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CONJUGATIONS OF UNITARY OPERATORS, I

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arXiv 2024年

作者： Mashreghi, Javad Ptak, Marek Ross, William T. Département de Mathématiques et de Statistique Université Laval QuébecQCG1K 0A6 Canada Department of Applied Mathematics University of Agriculture Ul. Balicka 253C Kraków30-198 Poland Department of Mathematics and Computer Science University of Richmond RichmondVA23173 United States

If U is a unitary operator on a separable complex Hilbert space H, an application of the spectral theorem says there is a conjugation C on H (an antilinear, involutive, isometry on H) for which CUC = U∗. In this paper, we fix a unitary operator U and describe all of the conjugations C which satisfy this property. As a consequence of our results, we show that a subspace is hyperinvariant for U if and only if it is invariant for any conjugation C for which CUC = U∗MSC Codes 47A465 © 2024, CC BY.

关键词： C (programming language)

Conjugations of Unitary Operators, II

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arXiv 2024年

作者： Mashreghi, Javad Ptak, Marek Ross, William T. Département dee Mathématiques et de Statistique Université Laval QuébecQCG1K 0A6 Canada Department of Applied Mathematics University of Agriculture UL. Balicka 253C Kraków30-198 Poland Department of Mathematics and Computer Science University of Richmond RichmondVA23173 United States

For a given unitary operator U on a separable complex Hilbert space H, we describe the set Cc(U) of all conjugations C (antilinear, isometric, and involutive maps) on H for which CUC = U. As this set might be empty, w... 详细信息

关键词： Hilbert spaces

Optimal Polynomial Approximation to Rational Matrix Functions Using the Arnoldi Algorithm

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arXiv 2023年

作者： Chen, Tyler Greenbaum, Anne Wellen, Natalie NYU Courant Department of Mathematics 251 Mercer St. New YorkNY10003 United States NYU Tandon Department of Computer Science and Engineering 370 Jay St. New YorkNY11201 United States University of Washington Applied Math Dept. Box 353925 SeattleWA98195 United States

Given an n by n matrix A and an n-vector b, along with a rational function R(z):= D(z)−1N (z), we show how to find the optimal approximation to R(A)b from the Krylov space, span(b, Ab, . . ., Ak−1b), using the basis vectors produced by the Arnoldi algorithm. To find this optimal approximation requires running max{deg(D), deg(N)} − 1 extra Arnoldi steps and solving a k + max{deg(D), deg(N)} by k least squares problem. Here optimal is taken to mean optimal in the D(A)∗D(A)-norm. Similar to the case for linear systems, we show that eigenvalues alone cannot provide information about the convergence behavior of this algorithm and we discuss other possible error bounds for highly nonnormal *** Codes 65F10 (Primary), 65F60, 47A12, 47A25 (Secondary) © 2023, CC BY.

关键词： Rational functions

Which Activation Function Works Best for Training Artificial Pancreas: Empirical Fact and Its Theoretical Explanation

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Which Activation Function Works Best for Training Artificial...

2023 IEEE Symposium Series on Computational Intelligence, SSCI 2023

作者： Denes-Fazakas, Lehel Szilagyi, Laszlo Eigner, Gyorgy Kosheleva, Olga Ceberio, Martine Kreinovich, Vladik Doctoral School Óbuda University Physiological Controls Research Center Appl. Informatics & Appl. Math. Budapest Hungary Sapientia University Computational Intelligence Res. Group Tg. Mures Romania Physiological Controls Res. Center Óbuda University Budapest Hungary Biomatics and Applied Ai Institute Obuda University Physiological Controls Res. Center John von Neumann Faculty of Informatics Budapest Hungary University of Texas at El Paso Department of Teacher Education El PasoTX United States University of Texas at El Paso Department of Computer Science El PasoTX United States

ISBN: (纸本)9781665430654

One of the most effective ways to help patients at the dangerous levels of diabetes is an artificial pancreas, a device that constantly monitors the patient's blood sugar level and injects insulin based on this level. Patient's reaction to insulin is highly individualized, so the artificial pancreas needs to be trained on each patient. It turns out that the best training results are attained when instead of the usual ReLU neurons, we use their minor modification known as Exponential Linear Units (ELU). In this paper, we provide a theoretical explanation for the empirically observed effectiveness of ELUs. © 2023 IEEE.

关键词： artificial pancreas diabetes Exponential Linear Unit (ELU) Leaky ReLU neural network Rectified Linear Unit (ReLU)

Beyond circular-arc graphs - recognizing lollipop graphs and medusa graphs

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arXiv 2022年

作者： Çağirici, Deniz Ağaoğlu Çağirici, Onur Derbisz, Jan Hartmann, Tim A. Hliněný, Petr Kratochvíl, Jan Krawczyk, Tomasz Zeman, Peter Masaryk University Brno Czech Republic Toronto Metropolitan University Toronto Canada Theoretical Computer Science Department Faculty of Mathematics and Computer Science Jagiellonian University Kraków Poland RWTH Aachen University Aachen Germany Department of Applied Mathematics Faculty of Mathematics and Physics Charles University Prague Czech Republic Institut de Mathématiques University of Neuchâtel Switzerland

In 1992 Biró, Hujter and Tuza introduced, for every fixed connected graph H, the class of H-graphs, defined as the intersection graphs of connected subgraphs of some subdivision of H. Such classes of graphs are related to many known graph classes: for example, K2-graphs correspond to interval graphs, K3-graphs correspond to circular-arc graphs, the union of T-graphs, where T ranges over all trees, correspond to chordal graphs. Recently, quite a lot of research has been devoted to understanding the tractability border for various computational problems, such as recognition or isomorphism testing, in classes of H-graphs for different graphs H. In this work we undertake this research topic, focusing on the recognition problem. Chaplick, Töpfer, Voborník, and Zeman showed an XP-algorithm testing whether a given graph is a T-graph, where the parameter is the size of the tree T. In particular, for every fixed tree T the recognition of T-graphs can be solved in polynomial time. Tucker showed a polynomial time algorithm recognizing K3-graphs (circular-arc graphs). On the other hand, Chaplick at al. showed also that recognition of H-graphs is NP-hard if H contains two different cycles sharing an edge. The main two results of this work narrow the gap between the NP-hard and P cases of H-graphs recognition. First, we show that recognition of H-graphs is NP-hard when H contains two different cycles. On the other hand, we show a polynomial-time algorithm recognizing L-graphs, where L is a graph containing a cycle and an edge attached to it (L is called a lollipop and L-graphs are called lollipop graphs). Clearly, the class of lollipop graphs extends the class of circular-arc graphs. Our work leaves open the recognition problems of M-graphs for every unicyclic graph M different from a cycle and a lollipop. Other results of this work, which shed some light on the cases that remain open, are as follows: • recognition of M-graphs, where M is a fixed unicyclic graph, admits a polynom

关键词： Polynomial approximation

A new trigonometric kernel function for support vector machine

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arXiv 2022年

作者： Hafshejani, Sajad Fathi Moaberfard, Zahra Department of Applied Mathematics Shiraz University of Technology Modarres Boulevard Fars Shiraz71557-13876 Iran Department of Math and Computer Science University of Lethbridge 4401 University Dr W LethbridgeABT1K 3M4 Canada

In the last few years, various types of machine learning algorithms, such as Support Vector Machine (SVM), Support Vector Regression (SVR), and Non-negative Matrix Factorization (NMF) have been introduced. The kernel approach is an effective method for increasing the classification accuracy of machine learning algorithms. This paper introduces a family of one-parameter kernel functions for improving the accuracy of SVM classification. The proposed kernel function consists of a trigonometric term and differs from all existing kernel functions. We show this function is a positive definite kernel function. Finally, we evaluate the SVM method based on the new trigonometric kernel, the Gaussian kernel, the polynomial kernel, and a convex combination of the new kernel function and the Gaussian kernel function on various types of datasets. Empirical results show that the SVM based on the new trigonometric kernel function and the mixed kernel function achieve the best classification accuracy. Moreover, some numerical results of performing the SVR based on the new trigonometric kernel function and the mixed kernel function are presented. Copyright © 2022, The Authors. All rights reserved.

关键词： Support vector machines

FURSTENBERG SYSTEMS OF PRETENTIOUS AND MRT MULTIPLICATIVE FUNCTIONS

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arXiv 2023年

作者： Frantzikinakis, Nikos Lemańczyk, Mariusz de la Rue, Thierry University of Crete Department of Mathematics and Applied Mathematics Voutes University Campus Heraklion71003 Greece Faculty of Mathematics and Computer Science Nicolaus Copernicus University Toruń Poland Université de Rouen Normandie CNRS Laboratoire de Mathématiques Raphaël Salem Avenue de L’Université Saint Étienne du Rouvray76801 France

We prove structural results for measure preserving systems, called Furstenberg systems, naturally associated with bounded multiplicative functions. We show that for all pretentious multiplicative functions these systems always have rational discrete spectrum and, as a consequence, zero entropy. We obtain several other refined structural and spectral results, one consequence of which is that the Archimedean characters are the only pretentious multiplicative functions that have Furstenberg systems with trivial rational spectrum, another is that a pretentious multiplicative function has ergodic Furstenberg systems if and only if it pretends to be a Dirichlet character, and a last one is that for any fixed pretentious multiplicative function all its Furstenberg systems are isomorphic. We also study structural properties of Furstenberg systems of a class of multiplicative functions, introduced by Matomäki, Radziwill, and Tao, which lie in the intermediate zone between pretentiousness and strong aperiodicity. In a work of the last two authors and Gomilko, several examples of this class with exotic ergodic behavior were identified, and here we complement this study and discover some new unexpected phenomena. Lastly, we prove that Furstenberg systems of general bounded multiplicative functions have divisible spectrum. When these systems are obtained using logarithmic averages, we show that trivial rational spectrum implies a strong dilation invariance property, called strong stationarity, but, quite surprisingly, this property fails when the systems are obtained using Cesàro *** Codes Primary: 11N37, Secondary: 37A44, 11K65 © 2023, CC BY.

关键词： Rational functions

Solving the Families In the Wild Kinship Verification Challenge by Program Synthesis

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arXiv 2021年

作者： Huang, Junyi Strome, Maxwell Benjamin Jenkins, Ian Williams, Parker Feng, Bo Wang, Yaning Wang, Roman Bagri, Vaibhav Cheng, Newman Drori, Iddo Department of Mathematics Columbia University United States Department of Computer Science Columbia University United States Department of Applied Physics and Math Columbia University United States Department of Electrical Engineering Columbia University United States Department of Electrical Engineering and Computer Science MIT United States

Kinship verification is the task of determining whether a parent-child, sibling, or grandparent-grandchild relationship exists between two people and is important in social media applications, forensic investigations, finding missing children, and reuniting families. We demonstrate high quality kinship verification by participating in the 2021 Recognizing Families in the Wild challenge which provides the largest publicly available dataset in the field. Our approach is among the top 3 winning entries in the competition. We ensemble models written by both human experts and a foundation model, OpenAI Codex, trained on text and code. We use Codex to generate model variants, and also demonstrate its ability to generate entire running programs for kinship verification tasks of specific relationships. © 2021, CC BY.

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HiPoNet: A Topology-Preserving Multi-View Neural Network For High Dimensional Point Cloud and Single-Cell Data

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arXiv 2025年

作者： Viswanath, Siddharth Madhu, Hiren Bhaskar, Dhananjay Kovalic, Jake Johnson, Dave Ying, Rex Tape, Christopher Adelstein, Ian Perlmutter, Michael Krishnaswamy, Smita Department of Computer Science Yale University New Haven United States Department of Genetics Yale University New Haven United States Department of Applied Physics Yale University New Haven United States Program in Computing Boise State University Idaho United States Cell Communication Lab Department of Oncology University College London Cancer Institute London United Kingdom Department of Mathematics Yale University New Haven United States Department of Mathematics Boise State University Idaho United States Computational Biology and Bioinformatics Program Yale University New Haven United States Wu-Tsai Institute Yale University New Haven United States Program for Applied Math Yale University New Haven United States

In this paper, we propose HiPoNet, an end-to-end differentiable neural network for regression, classification, and representation learning on high-dimensional point clouds. Single-cell data can have high dimensionality exceeding the capabilities of existing methods for point clouds which are tailored for 3D data. Moreover, modern single-cell and spatial experiments now yield entire cohorts of datasets (i.e. one on every patient), necessitating models that can process large, high-dimensional point clouds at scale. Most current approaches build a single nearest-neighbor graph, discarding important geometric information. In contrast, HiPoNet forms higher-order simplicial complexes through learnable feature reweighting, generating multiple data views that disentangle distinct biological processes. It then employs simplicial wavelet transforms to extract multiscale features—capturing both local and global topology. We empirically show that these components preserve topological information in the learned representations, and that HiPoNet significantly outperforms state-of-the-art point-cloud and graph-based models on single cell. We also show an application of HiPoNet on spatial transcriptomics datasets using spatial co-ordinates as one of the views. Overall, HiPoNet offers a robust and scalable solution for high-dimensional data analysis. © 2025, CC BY.

关键词： Network topology