The Powers of Precision: Structure-Informed Detection in Complex Systems -- From Customer Churn to Seizure Onset

TL;DR

The paper addresses early detection of emergent phenomena in complex, partially observed systems by leveraging structure-informed features that reflect latent interaction graphs. It introduces a one-parameter family of covariance/precision transforms, selects an optimal exponent $\beta^{\star}$ from data, and uses a supervised classifier on the resulting features, with test-time adaptation for single-shot observations. A central theoretical contribution is a structural-consistency theorem that guarantees recovery of latent graph structure from partial observations under mild cross-block conditions, and the methodology is validated on EEG seizure onset and IBM Telco churn, demonstrating competitive predictive performance, feature identifiability on the SPD manifold using the AIR metric, and interpretable structural signatures. The work offers a universal framework for structure-aware learning under partial observability, bridging SEMs, GGMs, and diffusion models, and enabling interpretable, real-time detection across domains without synthetic data balancing.

Abstract

Emergent phenomena -- onset of epileptic seizures, sudden customer churn, or pandemic outbreaks -- often arise from hidden causal interactions in complex systems. We propose a machine learning method for their early detection that addresses a core challenge: unveiling and harnessing a system's latent causal structure despite the data-generating process being unknown and partially observed. The method learns an optimal feature representation from a one-parameter family of estimators -- powers of the empirical covariance or precision matrix -- offering a principled way to tune in to the underlying structure driving the emergence of critical events. A supervised learning module then classifies the learned representation. We prove structural consistency of the family and demonstrate the empirical soundness of our approach on seizure detection and churn prediction, attaining competitive results in both. Beyond prediction, and toward explainability, we ascertain that the optimal covariance power exhibits evidence of good identifiability while capturing structural signatures, thus reconciling predictive performance with interpretable statistical structure.

Figures (14)

• Figure 1: Proposed approach. Top: theoretical regimes in which specific covariance powers are structurally consistent. Bottom: pipeline: data $\mapsto$ covariance $\mapsto$ power transform $C^{\beta}$$\mapsto$ supervised classifier; $\beta^\star$ is selected on a train--validation split.
• Figure 2: Sliding window to split the data for train-validation-test.
• Figure 3: Seizure early detection benchmark: i) Superior performance across all metrics (green background); ii) Superior performance on sensitivity (purple background).
• Figure 4: Accuracy and recall performance for churn prediction against benchmark references.
• Figure 5: Performance of ML models for seizure detection (left) and churn prediction (right) trained directly on the raw data.

Theorems & Definitions (17)

• Theorem 1: Structural consistency under partial observability
• Lemma 1: Bound on $R_{\mathcal{S}}(\lambda)$
• proof
• Lemma 2: Spectral relation between $C$ and $A$ or $\overline{A}$
• proof
• Lemma 3: Bound on cross--blocks of $M^k$
• proof
• Theorem 2: Structural consistency of $(C_{\mathcal{S}})^{-1/\alpha}$
• proof
• Lemma 4: Resolvent bound