Position Interpolation (PI)

• Position Interpolation is a mathematical framework that synthesizes continuous signals from discrete samples, balancing high-frequency details with smooth transitions.
• It employs analytic and learned interpolation weights with basis augmentation to accurately map features across spatial, temporal, or sequential domains.
• PI is key for applications in 3D geometry, neural rendering, and LLMs, enhancing positional encodings and stabilizing performance in extended context scenarios.

Position Interpolation (PI) is a family of mathematical and algorithmic frameworks for synthesizing continuous signals, mappings, or transformations at novel spatial, temporal, or sequential positions from finite, discrete samples or network outputs. PI appears across modalities, from geometric interpolation on Lie groups, to surface properties, to latent space aggregations in neural audio and vision models, and especially as a central mechanism for extending positional encodings in LLMs. In each case, PI balances the tradeoff between faithful high-frequency detail and smooth, physically or semantically coherent transitions, with the ultimate goal of generalizing models to domains, contexts, or input lengths far beyond those seen at training.

1. Theoretical Foundations and Mathematical Formulations

The core of Position Interpolation lies in directly mapping function values or model parameters at arbitrary positions through a combination of learned or analytic interpolation weights and basis functions.

Generic PI Framework. Given a set of “anchors” or reference points $\{p_j\} \subset \mathbb{R}^n$ with associated local models or features (e.g., MLP outputs, embeddings, basis coefficients), and a query location $p$, PI typically defines normalized weights $w_j(p)$ by a kernel, e.g., inverse distance,

$$w_j(p) = \frac{\phi(\|p-p_j\|)}{\sum_{k\in\mathcal{N}(p)} \phi(\|p-p_k\|)}, \quad \phi(r) = \frac{1}{r}$$

for $K$-nearest anchors $\mathcal{N}(p)$, enforcing $\sum_j w_j(p) = 1$. The desired property vector (e.g., geometric pose, color, SH/Fourier coefficients, embedding) is synthesized as a (possibly weighted) sum over anchor outputs, optionally modulated by freely learned basis functions: $$\theta(p) = \theta^0 + \sum_{k=1}^B \alpha_k(p) B_k$$ with $\alpha_k(p)$ typically a locally weighted blend of anchor-MLP coefficients, and $\{B_k\}$ a set of unconstrained “offset bases” enhancing representational capacity (Zhan et al., 17 Apr 2025).

In high-dimensional function interpolation and geometry, similar schemes appear in boundary-weighted mean value coordinates (Floater et al., 2019) and spline constructions in the tangent/algebra space of SE(3) for rigid body motions (Mueller, 21 Sep 2025).

Linear PI for Sequence Models. For position encodings (e.g., RoPE/ALiBi in LLMs), if models are pretrained for a maximum context length $L$, PI linearly rescales absolute or relative positions, squeezing long-range indices into their trained range: $$r' = \alpha r, \quad \alpha = \frac{L_\mathrm{orig}}{L_\mathrm{new}}$$ and applies embeddings, rotary phases, or bias slopes at $r' \leq L_\mathrm{orig}$. This guarantees all positional signals remain within trained operational bounds, sharply contrasting with naive extrapolation, which leads to instability and attention magnitude blowup (Chen et al., 2023, Al-Khateeb et al., 2023).

2. PI in 3D Geometry, Lie Groups, and Motion Planning

Position Interpolation is integral to geometric modeling, especially where the underlying space is nonlinear or a Lie group. On SE(3), PI is realized either through explicit polynomial synthesis in twist coordinates or via rational trajectories induced by ambient linear interpolation in dual-quaternion or matrix representations (Mueller, 21 Sep 2025, Schröcker, 2017).

SE(3) Polynomial PI. For boundary-value problems over rigid poses $C_0, C_T$ with prescribed initial/final twists, a cubic interpolant is given in exponential coordinates: $C(t) = C_0 \exp\left( \hat{\xi}(t) \right)$ where $\xi(t)$ is a cubic polynomial vector satisfying endpoint and twist conditions. More generally, higher-order interpolation matches both pose and derivatives, constructed through Magnus-series expansions.

Ambient Linear Interpolation. By embedding poses as points in $\mathbb{R}^{12}$ or $P^7$ (Study dual quaternions), one takes convex combinations and projects/interprets the path as a valid rigid motion:

• In $P^7$, linear interpolation yields a vertical Darboux motion—a planar elliptical (or linear) trajectory about and along a fixed axis.
• In the matrix embedding, cubic-circular motions arise, ensuring analytic, rational, and group-intrinsic interpolation (Schröcker, 2017).

PI on Lie groups preserves group structure and coupling between rotation and translation, unlike Euclidean splines that decompose the two.

3. Neural and Signal Processing Applications

Modern high-fidelity rendering, audio, and scientific models increasingly depend on PI variants. For Gaussian human avatars (Zhan et al., 17 Apr 2025), spatially distributed anchor-MLPs generate local pose-dependent coefficients. Each anchor is associated with a compact MLP $\mathcal{E}^j(\theta_\mathrm{pose})$, outputting basis coefficients, which are then blended for any Gaussian at $p$: $$\alpha_k(p) = \sum_{j \in \mathcal{N}(p)} w_j(p) \alpha_k^j$$ The final properties ($\mu$ (mean), $\Sigma$ (covariance), $c$ (color/SH)) are composed as

$\theta(p) = \theta^0 + \sum_k \alpha_k(p) B_k$

Surface control points with a second interpolation stage enforce geometric integrity and robustness for dynamic, pose-dependent rendering.

For head-related transfer function (HRTF) modeling (Ito et al., 2022), PI is reinterpreted as a position-conditioned autoencoder. Encoder and decoder weights are hypernetworks modulated by spatial position, while a prototype aggregation collapses position-dependent embeddings to subject-specific latent codes.

4. PI in Positional Encoding for LLMs

A primary influence of PI is in scaling positional encodings in transformers:

RoPE and ALiBi:

• For models pretrained using rotary position encoding, PI rescales all token position indices ($m \mapsto (L_\mathrm{orig}/L_\mathrm{new}) m$). Rotary phases and embedding rotations remain within original angular domains, preventing attention-score explosion on long contexts (Chen et al., 2023, Qiao et al., 17 Sep 2025).
• ALiBi, using linear bias slopes $m_j$, applies $m'_j = m_j (L/L')$ on extended inputs. This parametric stretch preserves recency bias and prevents underflow or over-regularization of attention weights (Al-Khateeb et al., 2023).

Stability and Empirics:

• Theoretical comparison reveals a $\sim600\times$ reduction in worst-case attention instability for PI versus extrapolation in RoPE (Chen et al., 2023).
• Empirically, PI-extended LLMs maintain training-quality perplexity out to $16$–$32$k context length, with minimal short-context regression.
• PI requires no architectural changes. Fine-tuning for only $\sim1000$ steps suffices to adapt the model, and all optimizer and kernel infrastructure can be maintained.

Quantization Failures and Remedies:

PI interacts nontrivially with post-training quantization, inducing aliasing, dynamic range dilation, axis-grid anisotropy, and outlier shifting. Diagnostics such as Interpolation Pressure (IP) and Tail Inflation Ratio (TIR) quantify these pathologies, and methods such as Q-ROAR use frequency-band-wise post-quantization scaling to recover lost accuracy and stabilize attention logits on long contexts (Qiao et al., 17 Sep 2025).

5. Transfinite and Mean Value Interpolation in Continuous Domains

Position Interpolation underpins generalized barycentric coordinate systems and transfinite interpolants on domains and manifolds. For example, mean value coordinates over arbitrary polygons construct continuous, boundary-aware interpolants for arbitrary edge data $f$ via

$$g(x) = \frac{\sum_{e \in E} \tau_e(x) I_e(x; f)}{\sum_{e \in E} \tau_e(x) I_e(x)}$$

with $I_e(x; f)$ edge-integrals and $\tau_e(x)$ signed orientations (Floater et al., 2019). These coordinates reproduce linear data exactly, extend to continuous boundary data (“transfinite”), and converge to $f$ on $\partial \Omega$.

6. Practical Implementation and Guideline Highlights

Position Interpolation designs, across domains, follow several universal implementation precepts:

• Reference locality: Interpolants use only nearby anchors (or $K$-nearest), keeping computation tractable and enabling high-frequency variation.
• Basis augmentation: Explicit basis expansion, often unconstrained, allows piecewise smooth blends to express fine detail.
• Decoupled coefficient and basis learning: Anchor-local MLPs predict coefficients (latent, pose, or spatial), amortizing complexity over anchors; bases can be high-frequency or global.
• No architectural or optimization overhaul: Especially in PI for LLMs, the same parameter set suffices, with interpolation schemes acting as lightweight runtime adapters.
• Quantization-aware stabilization: Diagnostics and per-frequency adaptation correct PI-PTQ mismatch for production systems (Qiao et al., 17 Sep 2025).

7. Limitations and Ongoing Research Directions

PI methods, while powerful, display intrinsic limitations:

• Finite extrapolation radius: For LLM encodings, stability degrades beyond $\sim2$–$16\times$ trained context, with failures in tasks like fine-grained retrieval (Al-Khateeb et al., 2023, Chen et al., 2023).
• Resolution loss: “Squeezing” longer contexts into pre-trained windows impoverishes the positional discriminability for short-range tokens.
• Underexplored generalization: Most theoretical and empirical results are for rotary/frequency-based position schemes; extension to learned absolute embeddings or novel parameterizations requires case-specific PI adaptation.
• Quantization effects: Interaction between PI and static quantizers necessitates corrective schemes, and further research is needed on integrated quantization-aware PI strategies.

A plausible implication is that future PI research will focus on adaptive, dynamically learned interpolation and diagnostic schemes—integrated with training and quantization—optimized for both model fidelity and deployment efficiency at extreme context sizes and multidomain signal settings.

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References:

• Real-time High-fidelity Gaussian Human Avatars with Position-based Interpolation of Spatially Distributed MLPs (Zhan et al., 17 Apr 2025)
• Geometric Interpolation of Rigid Body Motions (Mueller, 21 Sep 2025)
• From A to B: New Methods to Interpolate Two Poses (Schröcker, 2017)
• Extending Context Window of LLMs via Positional Interpolation (Chen et al., 2023)
• Position Interpolation Improves ALiBi Extrapolation (Al-Khateeb et al., 2023)
• Q-ROAR: Outlier-Aware Rescaling for RoPE Position Interpolation in Quantized Long-Context LLMs (Qiao et al., 17 Sep 2025)
• Transfinite mean value interpolation over polygons (Floater et al., 2019)
• Head-Related Transfer Function Interpolation from Spatially Sparse Measurements Using Autoencoder with Source Position Conditioning (Ito et al., 2022)