A survey on multidimensional persistence theory

Karagüler, Dilan
Persistence homology is one of the commonly used theoretical methods in topological data analysis to extract information from given data using algebraic topology. Converting data to a filtered object and analyzing the topological features of each space in the filtration, we will obtain a way of representing these features called the shape of data. This will give us invariants like barcodes or persistence diagrams for the data. These invariants are stable under small perturbations. In most applications, we need multiscaled analysis of data, which is done by multidimensional persistence. This thesis is a survey that contains how to produce invariants for multiscaled filtration obtained from data and the related stability results.


A survey on persistence landscape theory
Gürses, Selçuk; Pamuk, Semra; Department of Mathematics (2022-7)
Topological data analysis (TDA) consists of a growing collection of techniques that reveal the shape of data. These techniques may be especially useful for comprehend ing global features of high-dimensional data that are inaccessible via other methods. The usage of TDA has been constrained by the difficulties of merging the subject’s primary tool, the barcode or persistence diagram, with statistics and machine learning. The persistence landscape is a stable topological summary that is easily combinable with...
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Multi-modal data analysis methods often learn representations that align different modalities in a new common domain, while preserving the within-class compactness and within-modality geometry and enhancing the between-class separation. In this study, we present a theoretical performance analysis for multi-modal representation learning methods. We consider a quite general family of algorithms learning a nonlinear embedding of the data space into a new space via regular functions. We derive sufficient condit...
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Hansen, Mikkel B.; Sparta, Manuel; Seidler, Peter; Toffolı, Danıele; Christiansen, Ove (2010-01-01)
A new implementation of the vibrational self-consistent field (VSCF) method is presented on the basis of a second quantization formulation. A so-called active terms algorithm is shown to be a significant improvement over a standard implementation reducing the computational effort by one order in the number of degrees of freedom. Various types of screening provide even further reductions in computational scaling and absolute CPU time. VSCF calculations on large polyaromatic hydrocarbon model systems are pres...
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We present a Bayesian method for building scoring systems, which are linear models with coefficients that have very few significant digits. Usually the construction of scoring systems involve manual efforthumans invent the full scoring system without using data, or they choose how logistic regression coefficients should be scaled and rounded to produce a scoring system. These kinds of heuristics lead to suboptimal solutions. Our approach is different in that humans need only specify the prior over what the ...
A unified approach for the formulation of interaction problems by the boundary element method
Mengi, Y; Argeso, H (Wiley, 2006-04-30)
A unified formulation is presented, based on boundary element method, in a form suitable for performing the interaction analyses by substructure method for solid-solid and soil-structure problems. The proposed formulation permits the evaluation of all the elements of impedance and input motion matrices simultaneously at a single step in terms of system matrices of the boundary element method without solving any special problem, such as, unit displacement or load problem, as required in conventional methods....
Citation Formats
D. Karagüler, “A survey on multidimensional persistence theory,” M.S. - Master of Science, Middle East Technical University, 2021.