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In many modern applications, including analysis of gene expression and text documents, the data are noisy, high-dimensional, and unordered--with no particular meaning to the given order of the variables. Yet, successful learning is often…

Methodology · Statistics 2008-07-25 Ann B. Lee , Boaz Nadler , Larry Wasserman

Dimensionality reduction is an integral part of data visualization. It is a process that obtains a structure preserving low-dimensional representation of the high-dimensional data. Two common criteria can be used to achieve a dimensionality…

Computational Geometry · Computer Science 2018-06-25 Lin Yan , Yaodong Zhao , Paul Rosen , Carlos Scheidegger , Bei Wang

Merge trees are a type of topological descriptors that record the connectivity among the sublevel sets of scalar fields. They are among the most widely used topological tools in visualization. In this paper, we are interested in sketching a…

Computational Geometry · Computer Science 2021-06-01 Mingzhe Li , Sourabh Palande , Lin Yan , Bei Wang

Computational topology has recently known an important development toward data analysis, giving birth to the field of topological data analysis. Topological persistence, or persistent homology, appears as a fundamental tool in this field.…

Statistics Theory · Mathematics 2013-05-28 Frédéric Chazal , Marc Glisse , Catherine Labruère , Bertrand Michel

Topological structures such as the merge tree provide an abstract and succinct representation of scalar fields. They facilitate effective visualization and interactive exploration of feature-rich data. A merge tree captures the topology of…

Computational Geometry · Computer Science 2024-06-06 Raghavendra Sridharamurthy , Talha Bin Masood , Adhitya Kamakshidasan , Vijay Natarajan

Human subject studies that map-like visualizations are as good or better than standard node-link representations of graphs, in terms of task performance, memorization and recall of the underlying data, and engagement [SSKB14, SSKB15]. With…

Computational Geometry · Computer Science 2019-12-10 Felice De Luca , Iqbal Hossain , Kathryn Gray , Stephen Kobourov , Katy Börner

With the rapid adoption of machine learning techniques for large-scale applications in science and engineering comes the convergence of two grand challenges in visualization. First, the utilization of black box models (e.g., deep neural…

Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information,…

Computer Vision and Pattern Recognition · Computer Science 2018-02-19 Christoph Hofer , Roland Kwitt , Marc Niethammer , Andreas Uhl

This study introduces a new method of visualizing complex tree structured objects. The usefulness of this method is illustrated in the context of detecting unexpected features in a data set of very large trees. The major contribution is a…

Applications · Statistics 2012-02-14 Burcu Aydin , Gabor Pataki , Haonan Wang , Alim Ladha , Elizabeth Bullitt , J. S. Marron

Representation learning on graphs is a fundamental problem that can be crucial in various tasks. Graph neural networks, the dominant approach for graph representation learning, are limited in their representation power. Therefore, it can be…

Machine Learning · Computer Science 2025-01-17 Zuoyu Yan , Qi Zhao , Ze Ye , Tengfei Ma , Liangcai Gao , Zhi Tang , Yusu Wang , Chao Chen

We develop a language for describing the relationship among observations, mathematical models, and the underlying principles from which they are derived. Using Information Geometry, we consider geometric properties of statistical models for…

Data Analysis, Statistics and Probability · Physics 2016-07-14 Mark K. Transtrum , Gus Hart , Peng Qiu

Understanding and predicting thermal transport in disordered materials remains a significant challenge due to the absence of periodicity and the complex nature of medium-range structural motifs. In this work, we investigate amorphous…

Materials Science · Physics 2025-12-16 Kosuke Yamazaki , Takuma Shiga , Kumpei Shiraishi , Emi Minamitani

Tree-based methods are powerful nonparametric techniques in statistics and machine learning. However, their effectiveness, particularly in finite-sample settings, is not fully understood. Recent applications have revealed their surprising…

Statistics Theory · Mathematics 2024-10-04 Hengrui Luo , Meng Li

Topological data analysis uses tools from topology -- the mathematical area that studies shapes -- to create representations of data. In particular, in persistent homology, one studies one-parameter families of spaces associated with data,…

Machine Learning · Computer Science 2020-12-01 Guido Montúfar , Nina Otter , Yuguang Wang

Physical phenomena in science and engineering are frequently modeled using scalar fields. In scalar field topology, graph-based topological descriptors such as merge trees, contour trees, and Reeb graphs are commonly used to characterize…

Computational Geometry · Computer Science 2019-10-10 Lin Yan , Yusu Wang , Elizabeth Munch , Ellen Gasparovic , Bei Wang

In this work, we propose trait-based merge trees a generalization of merge trees to feature level sets, targeting the analysis of tensor field or general multi-variate data. For this, we employ the notion of traits defined in attribute…

Machine Learning · Computer Science 2023-08-21 Jochen Jankowai , Talha Bin Masood , Ingrid Hotz

In recent years, topological data analysis has been utilized for a wide range of problems to deal with high dimensional noisy data. While text representations are often high dimensional and noisy, there are only a few work on the…

Machine Learning · Computer Science 2020-04-21 Shafie Gholizadeh , Armin Seyeditabari , Wlodek Zadrozny

Comparative analysis of scalar fields is an important problem with various applications including feature-directed visualization and feature tracking in time-varying data. Comparing topological structures that are abstract and succinct…

Graphics · Computer Science 2024-06-06 Raghavendra Sridharamurthy , Vijay Natarajan

Topological Data Analysis (TDA) is a rising field of computational topology in which the topological structure of a data set can be observed by persistent homology. By considering a sequence of sublevel sets, one obtains a filtration that…

Methodology · Statistics 2020-03-17 Yu-Min Chung , William Cruse , Austin Lawson

The high performance of tree ensemble classifiers benefits from a large set of rules, which, in turn, makes the models hard to understand. To improve interpretability, existing methods extract a subset of rules for approximation using model…

Machine Learning · Computer Science 2025-01-03 Zhen Li , Weikai Yang , Jun Yuan , Jing Wu , Changjian Chen , Yao Ming , Fan Yang , Hui Zhang , Shixia Liu