scholarly journals Mixing in High-Dimensional Expanders

2017 ◽  
Vol 26 (5) ◽  
pp. 746-761 ◽  
Author(s):  
ORI PARZANCHEVSKI
Keyword(s):  
Arbitrary Dimension ◽  
Simplicial Complexes ◽  
Similar Type ◽  
High Dimensional ◽  
Hodge Laplacian ◽  
Laplace Spectrum ◽  
Open Questions

We establish a generalization of the Expander Mixing Lemma for arbitrary (finite) simplicial complexes. The original lemma states that concentration of the Laplace spectrum of a graph implies combinatorial expansion (which is also referred to asmixing, orpseudo-randomness). Recently, an analogue of this lemma was proved for simplicial complexes of arbitrary dimension, provided that the skeleton of the complex is complete. More precisely, it was shown that a concentrated spectrum of the simplicial Hodge Laplacian implies a similar type of pseudo-randomness as in graphs. In this paper we remove the assumption of a complete skeleton, showing that simultaneous concentration of the Laplace spectra in all dimensions implies pseudo-randomness in any complex. We discuss various applications and present some open questions.

scholarly journals Large random simplicial complexes, I

2016 ◽  
Vol 08 (03) ◽  
pp. 399-429 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Probability Measure ◽  
Simplicial Complex ◽  
Parameter Space ◽  
Simplicial Complexes ◽  
High Dimensional ◽  
Topological Properties ◽  
Convex Domains ◽  
Dimensional Parameter ◽  
Random Simplicial Complexes ◽  
Simply Connected

In this paper we introduce and develop the multi-parameter model of random simplicial complexes with randomness present in all dimensions. Various geometric and topological properties of such random simplicial complexes are characterised by convex domains in the high-dimensional parameter space (rather than by intervals, as in the usual one-parameter models). We find conditions under which a multi-parameter random simplicial complex is connected and simply connected. Besides, we give an intrinsic characterisation of the multi-parameter probability measure. We analyse links of simplexes and intersections of multi-parameter random simplicial complexes and show that they are also multi-parameter random simplicial complexes.

scholarly journals Swarm Intelligence Algorithms for Feature Selection: A Review

2018 ◽  
Vol 8 (9) ◽  
pp. 1521 ◽  
Author(s):  
Lucija Brezočnik ◽  
Iztok Fister ◽  
Vili Podgorelec
Keyword(s):  
Feature Selection ◽  
Swarm Intelligence ◽  
Optimization Problems ◽  
Relevant Information ◽  
High Dimensional ◽  
Comprehensive Literature Review ◽  
Open Questions ◽  
Common Application ◽  
High Dimensional Datasets ◽  
Taxonomic Categories

The increasingly rapid creation, sharing and exchange of information nowadays put researchers and data scientists ahead of a challenging task of data analysis and extracting relevant information out of data. To be able to learn from data, the dimensionality of the data should be reduced first. Feature selection (FS) can help to reduce the amount of data, but it is a very complex and computationally demanding task, especially in the case of high-dimensional datasets. Swarm intelligence (SI) has been proved as a technique which can solve NP-hard (Non-deterministic Polynomial time) computational problems. It is gaining popularity in solving different optimization problems and has been used successfully for FS in some applications. With the lack of comprehensive surveys in this field, it was our objective to fill the gap in coverage of SI algorithms for FS. We performed a comprehensive literature review of SI algorithms and provide a detailed overview of 64 different SI algorithms for FS, organized into eight major taxonomic categories. We propose a unified SI framework and use it to explain different approaches to FS. Different methods, techniques, and their settings are explained, which have been used for various FS aspects. The datasets used most frequently for the evaluation of SI algorithms for FS are presented, as well as the most common application areas. The guidelines on how to develop SI approaches for FS are provided to support researchers and analysts in their data mining tasks and endeavors while existing issues and open questions are being discussed. In this manner, using the proposed framework and the provided explanations, one should be able to design an SI approach to be used for a specific FS problem.

scholarly journals Directed Rooted Forests in Higher Dimension

10.37236/5819 ◽  
2016 ◽  
Vol 23 (4) ◽  
Author(s):  
Olivier Bernardi ◽  
Caroline J. Klivans
Keyword(s):  
Generating Function ◽  
Vector Fields ◽  
Arbitrary Dimension ◽  
Graph Laplacian ◽  
Connected Components ◽  
Higher Dimension ◽  
Cell Complexes ◽  
Open Questions ◽  
Higher Dimensional ◽  
Rooted Forest

For a graph $G$, the generating function of rooted forests, counted by the number of connected components, can be expressed in terms of the eigenvalues of the graph Laplacian. We generalize this result from graphs to cell complexes of arbitrary dimension. This requires generalizing the notion of rooted forest to higher dimension. We also introduce orientations of higher dimensional rooted trees and forests. These orientations are discrete vector fields which lead to open questions concerning expressing homological quantities combinatorially.

scholarly journals Interconnected High-Dimensional Landscapes of Epithelial-Mesenchymal Plasticity and Stemness

2021 ◽  
Author(s):  
Sarthak Sahoo ◽  
Atchuta Srinivas Duddu ◽  
Adrian Biddle ◽  
Mohit Kumar Jolly
Keyword(s):  
Decision Making ◽  
Stem Cells ◽  
Cancer Stem Cells ◽  
Therapy Resistance ◽  
High Dimensional ◽  
Self Renewal ◽  
Attrition Rates ◽  
Functional Changes ◽  
Open Questions ◽  
Proposed Model

Establishing macrometastases at distant organs is a highly challenging process for cancer cells, with extremely high attrition rates. A very small percentage of disseminated cells have the ability to dynamically adapt to their changing micro-environments through reversibly switching to another phenotype, aiding metastasis. Such plasticity can be exhibited along one or more axes – epithelial-mesenchymal plasticity (EMP) and cancer stem cells (CSCs) being the two most studied, and often tacitly assumed to be synonymous. Here, we review the emerging concepts related to EMP and CSCs across multiple cancers. Both processes are multi-dimensional in nature; for instance, EMP can be defined on morphological, molecular and functional changes, which may or may not be synchronized. Similarly, self-renewal, multi-lineage potential, and anoikis and/or therapy resistance may not all occur simultaneously in CSCs. Thus, arriving at rigorous functional definitions for both EMP and CSCs is crucial. These processes are dynamic, reversible, and semi-independent in nature; cells traverse the inter-connected high-dimensional EMP and CSC landscapes in diverse paths, each of which may exhibit a distinct EMP-CSC coupling. Our proposed model offers a potential unifying framework for elucidating the coupled decision-making along these dimensions and highlights a key set of open questions to be answered.

scholarly journals High Dimensional Hyperbolic Geometry of Complex Networks

Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 1861
Author(s):  
Weihua Yang ◽  
David Rideout
Keyword(s):  
Random Graph ◽  
Language Processing ◽  
Arbitrary Dimension ◽  
Named Entity Recognition ◽  
Hierarchical Structures ◽  
Entity Recognition ◽  
High Dimensional ◽  
Graph Models ◽  
Random Graph Models ◽  
Striking Success

High dimensional embeddings of graph data into hyperbolic space have recently been shown to have great value in encoding hierarchical structures, especially in the area of natural language processing, named entity recognition, and machine generation of ontologies. Given the striking success of these approaches, we extend the famous hyperbolic geometric random graph models of Krioukov et al. to arbitrary dimension, providing a detailed analysis of the degree distribution behavior of the model in an expanded portion of the parameter space, considering several regimes which have yet to be considered. Our analysis includes a study of the asymptotic correlations of degree in the network, revealing a non-trivial dependence on the dimension and power law exponent. These results pave the way to using hyperbolic geometric random graph models in high dimensional contexts, which may provide a new window into the internal states of network nodes, manifested only by their external interconnectivity.

scholarly journals Schur polynomials and matrix positivity preservers

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Alexander Belton ◽  
Dominique Guillot ◽  
Apoorva Khare ◽  
Mihai Putinar
Keyword(s):  
Variational Approach ◽  
Arbitrary Dimension ◽  
High Dimensional ◽  
Cell Type ◽  
Schur Polynomials ◽  
Fixed Dimension ◽  
Significant Interest ◽  
International Audience ◽  
Recent Attention

International audience A classical result by Schoenberg (1942) identifies all real-valued functions that preserve positive semidefi- niteness (psd) when applied entrywise to matrices of arbitrary dimension. Schoenberg's work has continued to attract significant interest, including renewed recent attention due to applications in high-dimensional statistics. However, despite a great deal of effort in the area, an effective characterization of entrywise functions preserving positivity in a fixed dimension remains elusive to date. As a first step, we characterize new classes of polynomials preserving pos- itivity in fixed dimension. The proof of our main result is representation theoretic, and employs Schur polynomials. An alternate, variational approach also leads to several interesting consequences including (a) a hitherto unexplored Schubert cell-type stratification of the cone of psd matrices, (b) new connections between generalized Rayleigh quo- tients of Hadamard powers and Schur polynomials, and (c) a description of the joint kernels of Hadamard powers.

scholarly journals Large random simplicial complexes, III the critical dimension

2017 ◽  
Vol 26 (02) ◽  
pp. 1740010 ◽  
Author(s):  
A. Costa ◽  
M. Farber
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
Betti Numbers ◽  
Critical Dimension ◽  
Simplicial Complexes ◽  
Topological Properties ◽  
Open Questions ◽  
Random Simplicial Complexes ◽  
Special Cases

In this paper, we study the notion of critical dimension of random simplicial complexes in the general multi-parameter model described in [Random simplicial complexes, preprint (2014), arXiv:1412.5805 ; Large random simplicial complexes, I, preprint (2015), arXiv:1503.06285 ; Large random simplical complexes, II, preprint (2015), arXiv:1509.04837 ]. This model includes as special cases the Linial–Meshulam–Wallach model [Homological connectivity of random 2-complexes, Combinatorica 26 (2006) 475–487; Homological connectivity of random [Formula: see text]-complexes, Random Struct. Alogrithms 34 (2009) 408–417.] as well as the clique complexes of random graphs. We characterize the concept of critical dimension in terms of various geometric and topological properties of random simplicial complexes such as their Betti numbers, the fundamental group, the size of minimal cycles and the degrees of simplexes. We mention in the text a few interesting open questions.

A GENERAL SMALL CANCELLATION THEORY

2000 ◽  
Vol 10 (01) ◽  
pp. 1-172 ◽  
Author(s):  
JONATHAN P. MCCAMMOND
Keyword(s):  
Simplicial Complexes ◽  
High Dimensional ◽  
Small Cancellation ◽  
Small Cancellation Theory

In this article, a generalized version of small cancellation theory is developed which is applicable to specific types of high-dimensional simplicial complexes. The usual results on small cancellation groups are then shown to hold in this new setting with only slight modifications. For example, arbitrary dimensional versions of the Poincaré construction and the Cayley complex are described.

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