Series Foreword |
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ix | |
Preface |
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xi | |
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1 Invariant Representations: Mathematics of Invariance |
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1 | (22) |
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1.1 Introduction and Motivation |
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1 | (3) |
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1.2 Invariance Reduces Sample Complexity of Learning |
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4 | (1) |
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1.3 Unsupervised Learning and Computation of an Invariant Signature (One-Layer Architecture) |
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5 | (4) |
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1.4 Partially Observable Groups |
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9 | (1) |
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1.5 Optimal Templates for Scale and Position Invariance Are Gabor Functions |
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10 | (1) |
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1.6 Quasi Invariance to Nongroup Transformations Requires Class-Specific Templates |
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11 | (3) |
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1.7 Two Stages in the Computation of an Invariant Signature: Extension of the HW Module to Hierarchical Architectures |
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14 | (7) |
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1.8 Deep Networks and i-Theory |
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21 | (2) |
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2 Biophysical Mechanisms of Invariance: Unsupervised Learning, Tuning, and Pooling |
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23 | (6) |
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2.1 A Single-Cell Model of Simple and Complex Cells |
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23 | (1) |
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2.2 Learning the Wiring in the Single-Cell Model |
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24 | (1) |
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2.3 Hebb Synapses and Principal Components |
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24 | (2) |
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2.4 Spectral Theory and Pooling |
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26 | (2) |
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2.5 Tuning of Simple Cells |
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28 | (1) |
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3 Retinotopic Areas: V1, V2, V4 |
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29 | (16) |
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29 | (6) |
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3.1.1 A New Model from i-Theory for Eccentricity Dependence of Receptive Fields |
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29 | (3) |
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32 | (2) |
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3.1.3 Scale and Position Invariance in V1 |
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34 | (1) |
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3.1.4 Tuning of Cells in V1 |
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35 | (1) |
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35 | (10) |
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35 | (1) |
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3.2.2 Predictions of Crowding Properties in the Foveola and Outside It (Bouma's Law) |
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36 | (3) |
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3.2.3 Scale and Shift Invariance Dictates the Architecture of the Retina and the Retinotopic Cortex |
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39 | (3) |
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3.2.4 Tuning of Simple Cells in V2 and V4 |
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42 | (3) |
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4 Class-Specific Approximate Invariance in Inferior Temporal Cortex |
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45 | (8) |
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4.1 From Generic Templates to Class-Specific Tuning |
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45 | (1) |
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4.2 Development of Class-Specific and Object-Specific Modules |
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45 | (3) |
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4.3 Domain-Specific Regions in the Ventral Stream |
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48 | (1) |
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4.4 Tuning in the Inferior Temporal Cortex |
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49 | (1) |
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4.5 Mirror-Symmetric Tuning in the Face Patches and Pooling over Principal Components |
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49 | (4) |
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53 | (12) |
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53 | (1) |
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5.2 i-Theory, Deep Learning Networks, and the Visual Cortex |
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54 | (1) |
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5.3 Predictions and Explanations |
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55 | (1) |
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56 | (5) |
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61 | (4) |
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65 | (44) |
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A.1 Invariant Representations and Bounds on Learning Rates: Sample Complexity |
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65 | (3) |
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66 | (1) |
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A.1.2 Scale and Translation: 1-D Affine Group |
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67 | (1) |
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A.2 One-Layer Architecture: Invariance and Selectivity |
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68 | (12) |
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A.2.1 Equivalence between Orbits and Probability Distributions |
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68 | (1) |
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A.2.2 Cramer-Wold Theorem and Random Projections for Probability Distributions |
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69 | (1) |
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A.2.3 Finite Random Projections Almost Discriminate among Different Probability Distributions |
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70 | (2) |
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A.2.4 Number of Templates Depends on Pooling Size |
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72 | (3) |
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A.2.5 Partially Observable Groups |
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75 | (4) |
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A.2.6 Approximate Invariance to Nongroup Transformations |
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79 | (1) |
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A.3 Multilayer Architecture: Invariance, Covariance, and Selectivity |
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80 | (6) |
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A.3.1 Recursive Definition of Simple and Complex Responses |
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80 | (2) |
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A.3.2 Inheriting Transformations: Covariance |
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82 | (4) |
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A.4 Complex Cells: Wiring and Invariance |
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86 | (1) |
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A.5 Mirror-Symmetric Templates Lead to Odd-Even Covariance Eigenfunctions |
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86 | (1) |
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A.6 Gabor-like Shapes from Translation Group |
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87 | (5) |
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A.6.1 Spectral Properties of Template Transformations Covariance Operator: Cortical Equation |
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87 | (2) |
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A.6.2 Cortical Equation: Derivation and Solution |
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89 | (3) |
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92 | (5) |
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A.7.1 Derivation from an Invariance Argument for 1-D Affine Group |
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92 | (2) |
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A.7.2 Wavelet Templates from Best Invariance and Heisenberg Principle |
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94 | (3) |
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A.8 Transformations with Lie Group Local Structure: HW Module |
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97 | (1) |
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A.9 Factorization of Invariances |
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98 | (3) |
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A.10 Invariant Representations and Their Cost in Number of Orbit Elements |
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101 | (5) |
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A.10.1 Discriminability Cost for Hierarchical versus Single-Layer Architecture |
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102 | (2) |
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A.10.2 Invariance Cost for Hierarchical versus Single-Layer architecture |
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104 | (2) |
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A.11 Nonlinearities Are Key in Gaining Compression from Repeated Random Projections |
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106 | (3) |
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A.11.1 Hierarchical (Linear) Johnson-Lindenstrauss Lemma |
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106 | (1) |
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A.11.2 Example of Nonlinear Extension of Johnson-Lindenstrauss Lemma |
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107 | (2) |
References |
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109 | (8) |
Index |
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117 | |