Information geometry and its applications / Shun-ichi Amari.
Series Applied mathematical sciences (Springer-Verlag New York Inc.): v. 194.Editor: [Tokyo] : Springer, [2016]Fecha de copyright: ©2016Descripción: xiii, 374 pages : illustrations ; 24 cmTipo de contenido: text Tipo de medio: unmediated Tipo de portador: volumeISBN: 9784431559771 (hbk : acidfree paper); 4431559779 (hbk : acid-free paper)Tema(s): Geometry, Differential | Mathematical statistics | Information theory in mathematics | Information theory -- Mathematics | Geometry, Differential | Information theory in mathematics | Information theory -- Mathematics | Mathematical statistics | Géométrie différentielle | Statistique mathématique | Information, Théorie de l'Otra clasificación: 62F12 (53B05 62B10 62H30 62M10 68T05 94A15 94A17)Part I Geometry of Divergence Functions: Dually Flat Riemannian Structure 1 Manifold, Divergence and Dually Flat Structure [3] 1.1 Manifolds [3] 1.1.1 Manifold and Coordinate Systems [3] 1.1.2 Examples of Manifolds [5] 1.2 Divergence Between Two Points [9] 1.2.1 Divergence [9] 1.2.2 Examples of Divergence [11] 1.3 Convex Function and Bregman Divergence [12] 1.3.1 Convex Function [12] 1.3.2 Bregman Divergence [13] 1.4 Legendre Transformation [16] 1.5 Dually Flat Riemannian Structure Derived from Convex Function [19] 1.5.1 Affine and Dual Affine Coordinate Systems [19] 1.5.2 Tangent Space, Basis Vectors and Riemannian Metric [20] 1.5.3 Parallel Transport of Vector [23] 1.6 Generalized Pythagorean Theorem and Projection Theorem [24] 1.6.1 Generalized Pythagorean Theorem [24] 1.6.2 Projection Theorem [26] 1.6.3 Divergence Between Submanifolds: Alternating Minimization Algorithm [27] 2 Exponential Families and Mixture Families of Probability Distributions [31] 2.1 Exponential Family of Probability Distributions [31] 2.2 Examples of Exponential Family: Gaussian and Discrete Distributions [34] 2.2.1 Gaussian Distribution [34] 2.2.2 Discrete Distribution [35] 2.3 Mixture Family of Probability Distributions [36] 2.4 Flat Structure: e-flat and m-flat [37] 2.5 On Infinite-Dimensional Manifold of Probability Distributions [39] 2.6 Kernel Exponential Family [42] 2.7 Bregman Divergence and Exponential Family [43] 2.8 Applications of Pythagorean Theorem [44] 2.8.1 Maximum Entropy Principle [44] 2.8.2 Mutual Information [46] 2.8.3 Repeated Observations and Maximum Likelihood Estimator [47] 3 Invariant Geometry of Manifold of Probability Distributions [51] 3.1 Invariance Criterion [51] 3.2 Information Monotonicity Under Coarse Graining [53] 3.2.1 Coarse Graining and Sufficient Statistics in Sn [53] 3.2.2 Invariant Divergence [54] 3.3 Examples of f-Divergence in Sn [57] 3.3.1 KL-Divergence [57] 3.3.2 X2-Divergence [57] 3.3.3 α-Divergence [57] 3.4 General Properties of f-Divergence and KL-Divergence [59] 3.4.1 Properties of f-Divergence [59] 3.4.2 Properties of KL-Divergence [60] 3.5 Fisher Information: The Unique Invariant Metric [62] 3.6 f-Divergence in Manifold of Positive Measures [65] 4 α-Geometry, Tsallis q-Entropy and Positive-Definite Matrices [71] 4.1 Invariant and Flat Divergence [71] 4.1.1 KL-Divergence Is Unique [71] 4.1.2 α-Divergence Is Unique in Rn+ [72] 4.2 α-Geometry in Sn and Rn+ [75] 4.2.1 α-Geodesic and α-Pythagorean Theorem in Rn+ [75] 4.2.2 α-Geodesic in Sn [76] 4.2.3 α-Pythagorean Theorem and α-Projection Theorem in Sn [76] 4.2.4 Apportionment Due to α-Divergence [77] 4.2.5 α-Mean [77] 4.2.6 α-Families of Probability Distributions [80] 4.2.7 Optimality of α-Integration [82] 4.2.8 Application to α-Integration of Experts [83] 4.3 Geometry of Tsallis q-Entropy [84] 4.3.1 q-Logarithm and q-Exponential Function [85] 4.3.2 q-Exponential Family (α-Family) of Probability Distributions [86] 4.3.3 q-Escort Geometry [87] 4.3.4 Deformed Exponential Family: X-Escort Geometry [89] 4.3.5 Conformal Character of q-Escort Geometry [91] 4.4 (u, v)-Divergence: Dually Flat Divergence in Manifold of Positive Measures [92] 4.4.1 Decomposable (u, v)-Divergence [92] 4.4.2 General (u, v) Flat Structure in Rn+ [95] 4.5 Invariant Flat Divergence in Manifold of Positive-Definite Matrices [96] 4.5.1 Bregman Divergence and Invariance Under Gl(n) [96] 4.5.2 Invariant Flat Decomposable Divergences Under O(n) [98] 4.5.3 Non-flat Invariant Divergences [101] 4.6 Miscellaneous Divergences [102] 4.6.1 ϓ-Divergence [102] 4.6.2 Other Types of (α,β-Divergences [102] 4.6.3 Burbea-Rao Divergence and Jensen-Shannon Divergence [103] 4.6.4 (p, t-Structure and (F, G, H)-Structure [104] Part II Introduction to Dual Differential Geometry 5 Elements of Differential Geometry [109] 5.1 Manifold and Tangent Space [109] 5.2 Riemannian Metric [111] 5.3 Affine Connection [112] 5.4 Tensors [114] 5.5 Covariant Derivative [116] 5.6 Geodesic [117] 5.7 Parallel Transport of Vector [118] 5.8 Riemann-Christoffel Curvature [119] 5.8.1 Round-the-World Transport of Vector [120] 5.8.2 Covariant Derivative and RC Curvature [122] 5.8.3 Flat Manifold [123] 5.9 Levi-Civita (Riemannian) Connection [124] 5.10 Submanifold and Embedding Curvature [126] 5.10.1 Submanifold [126] 5.10.2 Embedding Curvature [127] 6 Dual Affine Connections and Dually Flat Manifold [131] 6.1 Dual Connections [131] 6.2 Metric and Cubic Tensor Derived from Divergence [134] 6.3 Invariant Metric and Cubic Tensor [136] 6.4 α-Geometry [136] 6.5 Dually Flat Manifold [137] 6.6 Canonical Divergence in Dually Flat Manifold [138] 6.7 Canonical Divergence in General Manifold of Dual Connections [141] 6.8 Dual Foliations of Flat Manifold and Mixed Coordinates [143] 6.8.1 k-cut of Dual Coordinate Systems: Mixed Coordinates and Foliation [144] 6.8.2 Decomposition of Canonical Divergence [145] 6.8.3 A Simple Illustrative Example: Neural Firing [146] 6.8.4 Higher-Order Interactions of Neuronal Spikes [148] 6.9 System Complexity and Integrated Information [150] 6.10 Input-Output Analysis in Economics [157] Part III Information Geometry of Statistical Inference 7 Asymptotic Theory of Statistical Inference [165] 7.1 Estimation [165] 7.2 Estimation in Exponential Family [166] 7.3 Estimation in Curved Exponential Family [168] 7.4 First-Order Asymptotic Theory of Estimation [171] 7.5 Higher-Order Asymptotic Theory of Estimation [173] 7.6 Asymptotic Theory of Hypothesis Testing [175] 8 Estimation in the Presence of Hidden Variables [179] 8.1 EM Algorithm [179] 8.1.1 Statistical Model with Hidden Variables [179] 8.1.2 Minimizing Divergence Between Model Manifold and Data Manifold [182] 8.1.3 EM Algorithm [184] 8.1.4 Example: Gaussian Mixture [184] 8.2 Loss of Information by Data Reduction [185] 8.3 Estimation Based on Misspecified Statistical Model [186] 9 Neyman-Scott Problem: Estimating Function and Semiparametric Statistical Model [191] 9.1 Statistical Model Including Nuisance Parameters [191] 9.2 Neyman-Scott Problem and Semiparametrics [194] 9.3 Estimating Function [197] 9.4 Information Geometry of Estimating Function [199] 9.5 Solutions to Neyman-Scott Problems [206] 9.5.1 Estimating Function in the Exponential Case [206] 9.5.2 Coefficient of Linear Dependence [208] 9.5.3 Scale Problem [209] 9.5.4 Temporal Firing Pattern of Single Neuron [211] 10 Linear Systems and Time Series [215] 10.1 Stationary Time Series and Linear System [215] 10.2 Typical Finite-Dimensional Manifolds of Time Series [217] 10.3 Dual Geometry of System Manifold [219] 10.4 Geometry of AR, MA and ARMA Models [223] Part IV Applications of Information Geometry 11 Machine Learning [231] 11.1 Clustering Patterns [231] 11.1.1 Pattern Space and Divergence [231] 11.1.2 Center of Cluster [232] 11.1.3 k-Means: Clustering Algorithm [233] 11.1.4 Voronoi Diagram [234] 11.1.5 Stochastic Version of Classification and Clustering [236] 11.1.6 Robust Cluster Center [238] 11.1.7 Asmptotic Evaluation of Error Probability in Pattern Recognition: Chernoff Information [240] 11.2 Geometry of Support Vector Machine [242] 11.2.1 Linear Classifier [242] 11.2.2 Embedding into High-Dimensional Space [245] 11.2.3 Kernel Method [246] 11.2.4 Riemannian Metric Induced by Kernel [247] 11.3 Stochastic Reasoning: Belief Propagation and CCCP Algorithms [249] 11.3.1 Graphical Model [250] 11.3.2 Mean Field Approximation and m-Projection [252] 11.3.3 Belief Propagation [255] 1 L3.4 Solution of BP Algorithm [257] 11.3.5 CCCP (Convex-Concave Computational Procedure). [259] 11.4 Information Geometry of Boosting [260] 11.4.1 Boosting: Integration of Weak Machines [261] 11.4.2 Stochastic Interpretation of Machine [262] 11.4.3 Construction of New Weak Machines [263] 11.4.4 Determination of the Weights of Weak Machines [263] 11.5 Bayesian Inference and Deep Learning [265] 11.5.1 Bayesian Duality in Exponential Family [266] 11.5.2 Restricted Boltzmann Machine [268] 11.5.3 Unsupervised Learning of RBM [269] 11.5.4 Geometry of Contrastive Divergence [273] 11.5.5 Gaussian RBM [275] 12 Natural Gradient Learning and Its Dynamics in Singular Regions [279] 12.1 Natural Gradient Stochastic Descent Learning [279] 12.1.1 On-Line Learning and Batch Learning [279] 12.1.2 Natural Gradient: Steepest Descent Direction in Riemannian Manifold [282] 12.1.3 Riemannian Metric, Hessian and Absolute Hessian [284] 12.1.4 Stochastic Relaxation of Optimization Problem [286] 12.1.5 Natural Policy Gradient in Reinforcement Learning [287] 12.1.6 Mirror Descent and Natural Gradient [289] 12.1.7 Properties of Natural Gradient Learning [290] 12.2 Singularity in Learning: Multilayer Perceptron [296] 12.2.1 Multilayer Perceptron [296] 12.2.2 Singularities in M [298] 12.2.3 Dynamics of Learning in M [302] 12.2.4 Critical Slowdown of Dynamics [305] 12.2.5 Natural Gradient Learning Is Free of Plateaus [309] 12.2.6 Singular Statistical Models [310] 12.2.7 Bayesian Inference and Singular Model [312] 13 Signal Processing and Optimization [315] 13.1 Principal Component Analysis [315] 13.1.1 Eigenvalue Analysis [315] 13.1.2 Principal Components, Minor Components and Whitening [316] 13.1.3 Dynamics of Learning of Principal and Minor Components [319] 13.2 Independent Component Analysis [322] 13.2.3 Estimating Function of ICA: Semiparametric Approach [330] 13.3 Non-negative Matrix Factorization [333] 13.4 Sparse Signal Processing [336] 13.4.1 Linear Regression and Sparse Solution [337] 13.4.2 Minimization of Convex Function Under L1 Constraint [338] 13.4.3 Analysis of Solution Path [341] 13.4.4 Minkovskian Gradient Flow [343] 13.4.5 Underdetermined Case [344] 13.5 Optimization in Convex Programming [345] 13.5.1 Convex Programming [345] 13.5.2 Dually Flat Structure Derived from Barrier Function [347] 13.5.3 Computational Complexity and m-curvature [348] 13.6 Dual Geometry Derived from Game Theory [349] 13.6.1 Minimization of Game-Score [349] 13.6.2 Hyvärinen Score [353] References [359]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
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Libros
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Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 62 Am485 (Browse shelf) | Available | A-9287 |
Incluye referencias bibliográficas (p. 359-369) e índice.
Part I Geometry of Divergence Functions: Dually Flat Riemannian Structure --
1 Manifold, Divergence and Dually Flat Structure [3] --
1.1 Manifolds [3] --
1.1.1 Manifold and Coordinate Systems [3] --
1.1.2 Examples of Manifolds [5] --
1.2 Divergence Between Two Points [9] --
1.2.1 Divergence [9] --
1.2.2 Examples of Divergence [11] --
1.3 Convex Function and Bregman Divergence [12] --
1.3.1 Convex Function [12] --
1.3.2 Bregman Divergence [13] --
1.4 Legendre Transformation [16] --
1.5 Dually Flat Riemannian Structure Derived from Convex Function [19] --
1.5.1 Affine and Dual Affine Coordinate Systems [19] --
1.5.2 Tangent Space, Basis Vectors and Riemannian Metric [20] --
1.5.3 Parallel Transport of Vector [23] --
1.6 Generalized Pythagorean Theorem and Projection Theorem [24] --
1.6.1 Generalized Pythagorean Theorem [24] --
1.6.2 Projection Theorem [26] --
1.6.3 Divergence Between Submanifolds: Alternating Minimization Algorithm [27] --
2 Exponential Families and Mixture Families of Probability Distributions [31] --
2.1 Exponential Family of Probability Distributions [31] --
2.2 Examples of Exponential Family: Gaussian and Discrete Distributions [34] --
2.2.1 Gaussian Distribution [34] --
2.2.2 Discrete Distribution [35] --
2.3 Mixture Family of Probability Distributions [36] --
2.4 Flat Structure: e-flat and m-flat [37] --
2.5 On Infinite-Dimensional Manifold --
of Probability Distributions [39] --
2.6 Kernel Exponential Family [42] --
2.7 Bregman Divergence and Exponential Family [43] --
2.8 Applications of Pythagorean Theorem [44] --
2.8.1 Maximum Entropy Principle [44] --
2.8.2 Mutual Information [46] --
2.8.3 Repeated Observations and Maximum Likelihood Estimator [47] --
3 Invariant Geometry of Manifold of Probability Distributions [51] --
3.1 Invariance Criterion [51] --
3.2 Information Monotonicity Under Coarse Graining [53] --
3.2.1 Coarse Graining and Sufficient Statistics in Sn [53] --
3.2.2 Invariant Divergence [54] --
3.3 Examples of f-Divergence in Sn [57] --
3.3.1 KL-Divergence [57] --
3.3.2 X2-Divergence [57] --
3.3.3 α-Divergence [57] --
3.4 General Properties of f-Divergence and KL-Divergence [59] --
3.4.1 Properties of f-Divergence [59] --
3.4.2 Properties of KL-Divergence [60] --
3.5 Fisher Information: The Unique Invariant Metric [62] --
3.6 f-Divergence in Manifold of Positive Measures [65] --
4 α-Geometry, Tsallis q-Entropy and Positive-Definite Matrices [71] --
4.1 Invariant and Flat Divergence [71] --
4.1.1 KL-Divergence Is Unique [71] --
4.1.2 α-Divergence Is Unique in Rn+ [72] --
4.2 α-Geometry in Sn and Rn+ [75] --
4.2.1 α-Geodesic and α-Pythagorean Theorem in Rn+ [75] --
4.2.2 α-Geodesic in Sn [76] --
4.2.3 α-Pythagorean Theorem and α-Projection Theorem in Sn [76] --
4.2.4 Apportionment Due to α-Divergence [77] --
4.2.5 α-Mean [77] --
4.2.6 α-Families of Probability Distributions [80] --
4.2.7 Optimality of α-Integration [82] --
4.2.8 Application to α-Integration of Experts [83] --
4.3 Geometry of Tsallis q-Entropy [84] --
4.3.1 q-Logarithm and q-Exponential Function [85] --
4.3.2 q-Exponential Family (α-Family) of Probability Distributions [86] --
4.3.3 q-Escort Geometry [87] --
4.3.4 Deformed Exponential Family: X-Escort Geometry [89] --
4.3.5 Conformal Character of q-Escort Geometry [91] --
4.4 (u, v)-Divergence: Dually Flat Divergence in Manifold of Positive Measures [92] --
4.4.1 Decomposable (u, v)-Divergence [92] --
4.4.2 General (u, v) Flat Structure in Rn+ [95] --
4.5 Invariant Flat Divergence in Manifold of Positive-Definite Matrices [96] --
4.5.1 Bregman Divergence and Invariance Under Gl(n) [96] --
4.5.2 Invariant Flat Decomposable Divergences Under O(n) [98] --
4.5.3 Non-flat Invariant Divergences [101] --
4.6 Miscellaneous Divergences [102] --
4.6.1 ϓ-Divergence [102] --
4.6.2 Other Types of (α,β-Divergences [102] --
4.6.3 Burbea-Rao Divergence and Jensen-Shannon Divergence [103] --
4.6.4 (p, t-Structure and (F, G, H)-Structure [104] --
Part II Introduction to Dual Differential Geometry --
5 Elements of Differential Geometry [109] --
5.1 Manifold and Tangent Space [109] --
5.2 Riemannian Metric [111] --
5.3 Affine Connection [112] --
5.4 Tensors [114] --
5.5 Covariant Derivative [116] --
5.6 Geodesic [117] --
5.7 Parallel Transport of Vector [118] --
5.8 Riemann-Christoffel Curvature [119] --
5.8.1 Round-the-World Transport of Vector [120] --
5.8.2 Covariant Derivative and RC Curvature [122] --
5.8.3 Flat Manifold [123] --
5.9 Levi-Civita (Riemannian) Connection [124] --
5.10 Submanifold and Embedding Curvature [126] --
5.10.1 Submanifold [126] --
5.10.2 Embedding Curvature [127] --
6 Dual Affine Connections and Dually Flat Manifold [131] --
6.1 Dual Connections [131] --
6.2 Metric and Cubic Tensor Derived from Divergence [134] --
6.3 Invariant Metric and Cubic Tensor [136] --
6.4 α-Geometry [136] --
6.5 Dually Flat Manifold [137] --
6.6 Canonical Divergence in Dually Flat Manifold [138] --
6.7 Canonical Divergence in General Manifold of Dual Connections [141] --
6.8 Dual Foliations of Flat Manifold and Mixed Coordinates [143] --
6.8.1 k-cut of Dual Coordinate Systems: Mixed Coordinates and Foliation [144] --
6.8.2 Decomposition of Canonical Divergence [145] --
6.8.3 A Simple Illustrative Example: Neural Firing [146] --
6.8.4 Higher-Order Interactions of Neuronal Spikes [148] --
6.9 System Complexity and Integrated Information [150] --
6.10 Input-Output Analysis in Economics [157] --
Part III Information Geometry of Statistical Inference --
7 Asymptotic Theory of Statistical Inference [165] --
7.1 Estimation [165] --
7.2 Estimation in Exponential Family [166] --
7.3 Estimation in Curved Exponential Family [168] --
7.4 First-Order Asymptotic Theory of Estimation [171] --
7.5 Higher-Order Asymptotic Theory of Estimation [173] --
7.6 Asymptotic Theory of Hypothesis Testing [175] --
8 Estimation in the Presence of Hidden Variables [179] --
8.1 EM Algorithm [179] --
8.1.1 Statistical Model with Hidden Variables [179] --
8.1.2 Minimizing Divergence Between Model Manifold and Data Manifold [182] --
8.1.3 EM Algorithm [184] --
8.1.4 Example: Gaussian Mixture [184] --
8.2 Loss of Information by Data Reduction [185] --
8.3 Estimation Based on Misspecified Statistical Model [186] --
9 Neyman-Scott Problem: Estimating Function and Semiparametric Statistical Model [191] --
9.1 Statistical Model Including Nuisance Parameters [191] --
9.2 Neyman-Scott Problem and Semiparametrics [194] --
9.3 Estimating Function [197] --
9.4 Information Geometry of Estimating Function [199] --
9.5 Solutions to Neyman-Scott Problems [206] --
9.5.1 Estimating Function in the Exponential Case [206] --
9.5.2 Coefficient of Linear Dependence [208] --
9.5.3 Scale Problem [209] --
9.5.4 Temporal Firing Pattern of Single Neuron [211] --
10 Linear Systems and Time Series [215] --
10.1 Stationary Time Series and Linear System [215] --
10.2 Typical Finite-Dimensional Manifolds of Time Series [217] --
10.3 Dual Geometry of System Manifold [219] --
10.4 Geometry of AR, MA and ARMA Models [223] --
Part IV Applications of Information Geometry --
11 Machine Learning [231] --
11.1 Clustering Patterns [231] --
11.1.1 Pattern Space and Divergence [231] --
11.1.2 Center of Cluster [232] --
11.1.3 k-Means: Clustering Algorithm [233] --
11.1.4 Voronoi Diagram [234] --
11.1.5 Stochastic Version of Classification and Clustering [236] --
11.1.6 Robust Cluster Center [238] --
11.1.7 Asmptotic Evaluation of Error Probability in Pattern Recognition: Chernoff Information [240] --
11.2 Geometry of Support Vector Machine [242] --
11.2.1 Linear Classifier [242] --
11.2.2 Embedding into High-Dimensional Space [245] --
11.2.3 Kernel Method [246] --
11.2.4 Riemannian Metric Induced by Kernel [247] --
11.3 Stochastic Reasoning: Belief Propagation and CCCP Algorithms [249] --
11.3.1 Graphical Model [250] --
11.3.2 Mean Field Approximation and m-Projection [252] --
11.3.3 Belief Propagation [255] --
1 L3.4 Solution of BP Algorithm [257] --
11.3.5 CCCP (Convex-Concave Computational Procedure). [259] --
11.4 Information Geometry of Boosting [260] --
11.4.1 Boosting: Integration of Weak Machines [261] --
11.4.2 Stochastic Interpretation of Machine [262] --
11.4.3 Construction of New Weak Machines [263] --
11.4.4 Determination of the Weights of Weak Machines [263] --
11.5 Bayesian Inference and Deep Learning [265] --
11.5.1 Bayesian Duality in Exponential Family [266] --
11.5.2 Restricted Boltzmann Machine [268] --
11.5.3 Unsupervised Learning of RBM [269] --
11.5.4 Geometry of Contrastive Divergence [273] --
11.5.5 Gaussian RBM [275] --
12 Natural Gradient Learning and Its Dynamics in Singular Regions [279] --
12.1 Natural Gradient Stochastic Descent Learning [279] --
12.1.1 On-Line Learning and Batch Learning [279] --
12.1.2 Natural Gradient: Steepest Descent Direction in Riemannian Manifold [282] --
12.1.3 Riemannian Metric, Hessian and Absolute Hessian [284] --
12.1.4 Stochastic Relaxation of Optimization Problem [286] --
12.1.5 Natural Policy Gradient in Reinforcement Learning [287] --
12.1.6 Mirror Descent and Natural Gradient [289] --
12.1.7 Properties of Natural Gradient Learning [290] --
12.2 Singularity in Learning: Multilayer Perceptron [296] --
12.2.1 Multilayer Perceptron [296] --
12.2.2 Singularities in M [298] --
12.2.3 Dynamics of Learning in M [302] --
12.2.4 Critical Slowdown of Dynamics [305] --
12.2.5 Natural Gradient Learning Is Free of Plateaus [309] --
12.2.6 Singular Statistical Models [310] --
12.2.7 Bayesian Inference and Singular Model [312] --
13 Signal Processing and Optimization [315] --
13.1 Principal Component Analysis [315] --
13.1.1 Eigenvalue Analysis [315] --
13.1.2 Principal Components, Minor Components and Whitening [316] --
13.1.3 Dynamics of Learning of Principal and Minor Components [319] --
13.2 Independent Component Analysis [322] --
13.2.3 Estimating Function of ICA: Semiparametric Approach [330] --
13.3 Non-negative Matrix Factorization [333] --
13.4 Sparse Signal Processing [336] --
13.4.1 Linear Regression and Sparse Solution [337] --
13.4.2 Minimization of Convex Function Under L1 Constraint [338] --
13.4.3 Analysis of Solution Path [341] --
13.4.4 Minkovskian Gradient Flow [343] --
13.4.5 Underdetermined Case [344] --
13.5 Optimization in Convex Programming [345] --
13.5.1 Convex Programming [345] --
13.5.2 Dually Flat Structure Derived from Barrier Function [347] --
13.5.3 Computational Complexity and m-curvature [348] --
13.6 Dual Geometry Derived from Game Theory [349] --
13.6.1 Minimization of Game-Score [349] --
13.6.2 Hyvärinen Score [353] --
References [359] --
MR, MR3495836
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