True Manifold Research Articles

Distortion corrected kernel density estimator on Riemannian manifolds

Manifold learning obtains a low-dimensional representation of an underlying Riemannian manifold supporting high-dimensional data. Kernel density estimates of the low-dimensional embedding with a fixed bandwidth fail to account for the way manifold learning algorithms distort the geometry of the Riemannian manifold. We propose a novel distortion-corrected kernel density estimator (DC-KDE) for any manifold learning embedding, with a bandwidth that depends on the estimated Riemannian metric at each data point. Exploiting the geometric information of the manifold leads to more accurate density estimation, which subsequently could be used for anomaly detection. To compare our proposed estimator with a fixed-bandwidth kernel density estimator, we run two simulations including one with data lying in a 100 dimensional ambient space. We demonstrate that the proposed DC-KDE improves the density estimates as long as the manifold learning embedding is of sufficient quality, and has higher rank correlations with the true manifold density. Further simulation results are provided via a supplementary R shiny app. The proposed method is applied to density estimation in statistical manifolds of electricity usage with the Irish smart meter data.

GeneBasis: an iterative approach for unsupervised selection of targeted gene panels from scRNA-seq

scRNA-seq datasets are increasingly used to identify gene panels that can be probed using alternative technologies, such as spatial transcriptomics, where choosing the best subset of genes is vital. Existing methods are limited by a reliance on pre-existing cell type labels or by difficulties in identifying markers of rare cells. We introduce an iterative approach, geneBasis, for selecting an optimal gene panel, where each newly added gene captures the maximum distance between the true manifold and the manifold constructed using the currently selected gene panel. Our approach outperforms existing strategies and can resolve cell types and subtle cell state differences.

Musical key extraction using diffusion maps

We propose a method for automatic musical key extraction using a two-stage spectral dimensionality reduction (two consecutive mappings). First we build a data set representing the 24 Western musical keys, and then we use a nonlinear dimensionality reduction method, in order to understand the true manifold on which the musical keys lie. The order of the keys along the manifold is perfectly correlated with a cognitive model for the key space. We exploit this manifold in order to extract the musical key from a musical piece. Furthermore we propose three classifiers using the extracted manifold. The Classifiers work in two stages, by first estimating the mode and then by estimating the key within the estimated mode. Finally we examine our method on The Beatles data set and demonstrate its improved performance compared to various existing methods.

Nonlinear Dynamic Projection for Noise Reduction of Dispersed Manifolds.

The search for a low-dimensional structure in high-dimensional data is one of the fundamental tasks in machine learning and pattern recognition. Manifold learning algorithms have recently emerged as alternatives to traditional linear dimension reduction techniques. In this paper, we propose a novel projection method that can be combined with any manifold learning methods to improve their dimension reduction performance when applied to high-dimensional data with a high level of noise. The method first builds a dispersion function that describes the distribution of dispersed manifold where the data lie. It then projects the noisy data onto a region wrapping the true manifold sufficiently close to it by applying a dynamical projection system associated with the constructed dispersion function. The effectiveness of the proposed projection method is validated by applying it to some real-world data sets with promising results.

Efficient simplicial reconstructions of manifolds from their samples

An algorithm for manifold learning is presented. Given only samples of a finite-dimensional differentiable manifold and no a priori knowledge of the manifold's geometry or topology except for its dimension, the goal is to find a description of the manifold. The learned manifold must approximate the true manifold well, both geometrically and topologically, when the sampling density is sufficiently high. The proposed algorithm constructs a simplicial complex based on approximations to the tangent bundle of the manifold. An important property of the algorithm is that its complexity depends on the dimension of the manifold, rather than that of the embedding space. Successful examples are presented in the cases of learning curves in the plane, curves in space, and surfaces in space; in addition, a case when the algorithm fails is analyzed.

Computing inertial manifolds

This paper discusses two numerical schemes that can be used to approximate inertial manifolds whose existence is given by one of the standard methods of proof. The methods considered are fully numerical, in that they take into account the need to interpolate the approximations of the manifold between a set of discrete gridpoints. As all the discretisations are refined the approximations are shown to converge to the true manifold.