Related papers: Manifold Learning via Manifold Deflation
We present a novel framework for learning cost-efficient latent representations in problems with high-dimensional state spaces through nonlinear dimension reduction. By enriching linear state approximations with low-order polynomial terms…
Recently, there has been much interest in spectral approaches to learning manifolds---so-called kernel eigenmap methods. These methods have had some successes, but their applicability is limited because they are not robust to noise. To…
We reframe linear dimensionality reduction as a problem of Bayesian inference on matrix manifolds. This natural paradigm extends the Bayesian framework to dimensionality reduction tasks in higher dimensions with simpler models at greater…
Classical nonlinear dimensionality reduction (NLDR) techniques like t-SNE, Isomap, and LLE excel at creating low-dimensional embeddings for data visualization but fundamentally lack the ability to map these embeddings back to the original…
Manifold learning (ML), known also as non-linear dimension reduction, is a set of methods to find the low dimensional structure of data. Dimension reduction for large, high dimensional data is not merely a way to reduce the data; the new…
This paper considers the problem of nonlinear dimensionality reduction. Unlike existing methods, such as LLE, ISOMAP, which attempt to unfold the true manifold in the low dimensional space, our algorithm tries to preserve the nonlinear…
We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the…
In image set classification, a considerable advance has been made by modeling the original image sets by second order statistics or linear subspace, which typically lie on the Riemannian manifold. Specifically, they are Symmetric Positive…
We introduce novel estimators for computing the curvature, tangent spaces, and dimension of data from manifolds, using tools from diffusion geometry. Although classical Riemannian geometry is a rich source of inspiration for geometric data…
We consider the theory of regression on a manifold using reproducing kernel Hilbert space methods. Manifold models arise in a wide variety of modern machine learning problems, and our goal is to help understand the effectiveness of various…
This paper presents a novel non-linear model reduction method: Probabilistic Manifold Decomposition (PMD), which provides a powerful framework for constructing non-intrusive reduced-order models (ROMs) by embedding a high-dimensional system…
In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. In many classification scenarios, the data can be naturally represented by symmetric positive…
Dimensionality reduction is an effective method for learning high-dimensional data, which can provide better understanding of decision boundaries in human-readable low-dimensional subspace. Linear methods, such as principal component…
In the realm of robotics, numerous downstream robotics tasks leverage machine learning methods for processing, modeling, or synthesizing data. Often, this data comprises variables that inherently carry geometric constraints, such as the…
Manifold learning builds on the "manifold hypothesis," which posits that data in high-dimensional datasets are drawn from lower-dimensional manifolds. Current tools generate global embeddings of data, rather than the local maps used to…
Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by…
Over the past decades, the increasing dimensionality of data has increased the need for effective data decomposition methods. Existing approaches, however, often rely on linear models or lack sufficient interpretability or flexibility. To…
Many tasks require mapping continuous input data (e.g. images) to discrete task outputs (e.g. class labels). Yet, how neural networks learn to perform such discrete computations on continuous data manifolds remains poorly understood. Here,…
Nonlinear dimensionality reduction or, equivalently, the approximation of high-dimensional data using a low-dimensional nonlinear manifold is an active area of research. In this paper, we will present a thematically different approach to…
Manifold learning seeks a low dimensional representation that faithfully captures the essence of data. Current methods can successfully learn such representations, but do not provide a meaningful set of operations that are associated with…