DOCSLIB.ORG

Singular Value Decomposition Learning on Double Stiefel Manifold

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Singular Value Decomposition Learning on Double Stiefel Manifold International Journal of Neural Systems, Vol. 13, No. 2 (2003) 1{16 c World Scientific Publishing Company SINGULAR VALUE DECOMPOSITION LEARNING ON DOUBLE STIEFEL MANIFOLD SIMONE FIORI Faculty of Engineering, Perugia University, Loc. Pentima bassa, 21, I-05100 Terni (Italy) [email protected] Received 25 October 2002 Revised 5 March 2003 Accepted 5 March 2003 The aim of this paper is to present a unifying view of four SVD-neural-computation techniques found in the scientific literature and to present some theoretical results on their behavior. The considered SVD neural algorithms are shown to arise as Riemannian-gradient flows on double Stiefel manifold and their geometric and dynamical properties are investigated with the help of differential geometry. Keywords: Singular value decomposition; orthogonal matrix group; Stiefel manifold; differential geometry; Lyapunov stability. 1. Introduction unsupervised learning by non-gradient techniques: For instance, in Ref. 1 a new technique was intro- The computation of the singular value decomposi- duced to enhance the learning capabilities of lin- tion (SVD) of a non-square matrix, also referred ear and MLP-type neural networks by Riemannian to as Autonne{Eckart{Young decomposition18;30 gradient, in Refs. 4 and 13 a theoretical deriva- plays a central role in several signal/data au- tion/analysis of new principal/minor subspace rules tomatic processing. Originally developed in nu- has been carried out; also, in Refs. 9 and 12 a large merical algebra to provide quantitative informa- class of learning rules for MLP-type neural networks, tion about the structure of linear systems of based on first/second-order non-gradient dynamics equations, it has found widespread applications and Lie-group flows, has been introduced and dis- 7{8;17;31 e.g. in signal processing, pattern recogni- cussed by the present Author as a theoretical frame- 26 24;30 tion and classification, automatic control, dig- work for explaining many learning paradigms ap- 28 ital circuit design, time-series prediction, image peared on the scientific literature, while papers10;11 6;21;25 2 processing and connectionism. were devoted to a particular algorithm of this class, Recently, some efforts have been devoted based on the rational kinematics of rigid bodies and to SVD computation by neural networks in its applications to real- and complex-valued signal 5;27;33 the neural community; the related learn- processing. ing theories emerge as interesting extensions The aim of this paper is to present some the- of the well-known neural principal compo- oretical notes on parallel SVD computation by 26 nent/subspace analysis techniques, long inves- unsupervised non-gradient neural learning, with tigated during the last 15 years. Also, recently special reference to learning theories involving a new light has been shed on adaptive second- weight-flows on double Stiefel manifold. Par- order (as well as higher-order) statistical de- allel techniques are considered in opposition to composition theories by researchers interested in sequential ones that employ the deflation method, 1 2 S. Fiori implemented by laterally connected neural architec- all-zero values except for the first r diagonal entries, tures, to discard previously computed vectors from termed singular values. It is easily checked that the original data.5;8 In particular, we recall from the sci- columns of U coincide with the eigenvectors of ZZ∗ entific literature four neural SVD learning theories while V contains the eigenvectors of Z∗Z with the appeared independently; then, as novel contribution same eigenvalues. to this field, we present: Here we consider four parallel SVD learning algo- rithms, which allow to simultaneously compute the A unifying view of the mentioned theories, show- • SVD vectors. The considered neural algorithms have ing the main relationships among them; been developed by Weingessel and Hornik32 and by A stability analysis based on Lyapunov criterion, • Helmke and Moore.18 These algorithms are utilized aimed at ensuring the non-divergence of the differ- to train in an unsupervised way a three-layer neural ential equations governing the learning phases of network with the classical ‘butterfly’ topology (see the SVD neural networks trained via the consid- e.g. Refs. 8, 32 and 33): The first layer has con- ered methods; nection matrix A, the second one has connection- A computer-based analysis of the learning differ- • matrix B and the middle (hidden) layer provides ential equations carried out in order to assess their network's output. Properly learnt, the network is numerical properties. able to perform the mentioned signal/data process- Through the paper we use the following notation. ing tasks, such as noise filtering.8 Symbol Im;n denotes the pseudo-identity matrix of The aim of this section is to analytically show 0 size m n and Im = Im;n. Symbol X denotes that the algorithms proposed by Weingessel{Hornik × ∗ the transposition of the matrix X while X denotes and Helmke{Moore are equivalent to some extent. Hermitian-transposition; symbol tr(X) denotes the Also, it is showed that when proper initial condi- trace of the square matrix X, i.e. the sum of its tions are chosen the associated learning trajectories in-diagonal entries; the trace operator enjoys the lie on the double Stiefel manifold. following properties: tr(X 0) = tr(X), tr(ABC) = tr(CAB) = tr(BCA). We also define the two ma- 2.1. The WH2, WH3 and WH4 neural def 0 0 def trix operators X; Y = X Y Y X and [X; Y ] = SVD-subspace dynamical systems 0 0 f g − X Y +Y X. The following matrix set (termed Stiefel In Ref. 32 some new learning equations have been manifold) is also useful to our expository purposes: def m×n ∗ introduced by Weingessel and Hornik in order to St(m; n; K) = X K X X = In with m 1, f 2 j g − compute the SVD-subspace of a given matrix. Here n 1 N; the field K may be either R or C; when − 2 we investigate on three of them, expressed as m = n the manifold coincides with the orthogonal continuous-time differential equations. The deriva- K def Km×m ∗ group O(m; ) = X X X = Im . We tions presented below make use of the matrix f 2 K j K g refer to the product O(m; ) O(n; ) as double differential calculus: A source reference for this is × K orthogonal group and to the product St(m; p; ) Ref. 23. K × St(n; p; ) as double Stiefel manifold (some defini- Let us denote as A(t) Rm×p the network- 2 tions and notes on these geometrical entities are connection matrix-stream that should learn p left available in the appendix A.1). Also, the Frobe- singular vectors and as B(t) Rn×p the estimator Rn×n 2 nius norm of a matrix X is defined as for p right singular vectors of the SVD of matrix def ∗ 2 X F = tr(X X). Z Rm×n, with p r min m; n , where r de- k k 2 ≤ ≤ f g p notes again the rank of matrix Z. The algorithm 2. Four Parallel SVD Learning WH232 reads: Algorithms: A Unifying View 0 0 A_ = ZB AB Z A ; A(0) = A0 ; m×n − (1) Denoting as Z C the matrix whose SVD is to _ 0 0 2 ( B = Z A BA ZB ; B(0) = B0 : be computed and as r min m; n the rank of Z, − ≤ f g ∗ the singular value decomposition writes Z = UDV , It has been derived by extending Brockett's work where U Rm×m and V Rn×n are orthogonal ma- on isospectral flow systems3 from single to double 2 2 trices and D is a pseudo-diagonal matrix that has orthogonal group; the initial state A0; B0 of the Singular Value Decomposition Learning on Double Stiefel Manifold 3 dynamical equations may be freely chosen. Here conditions (2): we consider the particular choice A0 St(m; p; R) 2 def 0 0 0 0 0 0 0 and B0 St(n; p; R), as for instance A0 = Im;p and S = A ZB = A (AB Z A) = (A A)B Z A 2 B0 = In;p. = B0Z0A = S0 : Theorem 1 Now, as A belongs to the Stiefel manifold St(m; p; R) If the initial states of the WH2 system belong to the at any time and has thus rank p, the equilibrium so- Stiefel manifold, then the whole dynamics is double- lution may be parameterized as A = UpKa; the same Stiefel. holds for B = VpKb, where Ka and Kb are matrices in O(p; R). This ensures A and B span the SVD- Proof subspace of Z. On the basis of this parameterization, the We wish to prove that if A0 St(m; p; R) and 0 0 0 2 product A ZB writes Ka(UpZVp)Kb, where, by def- B0 St(n; p; R), then A(t) St(m; p; R) and B(t) 0 2 2 2 inition, UpZVp = D1, the diagonal matrix of p St(n; p; R) for all t 0. 0 ≥ singular values. On the other hand, S = K D1K To show this for matrix A, it is sufficient to prove a b must be symmetric and this may hold only if K = that the trajectory emanating from any point such a 0 0 Kb = K. that A A = Ip has differential d(A A) = 0. Since d(A0A) = (dA)0A + A0(dA) = (A_0A + A0A_)dt = This shows that the WH2 algorithm does not ac- [A;_ A]dt, this may be proven by computing the brack- tually compute the true SVD, but a SVD-subspace ets [A_; A]: of dimension p.
Load more

Recommended publications
  • Isospectralization, Or How to Hear Shape, Style, and Correspondence
    Isospectralization, or how to hear shape, style, and correspondence Luca Cosmo Mikhail Panine Arianna Rampini University of Venice Ecole´ Polytechnique Sapienza University of Rome [email protected] [email protected] [email protected] Maks Ovsjanikov Michael M. Bronstein Emanuele Rodola` Ecole´ Polytechnique Imperial College London / USI Sapienza University of Rome [email protected] [email protected] [email protected] Initialization Reconstruction Abstract opt. target The question whether one can recover the shape of a init. geometric object from its Laplacian spectrum (‘hear the shape of the drum’) is a classical problem in spectral ge- ometry with a broad range of implications and applica- tions. While theoretically the answer to this question is neg- ative (there exist examples of iso-spectral but non-isometric manifolds), little is known about the practical possibility of using the spectrum for shape reconstruction and opti- mization. In this paper, we introduce a numerical proce- dure called isospectralization, consisting of deforming one shape to make its Laplacian spectrum match that of an- other. We implement the isospectralization procedure using modern differentiable programming techniques and exem- plify its applications in some of the classical and notori- Without isospectralization With isospectralization ously hard problems in geometry processing, computer vi- sion, and graphics such as shape reconstruction, pose and Figure 1. Top row: Mickey-from-spectrum: we recover the shape of Mickey Mouse from its first 20 Laplacian eigenvalues (shown style transfer, and dense deformable correspondence. in red in the leftmost plot) by deforming an initial ellipsoid shape; the ground-truth target embedding is shown as a red outline on top of our reconstruction.
    [Show full text]
  • Isospectral Towers of Riemannian Manifolds
    New York Journal of Mathematics New York J. Math. 18 (2012) 451{461. Isospectral towers of Riemannian manifolds Benjamin Linowitz Abstract. In this paper we construct, for n ≥ 2, arbitrarily large fam- ilies of infinite towers of compact, orientable Riemannian n-manifolds which are isospectral but not isometric at each stage. In dimensions two and three, the towers produced consist of hyperbolic 2-manifolds and hyperbolic 3-manifolds, and in these cases we show that the isospectral towers do not arise from Sunada's method. Contents 1. Introduction 451 2. Genera of quaternion orders 453 3. Arithmetic groups derived from quaternion algebras 454 4. Isospectral towers and chains of quaternion orders 454 5. Proof of Theorem 4.1 456 5.1. Orders in split quaternion algebras over local fields 456 5.2. Proof of Theorem 4.1 457 6. The Sunada construction 458 References 461 1. Introduction Let M be a closed Riemannian n-manifold. The eigenvalues of the La- place{Beltrami operator acting on the space L2(M) form a discrete sequence of nonnegative real numbers in which each value occurs with a finite mul- tiplicity. This collection of eigenvalues is called the spectrum of M, and two Riemannian n-manifolds are said to be isospectral if their spectra coin- cide. Inverse spectral geometry asks the extent to which the geometry and topology of M is determined by its spectrum. Whereas volume and scalar curvature are spectral invariants, the isometry class is not. Although there is a long history of constructing Riemannian manifolds which are isospectral but not isometric, we restrict our discussion to those constructions most Received February 4, 2012.
    [Show full text]
  • Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
    Published in: Journal of Geometric Mechanics doi:10.3934/jgm.2017014 Volume 9, Number 3, September 2017 pp. 335{390 GEOMETRY OF MATRIX DECOMPOSITIONS SEEN THROUGH OPTIMAL TRANSPORT AND INFORMATION GEOMETRY Klas Modin∗ Department of Mathematical Sciences Chalmers University of Technology and University of Gothenburg SE-412 96 Gothenburg, Sweden Abstract. The space of probability densities is an infinite-dimensional Rie- mannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher{Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry|the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multi- variate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher{Rao geometries are discussed. The link to matrices is obtained by considering OMT and information ge- ometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent descrip- tion of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decomposi- tions (smooth and linear category), entropy gradient flows, and the Fisher{Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions.
    [Show full text]
  • Class Notes, Functional Analysis 7212
    Class notes, Functional Analysis 7212 Ovidiu Costin Contents 1 Banach Algebras 2 1.1 The exponential map.....................................5 1.2 The index group of B = C(X) ...............................6 1.2.1 p1(X) .........................................7 1.3 Multiplicative functionals..................................7 1.3.1 Multiplicative functionals on C(X) .........................8 1.4 Spectrum of an element relative to a Banach algebra.................. 10 1.5 Examples............................................ 19 1.5.1 Trigonometric polynomials............................. 19 1.6 The Shilov boundary theorem................................ 21 1.7 Further examples....................................... 21 1.7.1 The convolution algebra `1(Z) ........................... 21 1.7.2 The return of Real Analysis: the case of L¥ ................... 23 2 Bounded operators on Hilbert spaces 24 2.1 Adjoints............................................ 24 2.2 Example: a space of “diagonal” operators......................... 30 2.3 The shift operator on `2(Z) ................................. 32 2.3.1 Example: the shift operators on H = `2(N) ................... 38 3 W∗-algebras and measurable functional calculus 41 3.1 The strong and weak topologies of operators....................... 42 4 Spectral theorems 46 4.1 Integration of normal operators............................... 51 4.2 Spectral projections...................................... 51 5 Bounded and unbounded operators 54 5.1 Operations..........................................
    [Show full text]
  • Lecture Notes : Spectral Properties of Non-Self-Adjoint Operators
    Journées ÉQUATIONS AUX DÉRIVÉES PARTIELLES Évian, 8 juin–12 juin 2009 Johannes Sjöstrand Lecture notes : Spectral properties of non-self-adjoint operators J. É. D. P. (2009), Exposé no I, 111 p. <http://jedp.cedram.org/item?id=JEDP_2009____A1_0> cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ GROUPEMENT DE RECHERCHE 2434 DU CNRS Journées Équations aux dérivées partielles Évian, 8 juin–12 juin 2009 GDR 2434 (CNRS) Lecture notes : Spectral properties of non-self-adjoint operators Johannes Sjöstrand Résumé Ce texte contient une version légèrement completée de mon cours de 6 heures au colloque d’équations aux dérivées partielles à Évian-les-Bains en juin 2009. Dans la première partie on expose quelques résultats anciens et récents sur les opérateurs non-autoadjoints. La deuxième partie est consacrée aux résultats récents sur la distribution de Weyl des valeurs propres des opé- rateurs elliptiques avec des petites perturbations aléatoires. La partie III, en collaboration avec B. Helffer, donne des bornes explicites dans le théorème de Gearhardt-Prüss pour des semi-groupes. Abstract This text contains a slightly expanded version of my 6 hour mini-course at the PDE-meeting in Évian-les-Bains in June 2009. The first part gives some old and recent results on non-self-adjoint differential operators. The second part is devoted to recent results about Weyl distribution of eigenvalues of elliptic operators with small random perturbations. Part III, in collaboration with B. Helffer, gives explicit estimates in the Gearhardt-Prüss theorem for semi-groups.
    [Show full text]
  • On Quantum Integrability and Hamiltonians with Pure Point
    On quantum integrability and Hamiltonians with pure point spectrum Alberto Enciso∗ Daniel Peralta-Salas† Departamento de F´ısica Te´orica II, Facultad de Ciencias F´ısicas, Universidad Complutense, 28040 Madrid, Spain Abstract We prove that any n-dimensional Hamiltonian operator with pure point spectrum is completely integrable via self-adjoint first integrals. Furthermore, we establish that given any closed set Σ ⊂ R there exists an integrable n-dimensional Hamiltonian which realizes it as its spectrum. We develop several applications of these results and discuss their implications in the general framework of quantum integrability. PACS numbers: 02.30.Ik, 03.65.Ca 1 Introduction arXiv:math-ph/0406022v3 6 Jul 2004 A classical Hamiltonian h, that is, a function from a 2n-dimensional phase space into the real numbers, completely determines the dynamics of a classi- cal system. Its complexity, i.e., the regular or chaotic behavior of the orbits of the Hamiltonian vector field, strongly depends upon the integrability of the Hamiltonian. Recall that the n-dimensional Hamiltonian h is said to be (Liouville) integrable when there exist n functionally independent first integrals in in- volution with a certain degree of regularity. When a classical Hamiltonian is integrable, its dynamics is not considered to be chaotic. ∗aenciso@fis.ucm.es †dperalta@fis.ucm.es 1 Given an arbitrary classical Hamiltonian there is no algorithmic procedure to ascertain whether it is integrable or not. To our best knowledge, the most general results on this matter are Ziglin’s theory [1] and Morales–Ramis’ theory [2], which provide criteria to establish that a classical Hamiltonian is not integrable via meromorphic first integrals.
    [Show full text]
  • ASYMPTOTICALLY ISOSPECTRAL QUANTUM GRAPHS and TRIGONOMETRIC POLYNOMIALS. Pavel Kurasov, Rune Suhr
    ISSN: 1401-5617 ASYMPTOTICALLY ISOSPECTRAL QUANTUM GRAPHS AND TRIGONOMETRIC POLYNOMIALS. Pavel Kurasov, Rune Suhr Research Reports in Mathematics Number 2, 2018 Department of Mathematics Stockholm University Electronic version of this document is available at http://www.math.su.se/reports/2018/2 Date of publication: Maj 16, 2018. 2010 Mathematics Subject Classification: Primary 34L25, 81U40; Secondary 35P25, 81V99. Keywords: Quantum graphs, almost periodic functions. Postal address: Department of Mathematics Stockholm University S-106 91 Stockholm Sweden Electronic addresses: http://www.math.su.se/ [email protected] Asymptotically isospectral quantum graphs and generalised trigonometric polynomials Pavel Kurasov and Rune Suhr Dept. of Mathematics, Stockholm Univ., 106 91 Stockholm, SWEDEN [email protected], [email protected] Abstract The theory of almost periodic functions is used to investigate spectral prop- erties of Schr¨odinger operators on metric graphs, also known as quantum graphs. In particular we prove that two Schr¨odingeroperators may have asymptotically close spectra if and only if the corresponding reference Lapla- cians are isospectral. Our result implies that a Schr¨odingeroperator is isospectral to the standard Laplacian on a may be different metric graph only if the potential is identically equal to zero. Keywords: Quantum graphs, almost periodic functions 2000 MSC: 34L15, 35R30, 81Q10 Introduction. The current paper is devoted to the spectral theory of quantum graphs, more precisely to the direct and inverse spectral theory of Schr¨odingerop- erators on metric graphs [3, 20, 24]. Such operators are defined by three parameters: a finite compact metric graph Γ; • a real integrable potential q L (Γ); • ∈ 1 vertex conditions, which can be parametrised by unitary matrices S.
    [Show full text]
  • Isospectral Sets for Fourth-Order Ordinary Differential Operators Lester Caudill University of Richmond, [email protected]
    University of Richmond UR Scholarship Repository Math and Computer Science Faculty Publications Math and Computer Science 7-1998 Isospectral Sets for Fourth-Order Ordinary Differential Operators Lester Caudill University of Richmond, [email protected] Peter A. Perry Albert W. Schueller Follow this and additional works at: http://scholarship.richmond.edu/mathcs-faculty-publications Part of the Ordinary Differential Equations and Applied Dynamics Commons Recommended Citation Caudill, Lester F., Peter A. Perry, and Albert W. Schueller. "Isospectral Sets for Fourth-Order Ordinary Differential Operators." SIAM Journal on Mathematical Analysis 29, no. 4 (July 1998): 935-66. doi:10.1137/s0036141096311198. This Article is brought to you for free and open access by the Math and Computer Science at UR Scholarship Repository. It has been accepted for inclusion in Math and Computer Science Faculty Publications by an authorized administrator of UR Scholarship Repository. For more information, please contact [email protected]. SIAM J. MATH. ANAL. °c 1998 Society for Industrial and Applied Mathematics Vol. 29, No. 4, pp. 935–966, July 1998 008 ISOSPECTRAL SETS FOR FOURTH-ORDER ORDINARY DIFFERENTIAL OPERATORS∗ LESTER F. CAUDILL, JR.y , PETER A. PERRYz , AND ALBERT W. SCHUELLERx 4 0 0 Abstract. Let L(p)u = D u ¡ (p1u ) + p2u be a fourth-order differential operator acting on 2 2 2 00 L [0; 1] with p ≡ (p1;p2) belonging to LR[0; 1] × LR[0; 1] and boundary conditions u(0) = u (0) = u(1) = u00(1) = 0. We study the isospectral set of L(p) when L(p) has simple spectrum. In particular 2 2 we show that for such p, the isospectral manifold is a real-analytic submanifold of LR[0; 1] × LR[0; 1] which has infinite dimension and codimension.
    [Show full text]
  • On Isospectral Deformations of Riemannian Metrics Compositio Mathematica, Tome 40, No 3 (1980), P
    COMPOSITIO MATHEMATICA RUISHI KUWABARA On isospectral deformations of riemannian metrics Compositio Mathematica, tome 40, no 3 (1980), p. 319-324 <http://www.numdam.org/item?id=CM_1980__40_3_319_0> © Foundation Compositio Mathematica, 1980, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ COMPOSITIO MATHEMATICA, Vol. 40, Fasc. 3, 1980, pag. 319-324 @ 1980 Sijthoff &#x26; Noordhoff International Publishers - Alphen aan den Rijn Printed in the Netherlands ON ISOSPECTRAL DEFORMATIONS OF RIEMANNIAN METRICS Ruishi Kuwabara 1. Introduction Let M be an n-dimensional compact orientable Coo manifold, and g be a C°° Riemannian metric on M. It is known that the Laplace- Beltrami operator àg = -gij~i~j acting on C~ functions on M has an infinite sequence of eigenvalues (denoted by Spec(M, g)) each eigenvalue being repeated as many as its multiplicity. Consider the following problem [1, p. 233]: Let g(t) (- E t E, E &#x3E; 0) be a 1-parameter C~ deformation of a Riemannian metric on M. Then, is there a deformation g(t) such that Spec(M, g(t))= Spec(M, g(o)) for every t? Such a deformation is called an isospectral deformation. First, we give some definitions to state the results of this article.
    [Show full text]
  • Random Matrix Theory of the Isospectral Twirling Abstract Contents
    SciPost Phys. 10, 076 (2021) Random matrix theory of the isospectral twirling Salvatore F. E. Oliviero1?, Lorenzo Leone1, Francesco Caravelli2 and Alioscia Hamma1 1 Physics Department, University of Massachusetts Boston, 02125, USA 2 Theoretical Division (T-4), Los Alamos National Laboratory, 87545, USA ? [email protected] Abstract We present a systematic construction of probes into the dynamics of isospectral ensem- bles of Hamiltonians by the notion of Isospectral twirling, expanding the scopes and methods of ref. [1]. The relevant ensembles of Hamiltonians are those defined by salient spectral probability distributions. The Gaussian Unitary Ensembles (GUE) describes a class of quantum chaotic Hamiltonians, while spectra corresponding to the Poisson and Gaussian Diagonal Ensemble (GDE) describe non chaotic, integrable dynamics. We com- pute the Isospectral twirling of several classes of important quantities in the analysis of quantum many-body systems: Frame potentials, Loschmidt Echos, OTOCs, Entangle- ment, Tripartite mutual information, coherence, distance to equilibrium states, work in quantum batteries and extension to CP-maps. Moreover, we perform averages in these ensembles by random matrix theory and show how these quantities clearly separate chaotic quantum dynamics from non chaotic ones. Copyright S. F. E. Oliviero et al. Received 21-12-2020 This work is licensed under the Creative Commons Accepted 22-03-2021 Check for Attribution 4.0 International License. Published 25-03-2021 updates Published by the SciPost Foundation.
    [Show full text]
  • Two Lectures on Spectral Invariants for the Schrödinger Operator Séminaire De Théorie Spectrale Et Géométrie, Tome 18 (1999-2000), P
    Séminaire de Théorie spectrale et géométrie MIKHAIL V. NOVITSKII Two lectures on spectral invariants for the Schrödinger operator Séminaire de Théorie spectrale et géométrie, tome 18 (1999-2000), p. 77-107 <http://www.numdam.org/item?id=TSG_1999-2000__18__77_0> © Séminaire de Théorie spectrale et géométrie (Grenoble), 1999-2000, tous droits réservés. L’accès aux archives de la revue « Séminaire de Théorie spectrale et géométrie » implique l’ac- cord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utili- sation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Séminaire de théorie spectrale et géométrie GRENOBLE Volume 18 (2000) 77-107 TWO LECTURES ON SPECTRAL INVARIANTS FOR THE SCHRÖDINGER OPERATOR Mikhaïl V. NOVITSKII Abstract An introduction into spectral invariants for the Schrödinger operators with per- iodic and almost periodic potentials is given. The following problems are conside- red: a description for the fundamental series of the spectral invariants, a complete- ness problem for these collections, spectral invariants for the Hill operator as motion intégrais for the KdV équation, a connection of the spectrum of the periodic multi- dimensional Schrödinger operator with the spectrum of a collection of the Hill ope- rators obtained by averaging of the potential over a family of closed geodesie on a torus, the direct and inverse problems. Some open problems are formulated. Table of Contents Introduction 78 1 Spectral invariants for 1-D Schrödinger operators with periodic and almost periodic potentials 79 1.1 Periodic case - the Hill operator 79 1.1.1 Spectral theory.
    [Show full text]
  • Analysis Notes
    Analysis notes Aidan Backus December 2019 Contents I Preliminaries2 1 Functional analysis3 1.1 Locally convex spaces...............................3 1.2 Hilbert spaces...................................6 1.3 Bochner integration................................6 1.4 Duality....................................... 11 1.5 Vector lattices................................... 13 1.6 Positive Radon measures............................. 14 1.7 Baire categories.................................. 17 2 Complex analysis 20 2.1 Cauchy-Green formula.............................. 20 2.2 Conformal mappings............................... 22 2.3 Approximation by polynomials.......................... 24 2.4 Sheaves...................................... 26 2.5 Subharmonicity.................................. 28 2.6 Operator theory.................................. 30 II Dynamical systems 31 3 Elementary dynamical systems 32 3.1 Types of dynamical systems........................... 32 3.2 Properties of the irrational rotation....................... 34 4 Ergodic theory 37 4.1 The mean ergodic theorem............................ 37 4.2 Pointwise ergodic theorem............................ 41 4.3 Ergodic systems.................................. 45 4.4 Properties of ergodic transformations...................... 47 4.5 Mixing transformations.............................. 48 4.6 The Hopf argument................................ 54 1 5 Flows on manifolds 59 5.1 Ergodic theorems for flows............................ 59 5.2 Geodesic flows in hyperbolic space.......................
    [Show full text]