Coordinate-Indexed Representation

• Coordinate-indexed representation is a mapping that directly associates explicit coordinates with features or values, underpinning methods in neural implicit models and combinatorial coding.
• This approach is applied in graphics and vision for tasks like image reconstruction, 3D scene modeling, and heatmap-based localization, providing high-precision output with methods such as multi-resolution hashing and Fourier constraints.
• To balance scalability and accuracy, advanced architectures utilize split-MLPs, distribution-aware decoding, and matrix-based positional schemes that optimize computational efficiency and interpretability.

A coordinate-indexed representation is a functional or combinatorial mapping wherein the value, property, or coding of a mathematical or signal object is defined directly as a function of an explicit coordinate, typically in a continuous, discrete, or hybrid domain. This paradigm underlies explicit neural fields in graphics, positional coding in discrete mathematics, numeration systems on high-dimensional discrete objects, and the heatmap-based localization protocols in vision and inference. Coordinate-indexed frameworks establish bijections or encodings between locations in a domain (continuous or discrete coordinates) and features, intensities, or combinatorial data, making them foundational in applied mathematics, machine learning, signal processing, and combinatorial design.

1. Formal Definition and Types

In its general form, a coordinate-indexed representation models a target object or signal $v(x)$ as the output of a map $f_\theta: \mathcal{C} \rightarrow \mathcal{R}$, where $\mathcal{C}$ is a space of coordinates (e.g., $\mathbb{R}^d$, $\mathbb{Z}^n$, or a combinatorial domain), and $\mathcal{R}$ is a value space (e.g., colors, intensities, labels, combinatorial codes). The coordinate $x$ directly indexes information, as opposed to being a mere feature or latent input.

In neural implicit models (coordMLPs), this paradigm is formalized as $f_\theta: \mathbb{R}^K \to \mathbb{R}^O$, with $x\in\mathbb{R}^K$ representing the spatial-temporal (or geometric) index, and $f_\theta(x)$ predicting, for example, color, occupancy, or density at $x$ (Liang et al., 2022). In combinatorial position coding, subarrays of prescribed size uniquely identify coordinates, establishing a bijection between the local pattern and a global position (0706.0869). In numeration systems over posets and high-dimensional vectors, coordinate-indexed expansions leverage mixed-radix or matrix-based positional schemes, systematically encoding multidimensional data (0804.1728, Farkas et al., 2023).

2. Neural Coordinate-Indexed Representations

Coordinate-based neural networks, especially implicit neural representations, have become central for continuous signal modeling in image reconstruction, 3D scanning, volumetric rendering, and neural scene representations. The core model is an MLP $f_\theta$ queried at an input coordinate to produce the signal:

• Baseline MLP Formulation: For $x\in\mathbb{R}^K$, propagate through $L$ layers as $h_{j} = \phi(W_j h_{j-1} + b_j)$, where $h_0=x$. Each query is independent and compute scales as $O(N L M^2)$ for $N$ queries and width $M$ (Liang et al., 2022).
• Accelerated Schemes: The CoordX split-MLP architecture decomposes early layers along coordinate axes, processes each axis separately, and fuses features in later layers to reduce compute, achieving up to $2.9\times$ speedup with minimal accuracy compromise (Liang et al., 2022). In volumetric data compression, multi-resolution hash encodings map coordinates to compact feature vectors via learnable lookup tables at each resolution, allowing efficient and robust representation of complex fields (Devkota et al., 2024).
• Band-Limited Models: BACON provides explicit control of the mapping’s frequency content by imposing a known, fixed Fourier spectrum using multiplicative filter networks; each layer is associated with sinusoidal filters whose frequencies are fixed and prescribe the global function bandwidth (Lindell et al., 2021).

3. Combinatorial and Discrete Coordinate Encodings

Coordinate-indexed coding is foundational in discrete mathematics for position coding and numeration of multidimensional objects:

• Position Coding: In patterns such as De Bruijn sequences and Anoto codes, each fixed-size local window (e.g., $k$-tuple or $m\times n$ block) occurs at most once, guaranteeing that the observed window uniquely determines its global coordinate. Decoding thus amounts to reading the local window, mapping via a bijection to its unique position, and vice versa (0706.0869). The Anoto code uses cyclically shifted De Bruijn sequences and combinatorial arithmetic to assign bits per axis, building very large global grids with unique local identification.
• Mixed-Radix and Matrix-Based Numeration: In graded posets and multivariate numeration, sequences of digits indexed by levels or vector positions, with variable base per position (from a sequence $F$ or matrix $M$), uniquely encode chain positions or vector elements. These systems generalize ordinary base-$b$ and factoradic representations to infinite products or matrix expansions, crucial for encoding maximal chains in DAGs or vectors in $\mathbb{Z}^m$ (0804.1728, Farkas et al., 2023). Existence and uniqueness hinge on residue-system and expansiveness conditions in the case of matrices.

4. Coordinate-Indexed Signal Localization: Heatmaps and Statistical Decoding

In human pose estimation and vision, coordinate-indexed (heatmap-based) representations encode target locations as peaks on continuous or discrete spatial maps:

• Heatmap Encoding: For a ground-truth coordinate $(x^*,y^*)$, the label heatmap for training is a spatial Gaussian centered at $(x^*,y^*)$, softening the target and enabling convolutional localization (Zhang et al., 2019).
• Decoding Accuracy: Integer grid quantization and argmax-based decoding introduce bias and coarse localization. Distribution-aware decoding (DARK) employs a Taylor expansion of the log-heatmap at the peak to recover a subpixel-accurate coordinate (solving $\nabla f(\mu)=0$ via the Hessian). Skipping integer rounding in encoding removes bias, and together, these advances yield $\sim+3.8$ AP on COCO for low-res models (Zhang et al., 2019, Dai et al., 2020).
• Generality of Approach: This distribution-aware pipeline extends to volumetric maps, 3D landmarks, and general point localization, relying on the principle of representing output as a function of spatial coordinates and leveraging local second-order information for high-precision estimation.

5. Infinite-Dimensional and Advanced Indexing

Fractional Brownian fields indexed by product spaces such as $(h,f)\in (0,1/2]\times L^2(T,m)$ extend the coordinate-indexed paradigm into stochastic process theory. Here, $h$ is the Hurst parameter and $f$ an $L^2$ function, and the field $X(h,f)$ is constructed so that all known covariance and regularity properties are prescribed by coordinate arguments (Richard, 2013). Covariance is explicitly given by kernel functions of the coordinates, and the process admits sharp variance, continuity, and small-ball estimates in both coordinate directions. Multiparameter and set-indexed extensions naturally fit this framework, as do multifractional variants.

6. Trade-Offs, Limitations, and Interpretability

Coordinate-indexed representations, while offering universality and flexibility, present trade-offs:

• Scalability: Naive MLPs scale poorly to high query counts ($N$ large), necessitating architectural tricks (splitting, hashing, or bandlimiting) for feasible inference and training (Liang et al., 2022, Devkota et al., 2024).
• Numerical and Memory Overhead: Multi-resolution encodings and matrix-based positional systems trade memory for compression and convergence; increasing hash-table size or feature dimension increases memory, but sublinearly improves PSNR (Devkota et al., 2024). Full splitting in CoordX can degrade high-frequency fidelity, requiring fusion trickery or retention of shared layers (Liang et al., 2022).
• Interpretability: Band-limited coordinate networks (BACON) ameliorate traditional black-box opacity, permitting explicit derivation of Fourier content, artifact prediction, and spectral blending (Lindell et al., 2021). Unique window-based coding in discrete arrays guarantees injectivity, but comes at the cost of combinatorial design complexity (0706.0869).
• Extensibility: For domains with labile or high-dimensional coordinates (function spaces, set indices), coordinate-indexed representations generalize classical notions and encode continuity, regularity, or combinatorial structure via the argument’s coordinate (Richard, 2013, 0804.1728).

7. Applications and Broader Impact

Coordinate-indexed representations underpin applications in graphics (novel-view synthesis, neural rendering, volumetric compression), computational geometry (distance functions, mesh encoding), discrete mathematics (coding theory, design of position sequences), and statistical estimation (keypoint detection, field construction). Architectures and schemes are cross-pollinated: signal processing leverages both continuous and discrete coordinate encodings, while neural forecast models now routinely adopt positional encodings or coordinate-indexed fields for data with geometric or spatiotemporal structure. The unifying concept is the explicit, (almost always bijective or strongly invertible) mapping between coordinates and value or code.

The main research trajectory continues to optimize for speed, compression, interpretability, and unbiased precision in coordinate-indexed maps, with theoretical work extending the reach into infinite-dimensional and non-Euclidean domains and practical advances targeting low-latency, high-fidelity deployment in scientific and commercial systems (Liang et al., 2022, Lindell et al., 2021, Devkota et al., 2024, 0706.0869, Farkas et al., 2023, 0804.1728, Zhang et al., 2019, Dai et al., 2020, Richard, 2013).