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Linear Algebra in Data Science
This textbook explores applications of linear algebra in data science at an introductory level, showing readers how the two are deeply connected. The...
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Metric Algebraic Geometry
Metric algebraic geometry combines concepts from algebraic geometry and differential geometry. Building on classical foundations, it offers practical...
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Mathematical Optimization Theory and Operations Research 23rd International Conference, MOTOR 2024, Omsk, Russia, June 30–July 6, 2024, Proceedings
This book constitutes the refereed proceedings of the 23rd International Conference on Mathematical Optimization Theory and Operations Research,...
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Wasserstein Distance
A fundamental problem in metric algebraic geometry is distance minimization.
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Reach and Offset
In this chapter, we study the medial axis, bottlenecks, and offset hypersurfaces. These notions are intuitive and important for many applications....
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Critical Equations
The following optimization problem arises in many applications, and we shall revisit it again and again throughout this book.
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Condition Numbers
The concept of a condition number has its origin in numerical analysis. It measures how much the output value of a function we wish to evaluate can...
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Matrix Algebra
The need to understand matrix algebra in the context of underlying applications is paramount. Rank one projections are the building tools and they...
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Voronoi Cells
Every real algebraic variety X determines aVoronoi decomposition of its ambient Euclidean space...
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Computer Vision
The field of computer vision studies how computers can gain understanding from images and videos, similar to human cognitive abilities. One of the...
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Volumes of Semialgebraic Sets
In this chapter, we study the problem of computing the volume of a semialgebraic subset 𝑆 of...
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Polar Degrees
The notion of polar degrees is fundamental for assessing the algebraic complexity of polynomial optimization problems of a metric origin.We already...
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Haar Wavelets
An important example of an orthonormal basis is the Haar basis. We provide an introduction to Haar wavelets using the matrix algebra, involving...
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Frequency Filtering
Frequency filtering is a vast area in applied mathematics and electrical engineering. We introduce the foundations of Fourier analysis and touch on...
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Rotations and Quaternions
Rotations and quaternions are not a data science concept. However, they are an excellent excuse for examining the utility of matrix transformations....
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Convolution
Convolution and the resulting circulant matrices are fundamental tools used in signal processing and data analysis. It is a natural place for the use...
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Singular Value Decomposition
Singular value decomposition is a fundamental result in linear algebra that has far-reaching applications in applied mathematics, statistics, and...
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Projections
The concept of projection is a fundamental tool in data science. It is the idea behind data reduction and compression. We provide a careful...
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Some Wavelet Transforms
The Haar wavelet transform is a foundational technique. It is a first example of wavelet techniques to decompose data to its coarse and detail...
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Neural Networks
Neural networks and machine learning are forthcoming applications of linear algebra, real analysis, and advances in computing. We present the...