4D Positional Encoding Overview

• 4D positional encoding is a method to embed spatial-temporal coordinates into vector spaces using fixed Fourier, learnable, and biologically-inspired designs.
• It employs diverse techniques like Fourier feature projections, grid-cell-inspired schemes, and shifted-basis methods to preserve local detail and global spatial relations.
• Empirical studies demonstrate improved dynamic scene modeling, robust interpolation, and efficient integration into multi-dimensional neural architectures.

Positional encoding in four dimensions (4D) refers to mathematical and algorithmic frameworks that embed 4D coordinates—commonly spatial-temporal (xyz+t) or general high-dimensional indices—into vector spaces suitable for neural network consumption, preserving geometric, relational, and frequency content. 4D positional encoding is fundamental for tasks such as dynamic scene modeling, 4D view synthesis, high-dimensional Transformer attention, and spatially-aware generative models. Recent literature establishes both theoretically principled and learnable approaches, spanning fixed, biologically-inspired, and fully adaptive designs.

1. Mathematical Foundations and Design Criteria

Four-dimensional positional encoding schemes aim to address two primary objectives: (a) sufficient capacity to recover high-frequency local detail; and (b) preservation of global and local spatial relationships—particularly, distance and shift invariance—under neural network processing. In general, these encodings can be classified as fixed Fourier/sinusoidal mappings, learnable Fourier-feature projections, shifted-basis embeddings, grid-cell-inspired superpositions, or sequential symbolic encodings.

The canonical mathematical constructions extend from lower-dimensional cases. For instance, the random Fourier features approach for $x \in \mathbb{R}^4$ generates:

$$\phi(x) = [\cos(2\pi w_1 \cdot x), \sin(2\pi w_1 \cdot x), ..., \cos(2\pi w_m \cdot x), \sin(2\pi w_m \cdot x)] \in \mathbb{R}^{2m}$$

where $w_j$ are frequency vectors sampled from an isotropic Gaussian (Zheng et al., 2021). Similar expansions form the basis of GridPE (Li et al., 11 Jun 2024).

Central to design is the stable rank of the embedding matrix (controlling memorization capacity) and the induced inner-product kernel $D(x,x')$ (governing generalization and interpolation). Non-Fourier alternatives can also satisfy these properties, but the random Fourier mapping is a special case that ensures both, with kernel convergence to the Gaussian:

$$D(x,x') = \mathbb{E}_{w}[\cos(2\pi w \cdot (x-x'))] = \exp(-2\pi^2 \sigma^2 \|x-x'\|^2)$$

(Zheng et al., 2021). In the 4D context, practical implementation requires dimensionality reduction (separable embeddings per axis, random-direction samplings) to avoid exponential parameter growth.

2. Fourier-Based and Learnable Encodings in 4D

Learnable Fourier feature encodings for 4D data are realized by parameterizing the frequency selection itself. Given $x \in \mathbb{R}^4$, a trainable matrix $B \in \mathbb{R}^{m \times 4}$ is used:

$$\phi(x) = [\sin(2\pi Bx) \, \| \, \cos(2\pi Bx)] \in \mathbb{R}^{2m}$$

where $Bx$ is an $m$-vector of inner products and $\|$ denotes concatenation. An optional small MLP $g(\cdot)$ further modulates $\phi(x)$, giving a non-linear projection (Li et al., 2021). The resulting embeddings allow direct control over capacity and scale sensitivity—by tuning $m$, the embedding dimensionality, and initial scale $\gamma$.

Empirical results in 4D widget-captioning and object detection tasks using this approach demonstrate improved metrics compared to fixed-index or axis-separable encodings, with capacity for strong L₂ distance preservation and fast convergence. Ablations confirm that a moderate $m$ (e.g., half the model embedding width) and small hidden widths ($H=32$–64) suffice for nearly optimal performance.

3. Biologically-Inspired and Grid Cell-Based Encodings

Grid-cell-inspired positional encodings—GridPE—draw from computational neuroscience, superposing planar Fourier waves at geometrically spaced wavelengths. For any $x \in \mathbb{R}^4$, select $m$ scales using ratio $r = e^{1/4} \approx 1.284$, and for each scale, sample $D_i$ unit directions on the 4D sphere $S^3$:

$$\boldsymbol\omega_{i,j} = \frac{2\pi}{\lambda_i} \mathbf{u}_{i,j}$$

$$\mathrm{GridPE}(x) = [\cos(\boldsymbol\omega_{i,j}^T x), \sin(\boldsymbol\omega_{i,j}^T x)]_{i=1..m,\; j=1..D_i} \in \mathbb{R}^{2D}$$

(Li et al., 11 Jun 2024). The induced kernel is shift-invariant:

$$\langle \Phi(x), \Phi(y) \rangle = \sum_{i=1}^D e^{j \boldsymbol\omega_i^T(x-y)}$$

By design, this enables both absolute and relative spatial representation, flexible scaling, and optimal coverage of 4D Euclidean space at all granularities. Integration strategies include direct additive projections onto query and key vectors, block-diagonal rotations, or Hermitian inner products.

4. Conditional and Temporal 4D Encodings

For applications in dynamic radiance fields, e.g. novel view synthesis in time-varying scenes, positional encoding must account for the entanglement of spatial and temporal components. V4D introduces a 4D conditional positional encoding (CPE) that injects a global time index $t$ as a phase shift into high-frequency spatial features:

$$\gamma(p_n; t) = \begin{bmatrix} \sin(2^{L-1} \pi p_n + \frac{2\pi}{2^{L-1}\pi} t) \ \cos(2^{L-1} \pi p_n + \frac{2\pi}{2^{L-1}\pi} t) \end{bmatrix}$$

with $L=5$ bands for spatial frequencies up to $16$ cycles/unit (Gan et al., 2022). This mapping is parameter-free and doubles the feature channels of the spatial embedding. Notably, CPE is only applied to texture channels, leaving density untouched to preserve physical transmittance constraints.

Empirical ablation demonstrates that CPE yields consistent gains (0.2–0.4 dB PSNR, improved SSIM/LPIPS) over time-agnostic PEs and enables recovery of sharper, high-frequency appearance in dynamic scenes.

5. Shifted-Basis and General Kernel Embeddings

Generalizing beyond the Fourier domain, shifted-basis positional encodings sample a function $\psi$ over shifted anchors in $\mathbb{R}^4$:

$$\phi(x) = [\psi(x-\tau_1), \psi(x-\tau_2), ..., \psi(x-\tau_m)]^T$$

Selections include axis-separable grids (where $\phi(x)$ is a concatenation of 1D embeddings per coordinate) and random-direction samplings with radial or non-Fourier functions (Zheng et al., 2021). The design is governed by bandwidth and shift spacing, with capacity and generalization controlled via the stable rank of the resulting matrix. Properly designed, shifted-basis encoders approximate optimal kernel properties for regression and interpolation.

6. Fully Learnable Sequential Multi-Dimensional Encodings

SeqPE introduces a fully end-to-end learnable scheme for multi-dimensional discrete indices. Each dimension of a 4D index $p = (x_1, x_2, x_3, x_4) \in \mathbb{Z}^4$ is expanded to fixed-width sequences of base-$b$ digits, concatenated and marked with a special token. Digit, position, and dimension-specific embeddings are summed and processed through a lightweight Transformer encoder, yielding $e_p \in \mathbb{R}^d$ (Li et al., 16 Jun 2025).

Regularization is achieved through contrastive distance alignment and out-of-distribution (OOD) distillation losses:

• Contrastive: Embedding distances align with Euclidean $\|p-p'\|_2$
• OOD: Teacher embeddings anchor extrapolated positions for stability

This architecture supports arbitrary extension to new coordinate ranges and dimensions with negligible computational cost. Ablation indicates that joint use of contrastive and distillation terms yields the strongest performance on out-of-distribution indices.

7. Practical Implementation and Computational Considerations

For all frameworks, 4D positional encoding must balance expressivity, memory/compute cost, and extrapolation/generalization. Comparative analysis:

Encoding Method	Dimensionality Scaling	Extrapolation Properties	Parameterization
Fixed Fourier/Random	Linear in $m$	Kernel extrapolates naturally	Frequency vectors fixed/random
Learnable Fourier	Linear in $m$	Adaptively learns new ranges	$B$ matrix trainable
GridPE (biological)	$2 m D_0$	Shift-invariant kernel, robust	Module count, directions
Shifted-basis	Linear in #shifts	Kernel determined by $\psi$	Embedding function/sample scheme
SeqPE (symbolic, NNs)	Logarithmic in range	Full learnability, OOD robust	Transformer, token matrices

Implementation steps include:

• Choosing embedding dimensionality $m$ such that $2m$ or $2m D_0$ matches model head dimension
• Sampling frequency vectors/directions uniformly or from Gaussian/optimal distributions
• (SeqPE) Deciding base $b$ and digit width $k$ for coverage
• Modulating with small MLPs where flexibility is desired
• Integrating into downstream model via additive, multiplicative, or learned mixing functions

For all methods, parameter and computational overhead is a modest fraction compared to the main model. GridPE and Fourier-feature variants allow efficient vectorized batch calculation.

8. Impact and Empirical Results

All reviewed frameworks provide clear empirical superiority over naïve index or axis-aligned encodings in 4D tasks. V4D’s CPE recovers lost high-frequency details in dynamic NeRF scenes (Gan et al., 2022). Learnable Fourier encoding enables strong L₂ distance-aware transformations in widget-captioning and object detection (Li et al., 2021). SeqPE generalizes symbolic sequences in 4D to novel domains with robust extrapolation (Li et al., 16 Jun 2025). GridPE provides biologically efficient, shift-invariant representations, supporting both absolute and relative position awareness (Li et al., 11 Jun 2024). The shifted-basis formalism characterizes generalization via kernel analysis and stable rank (Zheng et al., 2021).

Collectively, 4D positional encoding stands as a cornerstone of high-dimensional neural modeling, enabling spatial, temporal, and relational induction in modern architectures with rigorously established mathematical and empirical properties.