Tensor Network Machine Learning for Wildfire Susceptibility Mapping: from Grokking Dynamics to Quantum Mixedness of Class Representations

Technical Summary: Tensor Network Machine Learning for Wildfire Susceptibility Mapping

Problem Statement
The study addresses the challenge of mapping wildfire susceptibility in the Gargano region of Southern Italy, a Mediterranean landscape characterized by high environmental heterogeneity, including diverse vegetation structures, land-use patterns, and anthropogenic pressures. Traditional wildfire modeling relies on manually harmonized predictor layers (e.g., fuel load, land cover, climate indices), which can be computationally intensive and difficult to scale. Furthermore, standard machine learning approaches often struggle to capture complex, non-linear spatial-spectral relationships inherent in large-scale Earth Observation (EO) data without incurring high computational costs or lacking interpretability regarding class separability. The authors aim to evaluate whether quantum-inspired tensor network models, specifically Matrix Product States (MPS), can effectively classify wildfire susceptibility using compact, learned geospatial representations while providing physically grounded insights into the learning dynamics and class distinguishability.

Methodology
The proposed framework integrates scalable geospatial data processing with quantum-inspired machine learning:

1. Data Representation (AlphaEarth Embeddings):

   • The study utilizes AlphaEarth Foundations, a global embedding field that transforms heterogeneous environmental data (multispectral imagery, vegetation indices, topography, climate) into a compact, 64-dimensional vector for each $10m \times 10m$ pixel.
   • This embedding serves as a unified, latent representation of the environmental state, reducing the need for manual feature engineering.
   • Boruta feature selection is applied to identify the most informative subset of features (30 dimensions) to reduce redundancy before model training.

2. Model Architecture (Matrix Product State Classifier):

   • The core learning architecture is a Matrix Product State (MPS) classifier, a tensor network model that encodes high-dimensional correlations through low-rank decompositions.
   • Quantum Encoding: Input vectors are mapped to a high-dimensional Hilbert space via a feature map $|\phi(x^{(j)})\rangle = (\cos x^{(j)}, \sin x^{(j)})^\top$.
   • Quantum Mask: A variational tensor object, termed a "quantum mask" ($W$), acts on the encoded state. $W$ is expressed in MPS form with a tunable bond dimension ($\chi$), allowing control over model complexity.
   • Training: The model is trained by minimizing a mean squared error loss between predicted amplitudes and target label encodings. Optimization uses a two-site update strategy with stochastic gradient descent (SGD) on mini-batches, executed on NVIDIA H100 GPUs.
   • Task Formulations: The model is evaluated on two tasks:
     • Binary Classification: Class 0 (no/negligible susceptibility) vs. Classes 1–3 (aggregated susceptibility).
     • Multiclass Classification: Distinguishing all four classes (0, 1, 2, 3).

3. Diagnostic Metrics:

   • Beyond standard accuracy, the study introduces level-resolved mixedness diagnostics based on reduced density matrices.
   • Key metrics include pairwise decoherence factors ($\Gamma_{ij}$), normalized triplet determinants ($\Delta_{ijk}$), and conditional purities. These quantify the degree of distinguishability and coherence between class representations within the learned tensor network.

Key Results

1. Predictive Performance:

   • Binary Task: The MPS classifier achieved an accuracy of 87.07%, with high sensitivity (91.13%) and specificity (83.02%).
   • Multiclass Task: The model reached an overall accuracy of 53.01%. Performance varied by class, with Class 0 (75.47%) and Class 3 (62.49%) performing better than the intermediate Classes 1 (35.88%) and 2 (38.21%).
   • While the Random Forest benchmark in the literature reports higher accuracy (>95%) for this specific region, the MPS results are presented as a complementary demonstration of the model's ability to learn from embeddings and reveal internal dynamics.

2. Grokking Dynamics:

   • The training trajectories revealed a distinct grokking transition, particularly in the binary classification task.
   • Initially, the model exhibited a gap between training and test performance, suggesting memorization of superficial correlations.
   • Between 400 and 600 training sweeps, a sharp transition occurred where test sensitivity and accuracy abruptly improved and aligned with training metrics. This indicates a reorganization of the internal representation from memorization to capturing the underlying structure of the classification problem.
   • In the multiclass setting, this transition was hierarchical and distributed: the model first learned to separate the dominant Class 0 from the rest, and only later refined its ability to distinguish among the minority classes (1, 2, 3).

3. Magnetization Patterns:

   • Monitoring the expectation values of the Pauli $\sigma_Z$ operator (magnetization) across feature sites showed that the quantum mask stabilizes into coherent patterns coinciding with the grokking transition.
   • The magnetization pattern for Class 0 in the multiclass setting was nearly identical to that in the binary setting, suggesting a stable, hierarchical representation where the binary separation is a coarse-grained projection of the multiclass structure.

4. Quantum Mixedness and Class Distinguishability:

   • Analysis of reduced density matrices revealed a fundamental geometric property of the learned representation: non-adjacent classes are more distinguishable than adjacent ones.
   • Decoherence Factors: Pairs of adjacent classes (e.g., 1-2, 2-3) retained higher coherence (higher $\Gamma_{ij}$), indicating they are harder to separate. Non-adjacent pairs (e.g., 0-2, 0-3) showed lower coherence, signifying better separation.
   • Triplet Determinants: Triplets involving non-adjacent classes (e.g., 0-1-3) stabilized at higher values than those with only adjacent classes (e.g., 1-2-3), confirming that the environment encodes more independent information for distant categories.
   • This hierarchy explains the observed classification errors, which are predominantly localized between neighboring susceptibility levels rather than extreme classes.

Significance and Claims
The paper claims that tensor network models offer more than just competitive classification accuracy; they provide a physically grounded framework to interpret complex environmental datasets.

• Interpretability: Unlike standard "black-box" models, the MPS approach allows for the direct quantification of class separability and coherence through quantum mechanical observables (magnetization, mixedness).
• Hierarchical Learning: The results demonstrate that the model naturally encodes a hierarchy of class distinguishability, where broad distinctions (low vs. high susceptibility) are learned before fine-grained distinctions (between moderate levels).
• Grokking in Environmental Data: The study identifies that the "grokking" phenomenon—delayed but abrupt generalization—is present in MPS classifiers applied to real-world geospatial data, linking this dynamic to the stabilization of internal quantum representations.
• Scalability: By leveraging AlphaEarth embeddings, the approach offers a scalable alternative to manual predictor engineering, capable of handling heterogeneous, multi-source EO data.

The authors conclude that while the MPS classifier does not yet surpass the predictive accuracy of established Random Forest models for this specific task, it provides a richer, geometrically interpretable description of the learning process and the intrinsic structure of wildfire susceptibility labels.