Vector Map as Language (VecLang)

• Vector Map as Language (VecLang) is a paradigm that represents maps, languages, and models as continuous vector spaces, allowing semantic operations and compositional reasoning.
• It is applied in diverse fields such as robot navigation, remote sensing, and multilingual model analysis, where spatial and language-like structures are manipulated via vector operations.
• VecLang underpins structured text generation, semantic mapping, and executable representations, supporting both geometric and logical interpretations for practical applications.

Searching arXiv for the cited VecLang-related papers to ground the article in current research. Vector Map as Language (VecLang) denotes a family of research programs in which maps, languages, or model collections are represented as vector-valued spaces and manipulated through vector operations rather than only through discrete symbols or category-specific structures. In the most explicit formulation, VecLang is described as “a unified paradigm that reformulates multiclass vector mapping as structured text generation” for remote sensing (Yan et al., 9 Jun 2026). In robot navigation, LAMP “can be read almost literally as an instance of ‘Vector Map as Language (VecLang)’” because it treats the environment as “a continuous field of language / CLIP-like embeddings” and uses that field directly for planning (Lee et al., 12 Feb 2026). Related work on language spaces, knowledge maps, model maps, and vector logic shows that the phrase spans several compatible but non-identical meanings: a map can be treated as language, a language can be treated as a vector space, and vector operations can function as semantic operators (Chen, 2024, Quigley, 2024).

1. Conceptual scope and core idea

Across the cited literature, VecLang is not a single canonical formalism. The remote-sensing formulation uses a GeoJSON-like textual representation for heterogeneous geospatial entities; the navigation formulation uses an implicit neural field over pose space; the language-theoretic formulation treats language as a high-dimensional vector space $V_L$; and model-mapping work represents each LLM as a point in a common Euclidean space defined by log-likelihoods on a reference corpus (Yan et al., 9 Jun 2026, Lee et al., 12 Feb 2026, Chen, 2024, Oyama et al., 22 Feb 2025). This suggests that VecLang is best understood as a paradigm in which semantic content is stored, queried, and composed in vector spaces, while language-like structure appears either as explicit syntax or as geometric regularity.

A compact comparison of major instantiations follows.

Setting	Representation	Primary operation
Robot navigation	Implicit language field $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$ plus sparse graph $\mathcal{G}$	Coarse graph planning and gradient-based pose refinement (Lee et al., 12 Feb 2026)
Remote sensing vector mapping	GeoJSON-like Structured Vector Language (SVL)	Autoregressive structured text generation from imagery (Yan et al., 9 Jun 2026)
Language theory	Language vector space $V_L$ with projections $P: V_L \to W$	Attribute extraction via subspaces and projections (Chen, 2024)
Multilingual modeling	Continuous language vectors $\mathbf{l}_i \in \mathbb{R}^d$	Conditioning a character-level LLM (Östling et al., 2016)
Model analysis	Doubly centered log-likelihood coordinates $q_i$	Euclidean comparison approximating KL divergence (Oyama et al., 22 Feb 2025)
Formal semantics	Typed vector spaces $S_{D_\tau}$ with injective maps $h_\tau$	Vector-space realization of extensional semantic functions (Quigley, 2024)

Two themes recur. First, the map is often a function rather than a table: LAMP defines $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$, while the remote-sensing formulation defines a reversible map-to-language conversion $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$0 and language-to-map conversion $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$1 (Lee et al., 12 Feb 2026, Yan et al., 9 Jun 2026). Second, geometry is semantic: Euclidean distance, cosine similarity, graph connectivity, or logical operator matrices encode meaning-bearing relations rather than merely storage layout (Filatov et al., 2015, Quigley, 2024, Oyama et al., 22 Feb 2025).

2. Formal foundations: vector spaces, projections, and semantic operators

A general theoretical substrate appears in “Vectoring Languages” (Chen, 2024). That paper introduces a language vector space $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$2, defines a word as a vector $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$3, and treats linguistic attributes through projections $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$4, where $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$5 is a subspace and, in the standard linear-algebraic case, $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$6. Attribute subspaces $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$7 support functions such as

$F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$8

with examples including “all meanings” and the “most common meaning” of a word. Existing semantic theories are then described as further projections of $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$9 (Chen, 2024).

A more rigorous logical bridge is provided by “A vector logic for extensional formal semantics” (Quigley, 2024). There, an extensional model

$\mathcal{G}$0

is embedded into vector spaces by injective maps $\mathcal{G}$1, with each

$\mathcal{G}$2

For every semantic function $\mathcal{G}$3, there exists a corresponding vector-space function $\mathcal{G}$4 satisfying

$\mathcal{G}$5

Truth values are represented as basis vectors in $\mathcal{G}$6, with $\mathcal{G}$7 and $\mathcal{G}$8; negation is the matrix

$\mathcal{G}$9

and conjunction is a $V_L$0 matrix acting on tensor products of truth vectors (Quigley, 2024). In this sense, vector maps are not only containers for meaning but also compositional semantic operators.

An earlier knowledge-representation precursor is the “Global Knowledge Map,” described as a multidimensional homogeneous mapping space in which concepts and documents are mapped to points and Euclidean distance expresses semantic difference (Filatov et al., 2015). Its feasibility is motivated by the Johnson–Lindenstrauss lemma, and it is proposed as a standard intermediary for semantic alignment across WWW documents, ontologies, e-libraries, and other knowledge bases (Filatov et al., 2015). The paper does not use the term VecLang, but its own description supports viewing the system as a proto-VecLang.

3. Spatial and embodied VecLang

In robot navigation, “LAMP: Implicit Language Map for Robot Navigation” defines the environment as a linguistic space over pose space $V_L$1, with a neural field

$V_L$2

that assigns to each pose $V_L$3 a CLIP-like embedding $V_L$4 describing “what you see from there” (Lee et al., 12 Feb 2026). In the Bayesian formulation, the network outputs $V_L$5, where $V_L$6 is a unit-norm mean direction and $V_L$7 is the concentration parameter of a von Mises–Fisher distribution. Training uses the vMF likelihood

$V_L$8

with a Gamma prior on $V_L$9 (Lee et al., 12 Feb 2026).

The map is

$P: V_L \to W$0

where graph nodes store only poses $P: V_L \to W$1, not explicit embeddings; embeddings are computed on demand as $P: V_L \to W$2 (Lee et al., 12 Feb 2026). The navigation pipeline is coarse-to-fine. A language query $P: V_L \to W$3 is embedded with CLIP as $P: V_L \to W$4, the most similar graph node is selected, A* produces a coarse path, and then a gradient-based optimization refines a correction $P: V_L \to W$5 around the coarse goal by maximizing semantic similarity while regularizing distance from the best sampled pose. The paper characterizes this as the first application of an implicit language map for precise path generation (Lee et al., 12 Feb 2026).

The remote-sensing paper “Vector Map as Language: Toward Unified Remote Sensing Vector Mapping” makes the VecLang label explicit (Yan et al., 9 Jun 2026). It observes that language offers a flexible and expressive representation for geometry, semantics, and topology, and therefore encodes vector maps into a GeoJSON-like Structured Vector Language. Each entity is represented as

$P: V_L \to W$6

The model first localizes vectorization units and then generates unit-level SVL sequences, which are merged into executable maps through a bottom-up parsing process (Yan et al., 9 Jun 2026). Autoregressive generation is formalized as

$P: V_L \to W$7

and a Progressive Vectorization Framework is paired with Hierarchical Vector Language Optimization using GRPO-based reinforcement learning (Yan et al., 9 Jun 2026).

These two systems realize distinct but related spatial VecLangs. LAMP treats the map as a continuous language field over pose space, whereas the remote-sensing system treats the map as structured text. This suggests that VecLang can refer either to implicit semantic geometry or to explicit textual serialization, provided that the representation supports cross-category or open-vocabulary reasoning (Lee et al., 12 Feb 2026, Yan et al., 9 Jun 2026).

4. Continuous language spaces and model maps

The multilingual work “Continuous multilinguality with language vectors” replaces discrete language IDs with continuous language vectors $P: V_L \to W$8, learned jointly with a character-based neural LLM over 990 languages and 1303 Bible translations (Östling et al., 2016). Each training language has three 64-dimensional embeddings injected into two stacked 1024-dimensional LSTM layers and the output layer; the model defines

$P: V_L \to W$9

and interpolation

$\mathbf{l}_i \in \mathbb{R}^d$0

produces intermediate varieties in a continuous language space (Östling et al., 2016). The learned vectors recover genealogical structure through hierarchical agglomerative clustering and support rapid adaptation to a new variety by optimizing only a new language vector (Östling et al., 2016).

A related but distinct VecLang-like construction appears in “Mapping 1,000+ LLMs via the Log-Likelihood Vector” (Oyama et al., 22 Feb 2025). For a fixed text set $\mathbf{l}_i \in \mathbb{R}^d$1, each model $\mathbf{l}_i \in \mathbb{R}^d$2 is represented by its log-likelihood vector

$\mathbf{l}_i \in \mathbb{R}^d$3

which is row- and column-centered to produce coordinates $\mathbf{l}_i \in \mathbb{R}^d$4 (Oyama et al., 22 Feb 2025). The main approximation is

$\mathbf{l}_i \in \mathbb{R}^d$5

so squared Euclidean distance in coordinate space approximates KL divergence between text-generation distributions (Oyama et al., 22 Feb 2025). Applied to 1,018 models and 10,000 texts, the resulting map reveals clusters by model family, domain specialization, and benchmark performance, and ridge regression on $\mathbf{l}_i \in \mathbb{R}^d$6 achieves $\mathbf{l}_i \in \mathbb{R}^d$7 for predicting 6-TaskMean on 996 models (Oyama et al., 22 Feb 2025).

“Likelihood Variance as Text Importance for Resampling Texts to Map LLMs” refines this model-map framework by resampling texts according to cross-model likelihood variance (Oyama et al., 21 May 2025). With the doubly centered matrix $\mathbf{l}_i \in \mathbb{R}^d$8, length-squared sampling uses

$\mathbf{l}_i \in \mathbb{R}^d$9

while the KL-targeted scheme uses

$q_i$0

Experiments show that LS and KL sampling achieve comparable performance to uniform sampling with about half as many texts and support efficient incorporation of new models into an existing map (Oyama et al., 21 May 2025). In this literature, the “map” is a vector space of models rather than an environmental or geographic structure, but the same VecLang intuition remains: texts define axes, models become points, and geometry encodes semantic or behavioral similarity.

5. Structured generation, internal algebra, and executable representations

The remote-sensing VecLang system is notable for making executability a first-class target (Yan et al., 9 Jun 2026). Its reward is

$q_i$1

with $q_i$2, $q_i$3, and $q_i$4, thereby emphasizing syntax validity, content fidelity, and map executability (Yan et al., 9 Jun 2026). For closed objects, executability includes polygon IoU and boundary alignment based on normalized Hausdorff distance; for roads, it includes buffered IoU, line alignment, and a connectivity score

$q_i$5

The result is a language whose well-formedness is judged not only by parseability but by whether it yields usable GIS objects (Yan et al., 9 Jun 2026).

A different internal algebra is proposed by “Generalizing Complex/Hyper-complex Convolutions to Vector Map Convolutions” (Gaudet et al., 2020). There, a vector map of dimension $q_i$6 is treated as a single entity, and output components are formed by circular permutations of a shared weight vector: $q_i$7 The authors argue that the benefits of complex and quaternion networks arise from weight sharing and from treating multidimensional data as a single entity rather than from fixed 2D or 4D algebras (Gaudet et al., 2020). This suggests a possible low-level mathematical analogue for VecLang: structured vector maps can possess an internal “syntax” governing how components interact, even when the system is not explicitly linguistic.

The vector-logic results strengthen this point at the semantic level. Since conjunction, disjunction, implication, and arbitrary $q_i$8-ary Boolean functions can be represented by matrices on tensor products of truth vectors, vector maps can serve simultaneously as data representations and as rule-bearing operators (Quigley, 2024). In that sense, VecLang spans both descriptive and operational roles.

6. Empirical performance, limitations, and research directions

The strongest explicit empirical evidence for VecLang comes from LAMP and the remote-sensing system. In LAMP’s simulation setting, the map uses approximately $q_i$9 GB, compared with $S_{D_\tau}$0 GB for a dense grid-based map and $S_{D_\tau}$1 GB for a dense node-based map (Lee et al., 12 Feb 2026). Under roughly equal memory, LAMP reports on easy tasks $S_{D_\tau}$2, $S_{D_\tau}$3, $S_{D_\tau}$4, and on hard tasks $S_{D_\tau}$5, $S_{D_\tau}$6, $S_{D_\tau}$7, outperforming sparse explicit baselines (Lee et al., 12 Feb 2026). On a real 28-floor building, LAMP and the explicit baseline both obtain $S_{D_\tau}$8 success, but LAMP reduces GDist from $S_{D_\tau}$9 m to $h_\tau$0 m at $h_\tau$1 s inference time (Lee et al., 12 Feb 2026).

In remote sensing, VecMap-Bench contains about 54K images and 800K instances and evaluates single-class, multi-class, cross-dataset, and open-vocabulary settings (Yan et al., 9 Jun 2026). On WHU buildings, VecLang reports mAP $h_\tau$2, IoU $h_\tau$3, C-IoU $h_\tau$4, and PoLiS $h_\tau$5; on Vec-WB water, mAP $h_\tau$6, IoU $h_\tau$7, C-IoU $h_\tau$8, and PoLiS $h_\tau$9; and on Cityscales roads, recall $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$0 and F1 $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$1 (Yan et al., 9 Jun 2026). On the multi-class IRSAMap setting, it reports for roads precision $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$2, recall $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$3, F1 $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$4, and APLS $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$5, and its open-vocabulary mean score is $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$6 (Yan et al., 9 Jun 2026). Framework ablations show the importance of all three components: base Qwen3-VL-4B yields building $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$7, road $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$8; adding SVL yields $\mathcal{M}=\{F_\Theta,\mathcal{G}\}$9 and $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$00; adding PVF yields $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$01 and $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$02; adding HVLO yields $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$03 and $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$04 (Yan et al., 9 Jun 2026).

The literature is equally explicit about limitations. LAMP states that performance depends strongly on VLM quality, that ambiguous or weak visual cues make distinct embeddings harder to produce, and that many pose–image pairs are still required for training (Lee et al., 12 Feb 2026). The remote-sensing VecLang system notes hallucinated road segments, remaining sequence-length complexity, lightweight connectivity metrics that do not capture all graph nuances, and heavier compute than single-purpose CNN systems in some settings (Yan et al., 9 Jun 2026). The model-map literature is corpus-dependent and base-model-dependent, so geometry reflects the chosen texts and may shift when the corpus or model population changes (Oyama et al., 22 Feb 2025, Oyama et al., 21 May 2025). The formal-semantics work is extensional and leaves intensionality, generalized quantification, and efficient representations for large $F_\Theta : \mathbb{R}^7 \to \mathbb{R}^d$05-ary operators as open directions (Quigley, 2024).

Taken together, these results support a broad but precise characterization. VecLang is a research paradigm in which vector maps are treated as semantic media: environments become language fields, geospatial outputs become structured text, languages become continuous coordinates, models become points in likelihood space, and logical meanings become vector-space operators. The literature does not reduce these usages to a single algebra. Instead, it shows that the “map as language” idea can be instantiated as continuous fields, executable textual schemas, semantic coordinate systems, and homomorphic vector logics, depending on the domain and the operational notion of meaning (Lee et al., 12 Feb 2026, Yan et al., 9 Jun 2026, Chen, 2024, Quigley, 2024).