Shared Gaussian Process Latent Variables Models - Oxford Brookes ...

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Contents 1 Introduction 10 1.1 Overview of the thesis . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3 Notations and Conventions . . . . . . . . . . . . . . . . . . . . . 13 2 Background 14 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.1.1 Curse of Dimensionality . . . . . . . . . . . . . . . . . . 15 2.2 Dimensionality Reduction . . . . . . . . . . . . . . . . . . . . . 17 2.3 Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Spectral Dimensionality Reduction . . . . . . . . . . . . . . . . . 21 2.5 Non-Linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5.1 Kernel-Trick . . . . . . . . . . . . . . . . . . . . . . . . 26 2.5.2 Proximity Graph Methods . . . . . . . . . . . . . . . . . 29 2.5.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.6 Generative Dimensionality Reduction . . . . . . . . . . . . . . . 35 2.7 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.7.1 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3
4 CONTENTS 2.7.2 Training . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2.7.3 Relevance Vector Machine . . . . . . . . . . . . . . . . . 45 2.8 GP-LVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.8.1 Latent Constraints . . . . . . . . . . . . . . . . . . . . . 49 2.8.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.9 Shared Dimensionality Reduction . . . . . . . . . . . . . . . . . 54 2.10 Feature Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 Shared GP-LVM 59 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.2 Shared GP-LVM . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . 63 3.2.2 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3 Subspace GP-LVM . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4 NCCA 80 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Shared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.4 Private . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Page 1 and 2: Shared Gaussian Process Latent Vari
Page 3: Acknowledgements 2
Page 7 and 8: 6 CONTENTS 5.10 Summary . . . . . .
Page 9 and 10: 8 LIST OF FIGURES 3.11 Toy data3: l
Page 11 and 12: Chapter 1 Introduction Information
Page 13 and 14: 12 CHAPTER 1. INTRODUCTION current
Page 15 and 16: Chapter 2 Background 2.1 Introducti
Page 17 and 18: 16 CHAPTER 2. BACKGROUND 1 0.9 0.8
Page 19 and 20: 18 CHAPTER 2. BACKGROUND X to its o
Page 21 and 22: 20 CHAPTER 2. BACKGROUND dimensions
Page 23 and 24: 22 CHAPTER 2. BACKGROUND approximat
Page 25 and 26: 24 CHAPTER 2. BACKGROUND diagonal c
Page 27 and 28: 26 CHAPTER 2. BACKGROUND unraveled.
Page 29 and 30: 28 CHAPTER 2. BACKGROUND apply an a
Page 31 and 32: 30 CHAPTER 2. BACKGROUND be connect
Page 33 and 34: 32 CHAPTER 2. BACKGROUND Local Line
Page 35 and 36: 34 CHAPTER 2. BACKGROUND 15 10 5 0
Page 37 and 38: 36 CHAPTER 2. BACKGROUND by W, Figu
Page 39 and 40: 38 CHAPTER 2. BACKGROUND Figure 2.4
Page 41 and 42: 40 CHAPTER 2. BACKGROUND X ∈ ℜ
Page 43 and 44: 42 CHAPTER 2. BACKGROUND 2.5 2 1.5
Page 45 and 46: 44 CHAPTER 2. BACKGROUND rupted by
Page 47 and 48: 46 CHAPTER 2. BACKGROUND where ˜ C
Page 49 and 50: 48 CHAPTER 2. BACKGROUND prior over
Page 51 and 52: 50 CHAPTER 2. BACKGROUND generative
Page 53 and 54: 52 CHAPTER 2. BACKGROUND the latent
Page 55 and 56:
54 CHAPTER 2. BACKGROUND 2.9 Shared
Page 57 and 58:
56 CHAPTER 2. BACKGROUND Y X Figure
Page 59 and 60:
58 CHAPTER 2. BACKGROUND Gaussian p
Page 61 and 62:
60 CHAPTER 3. SHARED GP-LVM sponden
Page 63 and 64:
62 CHAPTER 3. SHARED GP-LVM makes m
Page 65 and 66:
64 CHAPTER 3. SHARED GP-LVM where t
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66 CHAPTER 3. SHARED GP-LVM 4 3 2 1
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68 CHAPTER 3. SHARED GP-LVM 1 0.8 0
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70 CHAPTER 3. SHARED GP-LVM Example
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72 CHAPTER 3. SHARED GP-LVM 0.3 0.2
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74 CHAPTER 3. SHARED GP-LVM φ Y X
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76 CHAPTER 3. SHARED GP-LVM leading
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78 CHAPTER 3. SHARED GP-LVM servati
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Chapter 4 NCCA 4.1 Introduction In
Page 83 and 84:
82 CHAPTER 4. NCCA as u Y i = x S
Page 85 and 86:
84 CHAPTER 4. NCCA X Y Z X X S Y Z
Page 87 and 88:
86 CHAPTER 4. NCCA By pre-multiplyi
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88 CHAPTER 4. NCCA 1 0.8 0.6 0.4 0.
Page 91 and 92:
90 CHAPTER 4. NCCA 4.5 Extensions W
Page 93 and 94:
Chapter 5 Applications 5.1 Introduc
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94 CHAPTER 5. APPLICATIONS as optic
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96 CHAPTER 5. APPLICATIONS of an im
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98 CHAPTER 5. APPLICATIONS histogra
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100 CHAPTER 5. APPLICATIONS the dat
Page 103 and 104:
102 CHAPTER 5. APPLICATIONS Figure
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104 CHAPTER 5. APPLICATIONS the GP
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106 CHAPTER 5. APPLICATIONS varianc
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108 CHAPTER 5. APPLICATIONS locatio
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Page 113 and 114:
112 CHAPTER 5. APPLICATIONS Error (
Page 115 and 116:
Page 117 and 118:
116 CHAPTER 5. APPLICATIONS over th
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118 CHAPTER 5. APPLICATIONS age rep
Page 121 and 122:
120 CHAPTER 6. CONCLUSIONS vated an
Page 123 and 124:
122 CHAPTER 6. CONCLUSIONS a shared
Page 125 and 126:
124 BIBLIOGRAPHY [7] S. Belongie, J
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126 BIBLIOGRAPHY [24] K. Grochow, S
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128 BIBLIOGRAPHY [39] D. MacKay. Ba
Page 131 and 132:
130 BIBLIOGRAPHY [56] H. A. Simon.
Page 133 and 134:
132 BIBLIOGRAPHY [74] S. Wachter an
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