JacNet: Learning Functions with Structured Jacobians (2408.13237v1)

Abstract: Neural networks are trained to learn an approximate mapping from an input domain to a target domain. Incorporating prior knowledge about true mappings is critical to learning a useful approximation. With current architectures, it is challenging to enforce structure on the derivatives of the input-output mapping. We propose to use a neural network to directly learn the Jacobian of the input-output function, which allows easy control of the derivative. We focus on structuring the derivative to allow invertibility and also demonstrate that other useful priors, such as $k$-Lipschitz, can be enforced. Using this approach, we can learn approximations to simple functions that are guaranteed to be invertible and easily compute the inverse. We also show similar results for 1-Lipschitz functions.

• The paper proposes learning the Jacobian matrix directly to control function derivatives and enhance neural network training.
• It employs numerical integration and differentiable backpropagation to enforce regularity conditions like invertibility and Lipschitz continuity.
• Experimental results on exponential and absolute value functions validate the approach’s effectiveness in maintaining structural properties.

JacNet: Learning Functions with Structured Jacobians

The paper "JacNet: Learning Functions with Structured Jacobians" by Jonathan Lorraine and Safwan Hossain introduces an innovative approach toward neural network training that focuses on learning the Jacobian matrix of the input-output mapping rather than the traditional point-wise functional approximation. This methodology permits imposing explicit structures on the derivatives of the learned function, addressing common issues related to invertibility and Lipschitz continuity in learned models.

Background and Motivation

Neural networks (NNs) are ubiquitously employed as universal function approximators due to their flexibility in modeling complex input-output relationships. Despite their success in achieving precise point-wise approximations, controlling the derivatives of these approximations has remained a challenging task. This control is crucial for various applications where derivative information is necessary, such as generative adversarial networks (GANs), multi-agent learning algorithms, and hyperparameter optimization.

The authors propose to directly learn the Jacobian matrix using a neural network, facilitating the incorporation of prior knowledge about the true function’s derivatives during training. This approach allows for enforcing specific regularity conditions, making learned functions invertible or Lipschitz continuous by enforcing these properties on their Jacobians.

Methodology

The central idea of the JacNet approach is to use a neural network to learn the Jacobian matrix $J_f(x)$ of the target function $f(x)$. The target function is then obtained by integrating this learned Jacobian along a path in the input space, given an initial condition and using a numerical integrator. The choice of path, while theoretically flexible due to the fundamental theorem of line integrals, can be a simple linear path in practice.

Key features of the proposed method include:

1. Learning the Jacobian: Learning the Jacobian matrix rather than the function directly allows for better control over the properties of the function derivatives.
2. Enforcing Regularity Conditions: The authors demonstrate how the proposed method can enforce regularity conditions such as invertibility and Lipschitz continuity by structuring the Jacobian matrix.
   • Invertibility: Ensuring that the Jacobian determinant is non-zero everywhere can guarantee global invertibility.
   • Lipschitz Continuity: Applying appropriate activation functions, such as a scaled $\tanh$ function, on the Jacobian elements can ensure the function remains within desired Lipschitz bounds.
3. Computing the Inverse Function: Once the Jacobian is learned, the inverse function can be computed by integrating the inverse Jacobian, leveraging numerical integration techniques.
4. Backpropagation through Numerical Integration: The method utilizes differentiable numerical integrators to enable backpropagation through the integration process, allowing the network to be trained efficiently using standard gradient-based optimization methods.

Experimental Validation

To validate the proposed approach, the authors conducted experiments involving the learning of invertible and Lipschitz functions with simple neural network architectures. Specifically, they focused on:

1. Invertible Function: Learning the exponential function $f(x) = e^x$. The empirical results demonstrated that the trained network could accurately learn the exponential function and its inverse, maintaining the invertibility criterion by ensuring the Jacobian has positive eigenvalues.
2. Lipschitz Function: Learning the absolute value function $f(x) = |x|$, a canonical example of 1-Lipschitz function. The results showed that the trained network maintained the 1-Lipschitz condition, providing accurate approximations while guaranteeing derivative bounds through the use of a $\tanh$ activation function.

Implications and Future Directions

The JacNet approach provides a principled way to incorporate derivative constraints into neural network training, which has practical implications for various applications requiring controlled derivative properties of learned functions. By focusing on learning the Jacobian, this methodology paves the way for developing machine learning models with guaranteed structural properties, enhancing their interpretability and robustness.

Future developments may involve scaling up the experiments to more complex domains and exploring other regularity conditions beyond invertibility and Lipschitz continuity. Additionally, addressing challenges related to high-dimensional inputs and outputs, and improving the efficiency of the numerical integration process, will be critical steps in advancing this research.

In conclusion, the proposed technique for learning functions by structuring their Jacobians represents a significant advancement in machine learning, enabling the enforcement of important functional properties directly through the derivatives. This work holds promise for enhancing the theoretical foundations and practical applications of neural network models in areas requiring strict control over derivative behaviors.