دانلود کتاب Information Geometry and Its Applications

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Contents......Page 7
Introduction......Page 9
Part I Applications of Information Geometry......Page 11
1 Introduction......Page 12
2 Markovian Dynamical Systems......Page 13
3.1 Full Model......Page 15
3.2 Fully Split Model......Page 16
3.3 Diagonally Split Graphical Model......Page 19
3.4 Causally Split Model (Geometric Model)......Page 21
3.5 Mismatched Decoding Model......Page 22
4 Comparison of Various Measures of Integrated Information......Page 23
References......Page 25
1 Introduction......Page 27
2 Preliminaries......Page 29
3 Information Geometry......Page 30
4 The Nash Equilibrium Theorem......Page 32
5 Variations of QREs......Page 36
6 An Example......Page 44
7 Another Example – Not so Nice......Page 49
8 Controlling the Game......Page 50
9 Channel Dependence......Page 51
References......Page 53
1 Introduction......Page 55
2.1 Definitions and Conditions......Page 58
2.2 Main Theorem......Page 63
3 Example......Page 65
4 Basic Lemmas......Page 69
5 Proof of Theorem1......Page 71
5.1 Proof of Theorem1, Cross Validation......Page 72
5.2 Proof of Theorem1, WAIC......Page 73
5.3 Mathematical Relations Between Priors......Page 75
5.5 Proof of Theorem1, Random Generalization Loss......Page 77
6.2 Divergence Phenomenon of CV and WAIC......Page 79
References......Page 80
1 Introduction......Page 82
2 Boltzmann Machines......Page 84
3 Restricted Boltzmann Machines......Page 87
4 Basics of Training......Page 95
5 Dimension......Page 101
6 Representational Power......Page 104
7 Divergence Bounds......Page 111
8 Implicit Description......Page 115
9 Open Problems......Page 118
References......Page 119
Part II Infinite-Dimensional Information Geometry......Page 123
1 Introduction......Page 124
2.1 Generalities......Page 125
2.2 Entropy......Page 127
2.3 Orlicz and Lebesgue Spaces......Page 129
3.2 Densities of Exponential Form......Page 130
3.3 Poincaré-Type Inequalities......Page 131
4 Exponential Manifold on the Gaussian Space......Page 135
4.1 Maximal Exponential Manifold as an Affine Manifold......Page 139
4.2 Translations and Mollifiers......Page 141
4.3 Gaussian Statistical Bundle......Page 146
5.1 Orlicz-Sobolev Spaces with Gaussian Weight......Page 150
References......Page 159
Congruent Families and Invariant Tensors......Page 161
1 Introduction......Page 162
2.1 The Space of (Signed) Finite Measures and Their Powers......Page 165
2.2 Parametrized Measure Models......Page 166
2.3 Congruent Markov Morphisms......Page 167
2.4 Tensor Algebras......Page 168
2.5 Tensor Fields......Page 171
3 Congruent Families of Tensor Fields......Page 173
4 Congruent Families on Finite Sample Spaces......Page 177
5 Congruent Families on Arbitrary Sample Spaces......Page 187
References......Page 190
1 Introduction......Page 192
2 Nonlinear Filtering and Information Geometry......Page 196
3 Nonlinear Filtering on the Hilbert Manifold......Page 201
4 The Ψ Multi-objective Criterion......Page 205
References......Page 210
1 Introduction and Motivation......Page 212
1.1 Background: Power Euclidean Distances......Page 214
2 The Finite-Dimensional Case: Log-Euclidean Divergences......Page 215
2.1 Properties of the Log-Euclidean Divergences......Page 217
3 The Infinite-Dimensional Case: Log-Hilbert–Schmidt Divergences......Page 218
3.1 Log-Hilbert–Schmidt Divergences Between Positive Definite Trace Class Operators......Page 219
3.2 Log-Hilbert–Schmidt Divergences Between Positive Definite Hilbert–Schmidt Operators......Page 221
3.3 Properties of the Log-Hilbert–Schmidt Divergences......Page 222
3.4 The Log-Hilbert–Schmidt Divergences Between RKHS Covariance Operators......Page 224
4 Extended Power-Hilbert–Schmidt Distances......Page 228
4.1 The Case of RKHS Covariance Operators......Page 229
5.1 Proofs for the Power Euclidean Distances......Page 232
5.2 Proofs for the Log-Hilbert–Schmidt Divergences......Page 233
5.3 Proofs for the Extended Power-Hilbert–Schmidt Distances......Page 240
5.4 Miscellaneous Technical Results......Page 245
References......Page 246
Part III Theoretical Aspects of Information Geometry......Page 247
1 Introduction......Page 248
2 Convex Bodies of Rank 2......Page 250
3 Regret and Bregman Divergences......Page 254
4 Spectral Sets......Page 258
5 Sufficient Regret Functions......Page 260
6 Spin Factors......Page 263
7 Monotonicity Under Dilations......Page 266
8 Monotonicity of Bregman Divergences on Spin Factors......Page 272
9 Strict Monotonicity......Page 276
References......Page 279
1 Introduction......Page 280
2 Generalized m-Geodesic and e-Geodesic......Page 282
3 Maximum Entropy Distribution......Page 287
4 Divergence Geometry......Page 289
5 Discussion......Page 294
References......Page 295
1 Introduction......Page 297
2.1 Conjugate Transformation and Codazzi Coupling Associated to (h, )......Page 299
2.2 Tangent Bundle Isomorphism......Page 302
2.3 Klein Group of Transformations on......Page 305
2.4 Compatible Quadruple (g, ω, L, )......Page 306
2.5 Role of Connection......Page 307
2.6 Codazzi-(Para-)Kähler Structure......Page 309
3 Divergence Functions and (Para-)Kähler Structures......Page 310
3.1 Classical Divergence Functions and Statistical Structures......Page 311
3.2 Generalized Divergence Functions and Symplectic Structures......Page 312
3.3 Para-Kähler Structure on mathfrakMtimesmathfrakM......Page 313
3.4 Local Divergence Functions and Kähler Structures......Page 316
3.5 An Example: The Case of Analytic Kähler Manifold......Page 317
4 Discussions......Page 319
References......Page 320
1 Introduction......Page 322
2.1 Information geometry of calSn and R+n+1......Page 324
2.2 An Example......Page 326
2.3 Denormalization......Page 327
3 Main Results......Page 328
4 Concluding Remarks......Page 332
References......Page 333
1 Introduction......Page 334
2 Main Results......Page 336
3.1 Raising Indices Preserves Markov Invariance......Page 338
3.2 Lowering Indices Preserves Markov Invariance......Page 342
4 Concluding Remarks......Page 343
A Uniqueness of f......Page 344
References......Page 346
1 Introduction......Page 347
2 Statistical Manifolds of a Constant α-Curvature......Page 348
3.1 Normal Model......Page 350
3.2 Logistic Model......Page 351
4 Pareto Statistical Model......Page 352
5 Weibull Statistical Model......Page 355
6 Comparison of Curvatures......Page 358
References......Page 359
1 Introduction......Page 360
2 Entropy, Information and the Optimal Channel Problem......Page 362
3 Dual Formulation of the Optimal Transport Problem......Page 366
References......Page 369
Part IV Quantum Information Geometry......Page 371
1 Introduction......Page 372
2 Skew Information as Quantum Fisher Information......Page 373
3.2 Superadditivity......Page 378
3.3 Uncertainty Inequalities......Page 380
3.4 Monotonicity Under Operations......Page 381
4.1 Coherence......Page 384
4.3 Correlations......Page 386
5 Summary......Page 388
References......Page 389
1 Introduction......Page 394
2.1 Coherence as Complementarity Between State and Observable......Page 395
2.2 Quantum Phase Estimation......Page 396
3.1 An Observable Lower Bound to Asymmetry......Page 399
4.1 Detecting Asymmetry via Bell State Projections......Page 401
4.2 Alternative Scheme......Page 403
References......Page 404
A Monotonicity of Eq.(35) Under Local Unital Channels......Page 406
2 Quantifying Quantum Correlations Beyond Entanglement......Page 408
3.1 Metric-Adjusted Skew Informations......Page 411
3.2 Maximising Metric-Adjusted f–Correlations Over Pairs of Local Observables......Page 412
3.3 The Quantum f–Correlations Satisfy Q1 and Q2......Page 413
3.4 Quantum f–Correlations as Entanglement Monotones for Pure Qubit-Qudit States......Page 414
3.5 Analytical Expression of the Two-Qubit Quantum f–Correlations......Page 416
3.6 Behaviour of the Quantum f–Correlations Under Local Commutativity Preserving Channels......Page 417
4 Physical Interpretation and Applications......Page 418
5.1 Extending the Optimisation in the Definition of the Quantum f–Correlations......Page 420
References......Page 423
The Effects of Random Qubit-Qubit Quantum Channels to Entropy Gain, Fidelity and Trace Distance......Page 426
1 Introduction......Page 427
2 The Space of Qubit-Qubit Quantum Channels......Page 428
3 The Effect of Random Quantum Channels from Information Theoretical Point of View......Page 431
3.1 Entropy Gain......Page 433
3.2 Fidelity......Page 434
3.3 Trace and Hilbert–Schmidt Distances......Page 435
References......Page 437
1 Introduction......Page 439
2 Overview of Uncertainty Principles......Page 442
3 Generalized Covariances......Page 443
4 Robertson Uncertainty Principle......Page 448
References......Page 450