Researchers are increasingly focusing on dimensionality reduction techniques to accurately represent data present in complex nonlinear manifolds. Alaa Elichi and Khalide Jbilou of the LMPA at the Université de la Cote de Opal, along with colleagues, are presenting new research on a Riemannian geometry-based method to achieve this goal. Their work extends principal geodesic analysis and adapts discriminant analysis by utilizing geodesic distances and unique statistical measures to create more reliable low-dimensional embeddings. This study is important because it demonstrates improved representation quality and classification performance, especially for data constrained to curved spaces, and highlights the important role of geometry-aware dimensionality reduction in modern data science applications.
Researchers leverage geodesic distances, tangent spatial representations, and unique statistical measures to achieve higher fidelity, lower-dimensional embeddings. We also discuss a variety of relevant learning techniques, highlighting their theoretical foundations and practical benefits. Experimental results on representative datasets show that the Riemann method has improved representation quality and classification performance compared to the Euclidean method, especially for data constrained to a curved space such as a hypersphere or a symmetric positive definite manifold. This study highlights the importance of geometry-aware dimensionality reduction in modern machine learning and data analysis.
Modeling manifold structures using Riemann geodesics for dimensionality reduction
Scientists are increasingly applying Riemannian geometry to dimensionality reduction techniques in data analysis, machine learning, and pattern recognition. Classical techniques such as principal component analysis (PCA) and linear discriminant analysis (LDA) are widely used, but they rely on linear assumptions and may be inappropriate for data that exhibit nonlinear structure or are constrained to non-Euclidean spaces.
Many modern applications, such as computer vision, signal processing, medical image processing, and shape analysis, involve data that resides in nonlinear manifolds rather than planar Euclidean space. Ignoring the underlying geometry can lead to distorted representations and suboptimal performance. PGA captures modes of variation along geodesics and provides a geometry-aware alternative to linear projections. Supervised dimensionality reduction techniques have also been extended to the Riemannian setting, adapting classical criteria to manifold-valued data by replacing Euclidean distances and statistics with their unique counterparts.
These approaches have shown improved classification performance, especially in curved spaces like hyperspheres and SPD manifolds. Various learning methods such as isomaps, local linear embeddings (LLE), and Laplacian eigenmaps have also been proposed to reveal the low-dimensional structure of nonlinear data.
However, these methods are often external and may lack statistical interpretation for Riemannian manifolds. In this study, we study dimensionality reduction techniques based on Riemannian geometry, focusing on PGA and Riemannian adaptations of classical discriminant and projection-based techniques.
The proposed framework generates low-dimensional embeddings that better respect the manifold structure of the data. Experimental evaluations on representative datasets demonstrate that the Riemann method consistently outperforms the Euclidean method in terms of representation fidelity and classification accuracy.
These results highlight the importance of geometry-aware dimensionality reduction in modern machine learning and data science. The paper is structured as follows. Section 2 introduces the mathematical foundations of Riemannian geometry, including smooth manifolds, tangent spaces, Riemann metrics, geodesic distances, and examples such as Grassmann manifolds, Stiefel manifolds, and SPD manifolds.
Section 3 considers optimization on Riemannian manifolds and presents concepts such as Riemannian gradients, retraction, and convergence guarantees for first-order methods. Section 5 introduces Riemann robust principal component analysis (RRPCA) for handling outliers in manifold-valued datasets.
In Section 6, we extend orthogonal neighborhood-preserving projection (ONPP) to Riemannian manifolds and specialize in SPD and Grassmannian manifolds. Section 7 presents Riemann-Laplacian eigenmaps for nonlinear dimensionality reduction on manifolds. Section 8 discusses extensions to supervised learning, including linear discriminant analysis for Riemannian manifolds.
Section 9 introduces the Riemann Isomap method, and Section 10 describes the Riemann Support Vector Machine (RSVM). A Riemannian manifold M is a topological space in which each element has a neighborhood that is homeomorphic to Rd, satisfies the Hausdorff condition and the second countability condition, and is locally Euclidean.
It is equipped with a dot product gp that varies smoothly on each tangent space associated with a point p in M. Examples include the sphere Sn, the symmetric positive definite (SPD) matrix S d ++, and the Grassmann manifold Gr(p,n). The tangent space at a point p ∈M, denoted TpM, is the set of tangent vectors at p that represent infinitesimal directions on the manifold.
The Riemann metric g assigns an inner product to each tangent space: gp: TpM ×TpM →R. Curve length γ: [0,1] →M is computed as the integral of the norm of the tangent vector to the Riemann metric. The geodesic distance dM (p,q) is the lower bound of the length of all geodesics connecting points p and q on the manifold.
The Grassmann manifold, denoted Gr(p,n), is the set of all p-dimensional linear subspaces of Rn. A point U ∈Gr(p,n) can be represented by an orthonormal basis matrix U ∈Rn×p (U⊤U = Ip). The tangent space at a point U ∈Gr(p,n) is given by TUGr(p,n) = {Z ∈Rn×p | U⊤Z = 0}.
The canonical Riemannian metric of Gr(p,n) is defined as ⟨Z1,Z2⟩= tr(Z⊤ 1 Z2). Given two subspaces U1,U2 ∈Gr(p,n), the geodesic distance is dGr(U1,U2) = ∑ i=1 p θ i 2 1/2. Here, θi are the principal angles. Exponential and logarithmic maps of Gr(p,n) allow closed-form expressions, allowing efficient optimization and learning algorithms.
The Stiefel manifold, denoted St(p,n), is the set of all n × p matrices with orthonormal columns such that U⊤U = Ip. The tangent space at a point U ∈St(p,n) is given by TUSt(p,n) = {Z ∈Rn×p | U⊤Z +Z⊤U = 0}. The commonly used Riemannian metric for St(p,n) is the canonical metric: ⟨Z1,Z2⟩= tr Z⊤ 1 I −1 2UU⊤ Z2.
Retractions such as QR-based retractions are often used. Stiefel manifolds appear in problems with orthogonality constraints, such as principal component analysis and optimization problems with orthonormality constraints. Table 2.1 provides a glossary of Riemannian manifold terms that define concepts such as manifold, Riemannian manifold, tangent space, logarithmic map, exponential map, geodesic, Fréchet mean, PGA, Riemannian gradient, and Riemannian optimization.
Figures 2.1 and 2.2 show the tangent space and geodesics and the exponential and logarithmic maps on the sphere S2. Optimization over manifolds naturally arises in problems with geometric constraints such as orthogonality, low-rank structure, and positive definiteness.
Riemannian geometry improves manifold data representation and classification performance
Dimension reduction techniques based on Riemannian geometry improve the representation quality of manifold-valued data. Experimental evaluations demonstrate that these Riemann methods consistently outperform Euclidean methods in both representation fidelity and classification accuracy.
In this work, we focus on leveraging native geometric tools to generate low-dimensional embeddings that better respect the manifold structure of the data. Research includes Grassmann, Stiefel, and symmetric positive definite manifolds as examples of curved spaces where these methods have proven particularly effective.
Riemann robust principal component analysis addresses the handling of outliers in manifold-valued datasets and further refines the robustness of the approach. Orthogonal neighborhood-preserving projections have been extended to Riemannian manifolds, with special emphasis on both symmetric positive definite manifolds and Grassmannian manifolds.
Riemann-Laplacian eigenmaps provide a means of nonlinear dimensionality reduction directly on the manifold, providing an alternative to extrinsic methods. Extensions to supervised learning include linear discriminant analysis adapted to Riemannian manifolds, and Riemann support vector machines have also been developed.
These advances highlight the potential of geometry-aware techniques in modern machine learning and data science applications and provide a framework for more accurate and efficient data analysis. This study highlights the importance of considering the underlying geometry when working with data residing in nonlinear manifolds.
Riemannian geometry enhances data representation and classification of manifolds
Dimensionality reduction methods that incorporate Riemannian geometry offer significant improvements over classical Euclidean methods when analyzing data that resides in nonlinear spaces. Experimental results show that these Riemann methods achieve improved representation quality and classification performance, especially on datasets constrained to surface spaces such as hyperspheres and symmetric positive definite manifolds.
For example, Isomap performs well on datasets as diverse as Swiss rolls, S-curves, and circles, as evidenced by the embeddings produced. This study confirms the importance of being aware of the underlying data geometry in the analysis process and highlights the growing relevance of the Riemann method in fields such as machine learning and computer vision.
The authors acknowledge that the performance improvement is most noticeable when processing data that truly exhibits significant manifold structure. Although this method has demonstrated improvements across benchmark datasets, the magnitude of these improvements may vary depending on the specific characteristics of the data. Future research may focus on developing more efficient algorithms for calculating geodesic distances and considering the application of these techniques to even more complex and higher dimensional datasets, further strengthening their role in modern data science.
