Learning discriminative topological structure information representation for 2D shape and social network classification via persistent homology

Extracting topological structure information from data sources such as images and social networks remains a significant challenge. Drawing inspiration from the theory of topological structure in visual perception (topological perception theory), this study employs topological data analysis (TDA) to extract topological information, which is typically represented using persistence diagrams. To facilitate end-to-end learning, we introduce a topological set network (TSNet) that transforms topological information into vector representations through mixed entropy and self-attention mechanisms. Our approach first applies persistent homology to extract topological structure information from the data, followed by a 45-degree clockwise rotation of this information. We then design a topological set layer (TS-Layer) that creates vectorized representations by encoding persistence diagrams into topological set blocks (TS-Blocks) with diverse distributions. We provide theoretical proof that the TS-Layer maintains stability under input perturbations. To further enhance the discriminative power of the encoded topological features, we incorporate a residual attention layer (RA-Layer). Experimental results demonstrate that our proposed approach achieves superior performance compared to recent state-of-the-art methods. Specifically, our method achieves accuracy improvements of 1.2% (75.8% vs. 74.6%) and 1.7% (94.4% vs. 92.7%) on the Animal and MPEG-7 datasets respectively compared to the best existing methods. For social network classification tasks, our approach demonstrates improvements of 1.0% (55.9% vs. 54.9%) and 2.1% (48.5% vs. 46.4%) on the reddit-5k and reddit-12k datasets respectively, validating the effectiveness of our topological feature extraction and vectorization approach.