Dimensionality reduction and clustering are often used as preliminary steps for many complex machine learning tasks. The presence of noise and outliers can deteriorate the performance of such preprocessing and therefore impair the subsequent analysis tremendously. In manifold learning, several studies indicate solutions for removing background noise or noise close to the structure when the density is substantially higher than that exhibited by the noise. However, in many applications, including astronomical datasets, the density varies alongside manifolds that are buried in a noisy background. We propose a novel method to extract manifolds in the presence of noise based on the idea of Ant colony optimization. In contrast to the existing random walk solutions, our technique captures points that are locally aligned with major directions of the manifold. Moreover, we empirically show that the biologically inspired formulation of ant pheromone reinforces this behavior enabling it to recover multiple manifolds embedded in extremely noisy data clouds. The algorithm performance in comparison to state-of-the-art approaches for noise reduction in manifold detection and clustering is demonstrated, on several synthetic and real datasets, including an N-body simulation of a cosmological volume.

The intrinsic nature of noisy and complex data sets is often concealed in low-dimensional structures embedded in a higher dimensional space. Number of methodologies have been developed to extract and represent such structures in the form of manifolds (i.e. geometric structures that locally resemble continuously deformable intervals of Rj1). Usually a-priori knowledge of the manifold's intrinsic dimensionality is required. Additionally, their performance can often be hampered by the presence of a significant high-dimensional noise aligned along the low-dimensional core manifold. In real-world applications, the data can contain several low-dimensional structures of different dimensionalities. We propose a framework for dimensionality estimation and reconstruction of multiple noisy manifolds embedded in a noisy environment. To the best of our knowledge, this work represents the first attempt at detection and modelling of a set of coexisting general noisy manifolds by uniting two aspects of multi-manifold learning: the recovery and approximation of core noiseless manifolds and the construction of their probabilistic models. The easy-to-understand hyper-parameters can be manipulated to obtain an emerging picture of the multi-manifold structure of the data. We demonstrate the workings of the framework on two synthetic data sets, presenting challenging features for state-of-the-art techniques in Multi-Manifold learning. The first data set consists of multiple sampled noisy manifolds of different intrinsic dimensionalities, such as Möbius strip, toroid and spiral arm. The second one is a topologically complex set of three interlocked toroids. Given the absence of such unified methodologies in the literature, the comparison with existing techniques is organized along the two separate aspects of our approach mentioned above, namely manifold approximation and probabilistic modelling. The framework is then applied to a complex data set containing simulated gas volume particles from a particle simulation of a dwarf galaxy interacting with its host galaxy cluster. Detailed analysis of the recovered 1D and 2D manifolds can help us to understand the nature of Star Formation in such complex systems.

Topological data analysis tools enjoy increasing popularity in a wide range of applications, such as Computer graphics, Image analysis, Machine learning, and Astronomy for extracting information. However, due to computational complexity, processing large numbers of samples of higher dimensionality quickly becomes infeasible. This contribution is twofold: We present an efficient novel sub-sampling strategy inspired by Coulomb’s law to decrease the number of data points in d-dimensional point clouds while preserving its homology. The method is not only capable of reducing the memory and computation time needed for the construction of different types of simplicial complexes but also preserves the size of the voids in d-dimensions, which is crucial e.g. for astronomical applications. Furthermore, we propose a technique to construct a probabilistic description of the border of significant cycles and cavities inside the point cloud. We demonstrate and empirically compare the strategy in several synthetic scenarios and an astronomical particle simulation of a dwarf galaxy for the detection of superbubbles (supernova signatures).