Persistent Cross Entropy

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

Topological Data Analysis (TDA) provides powerful tools to explore the shape and structure of data through topological features such as clusters, loops, and voids. Persistence diagrams are a cornerstone of TDA, capturing the evolution of these features across scales. While effective for analyzing individual manifolds, persistence diagrams do not account for interactions between pairs of them. Cross-persistence diagrams (cross-barcodes), introduced recently, address this limitation by characterizing relationships between topological features of two point clouds. In this work, we present the first systematic study of the density of cross-persistence diagrams. We prove its existence, establish theoretical foundations for its statistical use, and design the first machine learning framework for predicting cross-persistence density directly from point cloud coordinates and distance matrices. Our statistical approach enables the distinction of point clouds sampled from different manifolds by leveraging the linear characteristics of cross-persistence diagrams. Interestingly, we find that introducing noise can enhance our ability to distinguish point clouds, uncovering its novel utility in TDA applications. We demonstrate the effectiveness of our methods through experiments on diverse datasets, where our approach consistently outperforms existing techniques in density prediction and achieves superior results in point cloud distinction tasks. Our findings contribute to a broader understanding of cross-persistence diagrams and open new avenues for their application in data analysis, including potential insights into time-series domain tasks and the geometry of AI-generated texts. Our code is publicly available at https://github.com/Verdangeta/TDA_experiments

Persistence diagrams (PDs) represent multiscale topological information as multisets of birth-death pairs $(b,d)$. How best to use them with Transformers, however, remains unclear. Connecting PDs to Transformers requires choosing how topological information is tokenized and exposed to self-attention. Prior methods use specialized designs for tokenization, aggregation, Transformer blocks, or attention, but are hard to compare since reported results rarely share PD inputs, splits, or model-selection rules. We instead confine the diagram-specific design to point representation, mapping each birth-death pair through fixed random Fourier features while leaving the Transformer backbone unchanged, and call this recipe Persistence Positional Encoding (PPE). In matched controls on eight TU graph benchmarks (TU8), changing only point representation shifts macro accuracy by 5.49 to 6.47 percentage points (pp), against 0.47 to 1.45 pp from changing the downstream module, identifying point representation as the dominant design variable tested. We further reevaluate vectorizations, kernels, learned representations, and Transformer-based models under one protocol with shared PD inputs, outer folds, and validation-only selection. Among all evaluated PD methods and downstream configurations, PPE achieves the highest mean on most benchmarks, including TU8, ORBIT5K, and MCB-C. Finally, we prove that grid averaging leaves a reconstruction error at any resolution, whereas a Transformer with fixed random Fourier frequencies is almost surely universal. PPE thus offers a minimal, Transformer-native baseline for persistence diagrams, and a common protocol for future comparison.