A topological measurement of protein compressibility Gameiro, Marcio; Hiraoka, Yasuaki; Izumi, Shunsuke ...
Japan journal of industrial and applied mathematics,
03/2015, Letnik:
32, Številka:
1
Journal Article
CONFIDENCE SETS FOR PERSISTENCE DIAGRAMS Fasy, Brittany Terese; Lecci, Fabrizio; Rinaldo, Alessandro ...
The Annals of statistics,
12/2014, Letnik:
42, Številka:
6
Journal Article
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Persistent homology is a method for probing topological properties of point clouds and functions. The method involves tracking the birth and death of topological features (2000) as one varies a ...tuning parameter. Features with short lifetimes are informally considered to be "topological noise," and those with a long lifetime are considered to be "topological signal." In this paper, we bring some statistical ideas to persistent homology. In particular, we derive confidence sets that allow us to separate topological signal from topological noise.
Persistent homology is a topological feature used in a variety of applications such as generating features for data analysis and penalizing optimization problems. We develop an approach to accelerate ...persistent homology computations performed on many similar filtered topological spaces which is based on updating associated matrix factorizations. Our approach improves the update scheme of Cohen-Steiner, Edelsbrunner, and Morozov for permutations by additionally handling addition and deletion of cells in a filtered topological space and by processing changes in a single batch. We show that the complexity of our scheme scales with the number of elementary changes to the filtration which as a result is often less expensive than the full persistent homology computation. Finally, we perform computational experiments demonstrating practical speedups in several situations including feature generation and optimization guided by persistent homology.
The last decade saw an enormous boost in the field of computational topology: methods and concepts from algebraic and differential topology, formerly confined to the realm of pure mathematics, have ...demonstrated their utility in numerous areas such as computational biology personalised medicine, and time-dependent data analysis, to name a few. The newly-emerging domain comprising topology-based techniques is often referred to as topological data analysis (TDA). Next to their applications in the aforementioned areas, TDA methods have also proven to be effective in supporting, enhancing, and augmenting both classical machine learning and deep learning models. In this paper, we review the state of the art of a nascent field we refer to as “topological machine learning,” i.e., the successful symbiosis of topology-based methods and machine learning algorithms, such as deep neural networks. We identify common threads, current applications, and future challenges.
Knot diagrams are among the most common visual tools in topology. Computer programs now make it possible to draw, manipulate and render them digitally, which proves to be useful in knot theory ...teaching and research. Still, an openly available tool to manipulate knot diagrams in a real-time, interactive way is yet to be developed. We introduce a method of operating on the geometry of the knot diagram itself without any underlying three-dimensional structure that can underpin such an application. This allows us to directly interact with vector graphics knot diagrams while at the same time computing knot invariants in ways proposed by previous work. An implementation of this method is provided.
An early research in solid modeling led by Herbert Voelcker at the University of Rochester and later at Cornell suggested that every solid representation scheme corresponds to an algebra, where the ...elements of the algebra are solid representations constructed and edited using operations in the algebra. For example, every CSG representation describes an element in a finite Boolean algebra of closed regular sets, whereas every boundary representation describes an element of a vector space of 2-chains in an algebraic topological chain complex. In this paper, we elucidate the precise relationships (functors) between all algebras used for CSG and boundary representations of solids. Based on these properties, we show that many solid modeling operations, including boundary evaluation, reduce to straightforward algebraic operations or application of identified functors that are efficiently implemented using point membership tests and sparse matrix operations. To fully exploit the efficacy of the new algebraic approach to solid modeling, all algorithms are fully implemented in Julia, the modern language of choice for numerical and scientific computing.
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•New research direction in solid modeling.•Algebraic relationships between CSG and boundary representations of PL solids.•Arrangements are isomorphic to the finite Boolean algebra of regularized sets.•This algebra is isomorphic to a linear vector space of 3-chains.•Operations on CSG and b-reps reduce to composition of algebraic operations. operations.
With the increasing use of black-box Machine Learning (ML) techniques in critical applications, there is a growing demand for methods that can provide transparency and accountability for model ...predictions. As a result, a large number of local explainability methods for black-box models have been developed and popularized. However, machine learning explanations are still hard to evaluate and compare due to the high dimensionality, heterogeneous representations, varying scales, and stochastic nature of some of these methods. Topological Data Analysis (TDA) can be an effective method in this domain since it can be used to transform attributions into uniform graph representations, providing a common ground for comparison across different explanation methods. We present a novel topology-driven visual analytics tool, Mountaineer, that allows ML practitioners to interactively analyze and compare these representations by linking the topological graphs back to the original data distribution, model predictions, and feature attributions. Mountaineer facilitates rapid and iterative exploration of ML explanations, enabling experts to gain deeper insights into the explanation techniques, understand the underlying data distributions, and thus reach well-founded conclusions about model behavior. Furthermore, we demonstrate the utility of Mountaineer through two case studies using real-world data. In the first, we show how Mountaineer enabled us to compare black-box ML explanations and discern regions of and causes of disagreements between different explanations. In the second, we demonstrate how the tool can be used to compare and understand ML models themselves. Finally, we conducted interviews with three industry experts to help us evaluate our work.
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