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arXivComputing the Quadratic Numerical Range Birgit Jacob, Lukas Vorberg, Christian Wyss... more(3) 20230525 A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the random vector sampling method to produce a point exceeding a neighborhood of the expectation value in dependence on norm and size of the matrix is given.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.16079view Article Title Computing the Quadratic Numerical RangeAuthors Birgit Jacob; Lukas Vorberg; Christian WyssAbstract A novel algorithm for the computation of the quadratic numerical range is presented and exemplified yielding much better results in less time compared to the random vector sampling method. Furthermore, a bound on the probability for the random vector sampling method to produce a point exceeding a neighborhood of the expectation value in dependence on norm and size of the matrix is given.Is Part Of 20230525 Identifier ISSN: Category math.NA cs.NA math.SPLicense 
arXivA Tutorial on Holographic MIMO CommunicationsPart II: Performance Analysis and Holographic Beamforming Jiancheng An, Chau Yuen, Chongwen Huang, ... more(6) 20230525 As Part II of a threepart tutorial on holographic multipleinput multipleoutput (HMIMO), this Letter focuses on the stateoftheart in performance analysis and on holographic beamforming for HMIMO communications. We commence by discussing the spatial degrees of freedom (DoF) and ergodic capacity of a pointtopoint HMIMO system, based on the channel model presented in Part I. Additionally, we also consider the sumrate analysis of multiuser HMIMO systems. Moreover, we review the recent progress in holographic beamforming techniques developed for various HMIMO scenarios. Finally, we evaluate both the spatial DoF and the channel capacity through numerical simulations.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.15730view Article Title A Tutorial on Holographic MIMO CommunicationsPart II: Performance Analysis and Holographic BeamformingAuthors Jiancheng An; Chau Yuen; Chongwen Huang; Merouane Debbah; H. Vincent Poor; Lajos HanzoAbstract As Part II of a threepart tutorial on holographic multipleinput multipleoutput (HMIMO), this Letter focuses on the stateoftheart in performance analysis and on holographic beamforming for HMIMO communications. We commence by discussing the spatial degrees of freedom (DoF) and ergodic capacity of a pointtopoint HMIMO system, based on the channel model presented in Part I. Additionally, we also consider the sumrate analysis of multiuser HMIMO systems. Moreover, we review the recent progress in holographic beamforming techniques developed for various HMIMO scenarios. Finally, we evaluate both the spatial DoF and the channel capacity through numerical simulations.Is Part Of 20230525 Identifier ISSN: DOI 10.1109/LCOMM.2023.3278682Category cs.IT eess.SP math.ITLicense 
arXivGeneral Rational Solutions and Soliton Solutions of the Nonlocal Resonant Nonlinear Schrodinger Equations Bo Wei, Zhenyun Qin, Gui Mu... more(3) 20230525 General rational solutions for the nonlocal resonant nonlinear Schrodinger equations are derived by using the Hirota bilinear method and the KP hierarchy reduction method. These rational solutions are presented in terms of determinants in which the elements are algebraic expressions. A weaker condition is given for KP reduction in the nonlocal case. The dynamics of firstorder solutions are investigated in details. As a special case, we studied rantional solutions of the RNLS equation. Moreover, soliton solutions of the RNLS equation are given by using the Backlund transformation and nonlinear superposition formula.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.16012view Article Title General Rational Solutions and Soliton Solutions of the Nonlocal Resonant Nonlinear Schrodinger EquationsAuthors Bo Wei; Zhenyun Qin; Gui MuAbstract General rational solutions for the nonlocal resonant nonlinear Schrodinger equations are derived by using the Hirota bilinear method and the KP hierarchy reduction method. These rational solutions are presented in terms of determinants in which the elements are algebraic expressions. A weaker condition is given for KP reduction in the nonlocal case. The dynamics of firstorder solutions are investigated in details. As a special case, we studied rantional solutions of the RNLS equation. Moreover, soliton solutions of the RNLS equation are given by using the Backlund transformation and nonlinear superposition formula.Is Part Of 20230525 Identifier ISSN: Category nlin.SI mathph math.MPLicense 
arXivA BurtonMillertype boundary element method based on a hybrid integral representation and its application to cavity scattering Riku Toshimitsu, Hiroshi Isakari 20230525 This study builds on a recent paper by Lai et al [Appl. Comput. Harmon. Anal., 2018] in which a novel boundary integral formulation is presented for scalar wave scattering analysis in twodimensional layered and halfspaces. The seminal paper proposes a hybrid integral representation that combines the Sommerfeld integral and layer potential to efficiently deal with the boundaries of infinite length. In this work, we modify the integral formulation to eliminate the fictitious eigenvalues by employing BurtonMiller's approach. We also discuss reasonable parameter settings for the hybrid integral equation to ensure efficient and accurate numerical solutions. Furthermore, we extend the modified formulation for the scattering from a cavity in a halfspace whose boundary is locally perturbed. To address the cavity scattering, we introduce a virtual boundary enclosing the cavity and couple the integral equation on it with the hybrid equation. The effectiveness of the proposed method is demonstrated through numerical examples.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.15815view Article Title A BurtonMillertype boundary element method based on a hybrid integral representation and its application to cavity scatteringAuthors Riku Toshimitsu; Hiroshi IsakariAbstract This study builds on a recent paper by Lai et al [Appl. Comput. Harmon. Anal., 2018] in which a novel boundary integral formulation is presented for scalar wave scattering analysis in twodimensional layered and halfspaces. The seminal paper proposes a hybrid integral representation that combines the Sommerfeld integral and layer potential to efficiently deal with the boundaries of infinite length. In this work, we modify the integral formulation to eliminate the fictitious eigenvalues by employing BurtonMiller's approach. We also discuss reasonable parameter settings for the hybrid integral equation to ensure efficient and accurate numerical solutions. Furthermore, we extend the modified formulation for the scattering from a cavity in a halfspace whose boundary is locally perturbed. To address the cavity scattering, we introduce a virtual boundary enclosing the cavity and couple the integral equation on it with the hybrid equation. The effectiveness of the proposed method is demonstrated through numerical examples.Is Part Of 20230525 Identifier ISSN: Category math.NA cs.NALicense 
arXivStrange Random Topology of the Circle Uzu Lim 20230525 We characterise highdimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers growing arbitrarily large over shrinking intervals of filtration radii. Using the fact that the homotopy type of the random Cech complex is either an odddimensional sphere or a bouquet of evendimensional spheres, we give probabilistic bounds of the homotopy types. By departing from the conventional practice of scaling down filtration radii as the sample size grows large, our findings indicate that the full breadth of filtration radii leads to interesting systematic behaviour that cannot be regarded as "topological noise".
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.16270view Article Title Strange Random Topology of the CircleAuthors Uzu LimAbstract We characterise highdimensional topology that arises from a random Cech complex constructed on the circle. Expected Euler characteristic curve is computed, where we observe limiting spikes. The spikes correspond to expected Betti numbers growing arbitrarily large over shrinking intervals of filtration radii. Using the fact that the homotopy type of the random Cech complex is either an odddimensional sphere or a bouquet of evendimensional spheres, we give probabilistic bounds of the homotopy types. By departing from the conventional practice of scaling down filtration radii as the sample size grows large, our findings indicate that the full breadth of filtration radii leads to interesting systematic behaviour that cannot be regarded as "topological noise".Is Part Of 20230525 Identifier ISSN: Category math.PR math.ATLicense 
arXivAn approach to the study of boundary actions Jacopo Bassi 20230525 Given an action of a discrete countable group $G$ on a countable set $\mathfrak{X}$, it is studied the relationship between properties of the associated Calkin representation and the dynamics of the group action on the boundary of the Stone\v{C}ech compactification of $\mathfrak{X}$. The first section contains results about amenability properties of actions of discrete countable groups on nonseparable spaces and is of independent interest. In the second section these results are applied in order to translate regularity properties of the Calkin representation and the topological amenability on the Stone{\v C}ech boundary within the common framework of measurable dynamics on certain extensions of the Stone{\v C}ech boundary of $\mathfrak{X}$.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.16277view Article Title An approach to the study of boundary actionsAuthors Jacopo BassiAbstract Given an action of a discrete countable group $G$ on a countable set $\mathfrak{X}$, it is studied the relationship between properties of the associated Calkin representation and the dynamics of the group action on the boundary of the Stone\v{C}ech compactification of $\mathfrak{X}$. The first section contains results about amenability properties of actions of discrete countable groups on nonseparable spaces and is of independent interest. In the second section these results are applied in order to translate regularity properties of the Calkin representation and the topological amenability on the Stone{\v C}ech boundary within the common framework of measurable dynamics on certain extensions of the Stone{\v C}ech boundary of $\mathfrak{X}$.Is Part Of 20230525 Identifier ISSN: Category math.OA math.DS math.GRLicense 
arXivTheoretical Guarantees of Learning Ensembling Strategies with Applications to Time Series Forecasting Hilaf Hasson, Danielle C. Maddix, Yuyang Wang, ... more(5) 20230525 Ensembling is among the most popular tools in machine learning (ML) due to its effectiveness in minimizing variance and thus improving generalization. Most ensembling methods for blackbox base learners fall under the umbrella of "stacked generalization," namely training an ML algorithm that takes the inferences from the base learners as input. While stacking has been widely applied in practice, its theoretical properties are poorly understood. In this paper, we prove a novel result, showing that choosing the best stacked generalization from a (finite or finitedimensional) family of stacked generalizations based on crossvalidated performance does not perform "much worse" than the oracle best. Our result strengthens and significantly extends the results in Van der Laan et al. (2007). Inspired by the theoretical analysis, we further propose a particular family of stacked generalizations in the context of probabilistic forecasting, each one with a different sensitivity for how much the ensemble weights are allowed to vary across items, timestamps in the forecast horizon, and quantiles. Experimental results demonstrate the performance gain of the proposed method.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.15786view Article Title Theoretical Guarantees of Learning Ensembling Strategies with Applications to Time Series ForecastingAuthors Hilaf Hasson; Danielle C. Maddix; Yuyang Wang; Gaurav Gupta; Youngsuk ParkAbstract Ensembling is among the most popular tools in machine learning (ML) due to its effectiveness in minimizing variance and thus improving generalization. Most ensembling methods for blackbox base learners fall under the umbrella of "stacked generalization," namely training an ML algorithm that takes the inferences from the base learners as input. While stacking has been widely applied in practice, its theoretical properties are poorly understood. In this paper, we prove a novel result, showing that choosing the best stacked generalization from a (finite or finitedimensional) family of stacked generalizations based on crossvalidated performance does not perform "much worse" than the oracle best. Our result strengthens and significantly extends the results in Van der Laan et al. (2007). Inspired by the theoretical analysis, we further propose a particular family of stacked generalizations in the context of probabilistic forecasting, each one with a different sensitivity for how much the ensemble weights are allowed to vary across items, timestamps in the forecast horizon, and quantiles. Experimental results demonstrate the performance gain of the proposed method.Is Part Of 20230525 Identifier ISSN: Category cs.LG math.ST stat.ML stat.THLicense 
arXivInformation loss, mixing and emergent type III$_1$ factors Keiichiro Furuya, Nima Lashkari, Mudassir Moosa, ... more(4) 20230525 A manifestation of the black hole information loss problem is that the twopoint function of probe operators in a large Antide Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the twopoint function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time "mixing" and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double (KMS state), if mixing operators form an algebra (close under multiplication) the resulting algebra must be a von Neumann type III$_1$ factor. In other words, the physically intuitive requirement that all nonconserved operators should diffuse is so strong that it fixes the observable algebra to be an exotic algebra called a type III$_1$ factor. More generally, for an arbitrary outofequilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra it is a type III$_1$ von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the twopoint function of GFF clusters in time all operators are mixing, and the algebra is a type III$_1$ factor. For instance, in $\mathscr{N=4}$ SYM, above the HawkingPage phase transition, clustering of the single trace operators implies that the algebra is a type III$_1$ factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C$^*$algebra and von Neumann subalgebras of GFF associated with time bands and more generally, open sets of the bulk spacetime using the HKLL reconstruction map.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.16028view Article Title Information loss, mixing and emergent type III$_1$ factorsAuthors Keiichiro Furuya; Nima Lashkari; Mudassir Moosa; Shoy OusephAbstract A manifestation of the black hole information loss problem is that the twopoint function of probe operators in a large Antide Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the twopoint function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity. We call operators that cluster in time "mixing" and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double (KMS state), if mixing operators form an algebra (close under multiplication) the resulting algebra must be a von Neumann type III$_1$ factor. In other words, the physically intuitive requirement that all nonconserved operators should diffuse is so strong that it fixes the observable algebra to be an exotic algebra called a type III$_1$ factor. More generally, for an arbitrary outofequilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra it is a type III$_1$ von Neumann factor. In a theory of Generalized Free Fields (GFF), we show that if the twopoint function of GFF clusters in time all operators are mixing, and the algebra is a type III$_1$ factor. For instance, in $\mathscr{N=4}$ SYM, above the HawkingPage phase transition, clustering of the single trace operators implies that the algebra is a type III$_1$ factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C$^*$algebra and von Neumann subalgebras of GFF associated with time bands and more generally, open sets of the bulk spacetime using the HKLL reconstruction map.Is Part Of 20230525 Identifier ISSN: Category hepth mathph math.MP quantphLicense 
arXivFast Online Node Labeling for Very Large Graphs Baojian Zhou, Yifan Sun, Reza Babanezhad... more(3) 20230525 This paper studies the online node classification problem under a transductive learning setting. Current methods either invert a graph kernel matrix with $\mathcal{O}(n^3)$ runtime and $\mathcal{O}(n^2)$ space complexity or sample a large volume of random spanning trees, thus are difficult to scale to large graphs. In this work, we propose an improvement based on the \textit{online relaxation} technique introduced by a series of works (Rakhlin et al.,2012; Rakhlin and Sridharan, 2015; 2017). We first prove an effective regret $\mathcal{O}(\sqrt{n^{1+\gamma}})$ when suitable parameterized graph kernels are chosen, then propose an approximate algorithm FastONL enjoying $\mathcal{O}(k\sqrt{n^{1+\gamma}})$ regret based on this relaxation. The key of FastONL is a \textit{generalized local push} method that effectively approximates inverse matrix columns and applies to a series of popular kernels. Furthermore, the perprediction cost is $\mathcal{O}(\text{vol}({\mathcal{S}})\log 1/\epsilon)$ locally dependent on the graph with linear memory cost. Experiments show that our scalable method enjoys a better tradeoff between local and global consistency.
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.16257view Article Title Fast Online Node Labeling for Very Large GraphsAuthors Baojian Zhou; Yifan Sun; Reza BabanezhadAbstract This paper studies the online node classification problem under a transductive learning setting. Current methods either invert a graph kernel matrix with $\mathcal{O}(n^3)$ runtime and $\mathcal{O}(n^2)$ space complexity or sample a large volume of random spanning trees, thus are difficult to scale to large graphs. In this work, we propose an improvement based on the \textit{online relaxation} technique introduced by a series of works (Rakhlin et al.,2012; Rakhlin and Sridharan, 2015; 2017). We first prove an effective regret $\mathcal{O}(\sqrt{n^{1+\gamma}})$ when suitable parameterized graph kernels are chosen, then propose an approximate algorithm FastONL enjoying $\mathcal{O}(k\sqrt{n^{1+\gamma}})$ regret based on this relaxation. The key of FastONL is a \textit{generalized local push} method that effectively approximates inverse matrix columns and applies to a series of popular kernels. Furthermore, the perprediction cost is $\mathcal{O}(\text{vol}({\mathcal{S}})\log 1/\epsilon)$ locally dependent on the graph with linear memory cost. Experiments show that our scalable method enjoys a better tradeoff between local and global consistency.Is Part Of 20230525 Identifier ISSN: Category cs.LG cs.AI math.SPLicense 
arXivOn the WeisfeilerLeman dimension of permutation graphs Jin Guo, Alexander L. Gavrilyuk, Ilia Ponomarenko... more(3) 20230525 It is proved that the WeisfeilerLeman dimension of the class of permutation graphs is at most 18. Previously it was only known that this dimension is finite (Gru{\ss}ien, 2017).
Subject 수학 Source arXiv URL https://arxiv.org/abs/2305.15861view Article Title On the WeisfeilerLeman dimension of permutation graphsAuthors Jin Guo; Alexander L. Gavrilyuk; Ilia PonomarenkoAbstract It is proved that the WeisfeilerLeman dimension of the class of permutation graphs is at most 18. Previously it was only known that this dimension is finite (Gru{\ss}ien, 2017).Is Part Of 20230525 Identifier ISSN: Category math.CO cs.CC cs.DMLicense