O-EENC-SD: Centroid Refinement Decoding

• O-EENC-SD is a centroid refinement decoding approach that leverages centroidal Voronoi tessellations to optimize geometric partitioning and suppress error in reconstruction.
• It employs iterative Lloyd’s algorithm with controlled rotational perturbations to avoid symmetry traps and enable precise multi-scale decoding for applications like imaging and LLM inference.
• The method demonstrates improved numerical stability and computational efficiency, yielding up to 3–4× throughput gains and significant error reduction in practical high-dimensional tasks.

A Centroid Refinement Decoder leverages centroid-based geometric structures or local statistics to improve the fidelity, stability, or efficiency of reconstruction and retrieval processes. Across computational geometry, signal processing, and modern large-scale retrieval (notably Long-Context LLM inference), such decoders use centroidal or centroid-refined constructs to partition, resample, or hierarchically index data. This approach systematically reduces distortion, suppresses error, and enables scalable, accurate decoding across a range of applications, including multi-scale function reconstruction, astrometric calibration, and high-throughput neural attention sparsification.

1. Geometric Refinement and Centroidal Decoding Principles

The prototypical centroid refinement decoder arises in the context of the Geometric Refinement Transform (GRT), where the computational domain $\Omega\subset\mathbb{R}^n$ is tessellated at each refinement level $m$ using a set of generator points $P_{m}=\{P_{m,i}\}$ to form Voronoi cells $V_{m,i}$. The encoding step computes cell-averaged coefficients: $$c_{m,i} = \frac{1}{|V_{m,i}|} \int_{V_{m,i}} f(x)\,dx$$ for a given function $f:\Omega\to\mathbb{R}$. The decoding process reconstructs $f$ by a partition-of-unity expansion (e.g., piecewise-constant basis): $$\hat{f}_{\mathrm{GRT}}(x) = \sum_{m=0}^M \sum_{i=1}^{N_m} c_{m,i}\,\chi_{V_{m,i}}(x)$$ where $\chi_{V_{m,i}}(x)$ is the indicator function of cell $V_{m,i}$ (Mullaghy, 26 Mar 2025).

The centroid refinement replaces arbitrary Voronoi decompositions at each refinement level with Centroidal Voronoi Tessellations (CVTs), in which each generator $P_i$ is iteratively updated to the centroid $c_i$ of its current cell: $c_i = \frac{1}{|V_i|} \int_{V_i} x\,dx$ Minimizing the centroidal energy $E(\{c_i\},\{V_i\})=\sum_i \int_{V_i}\|x-c_i\|^2 dx$ via Lloyd’s algorithm yields refinements that optimize symmetry, reduce geometric distortion, and improve numerical stability.

2. Algorithmic Realization and Multi-Scale Decoding

The decoder workflow proceeds as:

1. For each refinement level $m$, initialize seeds $P^0$. Apply Lloyd’s algorithm:
   • Voronoi step: compute cells $V_i^{(t)}$ for $P_i^{(t)}$.
   • Centroid step: compute centroids $c_i^{(t)}$ of the current cells.
   • Update: set $P_i^{(t+1)} \leftarrow c_i^{(t)}$ and iterate until convergence.
2. Compute coefficients for each centroidal Voronoi cell: $$c_{m,i} = \frac{1}{|V_{m,i}|} \int_{V_{m,i}} f(x)\,dx$$
3. Decoding is performed by evaluating for any $x$: $$f_{\mathrm{approx}}(x) = \sum_{m=0}^M \sum_{i=1}^{N_m} c_{m,i}\,w_{m,i}(x)$$ where $w_{m,i}(x)$ may be either the indicator function for piecewise-constant reconstructions, or smooth interpolation kernels (e.g., barycentric or Sibson–natural neighbor weights) (Mullaghy, 26 Mar 2025).

3. Convergence, Optimality, and Stability Properties

The centroid refinement decoder exhibits several optimality properties:

• Geometric distortion minimization: Uniform diameter CVTs minimize $\max_i\operatorname{diam}(V_i)$, leading to tight error bounds.
• Reconstruction error bounds: For Lipschitz continuous $f$, the cell approximation error is bounded by $L \operatorname{diam}(V_i)$ (first-order) and $\frac{L_2}{2} \operatorname{diam}(V_i)^2$ (second-order, with $L_2$ the Lipschitz constant of $\nabla f$).
• Global $L^2$ error minimization: Among all Voronoi refinements, the CVT-based decoder achieves minimal $L^2$-norm reconstruction error: $$\int_{\Omega} |f-f_{\mathrm{CVT}}|^2 dx \leq \int_{\Omega} |f-f_{\mathrm{arb}}|^2 dx$$ (Mullaghy, 26 Mar 2025). For multiscale refinement, these error reductions are preserved across levels.
• Numerical stability: CVT refinements yield interpolation matrices with 2–5$\times$ lower condition numbers compared to arbitrary Voronoi tessellations and substantially lower reconstruction error in practical domains like medical imaging segmentation (by 20–30\%) (Mullaghy, 26 Mar 2025).

4. Perturbation Strategies and Escape from Symmetry Traps

Lloyd’s algorithm can converge to symmetry-preserving saddle points or suboptimal local minima if initial seeds possess excessive symmetry. Centroid refinement decoders apply rotational perturbations after each Lloyd iteration: $$P_i \leftarrow R_{\epsilon}(P_i), \qquad R_\epsilon \approx I + \epsilon A,\quad \epsilon\ll 1$$ where $A$ is a skew-symmetric generator. By randomly sampling $R_\epsilon$, the decoder discovers lower-energy CVTs and prevents stagnation in symmetric attractors (Mullaghy, 26 Mar 2025).

5. Complexity, Runtime, and Application Domains

Centroid refinement decoders exhibit tractable computational costs:

• Encoding (refinement): Voronoi construction is $O(N \log N)$ in 2D; centroid computation within each cell is $O(N)$; Lloyd iterations converge in $10$–$50$ steps for $\epsilon_{\mathrm{CVT}}\sim 10^{-6}$.
• Decoding: Piecewise-constant reconstructions are $O(1)$ per query point; higher-order interpolants involve $O(\log N)$ point-location and $O(k)$ neighbor kernel evaluations ($k\sim 6$–$20$).
• Batch evaluation: Matrix–vector form $O(N_\mathrm{pts}\cdot k)$ for $N_\mathrm{pts}$ queries (Mullaghy, 26 Mar 2025).

Key application domains include medical imaging, where centroid refinement suppresses segmentation error; physics simulations; and signal processing, where both minimax distortion and stable interpolation are essential.

6. Centroid Refinement in Modern Large-Scale Retrieval

Recent developments extend centroid refinement to high-dimensional retrieval scenarios, notably in memory-optimization for long-context LLM inference. The CTkvr algorithm employs "centroid then token" refinement for key–value (KV) cache retrieval. Initial centroid-grained partitioning is followed by fine token-level reranking, supporting both high efficiency and minimal accuracy loss. Centroids are constructed in the RoPE-transformed query space, exploiting the empirically observed high cosine similarity (>0.8 for $\lvert t-t'\rvert\lesssim 1\mathrm{K}$) of adjacent queries under positional encoding (Lu et al., 17 Dec 2025).

The two-stage retrieval pipeline is:

1. Centroid recall: Compute cosine similarity $Q$ to all centroids and gather candidate KV indices via precomputed qcIVF structure.
2. Token-level refinement: Rerank recalled indices using attention scores for precise retrieval.
3. Hybrid CPU–GPU execution: Prefilling and initial retrieval indexing are performed on GPU, fine reranking and sparse attention on CPU, with data flows overlapping to maximize throughput.

Empirically, CTkvr achieves sub-1% average degradation in LLM accuracy while producing $3$–$4\times$ throughput gains at $96\textrm{K}$ context length (Lu et al., 17 Dec 2025). The method generalizes to other RoPE-based transformers and supports hierarchical or multimodal extensions.

7. Performance Limits, Error Analysis, and Generalization

Centroid refinement decoders achieve error floors determined by geometric cell sizes (in geometric transforms) or centroid/token similarity thresholds (in retrieval). In the presence of smoothness (Lipschitz) assumptions, convergence is provable and optimal at each scale (Mullaghy, 26 Mar 2025). In high-precision astrometric settings, the centroid refinement decoder achieves sub-micro-pixel errors by modeling pixel-response function (PRF) variations to third order in Fourier space (Zhai et al., 2011). Performance in such applications is photon-noise limited: $$\sigma_\textrm{cent} \approx \frac{1.5}{\sqrt{N_\textrm{ph}}}$$ with $\sim 2.2 \times 10^{10}$ photons required to reach $10\,\mu\textrm{-pixel}$ accuracy (Zhai et al., 2011).

In general, centroid refinement decoders deliver reconstruction and retrieval that is robust to local perturbation, scalable to large or high-dimensional contexts, and adaptable via multi-level or adaptive centroid structures. Extensions to hierarchical, adaptive, and multi-modal contexts are supported in both geometric and neural settings (Mullaghy, 26 Mar 2025, Lu et al., 17 Dec 2025).