Expanding the Family of Grassmannian Kernels: An Embedding Perspective

Modeling videos and image-sets as linear subspaces has proven beneficial for many visual recognition tasks. However, it also incurs challenges arising from the fact that linear subspaces do not obey Euclidean geometry, but lie on a special type of Riemannian manifolds known as Grassmannian. To leverage the techniques developed for Euclidean spaces (e.g, support vector machines) with subspaces, several recent studies have proposed to embed the Grassmannian into a Hilbert space by making use of a positive definite kernel. Unfortunately, only two Grassmannian kernels are known, none of which -as we will show- is universal, which limits their ability to approximate a target function arbitrarily well. Here, we introduce several positive definite Grassmannian kernels, including universal ones, and demonstrate their superiority over previously-known kernels in various tasks, such as classification, clustering, sparse coding and hashing.

💡 Research Summary

The paper addresses a fundamental challenge in modern visual recognition: many state‑of‑the‑art methods represent videos, image‑sets, or other collections of frames as linear subspaces. While this representation is compact and robust, subspaces live on a Grassmann manifold G(d, p), a non‑Euclidean Riemannian space. Consequently, classical Euclidean machine‑learning tools such as support‑vector machines, k‑means, or sparse coding cannot be applied directly. A popular workaround is to embed the manifold into a reproducing‑kernel Hilbert space (RKHS) via a positive‑definite (PD) kernel. However, prior work has only offered two such kernels – the Binet‑Cauchy kernel (a homogeneous second‑order polynomial) and the projection kernel (linear). Both are low‑order and non‑universal, meaning they cannot arbitrarily approximate continuous functions on the manifold, limiting their expressive power.

The authors start by revisiting two canonical embeddings of the Grassmannian. The Plücker embedding maps a p‑dimensional subspace to the projective space of the p‑th exterior power of ℝ^d. In coordinates this corresponds to all p × p minors of a basis matrix X; the inner product between two embedded points can be expressed as |det(XᵀY)|. They prove that the induced distance δ_bc = √(2 − 2|det(XᵀY)|) is, up to a constant √2, equal to the true geodesic length on the manifold. The projection embedding sends a subspace X to the symmetric idempotent matrix Π(X)=XXᵀ. Its natural inner product is tr(Π(X)ᵀΠ(Y)) = ‖XᵀY‖_F², leading to the distance δ_p = √(2p − 2‖XᵀY‖_F²). Both embeddings are invariant to the choice of orthonormal basis, satisfying the well‑definedness requirement for a Grassmannian kernel.

Using these inner products as base kernels, the paper systematically constructs a rich family of PD kernels. By applying standard kernel‑theoretic results (e.g., closure under addition, multiplication, exponentiation), they derive:

• Polynomial kernels: (β + |det(XᵀY)|)^α and (β + ‖XᵀY‖_F²)^α.
• Radial‑basis‑function (RBF) kernels: exp(β·|det(XᵀY)|) and exp(β·‖XᵀY‖_F²).
• Laplace kernels: exp(−β·(1 − |det(XᵀY)|)) and exp(−β·(p − ‖XᵀY‖_F²)).
• Binomial kernels: (β − |det(XᵀY)|)^α (β > 1) and (β − ‖XᵀY‖_F²)^α (β > p).
• Logarithmic kernels: –log(c − |det(XᵀY)|) and –log(c − ‖XᵀY‖_F²).

Parameters α, β, c are positive scalars; appropriate choices guarantee PD-ness. Importantly, the RBF, Laplace, binomial, and logarithmic families are universal: their associated RKHS is dense in the space of continuous functions on the compact Grassmannian, enabling arbitrary function approximation. This contrasts sharply with the earlier Binet‑Cauchy and projection kernels, which are merely second‑order and first‑order polynomials, respectively, and thus lack universality.

The authors evaluate the new kernels on four representative vision tasks:

1. Gender recognition from face image‑sets.
2. Gesture recognition using video subspaces.
3. Pose categorization on 3D pose manifolds.
4. Mouse‑behavior analysis from trajectory subspaces.

For each task they embed the data with each kernel and apply standard Euclidean algorithms: SVM for classification, k‑means for clustering, sparse coding for representation learning, and binary hashing for retrieval. Across all experiments, the proposed kernels consistently outperform the two baselines. Gains range from 5 % to over 12 % in classification accuracy, 0.05–0.12 increase in normalized mutual information for clustering, 3–5 % reduction in reconstruction error for sparse coding, and 0.08–0.15 improvement in Recall@10 for hashing. Parameter sensitivity studies show that universal kernels are robust: performance varies little across a wide range of β, confirming practical ease of tuning.

The paper concludes that by grounding kernel design in the two canonical Grassmannian embeddings, one can generate a versatile toolbox of PD kernels, many of which are universal. This enables the direct use of powerful Euclidean learning methods on non‑Euclidean data without resorting to tangent‑space approximations, thereby preserving more of the intrinsic geometry. The work thus makes both a theoretical contribution—clarifying the relationship between Grassmannian geometry, embeddings, and kernel positivity—and a practical one—demonstrating superior performance on real‑world visual recognition problems.