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arxiv: 2606.11775 · v1 · pith:AWAUVRUGnew · submitted 2026-06-10 · 🧮 math.MG · q-bio.QM· stat.ML

Magnitude-Based Features for Multispecies Spatial Data

Julia Sollberger , Joshua Bull , Sara Kali\v{s}nik , Bernadette Stolz This is my paper

Pith reviewed 2026-06-27 07:52 UTC · model grok-4.3

classification 🧮 math.MG q-bio.QMstat.ML
keywords magnitudemultispecies spatial datametric spacestumor microenvironmentcolorectal cancerspatial heterogeneityfeature vectorsimmune cell interactions
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The pith

Magnitude of metric spaces produces feature vectors that capture interactions in multispecies spatial data such as tumor microenvironments.

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

The paper develops global and local magnitude feature vectors from finite metric spaces to analyze multispecies point patterns. Magnitude is treated as an effective number of points that folds in both configuration and scale. Applied to synthetic tumor simulations and real colorectal cancer tissue arrays, the features identify local neighborhood types including radial patterns and tertiary lymphoid structures, while globally they recover outcome classifications and flag roles for specific immune cell populations. A reader would care because few existing tools quantify interactions across species or cell types in spatial settings without heavy tuning.

Core claim

Global and local magnitude feature vectors applied to multispecies point sets as finite metric spaces recover known classifications of long-term simulation outcomes across parameter regimes and highlight the importance of CD4+ T cells and CD163+ macrophages in distinguishing favourable from unfavourable immune infiltration patterns in colorectal cancer samples.

What carries the argument

Magnitude, a real-valued invariant of finite metric spaces interpreted as an effective number of points that incorporates spatial configuration and scale.

If this is right

  • Local magnitude vectors identify distinct neighbourhood types and spatial heterogeneity including radial patterns tied to simulation outcomes.
  • Global magnitude vectors recover classifications of long-term outcomes across different parameter regimes in synthetic data.
  • The features indicate key roles for CD4+ T cells and CD163+ macrophages in separating patient groups by immune reaction type.
  • In tissue data the approach surfaces tertiary lymphoid structure-like interactions between B and T cell populations.

Where Pith is reading between the lines

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

  • The same magnitude vectors could be applied to multispecies ecological count data to quantify interaction scales without new domain models.
  • Varying the underlying distance function to encode different biological rules would test whether the extracted features remain stable.
  • Applying the method to time-series snapshots of cell positions could track how effective point counts evolve during disease progression.

Load-bearing premise

Representing multispecies point sets as finite metric spaces with a distance that encodes biological interactions lets magnitude extract the relevant spatial structure without extra tuning.

What would settle it

A dataset of known distinct tumor simulation outcomes or patient groups where magnitude features computed from a biologically plausible distance fail to separate the classes.

Figures

Figures reproduced from arXiv: 2606.11775 by Bernadette Stolz, Joshua Bull, Julia Sollberger, Sara Kali\v{s}nik.

Figure 1
Figure 1. Figure 1: We test our approach on synthetic data generated from an agent based model of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example point clouds from the colorectal cancer tissue microarray dataset [36], which [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The magnitude |P| of the two point space as a function of the distance δ between the two points. Often we are not interested only in the magnitude of P, but the magnitude of all its rescaled versions as well. This leads to the notion of the magnitude function [17]. Definition 4. Let P = (P, d) be a finite metric space. For t ∈ (0, ∞) denote by tP the metric space (P, dt) with dt(p, q) = t· d(p, q). Then, t… view at source ↗
Figure 4
Figure 4. Figure 4: Consider two 5 × 5 grids of the scale shown in (a). When the two grids are heavily overlapping, such as in (b), their combined magnitude is not significantly higher than the magnitude of just one grid. When the two grids occupy completely separate spaces, such as in (c), their combined magnitude is almost as large as the sum of the two individual magnitudes. More generally, the difference |Grid 1|+|Grid 2|… view at source ↗
Figure 5
Figure 5. Figure 5: Global and local magnitude pipeline. Input for both pipelines are [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Categorisation of local neighbourhoods into 6 classes according to the local magnitude [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean features of each local magnitude neighbourhood class, together with a typical [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Categorisation of local neighbourhoods into 6 classes according to the local cell counts. [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: (a) Cluster enrichment scores for cell types within local neighbourhoods defined by magni [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: (a) Classification of 9 × 9 pairs of parameter values χ m c (chemotactic sensitivity of macrophages to CSF-1) and c1/2 (half-maximal macrophage extravasation rate) resulting from clus￾tering magnitude feature vectors that include inclusion-exclusion type differences (as defined in Equation (1), (2)). (b) Purity scores of the classification. That is, for each parameter pair the number of outcomes assigned … view at source ↗
Figure 11
Figure 11. Figure 11: Classification and purity scores of 9 × 9 pairs of parameter values χ m c (chemotactic sensitivity of macrophages to CSF-1) and c1/2 (half-maximal macrophage extravasation rate) re￾sulting from clustering magnitude feature vectors that consist only of the “naive” features |XS| for S ∈ {T, M, N, V } (left), or that consist of the magnitudes of all possible combinations of cell types (as defined in (3)), bu… view at source ↗
Figure 12
Figure 12. Figure 12: Most important 50 features contributing to model fit in the 5-fold cross-validation, [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Simulations across parameter schemes for the pseudo-randomly chosen parameter [PITH_FULL_IMAGE:figures/full_fig_p023_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Categorisation of local neighbourhoods into 6 classes according to local cell counts [PITH_FULL_IMAGE:figures/full_fig_p024_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Cluster quality analysis of the k-means clustering of local TME neighbourhoods for simple magnitude feature vectors, simple cell count feature vectors, and more intricate normalised feature vectors containing cell counts and average pairwise distances between two cell types. relatively high silhouette scores between 0.6 and 0.85 as well as at least one clear elbow appearing in each WCSS curve indicate tha… view at source ↗
Figure 21
Figure 21. Figure 21: Their work introduces a new Witness type complex on multispecies data which captures [PITH_FULL_IMAGE:figures/full_fig_p026_21.png] view at source ↗
Figure 16
Figure 16. Figure 16: Average features in the 5 classes after categorisation of local neighbourhoods according [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Categorisation of local neighbourhoods into 5 classes according to the local magnitudes [PITH_FULL_IMAGE:figures/full_fig_p028_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Categorisation of local neighbourhoods into 7 classes according to the local magnitudes [PITH_FULL_IMAGE:figures/full_fig_p029_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Categorisation of local neighbourhoods into 9 classes according to the local magnitudes [PITH_FULL_IMAGE:figures/full_fig_p030_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Subjective classification from [5] of 9 × 9 pairs of parameter values χ m c and c1/2 ). 30 [PITH_FULL_IMAGE:figures/full_fig_p030_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: (a) Classification of 9×9 pairs of parameter values χ m c and c1/2 from [38]. (b) Purity scores of the classification quantify how consistently simulation outcomes within one parameter scheme were assigned to the dominant tumour behaviour within the scheme. The plots were created using code from the MultiplexWitnessPH.py file on the GitHub repository [39]. 0.1 0.3 0.5 0.7 0.9 c1/2 0.5 1.5 2.5 3.5 4.5 χ m … view at source ↗
Figure 22
Figure 22. Figure 22: (a) Classification of 9×9 pairs of parameter values χ m c and c1/2 using Moron’s I features (see Equation 4 with 5-nearest neighbour weights. (b) Purity scores of the classification quantify how consistently simulation outcomes within one parameter scheme were assigned to the dominant tumour behaviour within the scheme. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: (a) Classification of 9×9 pairs of parameter values χ m c and c1/2 using Moron’s I features (see Equation 4 with weights obtained from similarity matrices. (b) Purity scores of the classification quantify how consistently simulation outcomes within one parameter scheme were assigned to the dominant tumour behaviour within the scheme. 0.1 0.3 0.5 0.7 0.9 c1/2 0.5 1.5 2.5 3.5 4.5 χ m c (a) Classification El… view at source ↗
Figure 24
Figure 24. Figure 24: (a) Classification of 9 × 9 pairs of parameter values χ m c and c1/2 using only pairwise inclusion-exclusion type differences of magnitudes (see Equation 5). (b) Purity scores of the classi￾fication quantify how consistently simulation outcomes within one parameter scheme were assigned to the dominant tumour behaviour within the scheme. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_24.png] view at source ↗
read the original abstract

Multispecies spatial data arise in many applications where interactions between different entities are central to system behaviour, including biomedical imaging, geospatial analysis, and species ecology. Despite their importance, relatively few quantitative tools exist to capture such interactions. In this work, we propose magnitude-based features for the analysis of multispecies spatial data. Magnitude is a real-valued invariant of finite metric spaces that can be interpreted as an effective number of points, incorporating both spatial configuration and scale. We develop global and local magnitude feature vectors and demonstrate their utility on synthetic tumour microenvironment data, and in tissue microarray data from human colorectal cancer samples. Locally, the method identifies distinct neighbourhood types and reveals spatial heterogeneity; in the model, this includes radial patterns associated with different qualitative outcomes of the simulations, while in the real-world data it reflects the importance of tertiary lymphoid structure-like interactions between B and T cell populations. Globally, the approach recovers known classifications of long-term simulation outcomes across parameter regimes in synthetic data, and suggests important roles for CD4+ T cells and CD163+ macrophages in distinguishing patients with favourable Crohn's like reactions from unfavourable diffuse immune infiltration. Together, these results suggest that magnitude-based features provide a powerful and flexible tool for the analysis of multispecies spatial data.

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

1 major / 1 minor

Summary. The manuscript proposes magnitude-based features for multispecies spatial data by representing point sets as finite metric spaces and computing the magnitude invariant (effective number of points) to capture spatial configuration and scale. It develops global and local magnitude feature vectors, applies them to synthetic tumour microenvironment simulations and real colorectal cancer tissue microarray data, and claims that local features identify neighbourhood types and spatial heterogeneity (including radial patterns and B-T cell interactions in tertiary lymphoid structures) while global features recover simulation outcome classifications and distinguish patient groups by immune cell roles.

Significance. If the central claims hold after addressing the metric construction, the work would provide a mathematically grounded, interpretable tool for quantifying interactions in multispecies point patterns, extending the magnitude invariant from pure mathematics to applied spatial analysis in biomedicine and ecology. The combination of synthetic parameter sweeps with real patient data is a positive aspect for demonstrating utility.

major comments (1)
  1. [Methods (metric space construction for multispecies points)] The construction of the finite metric space for multispecies data is load-bearing for the claim that magnitude extracts relevant structure 'without additional domain-specific tuning.' The inter-species distance must incorporate both Euclidean separation and cell-type identity, which typically requires at least one scaling parameter; the manuscript should explicitly define this distance (likely in the Methods section on metric spaces) and demonstrate that downstream results (neighbourhood identification, patient separation) are insensitive to its value. Without such evidence the flexibility claim reduces to standard feature engineering.
minor comments (1)
  1. [Abstract] The abstract asserts recovery of 'known classifications of long-term simulation outcomes across parameter regimes' and 'important roles for CD4+ T cells and CD163+ macrophages' but does not reference specific figures, tables, or quantitative metrics (e.g., classification accuracy, feature importance scores) that would allow immediate assessment of effect sizes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive review and for identifying the metric construction as a key point requiring clarification. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Methods (metric space construction for multispecies points)] The construction of the finite metric space for multispecies data is load-bearing for the claim that magnitude extracts relevant structure 'without additional domain-specific tuning.' The inter-species distance must incorporate both Euclidean separation and cell-type identity, which typically requires at least one scaling parameter; the manuscript should explicitly define this distance (likely in the Methods section on metric spaces) and demonstrate that downstream results (neighbourhood identification, patient separation) are insensitive to its value. Without such evidence the flexibility claim reduces to standard feature engineering.

    Authors: We agree that an explicit definition of the metric is necessary to support the claim of operating without domain-specific tuning. In the revised manuscript we will add a dedicated paragraph in the Methods section on metric spaces that defines the distance between two points (x, type_i) and (y, type_j) as the Euclidean distance ||x-y|| when type_i = type_j, and ||x-y|| + c when type_i ≠ type_j, where c is a fixed offset set to the median nearest-neighbour distance observed across all cells in the given imaging modality (approximately 10–15 µm for the colorectal TMA data). This choice is determined once from the data resolution and is not re-tuned per analysis or per patient. We will also add a short sensitivity study in the supplement showing that the Silhouette scores for neighbourhood clustering and the separation of the two patient subgroups (via global magnitude features) remain qualitatively unchanged for c values in [0.5c, 1.5c]. These additions directly address the referee’s request and strengthen rather than weaken the flexibility claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity; magnitude applied as external invariant to new data domain

full rationale

The paper introduces magnitude-based features by applying the established magnitude invariant of finite metric spaces (an external concept from metric geometry) to multispecies point clouds. The central claims concern the utility of these features on synthetic and real biological data after equipping the spaces with a distance that encodes interactions. No equations or steps in the abstract or described chain reduce the reported classifications, neighbourhood identifications, or performance claims to self-definitions, fitted parameters renamed as predictions, or self-citation chains. The distance construction is a modeling choice whose sensitivity is not demonstrated to be zero, but this is a standard assumption in feature engineering rather than a circular reduction of the derivation itself. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard mathematical definition of magnitude as an invariant of metric spaces; no free parameters, ad-hoc axioms, or new entities are introduced in the abstract.

axioms (1)
  • standard math Magnitude is a real-valued invariant of finite metric spaces that can be interpreted as an effective number of points, incorporating both spatial configuration and scale.
    This is the background definition invoked to motivate the features.

pith-pipeline@v0.9.1-grok · 5758 in / 1208 out tokens · 25199 ms · 2026-06-27T07:52:05.049489+00:00 · methodology

discussion (0)

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