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arxiv: 2604.24167 · v1 · submitted 2026-04-27 · 💻 cs.CV · cs.GR· cs.LG

PEPS: Positional Encoding Projected Sampling -- Extended

Guillaume Perez , Janarbek Matai , Takahiro Harada This is my paper

Pith reviewed 2026-05-08 04:44 UTC · model grok-4.3

classification 💻 cs.CV cs.GRcs.LG
keywords positional encodingprojected samplingimplicit neural representationsgrid-based encodingtexture compressionsigned distance functionsparameter efficiencyimage representation
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The pith

Positional encoding decomposes into projected points whose frequency motion patterns enable compact learned grid encodings that outperform prior methods with 25% fewer parameters.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that standard positional encodings in implicit neural representations can be reframed as a set of points obtained by projecting input coordinates at successive frequencies. These points trace distinctive trajectories as frequency varies, providing a natural basis that replaces the need for high-resolution grids. The authors introduce Positional Encoding Projected Sampling to exploit these trajectories for grid-based learning. Across image representation, texture compression, and signed distance function tasks, the resulting models achieve lower error or equivalent quality while using substantially fewer parameters than existing encoders. A sympathetic reader cares because this reduces the memory and compute burden of high-fidelity coordinate-based networks without sacrificing reconstruction fidelity.

Core claim

We propose the Positional Encoding Projected Sampling, where we treat the projection of the original coordinate at each frequency as a point of interest. We describe the motion of each point with respect to the frequencies and show that it follows a unique pattern. Finally, we use the unique motion of each point as a basis decomposition for doing learned positional encoding using grids. We prove, using three competitive applications—image representation, texture compression, and signed distance function—that the proposed approach outperforms the current state of the art methods, and often requires 25% less parameters for equivalent reconstruction error or rendering.

What carries the argument

Positional Encoding Projected Sampling (PEPS), which decomposes each frequency projection into a point and uses the resulting frequency-dependent motion trajectories as the basis for learned grid encodings.

If this is right

  • Image representation reaches lower reconstruction error using grids whose resolution is determined by the motion patterns rather than by brute-force upsampling.
  • Texture compression maintains visual quality with approximately 25 percent fewer network parameters than current grid or encoding baselines.
  • Signed distance function rendering achieves equivalent or better accuracy without the fitting artifacts that high-resolution grids typically produce.
  • Learned positional encodings become feasible on modest grids because the point-motion basis supplies the necessary high-frequency information.
  • The same decomposition applies uniformly to the three tested domains, suggesting the motion patterns are not task-specific.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The motion-pattern analysis could be applied to other coordinate-to-signal mappings such as neural radiance fields, potentially lowering memory use in 3D scene representation.
  • If the trajectories prove stable under input perturbations, the method may reduce sensitivity to coordinate scaling hyperparameters that plague many INR pipelines.
  • Extending the point-motion basis to time-varying signals might enable compact representations of dynamic scenes without increasing grid resolution.
  • Comparing the learned grids against analytically derived ratio-symmetric encodings could test whether the empirical patterns capture deeper geometric structure.

Load-bearing premise

The observed motion patterns of the projected points are consistent and general enough to form a reliable basis for grid-based learned encodings across different signals and without introducing new artifacts.

What would settle it

Run the three reported applications with the motion-pattern basis replaced by random trajectories or by standard positional encoding; if reconstruction error or parameter count is not worse than PEPS, the claim that the unique patterns are essential would be falsified.

Figures

Figures reproduced from arXiv: 2604.24167 by Guillaume Perez, Janarbek Matai, Takahiro Harada.

Figure 1
Figure 1. Figure 1: Grid-PEPS method applied to neural texture compression of a texture set of size k. (a) Given a coordinate point x ∈ R 2 , the method extracts Sϕ = sin xϕ and Cϕ = cos xϕ for all the targeted frequencies along the frequency curves. (b) Then for each point in P = (x, Sϕ1 , . . . , Cϕk ) it samples the grid to get latent values. (c) The result is the concatenation of the latent vector of each point. (d) The r… view at source ↗
Figure 2
Figure 2. Figure 2: Rotation of x and y axis at each frequency. On the first plot, the origi￾nal line of points. Then on each frequency i plot, the coordinates are P E(x) = (PE(x, 2i),PE(x, 2i + 1)) and normalized. Lissajous curves Even if the rotations of each axis is independent, linear and simple, the multi-dimensional resulting spatial information is more complex. Consider first the sinusoidal 2D case. Given a point (x, y… view at source ↗
Figure 3
Figure 3. Figure 3: (left) Power spectra of 10 textures. (right) Power spectra of the Kodak dataset. As we can see, both are following a 1 fα law [37]. For PEPS, aggregating by directly concatenating the latent vectors resulting from each frequency gives the same potential power to each frequency. There￾fore, we design an aggregation function to control the power allocated to each frequency sampled by the PEPS method. Intuiti… view at source ↗
Figure 4
Figure 4. Figure 4: Pink aggregation applied to a latent dimension (d) of 8, and using the Fourier encoding (ϕi = 2iπ). The simple concatenation would directly concatenate the 8*7=56 values (i.e., the blue, purple, orange, and gray ones), while the Pink-PEPS only con￾siders 8+2*4+2*2+2*1=22 values that are inversely spread given their frequency (i.e., the blue, purple, and orange ones). 5 Application: Implicit Image Represent… view at source ↗
Figure 5
Figure 5. Figure 5: Impact of the number of parameters on the image learning capability (PSNR↑). Dashed lines are grid resolution modifications. Regular lines are feature dimension modifications. parameters should always improve the model. Yet, view at source ↗
Figure 6
Figure 6. Figure 6: (Image) 100 pixels samples from the Kodak dataset. Bottom images are Flip error [3] view at source ↗
Figure 7
Figure 7. Figure 7: Dual scatters of PSNR results for the Kodak dataset. Each axis is representing the PSNR of a method. If a point is on the side of a method, its result is better for this method. Original BI Grid LPE NTNN G-PEPS G-P-PEPS NTNPEPS NTNP PEPS G-PEPS-25% G-P-PEPS-25% NTNPEPS-25% NTNP PEPS-25% view at source ↗
Figure 8
Figure 8. Figure 8: 100 pixels samples from the 4K Paving Stone texture set. Bottom images are Flip error [3]. 6 Application: Neural Texture Compression Textures and materials are the backbone of modern 3D engines and physically based rendering that are used in applications such as games. In those engines, many 4K textures must be loaded at the same time, creating a memory bot￾tleneck. That is the reason why neural texture co… view at source ↗
Figure 9
Figure 9. Figure 9: is one of those samples. Original PE LPE BI Grid Grid-PEPS Hash HashPEPS m-grid m-PEPS m-hash m-hashPEPS view at source ↗
Figure 10
Figure 10. Figure 10: Normalized absolute positional encoding for an interval of frequencies for point (0.35,0.2) on the left and point (0.2,0.3) on the right. On both curves, red dots are the points at frequencies 2 iπ for i in {1, 2, 3, 4}. PE(x) 1.000.750.500.250.000.250.500.751.00 PE(y) 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Frequency 0 5 10 15 20 25 30 view at source ↗
Figure 11
Figure 11. Figure 11: Normalized absolute positional encoding for an interval of frequencies for points (0.2,0.3) in red, (0.3,0.45) in green and (0.4,0.6) in blue. All those points have the same ratio and no difference in delta, yet they result in different curves. The cross (+) points are sampled at equivalent frequencies. We can see that they result is clearly different positions. Discrete coordinate case Consider for examp… view at source ↗
read the original abstract

Implicit neural representations (INRs) are increasingly being used as tools to map coordinates to signals, encompassing applications from neural fields to texture compression, shape representations, and beyond. Most INR methods are based on using high-dimensional projections of the initial coordinates through encoders such as grid or positional encoding. Nevertheless, positional encoding is often insufficient and grids, as we show in this paper, require high resolution for being able to learn. In this paper, we demonstrate that positional encoding can be used not only as a high-dimensional embedding but also decomposed as a series of meaningful points. We propose the Positional Encoding Projected Sampling, where we treat the projection of the original coordinate at each frequency as a point of interest. We describe the motion of each point with respect to the frequencies and show that it follows a unique pattern. Finally, we use the unique motion of each point as a basis decomposition for doing learned positional encoding using grids. We prove, using three competitive applications; image representation, texture compression, and signed distance function; that the proposed approach outperforms the current state of the art methods, and often requires 25\% less parameters for equivalent reconstruction error or rendering.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces PEPS (Positional Encoding Projected Sampling) for implicit neural representations. It decomposes standard positional encodings by projecting input coordinates at each frequency, treating these as points whose motion across frequencies is claimed to follow a unique pattern. This pattern serves as a basis decomposition for learned positional encoding on grids. The approach is evaluated empirically on image representation, texture compression, and signed distance function tasks, with the central claim that it outperforms current state-of-the-art methods while often requiring 25% fewer parameters for equivalent reconstruction error or rendering quality.

Significance. If the projected-point motion patterns indeed supply a reliable, artifact-free basis that avoids the high-resolution grid requirement while delivering consistent gains, the method could meaningfully advance parameter-efficient INRs across computer vision and graphics. The multi-task empirical evaluation is a positive feature, as is the explicit framing of positional encoding as a decomposable point-motion process rather than a black-box embedding.

major comments (2)
  1. [Abstract and method description of projected sampling] The abstract asserts that the motion of projected points 'follows a unique pattern' usable for learned positional encoding on grids without high-resolution requirements or new artifacts, yet no derivation, uniqueness proof, or explicit construction of the decomposition is referenced. This is load-bearing for the parameter-reduction claim, as the pattern must demonstrably differ from standard sinusoidal embeddings or implicit dense sampling.
  2. [Experimental evaluation sections] The headline result (outperformance + 25% parameter savings across three tasks) is stated without reference to specific baselines, quantitative tables, error bars, or ablation controls in the provided text. If §4 or the experimental section contains these, they must be cross-referenced to the uniqueness claim; otherwise the empirical 'proof' reduces to unsupported assertion.
minor comments (2)
  1. [Method] Notation for the frequency-wise projections and point-motion descriptors should be formalized with equations early in the method section to improve readability.
  2. [Introduction] The introduction should explicitly contrast PEPS with prior grid-based INR works (e.g., those using multi-resolution or hash grids) to clarify the claimed novelty.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments. We address each major point below and describe the revisions that will be incorporated.

read point-by-point responses

  1. Referee: [Abstract and method description of projected sampling] The abstract asserts that the motion of projected points 'follows a unique pattern' usable for learned positional encoding on grids without high-resolution requirements or new artifacts, yet no derivation, uniqueness proof, or explicit construction of the decomposition is referenced. This is load-bearing for the parameter-reduction claim, as the pattern must demonstrably differ from standard sinusoidal embeddings or implicit dense sampling.

    Authors: We agree that an explicit derivation strengthens the central claim. Section 3 already describes the projected-point trajectories and contrasts them visually with standard sinusoidal embeddings, but we will add a new subsection that derives the frequency-dependent motion analytically from the sinusoidal basis functions, shows the resulting unique trajectories, and formally distinguishes the decomposition from both dense sampling and conventional positional encodings. This addition will directly underpin the parameter-efficiency argument. revision: yes

  2. Referee: [Experimental evaluation sections] The headline result (outperformance + 25% parameter savings across three tasks) is stated without reference to specific baselines, quantitative tables, error bars, or ablation controls in the provided text. If §4 or the experimental section contains these, they must be cross-referenced to the uniqueness claim; otherwise the empirical 'proof' reduces to unsupported assertion.

    Authors: Section 4 already contains the quantitative tables, baseline comparisons (including grid-based and positional-encoding methods), error metrics with standard deviations across runs, and ablations on grid resolution and parameter count. We will insert explicit forward references from the abstract and from the method description in Section 3 to the relevant tables and figures in Section 4, making the link between the uniqueness of the decomposition and the reported gains unambiguous. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained.

full rationale

The abstract frames the method as an observation of projected-point motion patterns used to construct a basis for grid-based positional encoding, followed by empirical validation on three tasks. No equations, parameter-fitting steps, or self-citations are supplied in the given text that would reduce any claimed prediction or uniqueness result to the inputs by construction. The central claim of outperformance with fewer parameters is presented as an external empirical result rather than a definitional or fitted tautology. This satisfies the default expectation of non-circularity when no load-bearing reduction can be exhibited via direct quote.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that positional encoding projections exhibit unique, exploitable motion patterns across frequencies that can be turned into an effective learned basis; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (1)
  • domain assumption Positional encodings can be decomposed into meaningful points whose motion with respect to frequencies follows a unique pattern usable as basis decomposition for learned grid encodings
    Directly stated in the abstract as the foundation for the PEPS proposal
invented entities (1)
  • PEPS projected sampling no independent evidence
    purpose: To treat frequency projections as points of interest for improved positional encoding
    New sampling procedure introduced to address limitations of standard positional encoding and grids

pith-pipeline@v0.9.0 · 5507 in / 1303 out tokens · 44351 ms · 2026-05-08T04:44:42.378193+00:00 · methodology

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