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  1. Abstract

    Topological data analysis (TDA) is a tool from data science and mathematics that is beginning to make waves in environmental science. In this work, we seek to provide an intuitive and understandable introduction to a tool from TDA that is particularly useful for the analysis of imagery, namely, persistent homology. We briefly discuss the theoretical background but focus primarily on understanding the output of this tool and discussing what information it can glean. To this end, we frame our discussion around a guiding example of classifying satellite images from the sugar, fish, flower, and gravel dataset produced for the study of mesoscale organization of clouds by Rasp et al. We demonstrate how persistent homology and its vectorization, persistence landscapes, can be used in a workflow with a simple machine learning algorithm to obtain good results, and we explore in detail how we can explain this behavior in terms of image-level features. One of the core strengths of persistent homology is how interpretable it can be, so throughout this paper we discuss not just the patterns we find but why those results are to be expected given what we know about the theory of persistent homology. Our goal is that readers of this paper will leave with a better understanding of TDA and persistent homology, will be able to identify problems and datasets of their own for which persistent homology could be helpful, and will gain an understanding of the results they obtain from applying the included GitHub example code.

    Significance Statement

    Information such as the geometric structure and texture of image data can greatly support the inference of the physical state of an observed Earth system, for example, in remote sensing to determine whether wildfires are active or to identify local climate zones. Persistent homology is a branch of topological data analysis that allows one to extract such information in an interpretable way—unlike black-box methods like deep neural networks. The purpose of this paper is to explain in an intuitive manner what persistent homology is and how researchers in environmental science can use it to create interpretable models. We demonstrate the approach to identify certain cloud patterns from satellite imagery and find that the resulting model is indeed interpretable.

     
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  3. Abstract

    We study tight projective 2‐designs in three different settings. In the complex setting, Zauner's conjecture predicts the existence of a tight projective 2‐design in every dimension. Pandey, Paulsen, Prakash, and Rahaman recently proposed an approach to make quantitative progress on this conjecture in terms of the entanglement breaking rank of a certain quantum channel. We show that this quantity is equal to the size of the smallest weighted projective 2‐design. Next, in the finite field setting, we introduce a notion of projective 2‐designs, we characterize when such projective 2‐designs are tight, and we provide a construction of such objects. Finally, in the quaternionic setting, we show that every tight projective 2‐design for determines an equi‐isoclinic tight fusion frame of subspaces of of dimension 3.

     
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  4. It is often of interest to identify a given number of points in projective space such that the minimum distance between any two points is as large as possible. Such configurations yield representations of data that are optimally robust to noise and erasures. The minimum distance of an optimal configuration not only depends on the number of points and the dimension of the projective space, but also on whether the space is real or complex. For decades, Neil Sloane’s online Table of Grassmannian Packings has been the go-to resource for putatively or provably optimal packings of points in real projective spaces. Using a variety of numerical algorithms, we have created a similar table for complex projective spaces. This paper surveys the relevant literature, explains some of the methods used to generate the table, presents some new putatively optimal packings, and invites the reader to competitively contribute improvements to this table. 
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