Geometric Understanding of Deep Learning (1805.10451v2)

Abstract: Deep learning is the mainstream technique for many machine learning tasks, including image recognition, machine translation, speech recognition, and so on. It has outperformed conventional methods in various fields and achieved great successes. Unfortunately, the understanding on how it works remains unclear. It has the central importance to lay down the theoretic foundation for deep learning. In this work, we give a geometric view to understand deep learning: we show that the fundamental principle attributing to the success is the manifold structure in data, namely natural high dimensional data concentrates close to a low-dimensional manifold, deep learning learns the manifold and the probability distribution on it. We further introduce the concepts of rectified linear complexity for deep neural network measuring its learning capability, rectified linear complexity of an embedding manifold describing the difficulty to be learned. Then we show for any deep neural network with fixed architecture, there exists a manifold that cannot be learned by the network. Finally, we propose to apply optimal mass transportation theory to control the probability distribution in the latent space.

• The paper presents a novel framework explaining how autoencoders learn low-dimensional manifolds from high-dimensional data.
• The paper introduces rectified linear complexity to quantify manifold segmentation and establishes limits on the learnability of complex data structures.
• The paper employs optimal transport theory to simplify latent space distributions, paving the way for more robust generative model training.

Geometric Understanding of Deep Learning

The paper "Geometric Understanding of Deep Learning" provides a theoretical framework to comprehend deep learning through a geometric lens, addressing the predominantly opaque understanding of deep neural networks (DNNs). The authors argue that the success of deep learning in diverse applications, such as image recognition and machine translation, can be attributed to the manifold structure inherent in data. This paper presents a novel geometric approach to demystify how deep learning models encapsulate this manifold structure and the associated probability distributions.

The central thesis of the paper is that deep learning models, specifically autoencoders, are inherently learning the manifold embedded in high-dimensional data. The authors emphasize that such data are often clustered around non-linear, low-dimensional manifolds. The paper articulates the learning process of autoencoders, demonstrating that these models can approximately capture manifold structures through nonlinear mappings. Autoencoders achieve this via an encoding map that projects data onto a latent space, and a decoding map, which reconstructs the data from this representation. This process realizes a piecewise linear approximation of the latent manifold, governed by the network's architecture.

Key Concepts and Results

1. Rectified Linear Complexity: The authors introduce the concept of rectified linear complexity to quantify the learning capacity of ReLU-based DNNs. This complexity is defined as the maximum number of affine pieces into which the manifold can be segmented by the network. The mathematical formulation reveals that complexity is dependent on both the DNN architecture and the manifold's geometric properties.
2. Limits of Manifold Learnability: A significant finding of the paper is the establishment of a theoretical boundary on the learnability of manifolds by a fixed architecture DNN. The authors demonstrate that for every fixed architecture, there exists a manifold too complex to be learned. This suggests a fundamental limitation based on the manifold's topology and the network's structure, implying that DNNs can only effectively learn manifolds up to a certain complexity threshold.
3. Distribution Control via Optimal Transport: The paper also advances the idea of using optimal mass transportation (OMT) theory to manage probability distributions in the latent space. This approach aims to transform a complex distribution into a simpler one (e.g., Gaussian) by explicitly constructing a transformation map, contrasting with the adversarial training schemes typical of Generative Adversarial Networks (GANs).

Implications and Future Directions

This research contributes significantly to the theoretical understanding of deep learning models from a geometric perspective. By conceptualizing data as residing on a manifold, it offers insights into the structural capabilities and limitations of neural networks. The concept of rectified linear complexity serves as a bridge between the abstract geometry of manifolds and the practical architectures of neural networks, offering a metric to gauge the suitability of a network for a specific dataset.

Practically, these insights encourage the design of network architectures better aligned with the geometric properties of target data. The use of OMT to control latent space distributions suggests a path towards more stable and efficient generative models. Speculatively, future advancements in AI could build upon this geometric framework, exploring deeper the interplay between manifold complexity and network architecture to engineer more robust, efficient learning systems.

Furthermore, this work lays the groundwork for future research into rectifiable complexity estimates, potentially leading to more refined architectural recommendations based on the intrinsic symmetry and variability of data manifolds. Understanding the manifold constraints could also inspire new regularization techniques that enhance generalization by explicitly tying network architecture to the manifold characteristics of the data.

In summary, this paper offers a profound geometric interpretation of deep learning, laying a theoretical foundation that may inspire both algorithmic innovations and a deeper exploration of the mathematical structures underlying intelligent data representations.