26  Latent-Space Complexity: Within-Class Geometric Analysis

27 Purpose

This section applies the latent-space geometric framework defined in the Methods section to quantify within-class structure across cancer types.

Specifically, we evaluate how the geometric properties of latent representations differ between normal and tumor samples for each tissue.

All metrics used in this section—participation ratio, eigenvalue entropy, anisotropy, class radius, centroid distance, and neighborhood structure—are defined formally in the preceding Methods section.

The present analysis does not introduce new measures; it applies those definitions to compute per-cancer geometric differences.

28 Analytical Scope

This section focuses on within-class structure, evaluating how the internal geometry of each class changes from normal to tumor.

Between-class relationships (e.g., global geometry across cancer types) are addressed separately.

29 Data Inputs

The analysis operates on the aligned latent dataset:

  • latent coordinates (latent.npy)
  • metadata (metadata_aligned.csv)

Each sample is represented as a latent vector ( z_i ^k ), with associated labels for condition (normal vs tumor) and cancer type.

30 Computational Strategy

For each cancer type ( t ):

  1. Partition samples into:
    • normal: ( X_t^{(N)} )
    • tumor: ( X_t^{(T)} )
  2. Compute geometric measures for each group:
    • participation ratio (PR)
    • eigenvalue entropy (H)
    • anisotropy (A)
    • class radius (R)
  3. Compute differences:

[ M_t = M_t^{(T)} - M_t^{(N)} ]

These deltas constitute the primary outputs of this section.

31 Interpretation Framework

The quantities computed in this section describe geometric properties of the latent representation, not direct biological observables.

Interpretation follows these rules:

  • Higher participation ratio or entropy indicates variance distributed across more latent dimensions.
  • Higher radius indicates greater dispersion of samples within a class.
  • Higher anisotropy indicates stronger directional structure.
  • These measures do not directly imply biological complexity gain or loss.

Comparison with classical expression-space complexity measures is deferred to the integrated results section.

32 Within-Class Latent Geometry

32.1 Overview

32.2 Overview

This section evaluates how the internal geometric structure of each cancer class changes between normal and tumor states using the metrics defined in the Methods chapter.

For each cancer type, samples are partitioned into normal and tumor groups, and geometric measures are computed separately for each group. Differences between these groups quantify how the geometric structure of the latent representation changes under tumorigenesis.

The results are presented in aggregated form, combining multiple geometric measures into summary tables, structural visualizations, and ranked comparisons.

The metrics summarized in this section quantify complementary aspects of latent geometry:

  • Participation ratio and eigenvalue entropy reflect effective dimensionality.
  • Anisotropy captures directional concentration of variance.
  • Radius measures overall dispersion of samples within a class.
  • Centroid distance reflects global separation between normal and tumor states.

The results shown below summarize geometric properties for normal and tumor groups, together with the corresponding tumor-minus-normal differences for each cancer type.

32.3 Comparison Metrics Table

This table provides the primary quantitative summary of within-class latent geometry across matched normal–tumor comparisons.

comparison n_normal n_tumor pr_normal pr_tumor pr_delta eig_entropy_normal eig_entropy_tumor eig_entropy_delta anisotropy_normal anisotropy_tumor anisotropy_delta radius_normal radius_tumor radius_delta centroid_distance
BLAD/TCC 7.000 11.000 2.868 3.416 0.548 1.210 1.430 0.220 0.485 0.416 −0.070 20.937 20.135 −0.801 37.080
BR/BRAD 5.000 11.000 1.654 2.879 1.225 0.655 1.319 0.664 0.740 0.512 −0.229 25.817 30.179 4.361 35.518
Brain/GBM 8.000 10.000 2.592 3.902 1.310 1.141 1.579 0.438 0.540 0.376 −0.164 17.915 22.013 4.098 26.573
Brain/MB 8.000 10.000 2.592 3.435 0.844 1.141 1.428 0.288 0.540 0.403 −0.137 17.915 19.155 1.240 33.631
COL/COADREAD 11.000 11.000 2.534 2.815 0.280 1.152 1.353 0.201 0.546 0.541 −0.006 19.358 27.708 8.350 25.238
GC/FL 6.000 11.000 1.385 3.772 2.387 0.554 1.537 0.983 0.840 0.402 −0.438 17.688 17.191 −0.497 28.671
GC/LBCL 6.000 11.000 1.385 3.276 1.891 0.554 1.464 0.910 0.840 0.482 −0.359 17.688 12.687 −5.001 26.278
KID/RCC 12.000 11.000 3.348 3.133 −0.214 1.422 1.356 −0.065 0.407 0.450 0.043 16.744 27.882 11.138 17.164
LU/LUAD 7.000 11.000 1.931 2.761 0.830 0.884 1.319 0.435 0.677 0.544 −0.133 21.418 25.737 4.319 15.244
OV/OVAD 4.000 11.000 2.162 2.431 0.269 0.906 1.199 0.293 0.614 0.581 −0.034 14.985 25.331 10.347 25.847
PA/PAAD 10.000 10.000 2.796 2.384 −0.413 1.302 1.217 −0.086 0.537 0.607 0.070 21.426 23.132 1.706 24.021
PR/PRAD 9.000 10.000 2.858 3.756 0.898 1.285 1.469 0.184 0.516 0.380 −0.136 15.971 21.765 5.794 12.517
UT/EAC 6.000 10.000 1.407 3.804 2.397 0.584 1.532 0.948 0.834 0.406 −0.429 23.443 28.415 4.972 26.499

32.4 Structural-Change Space

The figure below places each matched comparison in a latent structural-change plane defined by the change in participation ratio (effective dimensionality) and the change in anisotropy (directional concentration of variance).

32.5 Ranked Summaries

To facilitate comparison across cancer types, selected metrics are ranked according to the magnitude of change between normal and tumor states.

These rankings provide a compact view of the most pronounced geometric shifts in latent space.

32.5.1 Largest decreases in participation ratio (ΔPR)

comparison pr_delta anisotropy_delta eig_entropy_delta radius_delta
PA/PAAD -0.413 0.070 -0.086 1.706
KID/RCC -0.214 0.043 -0.065 11.138
OV/OVAD 0.269 -0.034 0.293 10.347
COL/COADREAD 0.280 -0.006 0.201 8.350
BLAD/TCC 0.548 -0.070 0.220 -0.801
LU/LUAD 0.830 -0.133 0.435 4.319
Brain/MB 0.844 -0.137 0.288 1.240
PR/PRAD 0.898 -0.136 0.184 5.794
BR/BRAD 1.225 -0.229 0.664 4.361
Brain/GBM 1.310 -0.164 0.438 4.098

32.5.2 Largest increases in participation ratio (ΔPR)

comparison pr_delta anisotropy_delta eig_entropy_delta radius_delta
UT/EAC 2.397 -0.429 0.948 4.972
GC/FL 2.387 -0.438 0.983 -0.497
GC/LBCL 1.891 -0.359 0.910 -5.001
Brain/GBM 1.310 -0.164 0.438 4.098
BR/BRAD 1.225 -0.229 0.664 4.361
PR/PRAD 0.898 -0.136 0.184 5.794
Brain/MB 0.844 -0.137 0.288 1.240
LU/LUAD 0.830 -0.133 0.435 4.319
BLAD/TCC 0.548 -0.070 0.220 -0.801
COL/COADREAD 0.280 -0.006 0.201 8.350

32.5.3 Largest centroid shifts (distance)

comparison centroid_distance pr_delta eig_entropy_delta anisotropy_delta
BLAD/TCC 37.080 0.548 0.220 -0.070
BR/BRAD 35.518 1.225 0.664 -0.229
Brain/MB 33.631 0.844 0.288 -0.137
GC/FL 28.671 2.387 0.983 -0.438
Brain/GBM 26.573 1.310 0.438 -0.164
UT/EAC 26.499 2.397 0.948 -0.429
GC/LBCL 26.278 1.891 0.910 -0.359
OV/OVAD 25.847 0.269 0.293 -0.034
COL/COADREAD 25.238 0.280 0.201 -0.006
PA/PAAD 24.021 -0.413 -0.086 0.070

33 Summary

This section computes within-class geometric differences between normal and tumor samples in latent space.

The results provide a structured description of:

  • dimensionality (PR, entropy)
  • dispersion (radius)
  • directional structure (anisotropy)
  • local organization (neighborhood metrics)

These quantities form the basis for cross-representation comparison with classical complexity measures.

They also serve as inputs for subsequent analysis of between-class geometry and integrated biological interpretation.

33.1 Notes

These results are descriptive summaries of latent-space geometry and should not be interpreted as direct measures of biological complexity.

Interpretation should be integrated with the broader comparison framework, including classical complexity measures and pathway-level biological context.