On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology
This work provides insights into representation learning for researchers in deep learning, though it is incremental as it extends prior topological analyses to more architectures and datasets.
The study analyzed how the topology of feature embedding spaces evolves across layers in deep neural networks using Betti numbers and cubical homology, showing that datasets become topologically simpler with depth, and proposed a metric based on decay rate that correlates better with fine-tuning accuracy for ranking pre-trained models in transfer learning.
We study how the topology of feature embedding space changes as it passes through the layers of a well-trained deep neural network (DNN) through Betti numbers. Motivated by existing studies using simplicial complexes on shallow fully connected networks (FCN), we present an extended analysis using Cubical homology instead, with a variety of popular deep architectures and real image datasets. We demonstrate that as depth increases, a topologically complicated dataset is transformed into a simple one, resulting in Betti numbers attaining their lowest possible value. The rate of decay in topological complexity (as a metric) helps quantify the impact of architectural choices on the generalization ability. Interestingly from a representation learning perspective, we highlight several invariances such as topological invariance of (1) an architecture on similar datasets; (2) embedding space of a dataset for architectures of variable depth; (3) embedding space to input resolution/size, and (4) data sub-sampling. In order to further demonstrate the link between expressivity \& the generalization capability of a network, we consider the task of ranking pre-trained models for downstream classification task (transfer learning). Compared to existing approaches, the proposed metric has a better correlation to the actually achievable accuracy via fine-tuning the pre-trained model.