BézierFlow: Finite-Dimensional Optimization

• BézierFlow is a framework that uses Bézier curve parameterizations to encode trajectories, surfaces, and schedulers, reducing high-dimensional problems to finite-dimensional ones.
• It employs control-point representations to map continuous functions into tractable manifolds, enabling efficient gradient flows and stability in shape and multi-objective optimization.
• BézierFlow demonstrates competitive empirical performance with lower FID scores and real-time optimization, ensuring efficient, PAC-stable solutions across diverse applications.

BézierFlow is a family of algorithmic frameworks built on Bézier curve parameterizations, enabling the reduction of high- or infinite-dimensional optimization or generation problems to tractable, finite-dimensional problems in diverse domains such as shape optimization, multi-objective optimization, and generative modeling. Central to BézierFlow is the encoding of trajectories, surfaces, or schedulers as Bézier curves or simplices, where control-point-based finite parameterizations induce differentiable and structure-preserving evolution, gradient flow, or sampling. The approach encompasses methods from vector field-driven shape flows to surface-wise descent in optimization, to continuous scheduler parameterizations in stochastic interpolant-based few-step generative models (Ruatta, 2013, Sannai et al., 2022, Min et al., 15 Dec 2025).

1. Foundational Principles and Mathematical Parameterization

BézierFlow frameworks utilize the Bézier curve or Bézier simplex representation to encode objects of interest. In shape optimization, the geometry of planar Jordan curves is captured via piecewise-continuous Bézier curves, where the set of control points

$$P=\{P_{i,j}\}_{0 \leq i \leq N,\,0 \leq j \leq D} \in \mathbb{R}^2$$

fully determines the shape. The global parametrization $\Gamma(t)$ on $[0,1]$ is realized by assembling Bernstein basis pieces per patch, mapping the infinite-dimensional space of $C^0$-continuous loops to a finite-dimensional manifold constrained by $P_{0,0} = P_{N,D}$ to ensure closure (Ruatta, 2013).

In multi-objective optimization, BézierFlow employs the Bézier simplex model. The Pareto front is embedded as the image of a simplex

$$\Delta^{M-1} = \{ t \in \mathbb{R}^M : t_m \geq 0,\, \sum_m t_m = 1 \}$$

under a Bézier map parameterized by multi-indexed control points $\{p_d\}$ in $\mathbb{R}^L$,

$$b(t|P) = \sum_{d \in \mathbb{N}^M_D} \binom{D}{d} t^d p_d$$

(Sannai et al., 2022).

In the setting of generative modeling with stochastic interpolants, BézierFlow parameterizes scheduler functions as Bézier polynomials,

$$\bar\alpha(s) = (\alpha_1-\alpha_0)\sum_{i=0}^n b_{i,n}(s) C_i^{(\alpha)} + \alpha_0$$

with endpoint constraints and interior monotonicity enforced by cumulative softmax parameterization of control points (Min et al., 15 Dec 2025).

2. Finite-Dimensional Reduction and Vector Field Formulation

BézierFlow transforms infinite- or high-dimensional continuous optimization tasks into finite-dimensional problems by representing functionals, gradients, or schedules in the control point space. In shape optimization, the cost functional $F(\alpha)$ induces an Eulerian shape-gradient $\nabla F(\alpha)$ along the curve. By sampling the gradient at discrete points on the Bézier curve, the vector $\mathbf{T}_{N,F}(P)$ is computed and mapped back to a vector field $V_F(P)$ in control point space through the inverse Bernstein block-diagonal transformation. This yields a finite-dimensional ODE system,

$\frac{dP}{dt} = V_F(P)$

subject to closure, allowing for integration by explicit Euler or Runge-Kutta methods (Ruatta, 2013). The gradient descent in this finite-dimensional regime replaces the original problem while preserving equivalence at sampling points.

In the surface-wise gradient descent regime for multi-objective optimization, gradients of the scalarized loss (weighted sum) at sampled simplex points are computed, and the Bézier control points are updated by least-squares fitting:

$$P^{(k+1)} = P^{(k)} - \alpha^{(k)} (Z^{(k)\top} Z^{(k)})^{-1} Z^{(k)\top} G^{(k)}$$

where $G^{(k)}$ contains the weighted gradients (Sannai et al., 2022).

3. Algorithmic Realizations and Training Procedures

BézierFlow’s implementations involve explicit workflows:

• Shape Optimization (Ruatta, 2013):
  • Discretize the time domain, solve for updated control points via Euler or Runge-Kutta, enforcing closure constraints.
  • Each iteration involves evaluating the shape-gradient at curve sampling points, mapping to vector fields, and updating control points.
  • Complexity per step: $O(ND)$ gradient evaluations, $O(ND^3)$ for block-diagonal linear solves.
• Multi-objective Optimization (Sannai et al., 2022):
  • At each iteration, sample weights from the simplex, compute current Bézier points, obtain weighted gradients, and fit updated control points by least squares.
  • Terminate after convergence or fixed iterations.
  • Computational cost per iteration: $O(N \cdot \lvert \mathbb{N}_D^M \rvert \cdot L)$ for evaluation and $O(\lvert \mathbb{N}_D^M \rvert^3)$ for least-squares fitting.
• Few-Step Generation (Min et al., 15 Dec 2025):
  • Scheduler control points (for the interpolant functions) are learned via teacher-forcing objectives, optimizing for perceptual distance (LPIPS) between high-NFE and low-NFE generated samples.
  • Enforcement of monotonicity and boundary constraints is achieved with cumulative softmax and fixed endpoint control points in the Bézier parameterization.
  • Training typically requires $\sim$15 minutes, with $\leq$100 Bézier and decoupled timestep parameters.

4. Theoretical Properties and Stability Guarantees

BézierFlow for multi-objective optimization establishes strong generalization guarantees based on PAC uniform stability. An algorithm is PAC uniformly stable if perturbing a single training sample causes at most a small change in expected per-weight loss, with high probability. The surface-wise BézierFlow update is shown to be decomposable and PAC stable, which, via standard concentration arguments, guarantees a bound on the difference between empirical and expected risk (Sannai et al., 2022).

The reduction from an infinite-dimensional functional to a finite-dimensional one in shape optimization is justified by the diffeomorphic mapping between continuous curves and the Bézier manifold, ensuring that minimization in control space is equivalent to minimization in curve space at the sampled points (Ruatta, 2013).

5. Empirical Performance and Comparative Results

BézierFlow demonstrates substantial efficiency and performance gains across domains:

Table: Sample Performance Comparison on CIFAR-10 (Diffusion Models, FID metric) (Min et al., 15 Dec 2025)

Method	NFE=4	NFE=6	NFE=8	NFE=10
UniPC (base)	50.30	19.33	9.64	6.16
LD3	12.04	3.56	2.43	2.62
BézierFlow	9.55	3.13	2.40	2.09

Key comparative outcomes include:

• BézierFlow achieves 2–3× lower FID at $\leq$10 NFEs over prior timestep-learning baselines in pretrained diffusion and flow models.
• In shape optimization, explicit Euler integration with 50 patches and degree 3 yields real-time (30 fps) contour propagation at $0.5$–$2$ ms per step on standard CPUs (Ruatta, 2013).
• Multi-objective BézierFlow achieves MSE of $5.51 \times 10^{-5}$ on “scaled-MED” with $N=30$, versus $1.51 \times 10^{-1}$ for NSGA-II, and lower GD/IGD across skew problems, especially in low-sample regimes (Sannai et al., 2022).

6. Applications and Limitations

Applications of BézierFlow span:

• Shape Optimization: Image segmentation and geometric deformation in the plane under arbitrary energy functionals (Ruatta, 2013).
• Multi-objective Optimization: Accurate Pareto front approximation for multi-objective problems with PAC stability and superior generalization, e.g., “scaled-MED,” “skew-3MED,” and “skew-3MMD” (Sannai et al., 2022).
• Few-Step Generation: Efficient few-step sampling in pretrained diffusion and flow frameworks, cross-domain transfer (e.g., 3D point clouds, layout generation), and real-time applications with minimal hardware requirement (Min et al., 15 Dec 2025).

Limitations include:

• Polynomial Bézier bases may not be the optimal representation in all regimes. Alternative bases such as rational, spline, or orthogonal polynomials are proposed directions for further reduction in parameter count (Min et al., 15 Dec 2025).
• In generative modeling, the SI assumption restricts applicability; non-SI frameworks such as GANs currently fall outside the BézierFlow domain.
• Global convergence is not theoretically guaranteed in multi-objective optimization, though empirical stability is consistently observed (Sannai et al., 2022).

7. Future Directions

Research trajectories for BézierFlow include:

• Enhanced scheduler expressivity via alternative basis functions beyond polynomials.
• Integration with adaptive ODE solvers in generative modeling to allow for variable NFE inference.
• Extending the finite-dimensional reduction strategy to non-SI generative processes and adversarial regimes.
• Leveraging real-time, topology-preserving deformation and segmentation capabilities for interactive systems and vision tasks (Ruatta, 2013, Min et al., 15 Dec 2025).

BézierFlow thus provides a principled, parameter-efficient framework for encoding and optimizing trajectories, shapes, and schedulers across a spectrum of high-impact computational problems, grounded in Bézier geometry and modern finite-dimensional analysis.