Principal manifolds and graphs in practice: from molecular biology to dynamical systems

Abstract

We present several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach). These approaches are generalizations of the Kohonen's self-organizing maps, a class of artificial neural networks. On several examples we show advantages of using non-linear objects for data approximation in comparison to the linear ones. We propose four numerical criteria for comparing linear and non-linear mappings of datasets into the spaces of lower dimension. The examples are taken from comparative political science, from analysis of high-throughput data in molecular biology, from analysis of dynamical systems.

12 pages, 9 figures

References in corpus (3)

• Topological Grammars for Data Approximation
• Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization
• PCA Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes

Cited by in corpus (26)

• Blessing of dimensionality: mathematical foundations of the statistical physics of data
• Robust And Scalable Learning Of Complex Dataset Topologies Via Elpigraph
• Scikit-dimension: a Python package for intrinsic dimension estimation
• The unreasonable effectiveness of small neural ensembles in high-dimensional brain
• Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph
• High--Dimensional Brain in a High-Dimensional World: Blessing of Dimensionality
• Trajectories, bifurcations and pseudotime in large clinical datasets: applications to myocardial infarction and diabetes data
• Fractional norms and quasinorms do not help to overcome the curse of dimensionality
• Data complexity measured by principal graphs
• Testing of information condensation in a model reverberating spiking neural network
• General stochastic separation theorems with optimal bounds
• Piece-wise quadratic approximations of arbitrary error functions for fast and robust machine learning
• High-dimensional separability for one- and few-shot learning
• Computational diagnosis and risk evaluation for canine lymphoma
• Coloring Panchromatic Nighttime Satellite Images: Comparing the Performance of Several Machine Learning Methods
• Core collapse and horizontal-branch morphology in galactic globular clusters
• Geometrical complexity of data approximators
• Symphony of high-dimensional brain
• Minimal cover of high-dimensional chaotic attractors by embedded recurrent patterns
• Nonlinear Quality of Life Index
• Leveraging Interpolation Models and Error Bounds for Verifiable Scientific Machine Learning
• Exploring the impact of social stress on the adaptive dynamics of COVID-19: Typing the behavior of naïve populations faced with epidemics
• Dealing with complexity of biological systems: from data to models
• Is it possible to predict long-term success with k-NN? Case Study of four market indices (FTSE100, DAX, HANGSENG, NASDAQ)
• Rule-Mining based classification: a benchmark study
• DeDaL: Cytoscape 3.0 app for producing and morphing data-driven and structure-driven network layouts