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arxiv: 2505.14062 · v4 · submitted 2025-05-20 · 💻 cs.CV

FractalMamba++: Scaling Vision Mamba Across Resolutions via Hilbert Fractal Geometry

Bo Li , Haoke Xiao , Lv Tang This is my paper

Pith reviewed 2026-05-22 13:54 UTC · model grok-4.3

classification 💻 cs.CV
keywords hilbert curvevision mambafractal serializationresolution scalingstate space modelposition encodinghigh-resolution visionimage segmentation
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The pith

Hilbert fractal curves let Vision Mamba models keep spatial continuity when input resolution changes.

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

The paper claims that turning a 2D image grid into a 1D sequence for state-space models loses neighborhood information, and this loss grows worse at resolutions different from training. It proposes that the Hilbert curve's recursive self-similar path can dictate how patches are ordered, where shortcuts are added to the state, and how positions are encoded so that local 2D relations stay consistent across scales. If this holds, Mamba-based vision models could process high-resolution inputs for classification, detection, and segmentation without the usual drop in accuracy that comes from mismatched sequence statistics.

Core claim

The central claim is that a single geometric principle—the recursive structure of the Hilbert curve—determines patch serialization, derives deterministic state-injection routes, and augments position encoding so that feature interactions reflect actual spatial proximity rather than 1D order, enabling Vision Mamba to scale across resolutions while preserving local neighborhoods.

What carries the argument

The Hilbert curve, a space-filling path whose recursive subdivisions keep nearby 2D patches close in the 1D sequence, applied here to create fractal serialization, hierarchy skip connections, and fractal-aware rotary position encoding.

If this is right

  • Performance improves over prior Mamba vision models on ImageNet-1K classification, with larger gains at high resolutions.
  • Detection and instance segmentation accuracy rises on COCO when inputs exceed training resolution.
  • Semantic segmentation on ADE20K and change detection on LEVIR-CD+ benefit similarly from the resolution-consistent ordering.
  • The skip connections and position encoding require no learned search or specialized kernels because they follow directly from the curve's recursion levels.

Where Pith is reading between the lines

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

  • The same curve-based ordering could be tested in other linear-time sequence models to see whether locality preservation helps beyond Mamba.
  • Extending the recursion to three dimensions might allow similar scaling for video or volumetric data without retraining per resolution.
  • If neighborhood consistency is the key mechanism, replacing the Hilbert curve with other locality-preserving space-filling curves could be compared on the same tasks.

Load-bearing premise

Hilbert-curve serialization maintains consistent neighborhood statistics when the image grid size changes.

What would settle it

A direct measurement of average distance between originally adjacent patches after serialization at several resolutions, followed by an ablation showing that the claimed performance gains disappear when those distances vary sharply.

Figures

Figures reproduced from arXiv: 2505.14062 by Bo Li, Haoke Xiao, Lv Tang.

Figure 1
Figure 1. Figure 1: Visualization of correlation between the final state and state tokens [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Top-1 classification accuracy of Mamba-based models across different input resolutions on ImageNet-1K. Results are grouped by parameter scale: [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Architecture of the FractalMamba++ backbone. The design contains three Hilbert-geometry-driven components: Fractal-Aware 2D Rotary Position [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
read the original abstract

Vision Mamba offers linear complexity for long visual sequences, yet its performance depends critically on how a two-dimensional patch grid is serialized into a one-dimensional state-space recurrence. Raster-style scans disrupt spatial continuity, and the mismatch between 2D locality and 1D state propagation becomes increasingly severe when the inference resolution grows beyond the training grid. This paper presents FractalMamba++, a resolution-scalable vision backbone organized around a single geometric principle: the recursive self-similar structure of the Hilbert curve determines how patches are serialized, where long-range state shortcuts are inserted, and how positional relations are encoded. First, Hilbert-curve-based Fractal Serialization preserves local 2D neighborhoods more faithfully than linear scans and provides consistent neighborhood statistics across resolutions. Second, the Fractal Hierarchy Skip Connection (FHSC) derives a compact set of deterministic state-injection routes from Hilbert recursion levels, mitigating long-sequence information fading without runtime search, hand-derived gradients, or dedicated CUDA kernels. Third, Fractal-Aware 2D Rotary Position Encoding (FA-RoPE) combines normalized 2D coordinates with a fractal hierarchy level so that feature interactions depend on actual spatial proximity and recursive structural role rather than serialized 1D distance. Extensive experiments on ImageNet-1K classification, COCO detection and instance segmentation, ADE20K semantic segmentation, and LEVIR-CD+ remote sensing change detection show that FractalMamba++ improves over existing Mamba-based vision backbones, especially under high-resolution inputs.

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 / 1 minor

Summary. The paper introduces FractalMamba++, a resolution-scalable vision backbone for Mamba-based models that organizes serialization, state shortcuts, and positional encoding around the recursive self-similar structure of the Hilbert curve. It proposes three components—Hilbert-curve-based Fractal Serialization to preserve 2D neighborhoods with consistent statistics across resolutions, Fractal Hierarchy Skip Connection (FHSC) for deterministic state-injection routes derived from recursion levels, and Fractal-Aware 2D Rotary Position Encoding (FA-RoPE) that incorporates normalized 2D coordinates and fractal hierarchy levels—and reports empirical gains over prior Mamba vision backbones on ImageNet-1K classification, COCO detection/instance segmentation, ADE20K semantic segmentation, and LEVIR-CD+ change detection, with particular emphasis on high-resolution inputs.

Significance. If the neighborhood-consistency property of Hilbert serialization is quantitatively validated and the reported gains prove robust, the work would supply a deterministic, parameter-light geometric mechanism for scaling state-space vision models to arbitrary resolutions without retraining or custom kernels, addressing a recognized limitation in current Vision Mamba designs and offering a reproducible template for other long-sequence visual tasks.

major comments (2)
  1. [Abstract] Abstract: The central claim that Hilbert-curve serialization 'preserves local 2D neighborhoods more faithfully than linear scans and provides consistent neighborhood statistics across resolutions' is asserted as the geometric foundation for both serialization and position encoding, yet no locality metric (e.g., average 2D Euclidean distance of k-nearest serialized neighbors), ablation, or comparison across grid sizes is supplied; this assumption is load-bearing for the high-resolution gains claimed on COCO, ADE20K, and LEVIR-CD+.
  2. [Experiments] Experiments section: The manuscript reports improvements across four benchmarks but supplies no quantitative details on ablation controls, error bars, or the precise protocol used for training-to-inference resolution scaling; without these, the attribution of gains specifically to the three new components cannot be rigorously assessed.
minor comments (1)
  1. [Abstract] Abstract: The acronyms FHSC and FA-RoPE are introduced without a one-sentence parenthetical gloss, which would aid readers unfamiliar with the method.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. The comments highlight important aspects that can strengthen the presentation of our geometric approach and experimental validation. We address each major comment point by point below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that Hilbert-curve serialization 'preserves local 2D neighborhoods more faithfully than linear scans and provides consistent neighborhood statistics across resolutions' is asserted as the geometric foundation for both serialization and position encoding, yet no locality metric (e.g., average 2D Euclidean distance of k-nearest serialized neighbors), ablation, or comparison across grid sizes is supplied; this assumption is load-bearing for the high-resolution gains claimed on COCO, ADE20K, and LEVIR-CD+.

    Authors: We agree that a direct quantitative locality metric would provide stronger, more explicit support for the geometric foundation. In the revised manuscript we will add a dedicated analysis (new subsection or appendix figure) that reports the average 2D Euclidean distance of the k-nearest serialized neighbors (for k=4,8) under Hilbert versus raster serialization, computed on grids of varying sizes (14×14, 28×28, 56×56). We will also include a short ablation that isolates the contribution of this locality property to high-resolution downstream performance. These additions will make the load-bearing assumption directly verifiable. revision: yes

  2. Referee: [Experiments] Experiments section: The manuscript reports improvements across four benchmarks but supplies no quantitative details on ablation controls, error bars, or the precise protocol used for training-to-inference resolution scaling; without these, the attribution of gains specifically to the three new components cannot be rigorously assessed.

    Authors: We concur that additional experimental rigor is required. In the revision we will expand the Experiments section to include: (i) full ablation tables quantifying the incremental contribution of each component (Fractal Serialization, FHSC, FA-RoPE) on all four benchmarks; (ii) mean and standard deviation over at least three independent runs with different random seeds; and (iii) an explicit protocol subsection describing the training resolution (224²), the exact higher inference resolutions tested, and the deterministic interpolation/padding procedure used for resolution scaling without retraining. These details will allow readers to assess attribution of the reported gains. revision: yes

Circularity Check

0 steps flagged

No circularity: design choices are independent geometric constructions with external experimental validation

full rationale

The paper's core claims rest on three explicitly introduced components (Fractal Serialization, FHSC, FA-RoPE) whose definitions are derived directly from the known recursive properties of the Hilbert curve rather than from any fitted parameter or self-referential equation. The abstract states the neighborhood-consistency property as a geometric fact about Hilbert curves and then reports downstream empirical gains on ImageNet, COCO, ADE20K and LEVIR-CD+; none of these gains are shown to be algebraically forced by the same quantities used to define the components. No self-citations, uniqueness theorems, or ansatzes from prior author work appear in the provided text, and no prediction is obtained by fitting a subset of the target metric. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the untested premise that Hilbert recursion supplies consistent 2D neighborhood statistics at every scale and that the derived skip routes and position encoding inherit that property without additional fitting.

axioms (1)
  • domain assumption Hilbert-curve-based serialization preserves local 2D neighborhoods more faithfully than linear scans and provides consistent neighborhood statistics across resolutions
    Invoked in the abstract as the single geometric principle organizing serialization, skip connections, and position encoding.
invented entities (2)
  • Fractal Hierarchy Skip Connection (FHSC) no independent evidence
    purpose: Derives deterministic state-injection routes from Hilbert recursion levels to mitigate long-sequence information fading
    Newly introduced component whose routes are claimed to be free of runtime search or hand-derived gradients.
  • Fractal-Aware 2D Rotary Position Encoding (FA-RoPE) no independent evidence
    purpose: Combines normalized 2D coordinates with fractal hierarchy level so feature interactions depend on spatial proximity and recursive structural role
    Newly introduced encoding whose dependence on fractal level is presented as the key to resolution consistency.

pith-pipeline@v0.9.0 · 5802 in / 1490 out tokens · 35563 ms · 2026-05-22T13:54:44.433694+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    the recursive self-similar structure of the Hilbert curve determines how patches are serialized, where long-range state shortcuts are inserted, and how positional relations are encoded... provides consistent neighborhood statistics across resolutions

  • IndisputableMonolith/Foundation/ArithmeticFromLogic.lean embed_strictMono_of_one_lt echoes
    ?
    echoes

    ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

    FHSC selects one representative pair of spatially adjacent but sequentially distant sibling segments at each recursion level... E = union over l=1 to L of {(mid(S(1)_l), mid(S(4)_l))}

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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