A Riemannian View on Active Subspaces

Technical Summary: A Riemannian View on Active Subspaces

Problem Statement
Active subspaces provide a principled, eigenvalue-ordered method for dimension reduction in high-dimensional Euclidean domains. They identify a low-dimensional subspace where a scalar quantity of interest, $f: \mathbb{R}^n \to \mathbb{R}$, varies most significantly on average, while remaining nearly constant in the orthogonal complement. However, many modern applications involve parameters that naturally reside on Riemannian manifolds (e.g., statistical shape analysis, where preshape spaces are hyperspheres) rather than Euclidean spaces. The standard definition of active subspaces relies on the canonical isomorphism of Euclidean space, which allows for the global averaging of gradient vectors. On a curved manifold, tangent vectors at different points belong to distinct tangent spaces, making a direct global average ill-defined without a mechanism to compare them. The paper addresses the challenge of extending the active subspace framework to smooth functions defined on Riemannian manifolds while preserving the interpretability of the eigenvalues as mean-squared directional derivatives.

Methodology
The paper develops an intrinsic formulation of active subspaces on a Riemannian manifold $(M, g)$ by generalizing the gradient outer product. The core methodology involves three key steps:

1. Intrinsic Averaging via Parallel Transport: To define a global average of gradients, the authors transport each Riemannian gradient $\nabla_M f(p)$ from its local tangent space $T_p M$ to a central tangent space $T_{p_0} M$ along the unique geodesic connecting $p$ to a central point $p_0$. This is achieved using the parallel transport operator $P_p$. The resulting "Riemannian average tensor product of the gradient" is defined as:
   $$G_0 = \int_X P_p [\nabla_M f(p)] \otimes P_p [\nla_M f(p)] \, d\mu(p)$$
   where $\mu$ is a probability measure on the manifold (typically supported on a geodesic ball). This construction yields a symmetric positive semi-definite tensor on the single vector space $T_{p_0} M$.

2. Active Manifold-Geodesics (AMG): The eigendecomposition of $G_0$ yields eigenvalues $\lambda_i$ and eigenvectors $w_i$ in $T_{p_0} M$. The authors define "Active Manifold-Geodesics" as the images of the straight lines spanned by these eigenvectors under the exponential map: $W_{1,i} = \{ \exp_{p_0}(t w_i) \}$. These geodesics serve as the active directions on the manifold.

3. Intrinsic vs. Extrinsic Comparison: The paper contrasts this intrinsic approach with an extrinsic one, where the manifold is embedded in a higher-dimensional Euclidean space $\mathbb{R}^m$. The extrinsic method averages the ambient gradients and projects them to the central tangent space. The authors derive a precise relationship between the two, showing that the extrinsic spectrum is a "conflated" version of the intrinsic spectrum due to the partial isometry of the projection.

Key Contributions
The paper makes four primary contributions:

• Intrinsic Generalization: It introduces the parallel-transport gradient outer product (Equation 12) as a coordinate-free tensor whose eigenvalues provide an exact interpretation as mean-squared directional derivatives along a transported frame field (Lemma 1).
• Active Manifold-Geodesics: It defines an ordered set of geodesic directions (Definition 3) that generalize active subspaces. Unlike fully nonlinear reductions (e.g., active manifolds or neural networks) which can degenerate into space-filling curves, AMGs maintain low complexity and identifiability by restricting the search to exponential images of straight lines in a single tangent space.
• Quantitative Intrinsic-Extrinsic Agreement: The paper establishes that while the intrinsic and extrinsic perspectives differ fundamentally, they agree to second order ($O(R^2)$) in the radius $R$ of the sampled geodesic ball. Specifically, eigenvalues agree at $O(R^2)$, and dominant eigenspaces agree at $O(R^2/\eta)$ where $\eta$ is the spectral gap (Proposition 3). Crucially, it demonstrates that projecting the dominant ambient eigenvector (a common extrinsic practice) can fail if the direction lies in the null space of the projection, whereas the intrinsic method remains robust.
• Application to Shape Analysis: The theory is explicitly motivated by and applied to preshape spaces in statistical shape analysis, which are hyperspheres. The results provide a direct framework for response-driven dimension reduction in shape functions.

Results and Numerical Illustrations
The authors validate the formalism using numerical examples on the 2-sphere ($S^2$):

• Ridge Recovery: For an ambient linear function restricted to the sphere (a ridge function), the intrinsic AMG successfully identifies the active geodesic direction. The paper demonstrates that the ridge structure is recovered in normal coordinates with an error decaying at a curvature-limited quadratic rate ($O(R^2)$), consistent with Proposition 3.
• Degenerate Cases: The paper examines a scenario where the central point coincides with the gradient direction of an ambient ridge. In this case, the extrinsic method (projecting the dominant ambient eigenvector) fails, yielding a trivial zero vector. In contrast, the intrinsic method correctly identifies that no single direction is dominant, returning two equal eigenvalues, thereby avoiding the "projected dominance" failure mode.
• Non-Ridge Functions: For quadratic functions that are not ridges, the intrinsic method provides a faithful ranking of directions based solely on the function's variation, whereas the extrinsic method entangles this variation with the geometry of the embedding.

Significance and Claims
The paper claims that the intrinsic formulation offers a more faithful representation of the geometry of the domain, free from the artifacts of a specific embedding. It argues that while extrinsic methods are computationally convenient, they introduce a "partial isometry" that can distort the ordering of importance, particularly when extending activity beyond the central point.

The authors emphasize that the proposed method is geodesically local. The interpretation of eigenvalues as indicators of inactive submanifolds is strictly valid only within a normal neighborhood (a geodesic ball) centered at $p_0$. Extending this to a global domain would require an atlas of local analyses or a more complex foliation structure, which is noted as future work. The paper does not claim to solve the global dimension reduction problem for arbitrary manifolds but rather provides a rigorous, explainable, and locally optimal extension of active subspaces for smooth functions on Riemannian manifolds, with immediate applicability to hyperspherical data in shape analysis.