Convex Fitting Using B-splines

---

# Convex Fitting Using B-splines

@luk036 👨‍💻 · 2026 📅

---

#### 📋 Agenda

**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

]
---

### 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:#ff9800
    style B fill:#4caf50
    style C fill:#2196f3
    style D fill:#9c27b0
    style E fill:#4caf50
    style F fill:#2196f3
    style G fill:#9c27b0
</pre>
]

---

### B-spline Representation

A B-spline curve is represented as:

$$ f(x) = \sum_{i=1}^n c_i B_i(x) $$

Where:
- $c_i$ are control points
- $B_i(x)$ are B-spline basis functions

---

### Ensuring Convexity in B-splines

A sufficient condition for convexity:

$$ c_{i+1} - c_i \geq c_i - c_{i-1} \quad \forall i$$

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:#4caf50
    style B fill:#4caf50
    style C fill:#4caf50
    style D fill:#f44336
    style E fill:#f44336
</pre>
]

---

### Convex Fitting Problem

Given data points $(x_j, y_j)$, we want to:

Minimize: $\sum_j (f(x_j) - y_j)^2$

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:#ff9800
    style B fill:#2196f3
    style C fill:#4caf50
    style D fill:#9c27b0
    style E fill:#4caf50
</pre>
]

---

### Optimization Formulation

Minimize: $\sum_j (\sum_i c_i B_i(x_j) - y_j)^2$

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

---

### 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:#ff9800
    style B fill:#2196f3
    style C fill:#f44336
    style D fill:#f44336
    style E fill:#f44336
</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.

---

# Q&A 🎤