Principal Curves on Riemannian Manifolds

Probabilistic Principal Curves on Riemannian Manifolds

This paper studies a new curve-fitting approach to data on Riemannian manifolds. We define a principal curve based on a mixture model for observations and unobserved latent variables and propose a new algorithm to estimate the principal curve for given data points on Riemannian manifolds.

A Robust and Explainable Structure-Based Algorithm for Detecting the Organ Boundary From Ultrasound Multi-Datasets

Detecting the organ boundary in an ultrasound image is challenging because of the poor contrast of ultrasound images and the existence of imaging artifacts. In this study, we developed a coarse-to-refinement architecture for multi-organ ultrasound segmentation. First, we integrated the principal curve-based projection stage into an improved neutrosophic mean shift-based algorithm to acquire the data sequence, for which we utilized a limited amount of prior seed point information as the approximate initialization. Second, a distribution-based evolution technique was designed to aid in the identification of a suitable learning network. Then, utilizing the data sequence as the input of the learning network, we achieved the optimal learning network after learning network training. Finally, a scaled exponential linear unit-based interpretable mathematical model of the organ boundary was expressed via the parameters of a fraction-based learning network. The experimental outcomes indicated that our algorithm 1) achieved more satisfactory segmentation outcomes than state-of-the-art algorithms, with a Dice score coefficient value of 96.68 ± 2.2%, a Jaccard index value of 95.65 ± 2.16%, and an accuracy of 96.54 ± 1.82% and 2) discovered missing or blurry areas.

Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design

Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. Hyperbolic space is a homogeneous Riemannian manifold of the Lorentz group which is a semi-Riemannian manifold, i.e. a manifold equipped with an indefinite metric. Most existing methods (with some exceptions) use local linearization to define a variety of operations paralleling those used in traditional deep neural networks in Euclidean spaces. In this paper, we present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space. The novelty here lies in the projection which is designed to project data on to a lower-dimensional embedded hyperbolic space and hence leads to a nested hyperbolic space representation independently useful for dimensionality reduction. The main theoretical contribution is that the proposed embedding is proved to be isometric and equivariant under the Lorentz transformations, which are the natural isometric transformations in hyperbolic spaces. This projection is computationally efficient since it can be expressed by simple linear operations, and, due to the aforementioned equivariance property, it allows for weight sharing. The nested hyperbolic space representation is the core component of our network and therefore, we first compare this representation - independent of the network - with other dimensionality reduction methods such as tangent PCA, principal geodesic analysis (PGA) and HoroPCA. Based on this equivariant embedding, we develop a novel fully hyperbolic graph convolutional neural network architecture to learn the parameters of the projection. Finally, we present experiments demonstrating comparative performance of our network on several publicly available data sets.

Spherical Principal Curves

This paper presents a new approach for dimension reduction of data observed on spherical surfaces. Several dimension reduction techniques have been developed in recent years for non-euclidean data analysis. As a pioneer work, (Hauberg 2016) attempted to implement principal curves on Riemannian manifolds. However, this approach uses approximations to process data on Riemannian manifolds, resulting in distorted results. This study proposes a new approach to project data onto a continuous curve to construct principal curves on spherical surfaces. Our approach lies in the same line of (Hastie and Stuetzle et al. 1989) that proposed principal curves for data on euclidean space. We further investigate the stationarity of the proposed principal curves that satisfy the self-consistency on spherical surfaces. The results on the real data analysis and simulation examples show promising empirical characteristics of the proposed approach.

Principal Manifold Estimation via Model Complexity Selection

We propose a framework of principal manifolds to model high-dimensional data. This framework is based on Sobolev spaces and designed to model data of any intrinsic dimension. It includes principal component analysis and principal curve algorithm as special cases. We propose a novel method for model complexity selection to avoid overfitting, eliminate the effects of outliers, and improve the computation speed. Additionally, we propose a method for identifying the interiors of circle-like curves and cylinder/ball-like surfaces. The proposed approach is compared to existing methods by simulations and applied to estimate tumor surfaces and interiors in a lung cancer study.

Laplacian Eigenmaps Feature Conversion and Particle Swarm Optimization-Based Deep Neural Network for Machine Condition Monitoring

This work reports a novel method by fusing Laplacian Eigenmaps feature conversion and deep neural network (DNN) for machine condition assessment. Laplacian Eigenmaps is adopted to transform data features from original high dimension space to projected lower dimensional space, the DNN is optimized by the particle swarm optimization algorithm, and the machine run-to-failure experiment were investigated for validation studies. Through a series of comparative experiments with the original features, two other effective space transformation techniques, Principal Component Analysis (PCA) and Isometric map (Isomap), and two other artificial intelligence methods, hidden Markov model (HMM) as well as back-propagation neural network (BPNN), the present method in this paper proved to be more effective for machine operation condition assessment.