Multidimensional Scaling — Dynamical Cascades

9/5/2014

Multidimensional Scaling — Dynamical Cascades | Milan Jovovic's space

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Multidimensional Scaling — Dynamical Cascades Posted on October 12, 2012

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P=NP — A mathematical connection to the God particle

Abstract: A data decomposition methodology, based on computation of renormalized synergies, is proposed for analysis and coding. It is a scale-space computing approach, employing covariance differentiability and the scale-space wave information propagation. It decomposes data in the structure suitable for scalable storage, transmission and computation. We propose it within the quantum information theory that lays down a new perspective to networked systems dynamics and computation. It proposes also our approach to NP completeness solving, pervasive in computational problems, in life sciences especially.

Introduction: A multi-scale method of polynomial complexity has been derived for robust data analysis, coding and control [1]. This computational method captures a physical model for the generation of the underlying data. Despite its non-linear and dynamical nature, it aims for the simplest explanation, coding and control. In particular, we are interested in its application to the networked systems dynamics and computation. A computable set of spatio-temporal events, organized in graph structure, is proposed as a coding scheme. The coupling parameter β quantifies the synergy exchange, while traversing the graph. We therefore denote this methodology as a scale-space computing.

Method: The renormalization methodology, based on our scale-space computing approach is used here in computation of spatial data synergies, in 2D. Mathematical foundation of the multidimensional scaling is written in [1]. Computation of the temporal sequence synergies from the brain wave recordings is reported in [2]. Computation is carried out by the scale-space wave information propagation. At the point of a ‘wave collapse’, Δβ+ = Δβ- = 0, the information is split in two quanta. On the other hand, the critical point of dynamical stability, Δβ+ = Δβ- ≠ 0, defines the condition of ‘wave resonance’, and the information resonates http://milanjovovic.wordpress.com/2012/10/12/multidimensional-scaling-dynamical-cascades/

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Multidimensional Scaling — Dynamical Cascades | Milan Jovovic's space

within a ‘nucleon’. Conditions of the static stability are defined by the positive definitiveness of the free energy for the quanta of information, and the synergy coupling within -1 ≤ β ≤ +1. Various interrelated phenomena arise, however, between the information clusters depending on positive/negative definitiveness of its correlation matrices.

Results: In Figure 1, three random 2D Gaussians are generated to describe the decomposition scheme. They are of the dispersion value σ=1, and separated at the center points dx,y =5, dx,y =3, and dx,y =2. When at the center points: (-2,0), (0,2) and (2,0), a significant overlap can be observed for the 30 points large Gaussian clusters. In this decomposition methodology, two scale-space waves resonate information within a ‘nucleon’. Their positive/negative envelopes are depicted in the color graphs, where green/red belongs to ‘forward’, and yellow/blue to ‘backward’ waves. In Figure 2, three Gaussians separated by dx,y =5 decompose in 4 quanta at β= 0.000462. In Figure 3, three Gaussians separated by dx,y =3 decompose in 4 quanta at β= 0.000282. When overlapping Gaussian distributions, separated by dx,y =2, its summary distribution decomposes in 4 quanta at β= -0.000120, shown in Figure 4.

Discussion: The renormalized synergies are computed by the means of covariance differentiability and the scale-space wave information propagation that exchange energies in a coupled system of binary oscillators. The mass conservation principle makes this a conservative system. No particular ordering relations of data is required. The system converges to invariant results, up to the border condition of the data array used. The results shown here are obtained by processing the data form an 1D array. This transformation method clusters data points depending on its distribution, without a priori specifying a final number of clusters. For a general distribution of data, it may not however converge within the limits of static stability, -1 ≤ β ≤ +1, for which the polynomial decomposition structure guaranties the polynomial solution to a NP hard problem – with a polynomial time search. Outside of this decomposition, the search remains chaotic, but a deterministic – without strange attractors due to the conservative system’s decomposition. We envisage, therefore, this theory derived as a core to different applications from computational physics to bioinformatics and cognitive psychology, in life sciences. On the engineering side, this methodology is suitable for scalable data storage, transmission and computation. References: 1. http://grids.ucs.indiana.edu/ptliupages/publications/Milan_report.pdf 2. Jovovic, M., Brain wave synergies, analysis and coding

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Multidimensional Scaling — Dynamical Cascades | Milan Jovovic's space

Figure 1: 3 2D Gaussian clusters are generated with data dispersion σ=1. They are shown here separated by dx,y =3, the ‘x’s showing the centers of the clusters.

Nucleon, d=5.

Figure 2: Three Gaussians separated by dx,y =5 decompose in 4 quanta at β= 0.000426.

http://milanjovovic.wordpress.com/2012/10/12/multidimensional-scaling-dynamical-cascades/

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Multidimensional Scaling — Dynamical Cascades | Milan Jovovic's space

Nucleon, d=3.

Figure 3: Three Gaussians separated by dx,y =3 decompose in 4 quanta at β= 0.000282.

Nucleon, d=2.

Figure 4: Three Gaussians separated by dx,y =2 overlap. They decompose in 4 quanta at β= -0.000120.

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Multidimensional Scaling — Dynamical Cascades | Milan Jovovic's space

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About Milan Jovovic I live my life to the fullest. How I do it? -I am working on it. (A long time exiled by the criminalized leadership of Montenegro). View all posts by Milan Jovovic →

This entry was posted in Language of Mathematics, Quantum Information, Research Statement and tagged complexity analysis, dimensional analysis, polynomial complexity, quantum information theory, research, science, Stochastic resonance, Travelling salesman problem. Bookmark the permalink.

3 Responses to Multidimensional Scaling — Dynamical Cascades kosiarki spalinowe says: March 26, 2013 at 9:39 pm

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Milan Jovovic says: March 29, 2013 at 6:01 pm

Not paid since exiled from academia here in Montenegro. Written though with an intention also to explain my research background to a potential employer, abroad. Reply

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Multidimensional Scaling — Dynamical Cascades | Milan Jovovic's space

are too quick for newbies. May just you please lengthen them a little from next time? Thanks for the post. Reply

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