On the Geometry of Deep Learning (2408.04809v2)

Abstract: In this paper, we overview one promising avenue of progress at the mathematical foundation of deep learning: the connection between deep networks and function approximation by affine splines (continuous piecewise linear functions in multiple dimensions). In particular, we will overview work over the past decade on understanding certain geometrical properties of a deep network's affine spline mapping, in particular how it tessellates its input space. As we will see, the affine spline connection and geometrical viewpoint provide a powerful portal through which to view, analyze, and improve the inner workings of a deep network.

• The paper reveals that deep networks function as affine splines, tessellating the input space into convex polytopes that enhance approximation capabilities.
• It compares optimization landscapes, demonstrating that architectures with skip connections yield smoother loss surfaces for effective gradient-based learning.
• It shows that batch normalization refines tessellation alignment around data-dense regions, leading to improved initialization and reduced sampling biases.

On the Geometry of Deep Learning

The paper “On the Geometry of Deep Learning” by Balestriero, Humayun, and Baraniuk investigates the mathematical foundations of deep learning through the lens of affine splines, particularly focusing on piecewise linear function approximation facilitated by ReLU activations. Instead of treating neural networks as inscrutable black boxes, the authors delve into how these architectures tessellate input space into convex polytopes. They explore various implications of this inherent geometry on deep learning system design, optimization, generalization, and biases.

Affine Splines and Deep Network Tessellation

Deep networks can be viewed as multidimensional extensions of affine splines, where the composition of operations results in a tessellation of the input space into convex polytopes, each representing an affine spline region. The ReLU activations, in particular, create hyperplane arrangements, thereby partitioning the input space into distinct tiles, each with an associated affine transformation.

With increasing depth and width, the number of these tiles grows exponentially, enhancing the representational capacity of the network. Interestingly, deeper architectures result in a more intricate tiling of the input space, which has significant implications for their expressive power and generalization capabilities.

Insights from Input Space Tessellation

Several empirical observations and theoretical findings highlight how the tessellation properties influence different aspects of neural network performance:

1. Approximation Capability: The paper discusses how the self-similarity in the tiling configuration of deep networks can be linked to their superior approximation capabilities compared to shallow networks. The ability to replicate function parts with different orientations and scales underlines the efficiency of deep models in approximating complex functions.
2. Optimization: The authors compare the optimization landscapes of different network architectures, particularly ConvNets versus ResNets with skip connections. The loss landscapes of ResNets are shown to be smoother and exhibit better conditioning, due to the coupling requirements imposed by their tessellations, making them preferable for gradient-based optimization.
3. Initialization and Batch Normalization: The geometric interpretation of batch normalization elucidates how it adapts the tessellation to better align with the training data, effectively improving the initialization. By focusing the hyperplane density around data-dense regions, batch norm helps in achieving better initial alignment, resulting in faster and more effective training.
4. Training Dynamics and Grokking: The dynamic changes in the tessellation throughout the training process reveal how deep networks balance interpolation and generalization. The paper identifies the phenomenon of “delayed robustness” or “grokking”, where extended training beyond interpolation leads to a more stable and less sensitive functional mapping around training examples.
5. Generative Models: For generative models such as GANs and VAEs, the affine spline perspective provides a mechanism to address sampling biases. By understanding the volumetric deformations within the tessellation, a post-processing method (MaGNET) is introduced to ensure uniform sampling on the manifold, thereby mitigating inherent biases.

Future Directions

The paper posits that further research into the affine spline perspective can lead to developments in understanding and improving deep learning architectures. Specific open problems include:

• Extending these results to more complex activation functions beyond ReLU.
• Improving normalization schemes to better adapt the tessellation to varied data and task-specific requirements.
• Developing new metrics and visualization techniques to assess the dynamics of training beyond simple gradient descent optimization improvements.
• Addressing the limitations of existing models in capturing the true manifold and distribution of real-world data.

Conclusion

By framing deep networks as affine splines, the paper offers a geometrically grounded understanding of their operation, which has broad implications across learning, optimization, generalization, and generative modeling. This perspective not only promises to refine current deep learning practices but also to open avenues for novel architectures and methods that leverage geometric insights for enhanced performance and reliability. The work invites further exploration into the deep connections between spline theory and neural computation, challenging researchers to uncover more layers of understanding in the field of deep learning.