"pretend you’re a mathematics professor who can explain most deep concepts with ease. you’re righting a book about understanding the mathematics behind SVR where the relation is only linear. Give me a table of content for this book.
Table of Contents

Preface
Acknowledgements
About the Author

Chapter 1: Introduction to Support Vector Regression (SVR)
1.1 Overview of Supervised Learning
1.2 Regression Analysis
1.3 Support Vector Machines (SVMs)
1.4 The Need for Support Vector Regression (SVR)
1.5 Differences Between SVM and SVR

Chapter 2: Linear Support Vector Regression (LSVR)
2.1 LSVR in Parametric Models
2.2 Basic Concepts of LSVR
2.3 Hyperplanes and Decision Boundaries
2.4 Formulation of LSVR Problems

Chapter 3: Optimization (Convex and Non-convex) in LSVR
3.1 Objective Function of LSVR
3.2 Convex Optimization
3.3 Non-convex Optimization
3.4 Constraints in Optimization

Chapter 4: Methods for Solving LSVR Problems
4.1 General Overview of Numerical Methods
4.2 Quadratic Programming and Sequential Minimal Optimization (SMO)
4.3 Gradient Descent Methods
4.4 Primal-Dual Interior Point Methods

Chapter 5: Loss Functions in LSVR
5.1 Linear Loss Functions
5.2 The Epsilon-Insensitive Loss Function
5.3 Hinge Loss Functions
5.4 Huber Loss Functions
5.5 The Role of Loss Functions in Regularization

Chapter 6: Regularization and Model Selection in LSVR
6.1 Regularization Techniques
6.2 Bias-Variance Trade-off
6.3 Cross-validation Techniques
6.4 Model Selection in LSVR

Chapter 7: Applications of LSVR
7.1 Real-world Applications
7.2 Case Studies in Finance, Healthcare, and More
7.3 Limitations of LSVR and Non-linear Alternatives

Chapter 8: Interpreting and Evaluating LSVR Results
8.1 Prediction and Confidence Intervals
8.2 Performance Metrics in LSVR
8.3 Model Explanations and Understanding Model Complexity

Chapter 9: Implementing LSVR Using Python
9.1 Python Libraries for LSVR
9.2 Step-by-Step Guide to Implement LSVR
9.3 Troubleshooting Common Issues

Chapter 10: Conclusion and Future Perspectives
10.1 Recap of Key Concepts
10.2 Resources for Further Learning
10.3 Exploring Non-linear SVR and Deep Learning Techniques

Appendix A: Linear Algebra Refresher
A.1 Vectors, Matrices, and Tensors
A.2 Basic Operations and Properties
A.3 Eigenvalues and Eigenvectors

Appendix B: Basic Calculus and Optimization
B.1 Limits, Derivatives, and Integrals
B.2 Multivariate Calculus
B.3 Constrained Optimization and Lagrange Multipliers

Bibliography
Index
write a full outline of all the key ideas you gonna discuss in “1.5 Differences Between SVM and SVR”
1.5 Differences Between SVM and SVR

1.5.1 Introduction to SVM and SVR Differences
- Brief overview of SVM and SVR
- Importance of understanding their differences

1.5.2 Core Objectives and Problem Structures
- SVM: Classification of data points into discrete classes
- SVR: Estimation of continuous values
- Differences in formulation and optimization problems

1.5.3 Loss Functions and Cross-validation
- Different loss functions used in SVM (e.g., hinge loss) and SVR (e.g., epsilon-insensitive loss)
- Differences in cross-validation techniques for model selection

1.5.4 Decision Boundaries vs. Regression Functions
- SVM: Maximizing the margin between classes (the decision boundary)
- SVR: Minimizing the distance between predictions and true values (the regression function)
- Geometrical interpretations and differences

1.5.5 Regularization and Margin Constraints
- Role of regularization in SVM and SVR
- Differences in soft margin and slack variables between SVM and SVR

1.5.6 Extension to Non-linear Tasks
- SVM: Kernel trick for mapping data to higher dimensions
- SVR: Non-linear regression by means of transformed feature spaces
- Differences in kernel choices and parameter tuning

1.5.7 Software Implementation and Practical Considerations
- Differences in libraries and algorithms used for SVM and SVR
- Computational complexity and scalability issues
- Application domains where each method might be more suitable

1.5.8 Conclusion and Connecting Ideas
- Recap of key differences between SVM and SVR
- Guidance on selecting the appropriate method based on the specific problem setting
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write "1.5.2 Core Objectives and Problem Structures" section now