Persistent Entropy as a Detector of Phase Transitions

Persistent entropy (PE) is an information-theoretic summary statistic of persistence barcodes that has been widely used to detect regime changes in complex systems. Despite its empirical success, a general theoretical understanding of when and why persistent entropy reliably detects phase transitions has remained limited, particularly in stochastic and data-driven settings. In this work, we establish a general, model-independent theorem providing sufficient conditions under which persistent entropy provably separates two phases. We show that persistent entropy exhibits an asymptotically non-vanishing gap across phases. The result relies only on continuity of persistent entropy along the convergent diagram sequence, or under mild regularization, and is therefore broadly applicable across data modalities, filtrations, and homological degrees. To connect asymptotic theory with finite-time computations, we introduce an operational framework based on topological stabilization, defining a topological transition time by stabilizing a chosen topological statistic over sliding windows, and a probability-based estimator of critical parameters within a finite observation horizon. We validate the framework on the Kuramoto synchronization transition, the Vicsek order-to-disorder transition in collective motion, and neural network training dynamics across multiple datasets and architectures. Across all experiments, stabilization of persistent entropy and collapse of variability across realizations provide robust numerical signatures consistent with the theoretical mechanism.

💡 Research Summary

This paper establishes a rigorous, model‑independent theoretical foundation for using Persistent Entropy (PE) as a detector of phase transitions in stochastic, data‑driven systems. The authors first formalize a probabilistic framework: for each system size N and control parameter λ, the observed data (point clouds, graphs, embeddings, etc.) are random objects due to noise, random initial conditions, or finite‑size fluctuations. Applying a fixed filtration and a fixed homological degree k yields a random persistence diagram D_N(λ). These diagrams live in the standard space of locally finite multisets of off‑diagonal points equipped with a bottleneck or p‑Wasserstein metric.

The central assumptions are twofold. (i) For any fixed λ, the random diagrams converge in probability as N → ∞ to a deterministic limiting diagram D(λ). (ii) There exists a critical value λ_c such that the limiting diagrams on the two sides, D_-(λ) for λ < λ_c and D_+(λ) for λ > λ_c, differ by at least one macroscopic bar: one phase contains a bar whose lifetime stays bounded away from zero, while the other phase’s corresponding bars have lifetimes that vanish in the limit. This “macroscopic feature separation condition” captures the topological signature of a phase transition.

Persistent Entropy is defined as the Shannon entropy of the normalized lifetimes of the off‑diagonal points: if ℓ_i = d_i – b_i are the lifetimes and L = Σℓ_i, then p_i = ℓ_i/L and PE(D) = – Σ p_i log p_i (with PE = 0 when L = 0). Although PE is not uniformly continuous on the whole diagram space, it is continuous on subclasses where total persistence is finite and the contribution of arbitrarily short bars is negligible, or when an explicit lifetime truncation is applied. Under the high‑probability assumption that D_N(λ) lies in such a subclass, convergence of diagrams implies convergence of PE.

The main theorem proves that under the two assumptions above, the PE values on the two sides of the critical point remain separated by a non‑vanishing gap Δ > 0 with probability tending to one as N → ∞. Consequently, PE provides a deterministic scalar that distinguishes the two phases, independent of the specific filtration, embedding, or homological degree.

To translate this asymptotic result into a practical tool for finite data, the authors introduce a “topological stabilization” operationalization. One computes PE over a sliding window of length w along a time series or a sequence of control‑parameter values. When the PE values within a window fluctuate by less than a prescribed ε for a sustained interval, the system is said to have entered a stabilized topological regime, and the onset time is defined as the topological transition time. Simultaneously, a probability‑based estimator of the critical parameter λ̂_c is obtained by locating the point where the mean PE exhibits a sharp change while its variance collapses. This provides a finite‑horizon method for detecting criticality.

The theoretical framework is validated on three benchmark systems. (1) Kuramoto synchronization: as the coupling strength K crosses a critical K_c, a giant synchronized cluster emerges, creating a long‑lived 1‑dimensional bar; PE jumps upward and then stabilizes, while its variance sharply drops. (2) Vicsek collective motion: increasing the noise amplitude η destroys ordered flocks, eliminating long‑lived 0‑dimensional bars; PE drops and stabilizes at the disorder side, again with variance collapse. (3) Neural network training: during stochastic gradient descent on several datasets and architectures, the weight‑space topology simplifies. Early epochs show many short bars and high, noisy PE; later epochs retain only a few persistent bars, leading to lower, stable PE and reduced variability. In all cases, the empirical PE gap and stabilization precisely match the theorem’s predictions, and the estimated transition times/parameters align with known order‑parameter changes.

The paper concludes that Persistent Entropy, when viewed through the lens of diagram‑level convergence and macroscopic bar separation, is a theoretically justified, universally applicable indicator of phase transitions. The topological stabilization protocol bridges asymptotic theory and finite‑sample practice, enabling reliable detection of critical points in complex, noisy, high‑dimensional data. Future directions suggested include extensions to multi‑parameter systems, non‑equilibrium transitions, and the integration of PE as a regularizer or monitoring tool within machine‑learning pipelines.