p-Laplacian Encodings

• p-Laplacian encodings are a family of embeddings that generalize classical Laplacian methods by introducing a tunable parameter to control nonlinearity.
• They integrate nonconvex optimization techniques and continuation strategies to extract richer geometric and topological features in graph and transformer models.
• The approach enhances neural network performance by providing expressive, stable invariants and adjustable inductive biases for applications such as GNNs and persistent homology.

p-Laplacian encodings generalize classical Laplacian-based embeddings by introducing a tunable parameter $p$ that controls the nonlinearity of the association between nodes (or features) in graphs and related data structures. Unlike the case $p=2$, which yields standard spectral embeddings via eigenvectors of the (combinatorial or normalized) Laplacian, the $p$-Laplacian admits a non-linear operator whose stationary points define richer embeddings or regularizers. These encodings manifest in diverse applications, including positional encoding for graph neural networks (GNNs), vectorization of persistent homology, and adaptive regularization in transformer architectures.

1. Mathematical Foundations of the p-Laplacian

The $p$-Laplacian on (weighted) graphs and simplicial complexes generalizes the quadratic Laplacian form by penalizing $p$-norm deviations:

• For a graph with a weight matrix $A=(a_{ij})$, the objective for Laplacian positional encodings becomes

$$Q_p(f) = \frac{1}{2}\sum_{i,j} a_{ij}|f_i - f_j|^p,$$

subject to normalization constraints to avoid trivial solutions. For multi-coordinate embeddings $X\in\mathbb{R}^{n\times k}$, the optimization problem reads

$$J_p(X) = \sum_{\ell=1}^k \frac{Q_p(x^\ell)}{\|x^\ell\|_p^p}, \quad X^\top X = I_k.$$

• The Euler–Lagrange analysis yields the graph $p$-Laplacian operator:

$p=2$0

and its associated nonlinear eigenproblem:

$p=2$1

• In persistent topology, the persistent $p=2$2-Laplacian $p=2$3 is constructed on pairs of filtered chain complexes using boundary operators and their adjoints, encapsulating both topological lifetimes and geometric information (Jung et al., 5 Dec 2025).

2. Algorithmic Realizations and Numerical Optimization

The computation of $p=2$4-Laplacian positional encodings and related quantities requires specialized, often nonconvex optimization techniques:

• The $p=2$5 objective is minimized over the Stiefel manifold $p=2$6, taking advantage of Riemannian gradients and retractions onto the constraint set. In practice,

$p=2$7

with updates

$p=2$8

Libraries such as Geoopt provide ready implementations in PyTorch (Maskey et al., 2022).

• To address poor conditioning at small $p=2$9, continuation (homotopy) strategies are employed: start at $p$0 (the Laplacian case), and decrement $p$1 in small steps, initializing each stage from the solution at the previous $p$2.
• For persistent Laplacian encodings, explicit construction of block matrices, Schur complements, and orthogonal projections is needed to extract spectral signatures from persistence pairs (Jung et al., 5 Dec 2025).
• In transformer architectures, the $p$3-Laplacian regularization augments attention with entrywise factors $p$4 over value-vector differences, retaining explicit compatibility with multi-head and softmax-based attention (Nguyen et al., 2023).

3. p-Laplacian Encodings in Graph Representation Learning

p-Laplacian positional encodings (p-PEs) provide a parametrized family of node representations driven by the graph’s structural geometry:

• Each node $p$5 receives an encoding $p$6; these are typically concatenated to the original node features $p$7 for use in message passing schemes. This enables GNNs to access positional information beyond the 1-WL test’s reach.
• The sign ambiguity of $p$8-eigenvectors is handled via either random sign-flips per run or by SignNet, a dedicated network that computes coordinatewise sign-invariant statistics (Maskey et al., 2022).
• The expressive power of message passing neural networks is strictly enhanced by augmenting with at least two $p$9-Laplacian coordinates for any $p$0, exceeding the distinguishing ability of the 1-Weisfeiler–Lehman test.
• Empirical evaluation shows that p-PEs perform comparably to standard Laplacian eigenvector PEs on node classification and display varying behavior across graph regression tasks. For smaller $p$1, embeddings tend to become piecewise constant, mirroring soft clustering or graph cut structures; for large $p$2, embeddings move toward shortest-path-type distance representations (Maskey et al., 2022).

4. Persistent Laplacian Encodings and Vectorization in Topological Data Analysis

Persistent Laplacian encodings extend positional encodings to the field of persistent homology, incorporating not only homological birth–death intervals but also spectral or geometric information from p-Laplacian operators (Jung et al., 5 Dec 2025):

• For each persistence interval $p$3, the persistent Laplacian $p$4 associated to the filtered complex (up to degree $p$5) is computed.
• Signature functions—such as the spectral gap, spectral entropy, or geometric eigenvector profiles—are evaluated on each $p$6 to provide a rich set of real-valued invariants.
• The Persistent Laplacian Diagram (PLD) records weighted locations $p$7 with associated signature values, extending ordinary persistence diagrams.
• The Persistent Laplacian Image (PLI) transforms the PLD by smoothing with Gaussians and discretizing on a rectangular grid, yielding a finite vector stable under perturbations of the filtration. This construction strictly contains ordinary persistence images as a special case, and encodes both topological and fine geometric information inaccessible to plain PDs.
• Theoretical results guarantee stability of PLIs with explicit Lipschitz bounds in supremum, $p$8, and $p$9 norms, ensuring robustness to input noise (Jung et al., 5 Dec 2025).

5. Incorporation into Deep Learning Architectures

p-Laplacian encodings have been seamlessly integrated into contemporary deep architectures:

• In GNNs, p-PEs are used for feature augmentation. The concatenated node features propagate through graph convolutional layers, with message passing and local aggregation inheriting improved positional expressiveness (Maskey et al., 2022).
• In transformers, the $p$0-Laplacian regularizer generalizes standard softmax-based attention. Each multi-head block can be assigned independent $p$1 values, or $p$2 can be learned per-head; this adjustment allows the attention mechanism to interpolate between smoothing ($p$3), neutrality ($p$4), and sharpening/sparsification ($p$5). The added computational cost is at most $p$6 per layer, and empirical performance gains are noted for both vision and language tasks (Nguyen et al., 2023).
• Vectorized persistent Laplacian features (PLIs) serve as stable, discriminative plug-in vectors for classical statistical classifiers or as node enrichments within neural networks operating on topological data (Jung et al., 5 Dec 2025).

6. Empirical Insights, Strengths, and Limitations

• On standard graph benchmarks (Cora, Citeseer), p-PEs match—though do not consistently exceed—2-PEs in node classification. In molecular graph regression (ZINC), the best results were achieved with 2-PE plus SignNet, with p-PEs occasionally outperforming the no-PE baseline but not surpassing Laplacian eigenvector PEs in a stable fashion.
• In transformer models, allowing heads with mixed or intermediate $p$7 yields consistent improvements: on ImageNet-1K, p-LaT increases Top-1 accuracy from 71.97% (baseline) to 72.78%; on WikiText-103, perplexity drops from 34.10 (softmax-transformer) to 33.50 (p-LaT) (Nguyen et al., 2023).
• The computational overhead is modest for graphs, with gradient plus QR step per iteration scaling as $p$8. For transformers, the main cost increase arises from computing $p$9 value differences per head, which is tractable in modern environments.
• Sign ambiguity and local minimization remain practical concerns; continuation methods and sign-invariant postprocessing are standard remedies.
• Qualitative behaviors as $A=(a_{ij})$0 varies inform practical choices: $A=(a_{ij})$1 highlights cuts and community structures, $A=(a_{ij})$2 yields smooth global geometry, $A=(a_{ij})$3 approximates shortest-path distances (Maskey et al., 2022).

7. Theoretical and Practical Significance

p-Laplacian encodings unify and extend a spectrum of geometric and topological feature extraction paradigms, providing a flexible framework for:

• Tuning inductive biases in neural networks toward smooth, sparse, or heterophilic relationships.
• Extracting expressive, stable invariants from both graph-structured and topological data.
• Enhancing the expressive power and discriminatory capability of state-of-the-art learning models. The approach is underpinned by rigorous optimization theory, robust stability guarantees, and practical success across multiple modalities, establishing it as a fundamental tool in modern representation learning (Maskey et al., 2022, Jung et al., 5 Dec 2025, Nguyen et al., 2023).