Tensors for Data Processing: Theory, Methods and Applications presents both classical and state-of-the-art methods on tensor computation for data processing, covering computation theories, processing methods, computing and engineering applications, with an emphasis on techniques for data processing. This reference is ideal for students, researchers and industry developers who want to understand and use tensor-based data processing theories and methods.
As a higher-order generalization of a matrix, tensor-based processing can avoid multi-linear data structure loss that occurs in classical matrix-based data processing methods. This move from matrix to tensors is beneficial for many diverse application areas, including signal processing, computer science, acoustics, neuroscience, communication, medical engineering, seismology, psychometric, chemometrics, biometric, quantum physics and quantum chemistry.
1. 4.1 Introduction - p. 91 2. 4.2 A brief introduction to Riemannian optimization - p. 93 3. 4.3 Riemannian Tucker manifold geometry - p. 97 4. 4.4 Algorithms for tensor learning problems - p. 104 5. 4.5 Experiments - p. 107 6. 4.6 Conclusion - p. 117 7. References - p. 117 8. Chapter 5. Generalized thresholding for low-rank tensor recovery: approaches based on model and learning - p. 121 9. Chapter 6. Tensor principal component analysis - p. 153 10. Abstract - p. 153 11. Contents - p. 153 12. 6.1 Introduction - p. 153 13. 6.2 Notations and preliminaries - p. 155 14. 6.3 Tensor PCA for Gaussian-noisy data - p. 161 15. 6.4 Tensor PCA for sparsely corrupted data - p. 166 16. 6.5 Tensor PCA for outlier-corrupted data - p. 191 17. 6.6 Other tensor PCA methods - p. 207 18. 6.7 Future work - p. 208 19. 6.8 Summary - p. 208 20. References - p. 209 21. Chapter 7. Tensors for deep learning theory: Analyzing deep learning architectures via tensorization - p. 215 22. Abstract - p. 215 23. Contents - p. 215 24. 7.1 Introduction - p. 215 25. 7.2 Bounding a function's expressivity via tensorization - p. 217 26. 7.3 A case study: self-attention networks - p. 223 27. 7.4 Convolutional and recurrent networks - p. 242 28. 7.5 Conclusion - p. 245 29. References - p. 245 30. Chapter 8. Tensor network algorithms for image classification - p. 249 31. Abstract - p. 249 32. Contents - p. 249 33. 8.1 Introduction - p. 249 34. 8.2 Background - p. 251 35. 8.3 Tensorial extensions of support vector machine - p. 258 36. 8.4 Tensorial extension of logistic regression - p. 284 37. 8.5 Conclusion - p. 288 38. References - p. 289 39. Chapter 9. High-performance tensor decompositions for compressing and accelerating deep neural networks - p. 293 40. Abstract - p. 293 41. Contents - p. 294 42. 9.1 Introduction and motivation - p. 294 43. 9.2 Deep neural networks - p. 295 44. 9.3 Tensor networks and their decompositions - p. 306 45. 9.4 Compressing deep neural networks - p. 320 46. 9.5 Experiments and future directions - p. 333 47. References - p. 338 48. Chapter 10. Coupled tensor decompositions for data fusion - p. 341 49. Chapter 11. Tensor methods for low-level vision - p. 371 50. Abstract - p. 371 51. Contents - p. 371 52. Acknowledgements - p. 371 53. 11.1 Low-level vision and signal reconstruction - p. 371 54. 11.2 Methods using raw tensor structure - p. 378 55. 11.3 Methods using tensorization - p. 409 56. 11.4 Examples of low-level vision applications - p. 415 57. 11.5 Remarks - p. 419 58. References - p. 420 59. Chapter 12. Tensors for neuroimaging: A review on applications of tensors to unravel the mysteries of the brain - p. 427 60. Chapter 13. Tensor representation for remote sensing images - p. 483 61. Abstract - p. 483 62. Contents - p. 483 63. 13.1 Introduction - p. 488 64. 13.2 Optical remote sensing: HSI and MSI fusion - p. 487 65. 13.3 Polarimetric synthetic aperture radar: feature extraction - p. 517 66. References - p. 532 67. Chapter 14. Structured tensor train decomposition for speeding up kernel-based learning - p. 537 68. Index - p. 565
Tensors for Data Processing: Theory, Methods and Applications presents both classical and state-of-the-art methods on tensor computation for data processing, covering computation theories, processing methods, computing and engineering applications, with an emphasis on techniques for data processing. This reference is ideal for students, researchers and industry developers who want to understand and use tensor-based data processing theories and methods.
As a higher-order generalization of a matrix, tensor-based processing can avoid multi-linear data structure loss that occurs in classical matrix-based data processing methods. This move from matrix to tensors is beneficial for many diverse application areas, including signal processing, computer science, acoustics, neuroscience, communication, medical engineering, seismology, psychometric, chemometrics, biometric, quantum physics and quantum chemistry.
1. 4.1 Introduction - p. 91 2. 4.2 A brief introduction to Riemannian optimization - p. 93 3. 4.3 Riemannian Tucker manifold geometry - p. 97 4. 4.4 Algorithms for tensor learning problems - p. 104 5. 4.5 Experiments - p. 107 6. 4.6 Conclusion - p. 117 7. References - p. 117 8. Chapter 5. Generalized thresholding for low-rank tensor recovery: approaches based on model and learning - p. 121 9. Chapter 6. Tensor principal component analysis - p. 153 10. Abstract - p. 153 11. Contents - p. 153 12. 6.1 Introduction - p. 153 13. 6.2 Notations and preliminaries - p. 155 14. 6.3 Tensor PCA for Gaussian-noisy data - p. 161 15. 6.4 Tensor PCA for sparsely corrupted data - p. 166 16. 6.5 Tensor PCA for outlier-corrupted data - p. 191 17. 6.6 Other tensor PCA methods - p. 207 18. 6.7 Future work - p. 208 19. 6.8 Summary - p. 208 20. References - p. 209 21. Chapter 7. Tensors for deep learning theory: Analyzing deep learning architectures via tensorization - p. 215 22. Abstract - p. 215 23. Contents - p. 215 24. 7.1 Introduction - p. 215 25. 7.2 Bounding a function's expressivity via tensorization - p. 217 26. 7.3 A case study: self-attention networks - p. 223 27. 7.4 Convolutional and recurrent networks - p. 242 28. 7.5 Conclusion - p. 245 29. References - p. 245 30. Chapter 8. Tensor network algorithms for image classification - p. 249 31. Abstract - p. 249 32. Contents - p. 249 33. 8.1 Introduction - p. 249 34. 8.2 Background - p. 251 35. 8.3 Tensorial extensions of support vector machine - p. 258 36. 8.4 Tensorial extension of logistic regression - p. 284 37. 8.5 Conclusion - p. 288 38. References - p. 289 39. Chapter 9. High-performance tensor decompositions for compressing and accelerating deep neural networks - p. 293 40. Abstract - p. 293 41. Contents - p. 294 42. 9.1 Introduction and motivation - p. 294 43. 9.2 Deep neural networks - p. 295 44. 9.3 Tensor networks and their decompositions - p. 306 45. 9.4 Compressing deep neural networks - p. 320 46. 9.5 Experiments and future directions - p. 333 47. References - p. 338 48. Chapter 10. Coupled tensor decompositions for data fusion - p. 341 49. Chapter 11. Tensor methods for low-level vision - p. 371 50. Abstract - p. 371 51. Contents - p. 371 52. Acknowledgements - p. 371 53. 11.1 Low-level vision and signal reconstruction - p. 371 54. 11.2 Methods using raw tensor structure - p. 378 55. 11.3 Methods using tensorization - p. 409 56. 11.4 Examples of low-level vision applications - p. 415 57. 11.5 Remarks - p. 419 58. References - p. 420 59. Chapter 12. Tensors for neuroimaging: A review on applications of tensors to unravel the mysteries of the brain - p. 427 60. Chapter 13. Tensor representation for remote sensing images - p. 483 61. Abstract - p. 483 62. Contents - p. 483 63. 13.1 Introduction - p. 488 64. 13.2 Optical remote sensing: HSI and MSI fusion - p. 487 65. 13.3 Polarimetric synthetic aperture radar: feature extraction - p. 517 66. References - p. 532 67. Chapter 14. Structured tensor train decomposition for speeding up kernel-based learning - p. 537 68. Index - p. 565
