scholarly journals Phase transition for the volume of high‐dimensional random polytopes

2020 ◽  
Author(s):  
Gilles Bonnet ◽  
Zakhar Kabluchko ◽  
Nicola Turchi
Keyword(s):  
Phase Transition ◽  
High Dimensional ◽  
Random Polytopes

UNIVERSALITY OF FIRST AND SECOND ORDER PHASE TRANSITION IN SOLAR ACTIVITY: EVIDENCE FOR NONEXTENSIVE TSALLIS STATISTICS

2012 ◽  
Vol 22 (09) ◽  
pp. 1250209 ◽  
Author(s):  
L. P. KARAKATSANIS ◽  
G. P. PAVLOS ◽  
D. S. SFIRIS
Keyword(s):  
Phase Transition ◽  
Solar Flare ◽  
Transition Process ◽  
High Dimensional ◽  
Nonequilibrium Phase Transition ◽  
Intermittent Turbulence ◽  
Chaotic State ◽  
Phase Transition Process ◽  
Low Dimensional ◽  
Q Values

In this work, we present the coexistence of self-organized criticality (SOC) and low-dimensional chaos at solar activity with results obtained by using the intermittent turbulence theory, the nonextensive q-statistics of Tsallis as well as the singular value decomposition analysis. Particularly, we show the independent dynamics of sunspot system related to the convection zone of sun and the solar flare system related to the lower solar atmosphere. However, both systems reveal nonequilibrium phase transition process from a high-dimensional intermittent turbulence state with SOC profile to a low-dimensional and chaotic intermittent turbulence state. The high-dimensional SOC state in both dynamical systems underlying the sunspot and solar flare signal is related with low q-values and low Flatness values (F) while the low-dimensional chaotic state is related with higher q-values and Flatness F-values. The higher q- and F-values reveal strong character of long-range correlations corresponding to system-wide global process while the lower q- and F-values reveal scale invariant local avalanche process. Also, the high-dimensional SOC state corresponds to second order nonequilibrium critical phase transition process while the low-dimensional chaotic state corresponds to first order nonequilibrium phase transition process. Finally, for both dynamics underlying sunspot index and solar flare, at both states of phase transition process, the multiscale and multifractal character was found to exist but with different profile or strength.

scholarly journals Threshold phenomena for high-dimensional random polytopes

2019 ◽  
Vol 21 (05) ◽  
pp. 1850038 ◽  
Author(s):  
Gilles Bonnet ◽  
Giorgos Chasapis ◽  
Julian Grote ◽  
Daniel Temesvari ◽  
Nicola Turchi
Keyword(s):  
Space Dimension ◽  
Random Point ◽  
High Dimensional ◽  
Convex Hulls ◽  
Point Sets ◽  
Random Polytopes ◽  
Intrinsic Volumes ◽  
Dimension Formula ◽  
Threshold Phenomena ◽  
Random Point Sets

Let [Formula: see text], [Formula: see text] be independent random points in [Formula: see text], distributed according to the so-called beta or beta-prime distribution, respectively. We establish threshold phenomena for the volume, intrinsic volumes, or more general measures of the convex hulls of these random point sets, as the space dimension [Formula: see text] tends to infinity. The dual setting of polytopes generated by random halfspaces is also investigated.

scholarly journals Universality laws for randomized dimension reduction, with applications

2017 ◽  
Vol 7 (3) ◽  
pp. 337-446 ◽  
Author(s):  
Samet Oymak ◽  
Joel A Tropp
Keyword(s):  
Phase Transition ◽  
Dimension Reduction ◽  
Large Class ◽  
Linear Codes ◽  
Applied Mathematics ◽  
Numerical Linear Algebra ◽  
Estimation Methods ◽  
High Dimensional ◽  
Embedding Dimension ◽  
Data Set

Abstract Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This article studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the minimum RSV. These results can be viewed as new universality laws in high-dimensional stochastic geometry. Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs.

scholarly journals A Data-Driven Dimensionality Reduction Approach to Compare and Classify Lipid Force Fields

2021 ◽  
Author(s):  
Riccardo Capelli ◽  
Andrea Gardin ◽  
Charly Empereur-mot ◽  
Giovanni Doni ◽  
Giovanni M. Pavan
Keyword(s):  
Phase Transition ◽  
Force Fields ◽  
Coarse Grained ◽  
High Dimensional ◽  
Crucial Point ◽  
Structure And Dynamics ◽  
Intrinsic Resolution ◽  
Explicit Solvent ◽  
Gel Phase ◽  
Dynamics Simulations

<div><div><div><p>Molecular dynamics simulations of all-atom and coarse-grained lipid bilayer models are increasingly used to obtain insights useful for understanding the structural dynamics of these assemblies. In this context, one crucial point concerns the comparison of the performance and accuracy of classical force fields (FFs), which sometimes remains elusive. To date, the assessments performed on different classical potentials are mostly based on the comparison with experimental observables, which typically regard average properties. However, local differences of structure and dynamics, which are poorly captured by average measurements, can make a difference, but these are non-trivial to catch. Here we propose an agnostic way to compare different FFs at different resolutions (atomistic, united-atom, and coarse-grained), by means of a high-dimensional similarity metrics built on the framework of Smooth Overlap of Atomic Positions (SOAP). We compare and classify a set of 13 force fields, modeling 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC) bilayers. Our SOAP kernels-based metrics allows us to compare, discriminate and correlate different force fields at different model resolutions in an unbiased, high-dimensional way. This also captures differences between FFs in modeling non-average events (originating from local transitions), such as for example the liquid-to-gel phase transition in dipalmitoylphosphatidylcholine (DPPC) bilayers, for which our metrics allows to identify nucleation centers for the phase transition, highlighting some intrinsic resolution limitations in implicit vs. explicit solvent force fields.</p></div></div></div>

scholarly journals DETECTION OF WEAK SIGNALS IN HIGH-DIMENSIONAL COMPLEX-VALUED DATA

2014 ◽  
Vol 03 (01) ◽  
pp. 1450001 ◽  
Author(s):  
ALEXEI ONATSKI
Keyword(s):  
Phase Transition ◽  
Hypergeometric Function ◽  
High Dimensional ◽  
Difficult Case ◽  
Weak Signals ◽  
Largest Eigenvalue ◽  
Sample Covariance ◽  
Asymptotic Probability ◽  
Complex Valued ◽  
The Largest Eigenvalue

This paper considers the problem of detecting a few signals in high-dimensional complex-valued Gaussian data satisfying Johnstone's [On the distribution of the largest eigenvalue in principal components analysis, Ann. Statist.29 (2001) 295–327] spiked covariance model. We focus on the difficult case where signals are weak in the sense that the sizes of the corresponding covariance spikes are below the phase transition threshold studied in Baik et al. [Phase transition of the largest eigenvalue for non-null complex sample covariance matrices, Ann. Probab.33 (2005) 1643–1697]. In contrast to the majority of previous studies, we base the signal detection on the information contained in all the eigenvalues of the sample covariance matrix, as opposed to a few of the largest ones. This allows us to detect weak signals with non-trivial asymptotic probability when the dimensionality of the data and the number of observations go to infinity proportionally. We derive a simple analytical expression for the maximal possible asymptotic probability of correct detection holding the asymptotic probability of false detection fixed. To accomplish this derivation, we establish a novel representation for the hypergeometric function [Formula: see text] of two p × p matrix arguments, one of which has a deficient rank r < p, as a repeated contour integral of the hypergeometric function [Formula: see text] of two r × r matrix arguments.

scholarly journals Aspects of a Phase Transition in High-Dimensional Random Geometry

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 805
Author(s):  
Axel Prüser ◽  
Imre Kondor ◽  
Andreas Engel
Keyword(s):  
Phase Transition ◽  
Financial Risk ◽  
Risk Measures ◽  
Risk Measure ◽  
Linear Equations ◽  
Minimax Problem ◽  
Random Coefficients ◽  
High Dimensional ◽  
Random Geometry ◽  
Competition For Resources

A phase transition in high-dimensional random geometry is analyzed as it arises in a variety of problems. A prominent example is the feasibility of a minimax problem that represents the extremal case of a class of financial risk measures, among them the current regulatory market risk measure Expected Shortfall. Others include portfolio optimization with a ban on short-selling, the storage capacity of the perceptron, the solvability of a set of linear equations with random coefficients, and competition for resources in an ecological system. These examples shed light on various aspects of the underlying geometric phase transition, create links between problems belonging to seemingly distant fields, and offer the possibility for further ramifications.

scholarly journals Phase Transitions in Transfer Learning for High-Dimensional Perceptrons

Entropy ◽  
2021 ◽  
Vol 23 (4) ◽  
pp. 400
Author(s):  
Oussama Dhifallah ◽  
Yue M. Lu
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Transfer Learning ◽  
Negative Transfer ◽  
High Dimensional ◽  
Source Information ◽  
Generalization Performance ◽  
Target Task ◽  
Learning Tasks ◽  
Perceptron Learning

Transfer learning seeks to improve the generalization performance of a target task by exploiting the knowledge learned from a related source task. Central questions include deciding what information one should transfer and when transfer can be beneficial. The latter question is related to the so-called negative transfer phenomenon, where the transferred source information actually reduces the generalization performance of the target task. This happens when the two tasks are sufficiently dissimilar. In this paper, we present a theoretical analysis of transfer learning by studying a pair of related perceptron learning tasks. Despite the simplicity of our model, it reproduces several key phenomena observed in practice. Specifically, our asymptotic analysis reveals a phase transition from negative transfer to positive transfer as the similarity of the two tasks moves past a well-defined threshold.

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