# Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes

@article{Gorban2008BeyondTC, title={Beyond The Concept of Manifolds: Principal Trees, Metro Maps, and Elastic Cubic Complexes}, author={Alexander N. Gorban and Neil R. Sumner and Andrei Yu. Zinovyev}, journal={Scopus}, year={2008}, pages={219-237} }

Multidimensional data distributions can have complex topologies and variable local dimensions. To approximate complex data, we propose a new type of low-dimensional ``principal object'': a principal cubic complex. This complex is a generalization of linear and non-linear principal manifolds and includes them as a particular case. To construct such an object, we combine a method of topological grammars with the minimization of an elastic energy defined for its embedment into multidimensional… Expand

#### Figures and Topics from this paper

#### 17 Citations

Principal Graphs and Manifolds

- Mathematics
- 2010

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 Pearson… Expand

Principal Graphs and Manifolds

- Computer Science, Mathematics
- ArXiv
- 2008

This chapter gives a brief practical introduction into the methods of construction of general principal objects, i.e. objects embedded in the ‘middle’ of the multidimensional data set, using the family of expectation/maximisation algorithms with nearest generalisations. Expand

Robust and scalable learning of data manifolds with complex topologies via ElPiGraph

- Computer Science
- ArXiv
- 2018

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

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

- Computer Science, Medicine
- Int. J. Neural Syst.
- 2010

This work presents several applications of non-linear data modeling, using principal manifolds and principal graphs constructed using the metaphor of elasticity (elastic principal graph approach), and proposes four numerical criteria for comparing linear and non- linear mappings of datasets into the spaces of lower dimension. Expand

Principal Curve Algorithms for Partitioning High-Dimensional Data Spaces

- Mathematics, Computer Science
- IEEE Transactions on Neural Networks
- 2011

Comparisons presented in this paper confirm that the proposed PC-tree and PCR-tree approaches show a better performance than several other competing partitioning algorithms in terms of vector quantization error and nearest neighbor search and the proposed algorithms outperform competing linear methods in total average coverage which measures the nonlinear compactness of partitioning algorithm. Expand

Robust and Scalable Learning of Complex Intrinsic Dataset Geometry via ElPiGraph

- Medicine, Computer Science
- Entropy
- 2020

ElPiGraph exploits and further develops the concept of elastic energy, the topological graph grammar approach, and a gradient descent-like optimization of the graph topology, and is capable of approximating data point clouds via principal graph ensembles. Expand

Fast and user-friendly non-linear principal manifold learning by method of elastic maps

- Computer Science
- 2015 IEEE International Conference on Data Science and Advanced Analytics (DSAA)
- 2015

This work presents user-friendly implementation of the method of elastic maps in ViDaExpert software, a handy tool allowing to construct an interactive 3D-scene representing a table of data in multidimensional space and perform its quick and insightfull statistical analysis. Expand

Robust And Scalable Learning Of Complex Dataset Topologies Via Elpigraph

- Mathematics, Computer Science
- 2018

ElPiGraph is presented, a scalable and robust method for approximation of datasets with complex structures which does not require computing the complete data distance matrix or the data point neighbourhood graph and is capable of approximating complex topologies via principal graph ensembles that can be combined into a consensus principal graph. Expand

Lizard Brain: Tackling Locally Low-Dimensional Yet Globally Complex Organization of Multi-Dimensional Datasets

- Computer Science, Medicine
- Frontiers in Neurorobotics
- 2019

This work reviews modern machine learning approaches for extracting low-dimensional geometries from multi-dimensional data and their applications in various scientific fields. Expand

PCA and K-Means Decipher Genome

- Computer Science, Biology
- 2008

Students start with a fragment of genetic text of a bacterial genome and discover that the information in the genome is encoded by non-overlapping triplets, and learn how to find gene positions. Expand

#### References

SHOWING 1-10 OF 52 REFERENCES

Topological grammars for data approximation

- Mathematics, Computer Science
- Appl. Math. Lett.
- 2007

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

- Computer Science, Physics
- 2007

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 MANIFOLDS AND THEIR PRACTICAL APPLICATIONS

- 2004

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 fast… Expand

Elastic Principal Graphs and Manifolds and their Practical Applications

- Mathematics, Computer Science
- Computing
- 2005

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

Principal Curves

- 2007

Principal curves are smooth one-dimensional curves that pass through the middle of a p-dimensional data set, providing a nonlinear summary of the data. They are nonparametric, and their shape is… Expand

Self-organizing maps: ordering, convergence properties and energy functions

- Mathematics, Computer Science
- Biological Cybernetics
- 2004

It is proved that the learning dynamics cannot be described by a gradient descent on a single energy function, but may be described using a set of potential functions, one for each neuron, which are independently minimized following a stochastic gradient descent. Expand

Cubic complexes and finite type invariants

- Mathematics
- 2002

Cubic complexes appear in the theory of finite type invariants so often that one can ascribe them to basic notions of the theory. In this paper we begin the exposition of finite type invariants from… Expand

Self-organized formation of topologically correct feature maps

- Mathematics, Computer Science
- Biological Cybernetics
- 2004

In a simple network of adaptive physical elements which receives signals from a primary event space, the signal representations are automatically mapped onto a set of output responses in such a way that the responses acquire the same topological order as that of the primary events. Expand

Self-Organization as an Iterative Kernel Smoothing Process

- Computer Science, Mathematics
- Neural Computation
- 1995

A generalized self-organizing algorithm is proposed, where the kernel smoothing step is replaced with an arbitrary nonparametric regression method, which strengthens the algorithm's connection with the Principal Curve algorithm. Expand

Piecewise Linear Skeletonization Using Principal Curves

- Mathematics, Computer Science
- IEEE Trans. Pattern Anal. Mach. Intell.
- 2002

The results indicated that the proposed algorithm can find a smooth medial axis in the great majority of a wide variety of character templates and that it substantially improves the pixel-wise skeleton obtained by traditional thinning methods. Expand