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From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices
Mehrtash T. Harandi, Mathieu Salzmann, Richard Hartley
TL;DR
High-dimensional SPD manifolds preserve useful geometry but make recognition methods costly and limit practical use. The paper learns an orthonormal, label-aware projection to a lower-dimensional SPD manifold, achieving state-of-the-art recognition accuracy across several challenging classification datasets.
Problem
Riemannian computation on high-dimensional SPD matrices is costly, while low-dimensional descriptors can be less powerful and discriminative.
Method
The method learns an orthonormal projection by optimizing an affinity-weighted, metric-based similarity on a Grassmann manifold.
Results
The resulting low-dimensional SPD matrices achieve state-of-the-art recognition accuracy on several challenging datasets and outperform state-of-the-art techniques on three classification tasks.
Takeaways & Limitations
Directly mapping between SPD manifolds retains useful SPD-manifold properties while making existing manifold-based recognition techniques more practical and effective.
Takeaways & Limitations
The authors identify extension to unsupervised and semisupervised settings and to other Riemannian manifolds as future work.
Abstract
from arXiv · showhide
Representing images and videos with Symmetric Positive Definite (SPD) matrices and considering the Riemannian geometry of the resulting space has proven beneficial for many recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices --especially of high-dimensional ones-- comes at a high cost that limits the applicability of existing techniques. In this paper we introduce an approach that lets us handle high-dimensional SPD matrices by constructing a lower-dimensional, more discriminative SPD manifold. To this end, we model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. In particular, we search for a projection that yields a low-dimensional manifold with maximum discriminative power encoded via an affinity-weighted similarity measure based on metrics on the manifold. Learning can then be expressed as an optimization problem on a Grassmann manifold. Our evaluation on several classification tasks shows that our approach leads to a significant accuracy gain over state-of-the-art methods.
1 Introduction
SPD matrices provide useful representations for recognition, but Euclidean or manifold-based processing becomes costly and potentially distorted at high dimensions. The paper addresses this by learning a direct projection to a lower-dimensional, discriminative SPD manifold.
- SPD matrices represent data such as diffusion tensors and region covariances for images and videos.
- Euclidean manipulation can produce undesirable behavior, including physically unacceptable swelling in diffusion tensor interpolation.
- Riemannian methods preserve SPD geometry but become computationally expensive as matrix dimension increases.
- Low-dimensional descriptors are often used for efficiency, although they can be less powerful and discriminative than high-dimensional features.
- The proposed approach directly maps a high-dimensional SPD manifold to a lower-dimensional, more discriminative SPD manifold without initial flattening.
- The method learns an affinity-weighted projection using class labels and manifold metrics, and outperforms state-of-the-art techniques on three classification tasks.
2 Related Work
Earlier manifold dimensionality-reduction methods commonly flatten SPD data or support mainly unsupervised analysis. This paper instead learns a label-aware mapping directly between SPD manifolds while retaining their geometry.
- Principal Geodesic Analysis reduces manifold data through a tangent space at the Karcher mean, so it does not fully exploit manifold structure.
- Covariance Discriminative Learning uses labels but maps SPD matrices into symmetric matrices through the principal matrix logarithm and a single tangent space.
- Riemannian extensions of nonlinear dimensionality reduction methods were applied to unsupervised clustering and can be limited for supervised classification.
- The proposed mapping avoids tangent-space approximations while simultaneously exploiting class-label information on the original manifold.
- The resulting lower-dimensional SPD representation can preprocess data for other Riemannian recognition methods, including manifold sparse coding.
- The authors describe this as the first work transforming a high-dimensional SPD manifold into another SPD manifold with lower intrinsic dimension.
3 Riemannian Geometry of SPD Manifolds
SPD matrices form a Riemannian manifold whose distances can be measured using structures such as AIRM and the Stein metric. These metrics preserve geometry-related properties while supporting the paper’s dimensionality-reduction framework.
- SPD matrices are real symmetric matrices whose quadratic form is positive for every nonzero vector.
- The space of n × n SPD matrices is not a vector space but forms the interior of a convex cone and is studied as a Riemannian manifold.
- Geodesic distance measures manifold closeness through the length of the shortest connecting curve.
- AIRM is a principal Riemannian structure for analyzing SPD matrices, with matrix-logarithm notation used in its formulation.
- The Stein metric resembles the AIRM-induced geodesic while being less expensive to compute.
- Curves have the same length under the geodesic and Stein metrics up to a scale factor of 2√.
- Affine invariance means that distances between SPD matrices remain unchanged under affine transformations, including image-feature changes such as illumination variation.
4 Geometry-Aware Dimensionality Reduction
The method learns an orthonormal projection from a high-dimensional SPD manifold to a lower-dimensional SPD manifold by optimizing an affinity-weighted, geometry-aware objective. Class-label affinities and manifold metrics support discriminative embeddings, while Grassmann-manifold optimization and low-dimensional region covariance representations improve practicality.
- Embedding formulation: The method learns W ∈ R^n×m, with m < n, to map high-dimensional SPD matrices into a lower-dimensional SPD manifold.The projection must preserve valid SPD structure, requiring W to have full rank; orthonormality enforces this constraint.
- Geometry-aware objective: AIRM or Stein metrics measure pairwise similarities on the induced SPD manifold, and their affine invariance makes orthonormal projection constraints lossless.The resulting pairwise costs are combined into a global empirical objective minimized with respect to W.
- Optimization on Grassmann manifolds: Because the objective is invariant to right-side rotations, optimization with W^T W = I_m is formulated on the Grassmann manifold and solved using nonlinear conjugate gradients.The implementation uses manifold gradients, line search along geodesics, and parallel transport of tangent vectors.
- Scope and use: The framework also supports unsupervised or semisupervised affinity construction and can preprocess data for other Riemannian methods.The learned mapping preserves nearby pairs from the original manifold when the affinity is based on pairwise similarities, and it can make manifold sparse coding practical.
- Affinity design: Class-labeled nearest-neighbor graphs encode within-class and between-class affinities, producing a mapping that minimizes intra-class distances and maximizes inter-class distances.The affinity matrix uses same-label and different-label neighbors, with the number of between-class neighbors selected no greater than the within-class count and tuned by cross-validation.
- Region covariance application: For region covariance descriptors, projection yields a valid S^m representation with only m observations instead of at least n, reducing the quadratic cost of covariance computation.Applying W directly to observations creates the final low-dimensional region covariance matrix, whose computation is quadratic in feature dimensionality.
5 Empirical Evaluation
The evaluation tests geometry-aware SPD manifold learning on material, action, and face recognition tasks, using nearest-neighbor and sparse-representation classifiers. Across these datasets, learned low-dimensional SPD representations improve recognition accuracy while reducing computational cost in the material experiment.
- Experimental setup: Experiments cover two image datasets and one motion-capture dataset, with low-dimensional manifold dimensionality selected by cross-validation.The tasks are material categorization, action recognition, and face recognition.
- Classifiers: SPD-ML produces low-dimensional, more discriminative SPD manifolds evaluated with manifold-metric nearest neighbors and Riemannian sparse representation.RSR is applied only to the Stein-based version because it relies on a Stein-metric kernel.
- Material categorization: 66.6% accuracy is achieved by RSR-ML on the UIUC material dataset, while NN-AIRM-ML reaches 58.3% versus 42.1% with PCA-based RCMs.The learned representation also improves over the original feature-space NN-AIRM accuracy of 35.6%.
- Material categorization: 9.7s replaces 490s for NN-AIRM classification after dimensionality reduction on the UIUC material dataset.The comparison uses the same 3GHz Matlab machine.
- Action recognition: 81.9% is obtained by RSR-ML for HDM05 action recognition, while NN-AIRM-ML reaches 67.6% versus 62.8% for NN-AIRM.The action descriptors are 3K × 3K joint covariance matrices computed from 3D joint locations over time.
- Face recognition: SPD-ML yields significant accuracy gains for FERET face recognition across all tested viewing angles, exceeding 10% improvement for some poses.The face representation concatenates three 43 × 43 RCMs into a 129 × 129 RCM.
6 Conclusions and Future Work
The paper learns geometry-preserving mappings from high-dimensional SPD manifolds to lower-dimensional, more discriminative SPD manifolds. Experiments report state-of-the-art recognition accuracies, while future work targets unsupervised and semisupervised learning.
- The method uses an affinity matrix encoding intra-class and inter-class distances, with Stein divergence or AIRM defining SPD similarity.
- Affine-invariant metrics permit an orthonormal projection, reducing learning to optimization on a Grassmann manifold.
- The resulting low-dimensional SPD matrices achieve state-of-the-art recognition accuracies on several challenging datasets.
- Future work will extend the learning scheme to unsupervised and semisupervised scenarios and investigate other Riemannian manifolds.
A Proof of Length Equivalence
The appendix proves that geodesic and Stein metrics induce equivalent curve lengths up to scale. The proof reduces the comparison using affine invariance and analyzes the Stein metric near the identity.
- The proof uses intrinsic metrics, defined as infima of curve lengths, to establish curve-length equivalence.
- If two metrics have identical intrinsic metrics up to scale, every curve has corresponding length up to that scale.
- Affine invariance allows the analysis to reduce transformed SPD matrices to diagonal forms near the identity matrix.
- Because the transformed diagonal matrix approaches the identity as t approaches zero, the proof studies the local behavior of the Stein metric there.
- L’Hôpital’s rule is applied twice to evaluate the indefinite limit, completing the equivalence proof.
B Conjugate Gradient on Grassmann Manifolds
The projection is represented as a point on a Grassmann manifold, where its orthonormality constraints are handled naturally. Conjugate-gradient optimization updates the projection along manifold geodesics until convergence.
- The projection W is modeled on G(m, n), the set of m-dimensional linear subspaces of R^n, enforcing W^T W = I_m.
- Grassmann conjugate-gradient optimization computes the manifold gradient of the objective at the current projection.
- The search direction combines the current gradient with the previous direction after parallel transport.
- A line search proceeds along a Grassmann geodesic whose direction is represented using a compact singular value decomposition.
- Iterations continue until convergence to a local minimum or until the maximum iteration count is reached.
C.1 Parameter Sensitivity
The experiment studies how the number of nearest neighbors from different classes affects NN-Stein-ML accuracy on the UIUC material dataset.
- The sensitivity analysis varies ν_b, the number of nearest neighbors from different classes, while fixing the other parameter to its reported value.
- The evaluation uses the UIUC material dataset and measures accuracy with the NN-Stein-ML method.
C.2 Influence of the Number of Observations
The evaluation examines how classification performance changes as the number of observations used to construct RCMs decreases. SPD-ML remains substantially more stable than competing algorithms when observations fall below n.
- Experimental setup: All algorithms maintain stable performance when the number of observations used to build RCMs is sufficiently large.The experiment uses the UIUC material dataset, with training RCMs built from 9600 observations per image and test observations reduced on an octave basis.
- Figures: Fig. 6 reports UIUC material-dataset accuracy for varying values of νb, while Fig. 7 reports algorithm sensitivity to observation count.
- Results: 17%, 19% and 20%, respectively, are the performance drops for CDL, NN-Stein and NN-AIRM when observations fall below n.
- Results: Less than 7% is the performance drop for SPD-ML under the same reduction in observations.