Principal Manifolds and Graphs in Practice: from Molecular Biology to Dynamical Systems

@article{Gorban2010PrincipalMA,
  title={Principal Manifolds and Graphs in Practice: from Molecular Biology to Dynamical Systems},
  author={Alexander N. Gorban and A. Zinovyev},
  journal={International journal of neural systems},
  year={2010},
  volume={20 3},
  pages={
          219-32
        }
}
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… Expand
Dealing with complexity of biological systems: from data to models
TLDR
The fourth chapter summarizes the experience in studying cancer by computational methods (through combining integrative data analysis and mathematical modeling approaches) and describes the on-going research projects such as mathematical modeling of cell fate decisions and synthetic lethal interactions in DNA repair network. Expand
Overcoming Complexity of Biological Systems: from Data Analysis to Mathematical Modeling
The problem of dealing with complexity arises when we fail to achieve a desired behavior of biological systems (for example, in cancer treatment). In this review I formulate the problem of tacklingExpand
Data complexity measured by principal graphs
TLDR
A set of data complexity measures using universal approximators, principal cubic complexes, that generalize the notion of principal manifolds for datasets with non-trivial topologies are suggested. Expand
Minimum Spanning vs. Principal Trees for Structured Approximations of Multi-Dimensional Datasets
TLDR
A methodology to compare and benchmark these two graph-based data approximation approaches, as well as to define their hyperparameters, implemented in Python using the standard scikit-learn library which provides high speed and efficiency. Expand
Model reduction in chemical dynamics: slow invariant manifolds, singular perturbations, thermodynamic estimates, and analysis of reaction graph
The paper has two goals: It presents basic ideas, notions, and methods for reduction of reaction kinetics models: quasi-steady-state, quasi-equilibrium, slow invariant manifolds, and limiting steps.Expand
VISUAL ANALYSIS AND PROCESSING OF CLUSTERS STRUCTURES IN MULTIDIMENSIONAL DATASETS
TLDR
The article describes the results of elastic maps approach application to visual analysis of clusters for different multidimensional datasets including medical data. Expand
Geometrical Complexity of Data Approximators
TLDR
A measure of complexity (geometrical complexity) is proposed which is applicable to approximators of several types and which allows comparing data approximations of different types. Expand
Practical Approach to Construction of Internal Variables of Complex Self-organized Systems and Its Theoretical Foundation
We propose a method for characterizing the image—multidimensional projection—of complex, self-organising, system. The method is general and may be used for characterisation of any structured,Expand
Robust and scalable learning of data manifolds with complex topologies via ElPiGraph
TLDR
ElPiGraph is currently implemented in five programming languages and accompanied by a graphical user interface, which makes it a versatile tool to deal with complex data in various fields from molecular biology, where it can be used to infer pseudo-time trajectories from single-cell RNASeq, to astronomy, where its used to approximate complex structures in the distribution of galaxies. Expand
Graph-based Methods for Visualization and Clustering
The amount of data that we produce and consume is larger than it has been at any point in the history of mankind, and it keeps growing exponentially. All this information, gathered in overwhelmingExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 74 REFERENCES
Principal Graphs and Manifolds
In many physical, statistical, biological and other investigations it is desirable to approximate a system of points by objects of lower dimension and/or complexity. For this purpose, Karl PearsonExpand
Topological grammars for data approximation
TLDR
A method of topological grammars is proposed for multidimensional data approximation that factorization of the whole process onto one-dimensional transformations using minimization of quadratic energy functionals allows us to construct efficient algorithms. Expand
Elastic Maps and Nets for Approximating Principal Manifolds and Their Application to Microarray Data Visualization
TLDR
It is shown that the method of elastic maps outperforms linear PCA in terms of data approximation, representation of between-point distance structure, preservation of local point neighborhood and representing point classes in low-dimensional spaces. Expand
Elastic Principal Graphs and Manifolds and their Practical Applications
TLDR
An algorithm for fast construction of grid approximations of principal manifolds with given topology based on analogy of principal manifold and elastic membrane is proposed, which makes the algorithm very effective, especially for parallel implementations. Expand
Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes
TLDR
A new type of low-dimensional ``principal object'' is proposed: a principal cubic complex, a generalization of linear and non-linear principal manifolds and includes them as a particular case. Expand
Invariant Manifolds for Physical and Chemical Kinetics
Introduction.- The Source of Examples.- Invariance Equation in the Differential Form.- Film Extension of the Dynamics: Slowness as Stability.- Entropy, Quasi-Equilibrium and Projector Field.- NewtonExpand
Principal Manifolds for Data Visualization and Dimension Reduction
TLDR
The book starts with the quote of the classical Pearson definition of PCA and includes reviews of various methods: NLPCA, ICA, MDS, embedding and clustering algorithms, principal manifolds and SOM. Expand
ELASTIC PRINCIPAL MANIFOLDS AND THEIR PRACTICAL APPLICATIONS
Principal manifolds defined as lines or surfaces passing through “the middle” of the data distribution serve as useful objects for many practical applications. We propose a new algorithm for fastExpand
Robust simplifications of multiscale biochemical networks
TLDR
A systematic treatment of model reduction of multiscale biochemical networks that allows critical parameter identification and produces hierarchies of models and deals naturally with the presence of multiple time scales, which is a general property of systems biology models. Expand
Invariant grids for reaction kinetics
Abstract In this paper, we construct low-dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold (MIM). MIM is based onExpand
...
1
2
3
4
5
...