 · p. 6

4.2 VALIDATION VIA SYNTHETIC GEOMETRIC CONTROL

• The synthetic validation embeds a three-dimensional spherical distribution in symmetric 20 × 20 matrices, varying its radius relative to ambient noise.Spherical coordinates populate the top-left 2 × 2 submatrix, with isotropic noise in remaining dimensions and a matrix exponential mapping to SPD.Source: §4.2 · p. 7
• At r=1.5, NSM matches positive curvature (Test ORC: +0.010); at r=0.25, NHM improves separability under negative curvature (Test ORC: −0.213).The spherical structure dominates noise at the larger radius but collapses into it at the smaller radius.Source: §4.2 · p. 7

4.3 GEOMETRIC REGULARIZATION AGAINST ADVERSARIAL OUTLIERS

• The outlier experiment uses four linearly separable classes with 200 total samples and injects cross-boundary, heavy-tailed, and extreme anomalies.The anomaly patterns are designed to disrupt linear separability and can exert disproportionate influence on optimization.Source: §4.3 · p. 7
• RCM recovered 100% accuracy without explicit outlier filtering, while unbounded Log-Euclidean optimization misclassified dense core clusters.Its nonlinear radial compression bounds representations and anomaly gradient contributions.Source: §4.3 · pp. 7–8

5 RESULTS

• FPHA and Radar both show negative train and test ORC, with the FPHA values described as consistent with human hand-action hierarchy.FPHA: train −0.0969 and test −0.1197; Radar: train −0.1780 and test −0.0646.Source: §5 · p. 8

5.1 MAIN BENCHMARK RESULTS (TRACK A AND TRACK B)

• NHM gains +9.9% absolute accuracy over Log-Euclidean RMLR and +7.1% over AIRM on deep FPHA RResNet.The native RMLR results are reported to corroborate the datasets’ intrinsic-curvature findings; NSM shows substantial margin collapse.Source: Table 1 · p. 8, §5.1 · p. 8

5.2 VALIDATION VIA SYNTHETIC GEOMETRIC CONTROL RESULTS

• At r = 1.5, NSM test accuracy is 1.0000 versus NHM’s 0.9950; at r = 0.25, NHM reaches 0.8000 versus NSM’s 0.6875.The first regime has positive ORC; the second has negative ORC.Source: Table 5 · p. 9, §5.2 · p. 9

5.3 GEOMETRIC REGULARIZATION: OUTLIER RESULTS (TRACK B)

• RCM bounds representations and limits outlier gradients, restoring state-of-the-art core classification across the adversarial variants.The results report a minor regularization penalty on clean datasets in deeper architectures, alongside RCM’s robustness against severe anomalies.Source: Table 6 · p. 9, §5.1 · p. 9, §5.3 · p. 10
• Under cross-boundary outlier injection, the unbounded Log-Euclidean baseline’s core accuracy falls to 22.5%, while RCM achieves 100.0% core and 92.3% total accuracy.The proposed mapping is described as preserving core-distribution variance and angular separation while bounding outlier impact.Source: Table 6 · p. 9, §5.3 · p. 10
• L2 normalization reaches 100.0% core and 91.8% total accuracy but removes magnitude; Tanh reaches 92.5% core and 86.4% total accuracy while skewing inter-class angles.RCM reports 100.0% core and 92.3% total accuracy without dimensional reduction or angular distortion.Source: §5.3 · p. 10

• The nested inductive bias framework uses nested diffeomorphisms to pull back hyperbolic and spherical geometries and construct curvature-aligned Riemannian classifiers.Source: §6 · p. 10
• Deep manifold networks experience significant degradation in class separability unless metric curvature aligns with the intrinsic geometry of the data distribution.Source: §6 · p. 10
• For standard vectorized architectures, RCM bounds the representation space and yields state-of-the-art robustness against extreme spatial anomalies.Future work would extend the pullback framework beyond the terminal classification layer to fully nested Riemannian dense layers.Source: §6 · p. 10

• On directional heavy-tailed data, RCM bounds outlier magnitude while preserving primary-data variance and angular separation.Source: Figure 1 · p. 12
• L2 normalization restores core accuracy but collapses distributions into rigid 1D arcs, while tanh bounding distorts the space anisotropically and skews inter-class angles.Source: Figure 1 · p. 12
• Across three anomaly topologies, RCM confines outliers within a fixed radial envelope and restores perfect core accuracy, while the unbounded Log-Euclidean baseline shifts decision boundaries and degrades core separability.Source: Figure 2 · p. 13