---
title: 'Num2Space: Mapping Numbers to Space'
url: https://www.emergentmind.com/topics/num2space
type: topic
---

# Num2Space: Mapping Numbers to Space

Searching arXiv for papers on "Num2Space" and "SpaceNum" to ground the article with current sources.
Num2Space denotes the direction of inference in which numerical inputs are mapped to spatial outcomes. In the vision-language setting formalized by the SpaceNum framework, it is the task of starting from numbers, or number-encoded descriptions, and selecting the spatial state consistent with them, either as a future observation in embodied exploration or as a layout observation in spatial reasoning [2605.23898]. More broadly, the term has also been used perspectivally to describe mappings from compact numerical structure to organized spaces in other domains, including Banach-space invariants, empirical galaxy-gas models, and semi-analytic cosmological catalogs [2606.01675], [2409.08539], [1508.07215]. The core commonality is a numerical-to-structured-space transformation, although only the SpaceNum paper gives Num2Space a formal task definition [2605.23898].

## 1. Definition and formal role

In SpaceNum, spatial numerical understanding refers to a VLM’s ability to use and interpret numbers as metric quantities in space, rather than merely as tokens in text [2605.23898]. Within that framework, Num2Space is one of two bidirectional tasks, paired with Space2Num, and evaluates how well models map between vision-side spatial structure and language-side numerical representations [2605.23898].

Num2Space is the direction where the model starts from numbers and must pick the spatial outcome consistent with them [2605.23898]. In the dynamic transition setting, the input is \((o_t, a, n)\), where \(o_t\) is the initial observation, \(a\) is the action type, and \(n\) is the numerical parameter; the required output is the correct next observation \(o_{t+1}\) chosen from multiple candidate images [2605.23898]. In the static layout setting, the input is a cognitive map \(M\) in number form, consisting of object names together with coordinates and sizes under a defined reference frame, and the output is the layout observation \(o\) that realizes \(M\), again selected among visually similar distractors [2605.23898].

The paper characterizes the task conceptually as a mapping from a space of numerical descriptions to a space of spatial states, writing Num2Space as a function of the form
\[
f_{\mathrm{Num2Space}}:\mathcal{N}\to\mathcal{S},
\]
with dynamic and static specializations corresponding respectively to forecasting a next observation and instantiating a layout from a map [2605.23898]. This suggests that Num2Space is best understood not as OCR-like numeric recognition but as metric grounding: numbers must alter or determine spatial structure in a way that is geometrically consistent.

## 2. Dynamic-transition Num2Space

The dynamic version of Num2Space is implemented in AI2-THOR and treats numbers as transition magnitudes in embodied exploration [2605.23898]. Actions are parameterized primitives: Move F/B by step size \(0.2\) m over a range \(0.2\)–\(2.4\) m, Move L/R by step size \(0.2\) m over a range \(0.2\)–\(1.2\) m, Rotate U/D by step size \(10^\circ\) over a range \(10\)–\(70^\circ\), and Rotate L/R by step size \(10^\circ\) over the same angular range [2605.23898].

The task presents \((o_t,a,n)\) and asks the model to select the correct resulting observation \(o_{t+1}\) from a candidate set generated by fixing \(o_t\) and \(a\) while varying \(n\) [2605.23898]. The distractors therefore correspond to distinct numerical transition magnitudes under the same initial state and action type. The latent intuition stated in the paper is that each candidate corresponds to a different \(\Delta s_t\) in a spatial state update, even though the model sees only rendered images rather than the underlying state [2605.23898].

Transition generation is constrained to ensure enough overlap between \(o_t\) and \(o_{t+1}\), enough anchors, specified as at least \(3\) object instances, and valid poses through occupancy maps that avoid collisions or empty scenes [2605.23898]. The purpose is to keep the problem genuinely spatial rather than degenerate. The model must determine, for example, how different magnitudes of rotation or translation alter relative displacements and occlusions in the image [2605.23898].

A central empirical asymmetry appears here: in dynamic transitions, Space2Num exceeds Num2Space across actions and models [2605.23898]. The reported interpretation is that dynamic transitions are more vision-dependent, so models benefit from observing spatial changes directly but struggle to predict future visual outcomes from numerical actions alone [2605.23898]. This makes dynamic Num2Space the more stringent test of forward spatial grounding.

## 3. Static-layout Num2Space

The static version of Num2Space uses NVIDIA Isaac Sim and treats numbers as layout parameters in a cognitive map [2605.23898]. Two anchor objects define the reference frame: one anchor is the origin, and the vector from the origin anchor to the second anchor defines a direction, thereby fixing the axes up to scale [2605.23898]. Anchor positions are fixed across samples in the same scene so that the coordinate frame remains stable [2605.23898].

Layout generation varies the position of a target object, its size, or both, subject to no overlap between objects and distances in a reasonable range [2605.23898]. The benchmark includes both desktop-scale and room-scale scenes, and for each layout it constructs number-based maps in 1D, 2D, and 3D representations [2605.23898]. These are explicitly described as different coordinate encodings of the same layout, ordered from simpler to richer [2605.23898].

Static Num2Space gives the model a cognitive map \(M\) and asks it to select the observation \(o\) consistent with the specified layout, with distractors created by changing object positions or sizes while keeping the same reference frame [2605.23898]. The underlying rendering function is described as \(f_{\mathrm{render}}:M\mapsto o\), and the benchmark asks models to approximate that mapping in multiple-choice form [2605.23898].

The task is intended to test whether a model has, or can build, a coordinate-aware internal representation that aligns numerical relations in \(M\) with image-space relations such as left/right, nearer/farther, and larger/smaller in candidate scenes [2605.23898]. In contrast to the dynamic case, the principal asymmetry reverses here: Num2Space exceeds Space2Num in static layouts, meaning that models more easily project a numeric map into a plausible spatial configuration than recover a precise numeric map from an image [2605.23898]. Even so, the paper states that performance remains far from high accuracy.

## 4. Benchmark construction, models, and metrics

The overall SpaceNum evaluation set contains **3,800** samples, with an additional training set of **77,412** samples used for fine-tuning experiments [2605.23898]. Dynamic-transition subsets are organized by action type, and static-layout subsets are organized by 1D-Map, 2D-Map, and 3D-Map, each further divided into desktop-scale and room-scale instances; both Num2Space and Space2Num are defined for each [2605.23898].

A typical Num2Space sample in the dynamic setting consists of one image \(o_t\), an action such as “Move Forward by 1.0 m,” and four candidate next-frame images, only one of which corresponds to the specified magnitude [2605.23898]. A static sample consists of a structured numeric map together with four images of similar scenes, of which the correct one implements the given coordinates and sizes [2605.23898]. Difficulty is controlled dynamically by magnitude ranges and staticly by map dimensionality, scene scale, and whether position, size, or both vary [2605.23898].

Eighteen VLMs, ranging from 2B to 72B, are evaluated, including Qwen2.5-VL, Qwen3-VL, InternVL3.5, Gemma-3, Ovis2.5, and Cosmos-Reason2 [2605.23898]. All use the same prompt format requiring output of only the option letter, and inference settings are specified as temperature \(0.7\), top-p \(0.9\), top-k \(50\), and bfloat16 [2605.23898].

The primary evaluation metric is multiple-choice accuracy [2605.23898]. For dynamic-transition Num2Space, \(K=3\) and the random baseline is **30.0%**; for static-layout Num2Space, \(K=4\) and the random baseline is **25.0%** [2605.23898]. The paper also introduces a semantic proximity diagnostic for dynamic transitions that scores candidate choices as Exact \(=100\), Near \(=70\), Moderate \(=40\), and Far \(=0\), and reports an answer proximity score as a weighted average [2605.23898]. For static-layout errors, it classifies wrong answers according to whether the mismatch is in position only, size only, or both position and size [2605.23898].

## 5. Empirical behavior and failure modes

The global result reported by SpaceNum is that current VLMs largely fail to ground numbers in spatial meaning and often perform close to random guess across dynamic transitions and static layouts [2605.23898]. The best model in Table 2 is **Qwen2.5-VL-72B**, with **39.8% average** over all SpaceNum tasks, while many models score near or even below random [2605.23898]. This establishes Num2Space as a difficult regime for current systems.

Complexity effects are task-dependent. In dynamic Num2Space, all action types are comparably hard, and even the strongest models remain around \(\sim 40\%\) accuracy [2605.23898]. In static Num2Space, 1D desk-scale tasks are relatively easier, whereas higher-dimensional 2D or 3D maps and room-scale scenes degrade sharply, often leaving performance only slightly above the \(25\%\) random baseline [2605.23898]. The paper explicitly concludes that layout complexity and dimensionality directly degrade Num2Space performance [2605.23898].

Error analysis indicates that larger models do not improve much in exact dynamic Num2Space accuracy, but they make numerically closer mistakes, reflected in higher proximity scores [2605.23898]. The interpretation given is that larger models have coarser sensitivity to magnitude, such as confusing \(20^\circ\) with \(30^\circ\) rather than with \(70^\circ\), while still failing to identify the exact value reliably [2605.23898].

For static Num2Space, errors overwhelmingly correspond to joint position-and-size mistakes rather than isolated errors in a single factor [2605.23898]. The paper takes this to suggest that when models mis-ground the layout from \(M\), they mis-ground the entire structure rather than performing disentangled reasoning [2605.23898]. Reasoning traces reinforce that diagnosis: models stop at coarse cues, avoid counterfactual magnitude comparisons, and sometimes substitute image-space left/right for the task-defined coordinate system anchored by objects [2605.23898]. The paper’s broader conclusion is that models rely heavily on shallow spatial cues, struggle to build stable coordinate-aware representations, and fail to abstract structured spatial layouts from visual observations [2605.23898].

## 6. Interventions, tuning, and broader interpretations

Controlled interventions show limited gains for Num2Space. In dynamic transitions, adding visual anchors changes performance only by a few percent and often inconsistently across models [2605.23898]. In static layouts, reducing the number of objects again yields only minor and inconsistent effects [2605.23898]. The stated conclusion is that the main limitations are not due to missing anchors or clutter [2605.23898].

Changes to numerical surface form also have weak effects. Transforming how magnitudes are expressed yields negligible gains for layouts and only small, model-specific gains for transitions [2605.23898]. The paper therefore argues that the difficulty is not parsing numeric forms but grounding them in space [2605.23898]. Replacing photorealistic layout images with structured abstractions such as points, 2D boxes, and 3D boxes improves Space2Num substantially more than Num2Space, which the authors interpret as evidence that the main bottleneck lies in vision-to-structure abstraction rather than in the abstract relation between structure and numbers [2605.23898].

Blind testing reveals a modality asymmetry. Replacing images with black inputs causes dynamic Num2Space accuracy to drop significantly, indicating that the task is genuinely vision-dependent [2605.23898]. Static Num2Space drops only slightly, suggesting heavier reliance on language-side priors and structural shortcuts [2605.23898]. Rotational symmetry analysis further shows degraded performance under equivalent transformations such as rotate_left\((20^\circ)\) versus rotate_right\((340^\circ)\), indicating a lack of geometric consistency in the learned action space [2605.23898].

Explicit reasoning yields little improvement: “think” versus “non-think” variants differ by typically **\(\le 1\%\)** on Num2Space accuracy [2605.23898]. Supervised LoRA fine-tuning on SpaceNum data partially improves performance, with partial cross-dimension transfer and the best overall mixture reported at approximately **25% dynamic transitions** and **75% static layouts** [2605.23898]. RL with GRPO produces only modest gains, with Num2Space improving by about **8–9 percentage points** over baseline for the 4B model and graded reward slightly outperforming strict reward [2605.23898]. The paper also reports transfer gains on external spatial reasoning benchmarks including OmniSpatial Motion, SAT Action Consequence, and SAT Object Movement after SpaceNum fine-tuning [2605.23898].

A broader interpretation of Num2Space appears in several other papers included in the record. In "The Numerical Index of Two-Dimensional Real \(\ell_p\) Spaces," the phrase “Num2Space perspective” refers to assigning to each Banach space a single numerical invariant, \(n(X)\in[0,1]\), with the specific result
\[
n(\ell_p^2)=v(J)=\max_{0\le t\le1}\frac{t-t^{p-1}}{1+t^p},
\]
thereby encoding the two-dimensional real \(\ell_p\) spaces by one number [2606.01675]. In "NeutralUniverseMachine," the same perspective describes a mapping from halo space and assembly history to gas space, with explicit formulae for \(M_{\rm HI}\) and \(M_{\rm H_2}\) as functions of halo and galaxy properties [2409.08539]. In "\(\nu^2\)GC," it denotes a semi-analytic pipeline that takes dark-matter simulation outputs and maps them into a cosmological space populated with galaxies and AGNs having positions, masses, sizes, luminosities, and spectra [1508.07215]. These usages do not redefine the benchmark task, but they suggest a wider conceptual family in which Num2Space designates a mapping from compact numerical structure to organized, interpretable spatial or state structure.

## 7. Conceptual significance

As formalized in SpaceNum, Num2Space isolates the problem of using numbers as metric controls over spatial states rather than merely as labels [2605.23898]. It requires an internal spatial model, metric sensitivity so that numerical changes induce proportional spatial differences, and geometric consistency under equivalent transformations [2605.23898]. The benchmark’s results indicate that current VLMs can often produce plausible numbers and plausible spatial descriptions without binding the two in a stable metric representation [2605.23898].

The SpaceNum study therefore positions Num2Space as a diagnostic for true spatial numerical grounding [2605.23898]. A plausible implication is that progress on Num2Space will require architectural mechanisms that maintain consistent coordinate systems or world models, together with training regimes in which numeric values directly control spatial transformations and evaluation protocols that test geometric consistency beyond multiple-choice recognition [2605.23898]. In that sense, Num2Space names both a specific benchmark direction and a broader research problem: the faithful realization of numerical structure as spatial consequence.

Source: https://www.emergentmind.com/topics/num2space