Learning Smooth Neural Functions via Lipschitz Regularization (2202.08345v2)

Abstract: Neural implicit fields have recently emerged as a useful representation for 3D shapes. These fields are commonly represented as neural networks which map latent descriptors and 3D coordinates to implicit function values. The latent descriptor of a neural field acts as a deformation handle for the 3D shape it represents. Thus, smoothness with respect to this descriptor is paramount for performing shape-editing operations. In this work, we introduce a novel regularization designed to encourage smooth latent spaces in neural fields by penalizing the upper bound on the field's Lipschitz constant. Compared with prior Lipschitz regularized networks, ours is computationally fast, can be implemented in four lines of code, and requires minimal hyperparameter tuning for geometric applications. We demonstrate the effectiveness of our approach on shape interpolation and extrapolation as well as partial shape reconstruction from 3D point clouds, showing both qualitative and quantitative improvements over existing state-of-the-art and non-regularized baselines.

• The paper introduces a Lipschitz regularization technique that ensures smoother neural implicit fields for improved 3D shape interpolation.
• It achieves computational efficiency with a simple four-line implementation and minimal hyperparameter tuning.
• Experimental results show lower Jacobian norms and enhanced robustness against adversarial perturbations in shape manipulation tasks.

Overview of "Learning Smooth Neural Functions via Lipschitz Regularization"

The paper presents a novel approach to ensuring smoothness in neural implicit fields, an increasingly prevalent representation for 3D shape modeling. Utilizing Lipschitz regularization, the authors focus on creating smoother latent spaces, which are critical for effectively manipulating and editing 3D shapes. This paper is especially relevant to the domain of geometric learning, where neural networks map 3D coordinates to scalar function values that describe these shapes.

Key Contributions

1. Lipschitz Regularization for Smoothness: The central contribution is the introduction of a Lipschitz regularization method tailored to improve the smoothness of latent spaces in neural implicit representations. By penalizing the Lipschitz constant of the neural function, the authors ensure that the interpolations and extrapolations across 3D shapes are smooth and free from unwanted artifacts.
2. Implementation Efficiency: The proposed method stands out due to its computational efficiency. It can be implemented in merely four lines of code and does not necessitate extensive hyperparameter tuning, making it an attractive alternative to prior weight normalization or spectral normalization techniques often used to enforce Lipschitz continuity.
3. Shape Manipulation Applications: The authors exhibit the method's utility through experiments on shape interpolation and extrapolation, showcasing qualitative and quantitative improvements. The regularization method enhances the robustness of neural networks against adversarial inputs, benefiting scenarios that demand resilience to imprecise or incomplete data, such as shape reconstruction from partial point clouds.

Strong Numerical Results

The paper demonstrates significant improvements over baseline methods, particularly in tasks demanding high-quality, smooth deformations between predefined shapes. The regularization effectively stabilizes the behavior of neural networks beyond the training data, leading to smooth interpolations visually and numerically, as evidenced by lower squared Jacobian norms.

Implications and Speculations for Future AI Developments

The introduction of Lipschitz regularization presents theoretical and practical implications for AI, particularly in the field of geometric learning. Theoretically, this approach contributes to a better understanding of how neural networks can maintain robust, smooth outputs amidst varying input conditions. Practically, the utility of such networks in graphics and 3D modeling tools could expand, enabling designers and engineers to manipulate shapes and interpolations with greater precision. Furthermore, this paper may inform future research on robust neural network architectures that inherently resist adversarial perturbations, thus enhancing security and reliability in critical applications.

Future research directions could involve exploring tighter bounds on the Lipschitz constant or integrating this method with other techniques, such as generative modeling or supervised learning tasks beyond geometry processing. Additionally, extending this work to more complex neural architectures and applications may uncover further advantages and insights.

In summary, this paper offers a precise and efficient approach to ensuring smoothness in neural functions, with promising results for 3D geometric learning and beyond. Its application could see substantial influence in fields that require intricate modeling and shape interpolation, promoting further developments in the intersection of deep learning and computer graphics.