layout: true class: typo, typo-selection --- count: false class: nord-dark, middle, center # 📈 Convex Fitting Using B-splines @luk036 👨‍💻 · 2026 📅 --- ### 📋 Agenda .pull-left[ **Part 1: B-spline Basics** 📐 - B-spline Representation - Convexity in B-splines **Part 2: Optimization** ⚙️ - Convex Fitting Problem - Solving Methods ] .pull-right[ **Part 3: Applications** 🔬 - Examples - Extensions ] --- class: nord-light, middle, center ## Part 1: B-spline Basics 📐 --- ### 🎯 Introduction - Convex fitting: Approximating data with a convex function 🎯 - B-splines: Flexible and powerful basis functions 🧩 - Combining convexity constraints with B-spline representation ⚖️ --- ### 📐 B-splines Basics - Piecewise polynomial functions 🧩 - Defined by: - Degree of polynomials - Knot sequence - Control points - Properties: - Local support - Smoothness - Easy to manipulate .mermaid[ <pre> graph LR A[B-spline] --> B[Degree k] A --> C[Knot Sequence] A --> D[Control Points] B --> E[Local Support] C --> F[Smoothness Cʲ] D --> G[Convex Combination] style A fill:#fff3e0,stroke:#e65100,stroke-width:3px style B fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style C fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style D fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px style E fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style F fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style G fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px </pre> ] --- ### 📝 B-spline Representation A B-spline curve is represented as: $$ f(x) = \sum_{i=1}^n {\color{orange} c_i} \, {\color{blue} B_i(x)} $$ Where: - ${\color{orange} c_i}$ are control points - ${\color{blue} B_i(x)}$ are B-spline basis functions --- ### ✅ Ensuring Convexity in B-splines A sufficient condition for convexity: $$ {\color{orange} c_{i+1} - c_i} \geq {\color{orange} c_i - c_{i-1}} \quad \forall i$$ This ensures that the second derivative is non-negative everywhere. .mermaid[ <pre> graph TD A[Control Points Condition] --> B["c_{i+1} - c_i ≥ c_i - c_{i-1}"] B --> C[Convex Curve ✅] D[Violated Condition] --> E[Non-Convex Curve ❌] style A fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style B fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style C fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style D fill:#ffcdd2,stroke:#c62828,stroke-width:3px style E fill:#ffcdd2,stroke:#c62828,stroke-width:3px </pre> ] --- class: nord-light, middle, center ## Part 2: Optimization ⚙️ --- ### 🎯 Convex Fitting Problem Given data points $(x_j, y_j)$, we want to: Minimize: $\sum_j (f(x_j) - {\color{coral} y_j})^2$ Subject to: - $f$ is convex - $f$ is a B-spline .mermaid[ <pre> graph LR A[Data Points] --> B[Select Basis] B --> C[Formulate QP] C --> D[Solve] D --> E[Fitted Curve] style A fill:#fff3e0,stroke:#e65100,stroke-width:3px style B fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style C fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px style D fill:#f3e5f5,stroke:#7b1fa2,stroke-width:3px style E fill:#c8e6c9,stroke:#2e7d32,stroke-width:3px </pre> ] --- ### 📐 Optimization Formulation Minimize: $\sum_j (\sum_i {\color{orange} c_i} {\color{blue} B_i(x_j)} - {\color{coral} y_j})^2$ Subject to: - $c_{i+1} - c_i \geq c_i - c_{i-1}$ forall i - Additional constraints (e.g., boundary conditions) --- ### ⚙️ Solving the Optimization Problem - Quadratic Programming (QP) problem 📈 - Efficient solvers available (e.g., CVXOPT, MOSEK) 🚀 - Can handle large datasets and many control points ⚡ --- ### 🏆 Advantages of B-spline Convex Fitting 1. Flexibility in representing complex convex shapes 🎨 2. Smoothness control 📈 3. Local control of the fit 🎯 4. Efficient computation and evaluation ⚡ --- class: nord-light, middle, center ## Part 3: Applications 🔬 --- ### 🌍 Applications 1. Economics: Production functions, utility functions 💹 2. Statistics: Density estimation 📊 3. Machine Learning: Regularization, model fitting 🤖 4. Computer Graphics: Shape modeling 🖼️ 5. Operations Research: Convex cost functions 📐 --- ### 📚 Example: Production Function Estimation - Data: Input-output pairs from a production process - Goal: Estimate a convex production function - Approach: Fit a convex B-spline to the data - Result: Smooth, convex estimate of the production function --- ### ⚠️ Challenges and Considerations 1. Choosing the number and placement of knots 📏 2. Balancing fit quality and convexity constraints ⚖️ 3. Handling noisy data 🎲 4. Extrapolation behavior 🔭 .mermaid[ <pre> graph LR A[Knot Placement] --> B[Few Knots] A --> C[Many Knots] B --> D[Underfitting] C --> E[Overfitting] D --> F[High Bias] E --> G[High Variance] style A fill:#fff3e0,stroke:#e65100,stroke-width:3px style B fill:#e3f2fd,stroke:#1565c0,stroke-width:3px style C fill:#ffcdd2,stroke:#c62828,stroke-width:3px style D fill:#ffcdd2,stroke:#c62828,stroke-width:3px style E fill:#ffcdd2,stroke:#c62828,stroke-width:3px style F fill:#ffcdd2,stroke:#c62828,stroke-width:3px style G fill:#ffcdd2,stroke:#c62828,stroke-width:3px </pre> ] --- ### 🔄 Extensions and Variations 1. Multivariate convex fitting 📊 2. Monotone convex fitting 📈 3. Adaptive knot placement 🎯 4. Incorporating shape constraints (e.g., asymptotes) ⛓️ --- ### 🛠️ Implementation Tools - Python: SciPy (B-splines), CVXPY (optimization) 🐍 - MATLAB: Curve Fitting Toolbox, Optimization Toolbox 🧮 - R: splines package, CVXR for optimization 📊 - Specialized software: SCOP (Shape Constrained Optimization) 🔬 --- ### 🔮 Future Directions 1. Integration with machine learning models 2. Real-time convex fitting for streaming data 3. Theoretical advances in adaptive knot placement 4. Applications in high-dimensional spaces --- ### 🏁 Conclusion - B-spline convex fitting: powerful tool for data approximation 🏆 - Combines flexibility of B-splines with convexity constraints 🧩 - Wide range of applications across various fields 🌍 - Active area of research with promising future directions 🔮 --- ### 📑 References - De Boor, C. (1978). A Practical Guide to Splines. Springer-Verlag. - Dierckx, P. (1993). Curve and Surface Fitting with Splines. Oxford University Press. - Relevant papers on convex fitting and B-splines applications. --- count: false class: nord-dark, middle, center ## Q&A 🎤