There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

In recent years, spectral clustering has become one of the most popular modern clustering algorithms. It is simple to implement, can be solved efficiently by standard linear algebra software, and very often outperforms traditional clustering algorithms such as the k-means algorithm. On the first glance spectral clustering appears slightly mysterious, and it is not obvious to see why it works at all and what it really does. The goal of this tutorial is to give some intuition on those questions. We describe different graph Laplacians and their basic properties, present the most common spectral clustering algorithms, and derive those algorithms from scratch by several different approaches. Advantages and disadvantages of the different spectral clustering algorithms are discussed.

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A tutorial on spectral clustering

The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.

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Community detection in graphs

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Convex Analysisの二,三の進展について

Matrix Analysis

This book represents the first synthesis of the considerable body of new research into positive definite matrices. These matrices play the same role in noncommutative analysis as positive real numbers do in classical analysis. They have theoretical and computational uses across a broad spectrum of disciplines, including calculus, electrical engineering, statistics, physics, numerical analysis, quantum information theory, and geometry. Through detailed explanations and an authoritative and inspiring writing style, Rajendra Bhatia carefully develops general techniques that have wide applications in the study of such matrices. Bhatia introduces several key topics in functional analysis, operator theory, harmonic analysis, and differential geometry--all built around the central theme of positive definite matrices. He discusses positive and completely positive linear maps, and presents major theorems with simple and direct proofs. He examines matrix means and their applications, and shows how to use positive definite functions to derive operator inequalities that he and others proved in recent years. He guides the reader through the differential geometry of the manifold of positive definite matrices, and explains recent work on the geometric mean of several matrices. Positive Definite Matrices is an informative and useful reference book for mathematicians and other researchers and practitioners. The numerous exercises and notes at the end of each chapter also make it the ideal textbook for graduate-level courses.

Positive Definite Matrices

The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated. 1991 Mathematics Subject Classification 15A24, 47A10, 47A62, 47B47, 47B49, 65F15, 65F30.

How and Why to Solve the Operator Equation AX−XB = Y

Despite its short history, the use of Riemannian geometry in brain-computer interface (BCI) decoding is currently attracting increasing attention, due to accumulating documentation of its simplicity, accuracy, robustness and transfer learning capabilities, including the winning score obtained in five recent international predictive modeling BCI data competitions. The Riemannian framework is sharp from a mathematical perspective, yet in practice it is simple, both algorithmically and computationally. This allows the conception of online decoding machines suiting real-world operation in adverse conditions. We provide here a review on the use of Riemannian geometry for BCI and a primer on the classification frameworks based on it. While the theoretical research on Riemannian geometry is technical, our aim here is to show the appeal of the framework on an intuitive geometrical ground. In particular, we provide a rationale for its robustness and transfer learning capabilities and we elucidate the link between a simple Riemannian classifier and a state-of-the-art spatial filtering approach. We conclude by reporting details on the construction of data points to be manipulated in the Riemannian framework in the context of BCI and by providing links to available open-source Matlab and Python code libraries for designing BCI decoders.

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Rajendra Bhatia

Papers

Matrix Analysis

Positive Definite Matrices

How and Why to Solve the Operator Equation AX−XB = Y

Riemannian geometry for EEG-based brain-computer interfaces; a primer and a review

Riemannian geometry and matrix geometric means