Learnable Position Matrix

• Learnable position matrices are trainable constructs that encode positional relationships in neural architectures, adapting flexibly to data modalities and downstream tasks.
• They employ diverse mechanisms such as permutation matrices, rotation/kernels, Fourier features, and ODE solvers to optimize token ordering and multi-dimensional embeddings.
• Empirical studies show these matrices boost model performance, robustness, and generalization across vision, language, multimodal, and graph learning domains.

A learnable position matrix refers to an end-to-end trainable construct that parameterizes position or order relationships in neural architectures, most pervasively in attention-based models. Its goal is to supersede manually designed positional encodings or static templates by enabling position information to adapt to data modality, task, or downstream context. Approaches vary from explicit parameter matrices indexing positions to dynamic mechanisms that influence token interleaving, rotation, or kernel-based feature transformations. Learnable position matrices address both adaptability and generalization, and now underpin state-of-the-art schemes across vision, language, multimodal, and graph learning domains.

1. Mathematical Formulations and Design Patterns

Learnable position matrices manifest in multiple forms, driven by architectural needs and inductive biases. Broadly, key variants include:

• Permutation matrices for token reordering/interleaving: A core recent example is the differentiable position matrix $W\in\mathbb{R}^{L\times L}$ in AnchorOPT, where $L$ is the prompt length. Each row specifies one-hot (hard) or Gumbel-softmax (soft, differentiable) distributions over which original prompt token occupies slot $i$ after reordering. Logit matrices $P\in\mathbb{R}^{L\times L}$ are trained, and $W$ is constructed as described:

$$w_i = \text{one\_hot}\left(\underset{j}{\mathrm{argmax}}(\log p_{ij} + \epsilon_j)\right),\ \epsilon_j \sim \text{Gumbel}(0,1)$$

and, for differentiability during training:

$$\hat w_{ij} = \frac{\exp((\log p_{ij}+\epsilon_j)/\tau)}{\sum_{k=1}^L \exp((\log p_{ik}+\epsilon_k)/\tau)}$$

(Li et al., 26 Nov 2025)

• Rotation or kernel parameter matrices:
  • In ComRoPE, rotation matrices $R(x) = \exp\left(\sum_{i=1}^{N} A_i x_i\right)$ with commuting, trainable skew-symmetric $A_i$ serve as a learnable generalization of RoPE. The learnability and commutativity of $A_i$ are critical for preserving positional robustness and allowing end-to-end adaptation to data (Yu et al., 4 Jun 2025).
  • Learnable Fourier Feature methods replace fixed $W$ in $[\cos(W_r x), \sin(W_r x)]$ with fully trainable $W_r\in\mathbb{R}^{D/2\times M}$, optionally followed by an MLP; this enables encoding of multi-dimensional position via a compact, expressive parametric mapping (Li et al., 2021).
• MLP/Graph Laplacian-based coordinate embedders:
  • Structured positional embedding matrices $P\in\mathbb{R}^{n\times d}$ are optimized jointly with downstream MLPs, often regularized by Laplacian smoothness. Per-coordinate or per-instance hyperparameters can induce instance-adaptive smoothness (e.g. via analytic or regression-tuned lengthscales) (Ramasinghe et al., 2021).
• Sequential encoder approaches:
  • SeqPE builds the embedding for multi-dimensional position $i=(i_0, ..., i_{n-1})$ as a symbolic sequence of digits, then feeds this token sequence through a compact Transformer encoder, yielding $p_i = g_\theta(\text{Seq}(i))$ (Li et al., 16 Jun 2025).
• Continuous-time ODE solvers:
  • FLOATER models $X(t)$ via $\dot X(t) = f(t, X(t); \Theta)$, where $f$ is a neural vector field. Stepwise solves yield the learnable position matrix $\{X(1), ..., X(L)\}$ (Liu et al., 2020).

2. Integration into Downstream Architectures

Injection of the learnable position matrix varies by application:

• Token interleaving: In AnchorOPT, the learned $W$ is applied to form $$t_{\text{norm}} = \operatorname{concat}(\hat W \odot t_v, [\mathrm{CLS}])$$, providing adaptive and stage-conditional prompt templates for CLIP text encoders (Li et al., 26 Nov 2025).
• Rotary and kernel-based mechanisms: In ComRoPE, learned angle matrices rotate key and query pairs, with

$Q' = R(P) Q,\quad K' = R(P) K,$

where $P$ encodes multidimensional position. The rotation is realized blockwise for computational efficiency and expressivity (Yu et al., 4 Jun 2025). Fourier-based embedders concatenate or sum learned position codes with token embeddings before the Q/K/V projections in attention (Li et al., 2021).

• MLP/Task-specific modules: In coordinate-MLPs, the position matrix $P$ is directly the input to MLPs solving regression or fitting tasks, sometimes regularized by graph Laplacians to enforce smoothness and avoid overfitting (Ramasinghe et al., 2021).
• Sequential encoders: In SeqPE, embeddings are produced by passing digit-tokenized index sequences through a lightweight Transformer with a compositional token/place/dimension embedding table, before concatenation or addition to the input pipeline (Li et al., 16 Jun 2025).
• Cognitive map architectures: In MapFormer, the learnable position matrix is realized dynamically by input-dependent block-diagonal rotations that update key/query projections, achieving disentanglement of episodic or working memory in self-supervised navigation tasks (Rambaud et al., 24 Nov 2025).

3. Training Objectives and Optimization Schemes

Learnable position matrices are updated by standard backpropagation, integrated with one or more loss functions:

• Joint downstream and auxiliary objectives: For prompt learning, cross-entropy on class labels is combined with KL-divergence-based distillation (ensemble knowledge transfer) to refine both soft-tokens and position matrix (Li et al., 26 Nov 2025).
• Contrastive and regularization loss:
  • SeqPE employs a contrastive loss to ensure that similarities of position embeddings reflect geometric proximity, and a KL-divergence-based distillation loss to regularize OOD extrapolation (Li et al., 16 Jun 2025).
  • Graph Laplacian regularization in coordinate-MLPs penalizes rapid local variation of the embedding matrix, yielding $$L = L_{\text{task}} + \lambda \operatorname{trace}(P^\top L P)$$ for stability and generalization (Ramasinghe et al., 2021).
• Spectral-view/temporal filtering: In dynamic graphs (L-STEP), a learnable complex filter in the frequency domain evolves node positions, regularized by spectral approximation guarantees, and refined via task error and position-disagreement penalties (Tieu et al., 10 Jun 2025).
• Gradient flow through differentiable surrogates: Hard permutation matrices are approximated by soft, differentiable relaxations (Gumbel-softmax, temperature annealing) to facilitate gradient-based learning, especially when the end-task is discrete in nature (Li et al., 26 Nov 2025).

4. Empirical Impact and Comparative Analysis

Quantitative ablation and benchmarking across modalities demonstrate:

Method / Setting	Metric	Key Result	Reference
AnchorOPT adaptive vs fixed order	HM (ImageNet)	+0.28 to +1.8 HM gain	(Li et al., 26 Nov 2025)
ComRoPE vs. RoPE/LieRE (ViT-B/16)	Top-1 acc	65.3-65.5% (+1.6%)	(Yu et al., 4 Jun 2025)
LMEC (LM-APE) vs. Conformer (4h)	Test WER	8.32 vs 8.51 (−0.63%)	(Yang et al., 2022)
Fourier+MLP vs. 2D sinusoid (DETR)	AP	+0.1~+0.5 absolute gain	(Li et al., 2021)
SeqPE vs. ALiBi/RoPE (ViT-S)	Acc	80.1% vs. 77–78%	(Li et al., 16 Jun 2025)
L-STEP vs. DyGFormer (UN Trade)	AP	75.84 vs. 65.39	(Tieu et al., 10 Jun 2025)
FLOATER vs. fixed/sinusoid (MT)	BLEU	Δ+0.4 BLEU, better OOD	(Liu et al., 2020)
LapReg-embedder vs. RFF (1D PSNR)	PSNR (test)	31.2 vs. 26.0	(Ramasinghe et al., 2021)

Learnable position matrices consistently outperform fixed or heuristic baselines, especially under extrapolation (long-context, out-of-distribution, or variable resolution). Their plug-and-play nature allows upgrades of legacy models to benefit from inductive generalization without architectural overhaul (Yu et al., 4 Jun 2025, Li et al., 16 Jun 2025).

5. Theoretical Guarantees and Regularization Principles

Several approaches provide formal guarantees:

• Commutativity and positional robustness: In ComRoPE, pairwise commutativity of angle matrices is required and sufficient for scaling and robustness to coordinate shifts, founded on the Baker–Campbell–Hausdorff lemma for matrix exponentials. This ensures that the inner-product attention is invariant to global coordinate shifts, which is critical for scalable and stable extrapolation (Yu et al., 4 Jun 2025).
• Spectral and smoothness bounds: The L-STEP approach links the learnable position evolution to ring-graph Fourier bases, guaranteeing low error under slowly-varying dynamics. Laplacian-regularized embedders ensure stable manifolds, naturally trading memorization and generalization (Ramasinghe et al., 2021, Tieu et al., 10 Jun 2025).
• Continuous-time solution coverage: FLOATER’s ODE-based parameterization allows position encodings of arbitrary length without reparameterization, theoretically covering the infinite-horizon space and yielding empirical gains on long sequences (Liu et al., 2020).

6. Generalization, Extrapolation, and Multimodal Extensions

Recent advances in learnable position matrices enable model generalization across:

• Sequence lengths/Resolutions: ODE-based and Fourier-based encoding mechanisms are inherently inductive and can be extended far beyond the training regime without learned parameter blowup (Li et al., 2021, Liu et al., 2020).
• Multi-dimensional spaces: Both MLP-based Fourier-feature learnable matrices and digit-sequence sequential encoding (SeqPE) natively generalize from 1D to 2D/3D or even higher; the complexity of the position matrix does not grow with the support size, only with input dimension (Li et al., 2021, Li et al., 16 Jun 2025).
• Graph and temporal domains: Spatial-temporal evolution in dynamic graphs is now handled by frequency-domain position matrices, with empirical scaling to large benchmark graphs (e.g., TGB) (Tieu et al., 10 Jun 2025).
• Cognitive mapping and structure-content disentanglement: Input-dependent position matrices (MapFormer) allow path integration, structure-content separation, and cognitive map formation with OOD generalization for navigation and memory tasks (Rambaud et al., 24 Nov 2025).

7. Limitations and Open Directions

Despite demonstrated robustness, several open technical problems remain:

• Commutativity-enforced parameterization vs. expressiveness: While blockwise or globally commuting angle matrices ensure stability, they restrict the possible transformations; research into relaxed commutativity or alternate Lie-group parameterizations (e.g., block, Clifford, quaternion) may unlock greater flexibility (Yu et al., 4 Jun 2025).
• Optimization difficulties: Gumbel-softmax relaxations and Laplacian-regularized learning can be sensitive to initialization scale, temperature schedules, and regularization weights (Li et al., 26 Nov 2025, Ramasinghe et al., 2021).
• Integration with deep intermediate layers: Whether joint optimization of positional parameters with content weights remains stable in very deep and large-scale models, or under fine-tuning regimes, is still under investigation. Empirical evidence suggests end-to-end updating without proper regularization can cause overfitting (Ramasinghe et al., 2021).
• Computational scaling for extremely large domains: Approximating or learning over very high-dimensional or sparse spatial/temporal graphs remains challenging, but blockwise and spectral approximations have shown promise (Tieu et al., 10 Jun 2025).

Continued research focuses on enhancing scalability, adaptability, and inductive biases in both generative and discriminative settings, with learnable position matrices now a central component in high-performing, generalizing neural architectures.