Related papers: Dimensionality Reduction on Riemannian Manifolds i…
This paper proposes an innovative extension of Principal Component Analysis (PCA) that transcends the traditional assumption of data lying in Euclidean space, enabling its application to data on Riemannian manifolds. The primary challenge…
There is growing interest in using the close connection between differential geometry and statistics to model smooth manifold-valued data. In particular, much work has been done recently to generalize principal component analysis (PCA), the…
Linear discriminant analysis (LDA) is a widely used algorithm in machine learning to extract a low-dimensional representation of high-dimensional data, it features to find the orthogonal discriminant projection subspace by using the Fisher…
This report concerns the problem of dimensionality reduction through information geometric methods on statistical manifolds. While there has been considerable work recently presented regarding dimensionality reduction for the purposes of…
Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…
In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man…
Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks.…
Dimensionality reduction on Riemannian manifolds is challenging due to the complex nonlinear data structures. While probabilistic principal geodesic analysis~(PPGA) has been proposed to generalize conventional principal component analysis…
Dimensionality reduction is critical across various domains of science including neuroscience. Probabilistic Principal Component Analysis (PPCA) is a prominent dimensionality reduction method that provides a probabilistic approach unlike…
In the recent past, nested structures in Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested…
This paper proposes a generalized framework with joint normalization which learns lower-dimensional subspaces with maximum discriminative power by making use of the Riemannian geometry. In particular, we model the similarity/dissimilarity…
Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…
We propose a geometric latent-subspace framework for generative modeling of discrete data. Specifically, we introduce latent subspaces in the exponential parameter space of product manifolds of categorical distributions as a novel method…
Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches…
An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The…
Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient…
Dimensionality reduction is a fundamental technique in machine learning and data analysis, enabling efficient representation and visualization of high-dimensional data. This paper explores five key methods: Principal Component Analysis…
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-valued datasets are widely encountered in many computer vision tasks. A non-linear analog of the PCA, called the Principal Geodesic Analysis (PGA) suited for data lying on Riemannian manifolds was reported in literature a decade…
Dimensionality reduction techniques play important roles in the analysis of big data. Traditional dimensionality reduction approaches, such as principal component analysis (PCA) and linear discriminant analysis (LDA), have been studied…