Geodesic Flow Kernel: Theory & Applications

• GFK is a kernel that integrates geodesic paths on Grassmann manifolds to quantify similarity between feature subspaces.
• It leverages principal angles and closed-form integration, enabling smooth interpolation and effective domain adaptation under data corruption.
• Empirical studies show that GFK enhances robustness and accuracy in semi-supervised tabular learning, outperforming traditional methods in noisy settings.

A Geodesic Flow Kernel (GFK) is a mathematical construct designed to define and exploit geometric paths—specifically, geodesics—through latent representation or kernel spaces. Its principal application in the machine learning literature has been to encode similarity or correlation between data representations that inhabit non-Euclidean manifolds, most notably the Grassmannian of linear subspaces. GFKs enable alignment, interpolation, or integration of features by integrating information along geodesic trajectories, allowing for effective domain adaptation, structured similarity computation, and manifold-informed statistical learning.

1. Mathematical Foundations and Geodesic Flow Construction

The GFK is formally defined by leveraging geodesics between subspaces, typically on the Grassmann manifold $\mathbf{G}(D, d)$ parametrizing $d$-dimensional linear subspaces of $\mathbb{R}^D$. Given two orthonormal basis matrices, $P_\mathrm{hard}$ and $P_\mathrm{soft}$ (each $d_\mathrm{lin} \times D$), the geodesic path between them, $\mathbf{GF}(\pi)$ for $\pi \in [0,1]$, is characterized as: $$\mathbf{GF}(\pi) = \left[P_\mathrm{hard} \quad R_\mathrm{hard}\right] \begin{bmatrix} U_1 \boldsymbol{\Gamma}(\pi) & 0 \ 0 & -U_2 \boldsymbol{\Sigma}(\pi) \end{bmatrix}$$ where $U_1$, $U_2$, and $V$ derive from generalized SVD of the pair, and $\boldsymbol{\Gamma}(\pi), \boldsymbol{\Sigma}(\pi)$ are diagonal matrices of $\cos(\theta_i\pi)$ and $\sin(\theta_i\pi)$, with $\theta_i$ the principal angles between the subspaces. The kernel itself integrates the inner product between representations projected along this geodesic: $$A = \int_0^1 \mathbf{GF}(\pi) \mathbf{GF}(\pi)^T d\pi$$ which, through closed-form manipulation, decomposes into blocks parameterized by the principal angles:

• $D$, $E$, $G$ are diagonal matrices with $d_{1i} = 1 + \frac{\sin(2\theta_i)}{2\theta_i}$, $e_{2i} = \frac{\cos(2\theta_i)-1}{2\theta_i}$, $g_{3i}=1-\frac{\sin(2\theta_i)}{2\theta_i}$.

This formulation encodes a manifold-aware, continuous interpolation between subspaces, accounting for the intrinsic geometry of the latent representation space.

2. Algorithmic Realizations in Learning Architectures

The GFK has historically been deployed as a similarity kernel for domain adaptation and representation alignment. In the context of the GFTab framework (Hwang et al., 17 Dec 2024), GFKs are used for semi-supervised learning on mixed-variable tabular data as follows:

1. Input data undergoes variable-type-specific corruption to generate two augmented views ($x_\mathrm{soft}$ and $x_\mathrm{hard}$).
2. Both are passed through feature encoders, and their representations are concatenated with tree-based embeddings.
3. These features are projected onto corresponding subspaces to obtain $P_\mathrm{soft}$ and $P_\mathrm{hard}$.
4. The geodesic flow kernel $A$ is computed, and the normalized similarity between $z_\mathrm{soft}$ and $z_\mathrm{hard}$ is measured:

$$\mathcal{L}_\mathrm{sim} = 1 - \frac{z_\mathrm{soft}^T A z_\mathrm{hard}}{\|\sqrt{A}z_\mathrm{soft}\| \cdot \|\sqrt{A}z_\mathrm{hard}\|}$$

1. The total objective combines this geometric similarity loss with a supervised cross-entropy loss.

Minimizing $\mathcal{L}_\mathrm{sim}$ enforces that representations under type-dependent, realistic corruptions remain geodesically aligned, reflecting robust invariance to data heterogeneity.

3. Theoretical Guarantees and Limitations of Geodesic-Based Kernels

The foundational work on geodesic exponential kernels (Feragen et al., 2014) establishes that positive definiteness of Gaussian kernels derived from geodesic distances is only preserved on flat (Euclidean) or certain conditionally negative definite (CND) geodesic spaces. For instance:

• A geodesic Gaussian kernel $k(x,y) = \exp(-\lambda d^2(x,y))$ is PD for all $\lambda > 0$ if and only if the space is Euclidean (zero curvature).
• The geodesic Laplacian kernel $k(x,y) = \exp(-\lambda d(x,y))$ is more widely applicable, being PD if $d$ is CND; this holds for spheres and hyperbolic spaces, but not for most curved manifolds used in learning (e.g., affine-invariant metrics on $\operatorname{Sym}^+_d$, Grassmannians with intrinsic metric).

A practical consequence is that many GFK constructions, if based on Gaussian geodesic kernels, cannot generally be used as PD kernels on nonlinear manifolds, restricting their application or necessitating alternative formulations that incorporate linearization or work under Laplacian-like constructions.

4. GFK in Tabular Data: Integration with Variable Corruption and Tree-Based Embedding

The use of GFKs within GFTab (Hwang et al., 17 Dec 2024) incorporates additional mechanisms to address challenges of mixed discrete-continuous tabular data:

• Variable-specific corruption ensures that "soft" and "hard" views mimic realistic data perturbations reflective of continuous/categorical structure. For categorical variables, permutations and neighborhood perturbations are customized to category properties and class imbalance; for continuous ones, row-shuffling and masking are used.
• Tree-based embeddings (from methods like GBDT) capture hierarchical and relational priors from labeled data, which are then fused with deep features prior to GFK computation.
• The GFK not only measures geometric alignment under corruptions but serves as an inductive bias that explicitly respects the Grassmannian structure of feature variation, enhancing robustness to both label noise and label scarcity.

5. Empirical Performance and Comparative Evaluation

Empirical evaluation of the GFK's role in GFTab demonstrates:

• Across 21 tabular datasets, GFK-based similarity loss outperforms InfoNCE, Barlow Twins, and Uniform Alignment in clean and label-noisy regimes.
• In low-label settings ($10$–$20\%$ labeled data), GFTab with GFK matches or exceeds classic (XGBoost, CatBoost) and neural (SCARF, VIME, SubTab) baselines, particularly when categorical variables dominate.
• Ablation studies attribute gains in both accuracy and robustness specifically to the inclusion of the geodesic similarity loss.
• Improved sample efficiency and invariance stem from the kernel's respect for subspace geometry and the heterogeneous structure of tabular variables.

Component	Role of GFK
Variable-Specific Corruption	Exposes meaningful, type-adaptive feature noise
Tree-Based Embedding	Provides strong tabular relational priors
Geodesic Flow Kernel (GFK)	Measures similarity between soft/hard views via subspace geometry
Effect	Superior semi-supervised learning and robustness to noise

6. Broader Context and Alternative Geodesic Kernel Approaches

Alternative geodesic-informed kernels and related methodologies include:

• Heat diffusion-based embeddings (Huguet et al., 2023), which recover geodesic distances via the heat kernel (Varadhan's formula), allowing robust, denoised distance estimation and improved manifold preservation compared to GFK, especially in nonlinear data scenarios. This approach is fundamentally different from GFK, as it operates on the manifold of the data distribution using diffusion and spectral techniques rather than subspace geodesics.
• Spectral flow on manifolds of SPD matrices (Katz et al., 2020), which interpolates between kernel matrices via geodesics in the space of SPD matrices, focuses on analysis of spectral evolution along these paths, and provides tools to isolate shared versus measurement-specific latent components in multimodal data.
• Fisher-Rao geodesic flows (Maurais et al., 8 Jan 2024), which define dynamic transport between probability measures along Fisher-Rao geodesics parameterized in an RKHS. While both GFK and these flows exploit geodesic structures, the former operates in feature or subspace geometry, while the latter is situated in the space of distributions.

A key limitation identified in (Feragen et al., 2014) is that for most curved (i.e., intrinsically non-Euclidean) manifolds, PD geodesic Gaussian kernels do not exist, and even PD Laplacian kernels can only be defined in restricted cases, often effectively linearizing the geometry. Alternatives based on flows, heat processes, or spectral analysis may better accommodate highly nonlinear structure, but typically entail distinct methodological or computational trade-offs.

7. Theoretical and Practical Significance

The geodesic flow kernel:

• Offers a rigorous, manifold-aware measure of similarity sensitive to the underlying geometry of high-dimensional representations, bridging non-Euclidean subspace interpolation and practical, label-efficient learning.
• Enables systematic exploitation of geometric invariances, particularly in the context of variable-type heterogeneity and data corruption regimes.
• Establishes performance gains and robust statistical properties in both theoretical and empirical settings, provided the kernel’s mathematical properties (notably, positive definiteness and respect of true geometry) are satisfied.
• Illuminates intrinsic limitations of kernel methods on curved spaces, motivating exploration of alternative constructions when nonlinear geometry cannot be faithfully encoded while preserving computational tractability and positive-definite structure.

The continued paper and deployment of GFKs across scientific domains highlights the importance—and ongoing challenge—of reconciling geometric fidelity, computational feasibility, and statistical efficacy in kernel-based machine learning.