Strips as Tokens (SATO)

• Strips as Tokens (SATO) are structured groupings of contiguous and semantically coherent sequences that serve as atomic units in generative models.
• SATO is applied in mesh generation, visual tokenization, and symbolic planning, delivering enhanced compression, structural fidelity, and efficient decoding.
• The methodology employs hierarchical quantization, unified decoding, and overlapping token schemes to support both autoregressive and masked modeling objectives.

Strips as Tokens (SATO) refers to a methodological paradigm in which structured sequences, known as "strips," serve as atomic units or tokens for autoregressive or non-autoregressive generative modeling. Originating in geometric deep learning for 3D mesh synthesis, the SATO abstraction has informed advances across mesh generation, visual tokenization, and symbolic world modeling. The central innovation is to leverage contiguous or semantic groupings—be they mesh face strips, visual latent strips, spatial strips in vision MLPs, or sequences of grounded actions in STRIPS planning—as the tokens provided to sequence models. This preserves structural coherence, improves compression, and aligns the generation process with natural semantics or workflow patterns (Xu et al., 10 Apr 2026, Qu et al., 17 Mar 2026, Cao et al., 2023, Núñez-Molina et al., 16 Sep 2025).

1. Foundational Principles and Definitions

In the SATO framework, a "strip" is a contiguous chain (or semantically grouped sequence) that encodes domain structure far beyond trivial coordinate-based or per-element tokenizations:

• In mesh generation, strips are vertex sequences defining triangle or quad strips, with explicit encoding of UV (parametrization) boundaries (Xu et al., 10 Apr 2026).
• In visual tokenization, SATO compresses a 2D grid of latent features into a 1D token sequence ("stripifying"), each token capturing global semantic context (Qu et al., 17 Mar 2026).
• In vision MLPs, horizontal or vertical strips comprise contiguous rows/columns, enabling overlapping and multi-scale spatial mixing as single tokens (Cao et al., 2023).
• In STRIPS planning world modeling, SATO treats grounded action instances—ordered in applied strips—as token sequences, encapsulating causal dependencies (Núñez-Molina et al., 16 Sep 2025).

Tokenization in SATO thus shifts from atomic, locality-agnostic units (e.g., single faces, pixels, independent actions) to structural entities that preserve local flow, boundary semantics, and global compositionality.

2. Canonical Implementations by Domain

2.1 Artist-Quality Mesh Generation

SATO for triangular and quadrilateral mesh generation proceeds as follows (Xu et al., 10 Apr 2026):

• Hierarchical quantization: Vertices are mapped to a three-level hierarchical code $(c_1, c_2, c_3)$, yielding high precision with concise vocabularies.
• Strip extraction: Triangle or quad strips grow sequentially by traversing face adjacencies with a stride parameter $\delta \in \{1,2\}$ and zipper-like connectivity.
• Structural token vocabulary: Specialized tokens denote the start of new strips ($C_1^t$) and UV island boundaries ($C_1^{uv}$); prefix-sharing compression emits only the minimal code difference per vertex.
• Unified decoding: The token sequence can be interpreted with either triangle or quad decoders by adjusting stride, supporting joint training and seamless decoding into either mesh type.

2.2 Semantic Visual Tokenization

In image modeling, SATO is implemented by compressing $h \times w$ latent patches into a short $L$-length 1D "strip" token sequence using a dual-branch transformer encoder with cross-attention (Qu et al., 17 Mar 2026). Token quantization utilizes a Binary Spherical Quantizer so that high-level semantics are encoded in short, information-rich strips. Semantic alignment constraints enforce that each strip token corresponds to meaningful global or regional content.

2.3 Cross-Strip MLP Mixing

For MLP-based vision architectures, SATO defines a strip as a multi-token row or column whose aggregation supports both within-strip and cross-strip information flow. This is realized through overlapping strip tokens, channel grouping, and a Cascade Group Strip Mixing Module (CGSMM) that operates efficiently even with highly reduced spatial resolutions. Local Strip Mixing Modules (LSMM) further enhance local context integration (Cao et al., 2023).

2.4 Symbolic World Model Learning

In propositional STRIPS planning, the SATO formalism treats each grounded action $a \in A$ as a token. Learning is cast as next-token prediction over action strips (valid or invalid traces). A domain-specific transformer with a dedicated attention head per fluent $p_\ell$ and QKV projections maps histories to the learnable precondition and (add, delete) action lists, reconstructing the symbolic world model in the process (Núñez-Molina et al., 16 Sep 2025).

3. Training Methodologies and Objectives

SATO models typically employ autoregressive, sequence-to-sequence, or masked modeling objectives adapted for their structural tokens:

• Mesh generation: SATO models undergo a three-stage curriculum—mass pretraining on triangle meshes, UV-segmentation post-training (with $C_1^{uv}$ tokens), and quad-specific fine-tuning (Xu et al., 10 Apr 2026).
• Visual tokenization: Training is staged: a diffusion-matched objective is used for initial token learning, augmented with semantic distillation and contrastive alignment constraints, followed by a one-step refiner with pixel- and perceptual-space losses (Qu et al., 17 Mar 2026).
• Vision MLPs: Training proceeds as usual for spatial MLPs but with SATO modules replacing patch- or pixel-level mixing for layers, leveraging standard image classification or reconstruction losses (Cao et al., 2023).
• STRIPS world models: Cross-entropy or focal loss optimizes the next-token prediction task from labeled valid/invalid action traces, with transformer weights directly reconstructing precondition and effect lists (Núñez-Molina et al., 16 Sep 2025).

Optimization leverages scaled AdamW, tranche-wise or random subsequence sampling, temperature sampling for diversity, and curriculum transfer in multi-stage scenarios.

4. Empirical Performance and Comparisons

SATO yields substantial gains in structural fidelity, compression, and efficiency across tasks:

Domain	Compression/Speed	Geometric/Semantic Metrics	User/Task Study
Mesh	Token rate: 0.283	F1 (Objaverse): 0.503	SATO artist rank: 2.61 > BPT 1.40
UV (Mesh)	—	L2 stretch: 0.979 vs 0.921	SATO (UV): 2.60 > PartUV 2.04
Quad Mesh	—	F1: 0.857	SATO: 1.80 > CrossGen: 1.24
Vision MLP	—	+1–3% Top-1 acc	Substantial gains on small datasets (Cao et al., 2023)
Image Gen	bpp 0.07–0.125	rFID: 0.67–0.88, SOTA	Outperforms MaskGIT, VQGAN AR
STRIPS	—	100% model recovery (N≥500)	Perfect generalization to long traces (Núñez-Molina et al., 16 Sep 2025)

SATO in mesh modeling consistently delivers superior geometric quality (F1, CD, NC) and UV chart regularity relative to prior patch- or coordinate-based transformers (Xu et al., 10 Apr 2026). In visual tokenization, SATO compresses images to 1D semantically dense strips, outperforming prior grid/patch-based tokenizers in FID, PSNR, and LPIPS at low bpp (Qu et al., 17 Mar 2026). In MLP-based vision, SATO recovers information flow under strong spatial collapsing, outperforming classic MLPs and even rivaling convolutional and transformer approaches (Cao et al., 2023). In symbolic STRIPS modeling, the SATO architecture identifies the operator schema with perfect fidelity given enough training data (Núñez-Molina et al., 16 Sep 2025).

5. Advantages, Insights, and Limitations

The SATO paradigm confers several structural and practical advantages:

• Edge-flow and semantic coherence: Strips align with natural boundaries and artist workflows, preserving mid-level flow and semantic UV island structure (Xu et al., 10 Apr 2026).
• Unified tokenization and multi-geometry decoding: A single representation suffices for both triangles and quads, enabling scalable and transferable training (Xu et al., 10 Apr 2026).
• Compression and efficiency: Long strips enable amortized $O(1)$ overhead per triangle/region; prefix-sharing and overlap reduce redundancy (Xu et al., 10 Apr 2026, Qu et al., 17 Mar 2026).
• Preservation of information under spatial reduction: Strip tokens and cross-group mixing maintain channel and spatial diversity even for small $\delta \in \{1,2\}$0 in vision MLPs (Cao et al., 2023).
• Symbolic interpretability and model extraction: In STRIPS world learning, SATO enables direct extraction of the symbolic transition model from transformer weights (Núñez-Molina et al., 16 Sep 2025).

Empirical limitations include reliance on quad corpus quality for mesh regularity, rare fallback triangles in odd-length strips, and potential suboptimality on near-spherical or highly non-axis-aligned data (Xu et al., 10 Apr 2026). In vision MLPs, SATO requires careful tuning of strip width and grouping per task (Cao et al., 2023). For semantically aligned visual strips, gains are maximized when semantic constraints are well matched; less structured domains may see diminished benefit (Qu et al., 17 Mar 2026).

6. Broader Applications and Theoretical Implications

SATO is not restricted to mesh, vision, or symbolic planning. The generalization of strips as structured, context-sensitive tokens implies a broader design principle: by aligning tokenization with the compositional, hierarchical, or semantic regularities of the data, sequence models can achieve improved efficiency, robustness, and interpretability. A plausible implication is that future generative and world modeling systems will increasingly adopt dynamic or learned strip-like tokenizations to unify structure and semantics, scaling across visual, geometric, and symbolic domains (Xu et al., 10 Apr 2026, Qu et al., 17 Mar 2026, Cao et al., 2023, Núñez-Molina et al., 16 Sep 2025).