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Heat Field Signatures: From Point Clouds to Smooth Geometry

Yuanqing Wang, Yapeng Tian, Baris Coskunuzer

arXiv:2609.07975v1cs.LGmath.DG

TL;DR

Irregular point clouds lack a standard interface to multiscale geometric analysis, so key structure is often inferred implicitly from coordinates or neighborhoods. HFS lifts point clouds to smooth ambient heat fields and extracts global, local, and shape-level signatures. Across seven benchmarks, it achieves strong performance with lower computational cost and exact rotation invariance, including a nearly 24-percentage-point improvement on SCOP over the strongest deep baseline.

  • Problem

    For irregular point clouds, class-relevant local dimension, anisotropy, curvature, density variation, and geometric transitions are often learned only implicitly from coordinates and neighborhoods.

    Source: §1 · p. 1

  • Method

    HFS lifts point clouds to smooth multiscale ambient heat fields and extracts global and local geometric signatures plus HDS, with closed-form, learned, or network-integrated representations.

    Source: §1 · pp. 1–2

  • Results

    Across seven benchmarks, HFS combines strong classification performance with substantially lower computational cost and exact geometric invariance; on SCOP, it improves over the strongest deep baseline by nearly 24 percentage points.

    Source: §1 · p. 2, §4.3 · p. 9

  • Takeaways & Limitations

    When class identity is carried by intrinsic multiscale geometry, computing that geometry can be advantageous over asking a network to discover it from coordinates.

    Source: §4.3 · p. 9

    HFS is not universally superior: coordinate-based networks remain competitive or stronger on tasks dominated by rigid global shape or localized appearance.

    Source: §4.3 · p. 9

Abstract

from arXiv · show

Bringing multiscale geometric analysis directly to irregular point clouds remains difficult: quantities such as local dimension, anisotropy, density variation, and geometric transitions are typically estimated through explicit neighborhood, manifold, or graph constructions, or left for neural networks to infer from coordinates. We introduce Heat Field Signatures (HFS), which lift a point cloud to a multiscale family of smooth ambient heat fields, providing a direct interface from discrete samples to geometric analysis. From this field, HFS computes closed-form global and local signatures directly from pairwise distances, capturing heat concentration, intrinsic dimension, anisotropy, and scale transitions. We further introduce the Heat Dimension Spectrum (HDS), a compact summary of multiscale geometric composition. HFS can be used as a closed-form descriptor, a lightweight learned representation, or a geometric feature channel for neural point-cloud models. Across synthetic and real-world benchmarks spanning subcellular, neuronal, tree, and protein data, HFS outperforms strong point-cloud and multiparameter-persistence baselines while substantially reducing end-to-end cost. On SCOP protein-fold classification, HFS improves over the strongest deep baseline by nearly $24$ percentage points using coordinates alone, while standalone HFS representations are exactly rotation-invariant by construction. More broadly, HFS turns a classical heat field into a practical interface for multiscale geometric analysis in modern point-cloud learning.

1 INTRODUCTION

Irregular point clouds lack a standard interface to differential-geometric analysis, leaving important geometric structure implicit in many learning methods. HFS lifts clouds to smooth multiscale heat fields and extracts complementary geometric signatures, with strong results across seven benchmarks.

1 INTRODUCTION

  • Point-cloud networks typically learn class-relevant local dimension, anisotropy, curvature, density variation, and geometric transitions implicitly from coordinates and neighborhoods.These quantities often distinguish classes.Source: §1 · p. 1
  • HFS lifts a discrete point cloud to a multiscale family of smooth ambient heat fields, making geometric quantities directly available for analysis.The field graph is a smooth hypersurface at every diffusion scale, with derivatives and other geometric quantities available.Source: §1 · p. 1
  • The ambient heat field is generated directly from samples, without requiring a mesh, graph Laplacian, normal estimation, or spectral decomposition.This makes the approach applicable to data whose identity depends on how geometry changes across space and scale.Source: §1 · p. 2
  • Global and local signatures capture concentration, intrinsic dimension, anisotropy, and scale transitions, while HDS summarizes the multiscale distribution of geometric types and sampling density.The measurements illustrate the information exposed by the heat-field interface rather than exhaust it.Source: §1 · p. 2
  • Across seven benchmarks, HFS achieves the best performance while reducing computational cost and providing exact rotation invariance; on SCOP, it improves over the strongest deep baseline by nearly 24 percentage points.The SCOP result uses coordinates alone.Source: §1 · p. 2

2 RELATED WORK

Prior point-cloud methods either learn geometric structure from coordinates and neighborhoods or use fixed-scale descriptors and estimates. HFS instead provides multiscale local geometric measurements directly from unstructured samples.

Point-cloud learning

  • Point-cloud networks, including density-adaptive and self-supervised approaches, learn geometric structure implicitly, whereas HFS makes local geometry explicit and its logarithmic scale derivatives are approximately invariant to multiplicative density variation by construction.The cited methods include PointNet, DGCNN, Point Transformer, and density-adaptive variants.Source: §2 · p. 2

Handcrafted descriptors and dimension estimation

  • Handcrafted descriptors rely on fixed scales and oriented normals, while common intrinsic-dimension estimators return a single fixed-scale scalar; HFS provides a multiscale profile without normal estimation.HFS aggregates the joint dimension, codimension, and scale-transition signals into HDS.Source: §2 · p. 3

Heat kernels and spectral methods

  • Mesh- or graph-based heat-kernel signatures require a Laplacian, whereas HFS analyzes ambient-field differential geometry on unstructured point clouds and extracts local dimension, codimension, and scale transitions.HKS and WKS use spectral methods; diffusion maps target global dimensionality reduction.Source: §2 · p. 3

Topological and geometric data analysis

  • Persistent-homology methods capture connectivity across scale, while heat-geometric methods measure local dimension, codimension, and anisotropy; the frameworks are complementary and could be combined.The paper adopts density–distance bifiltration benchmarks for evaluation.Source: §2 · p. 3

3 HEAT FIELD SIGNATURES

HFS converts a point cloud into a multiscale ambient heat field and reads it through global, local, and shape-level measurements. These are pooled into fixed-dimensional or learned representations for classification.

3.1 OVERVIEW

  • HFS transforms a discrete point cloud into a smooth multiscale heat field, then combines global responses, local geometric signatures, and HDS to capture dominant morphology and localized structures.The pipeline aggregates these measurements into a representation for classification.Source: §3.1 · p. 3

3.2 HEAT FIELD AND GLOBAL SIGNATURES

  • Diffusion time sets geometric scale: small values resolve fine neighborhoods, intermediate values reveal branches and junctions, and coarse values retain global structure.Because the field is a Gaussian mixture, global functionals can be computed directly from pairwise distances; these measure heat concentration, scale response, and spatial and normalized roughness.Source: §3.2 · p. 4

3.3 LOCAL HEAT-GEOMETRIC SIGNATURES

  • Local logarithmic derivatives yield heat dimension, log-Hessian eigenvalues, and scale-transition rate, jointly encoding effective dimension, tangent–normal structure, anisotropy, and changing geometry.In R3, heat-dimension values near 1, 2, and 3 ideally indicate filament-, surface-, and volume-like regions, respectively; large absolute transition rates highlight boundaries, junctions, protrusions, and mixed-dimensional transitions.Source: §3.3 · p. 4

3.4 HEAT DIMENSION SPECTRUM

  • HDS summarizes the distribution of local heat dimensions across scales, including dimensional composition and how it changes in heat-concentrated regions.This captures clouds that mix geometric types, such as a neuron with filamentary branches and a compact soma.Source: §3.4 · p. 5

3.5 VECTORIZATION AND HFS VARIANTS

  • Per-point features concatenate density, heat dimension, transition rate, and log-Hessian eigenvalues across scales; HFS-simple and HFS-desc use fixed pooling, while HFS-full uses a shared MLP and mean/max aggregation.In R3, each per-point feature has 6T components; dense HFS computation costs O(m^2T) for fixed ambient dimension.Source: §3.5 · pp. 5–6

4 EXPERIMENTS

The experiments test whether explicit multiscale geometry improves point-cloud classification, its computational cost, and rotation behavior. Across seven benchmarks, HFS combines strong classification performance with lower cost and exact geometric invariance, but its advantage is not universal.

4.1 EXPERIMENTAL SETUP

  • The evaluation spans three synthetic density/topology tasks and four real morphology benchmarks, using shared splits or folds and common descriptor-classifier settings.HFS variants separate closed-form extraction, learned pooling, and integration with a DGCNN backbone.Source: §4.1 · p. 6

4.2 RESULTS

  • HFS variants achieve the best performance on all seven benchmarks, with HFS-full reaching 59.6% versus 35.7% for the strongest deep baseline on SCOP.The SCOP result is a 23.9-point gain using Cα coordinates alone.Source: §4.2 · p. 6, Table 1 · p. 7
  • HFS-full is about 8.5× faster than DGCNN in median runtime, while HFS-simple is about 25× faster and requires no learned feature extractor.HFS-desc is 2.4–5.5× cheaper than PointNet++, DGCNN, and Point Transformer; HFS+DGCNN trades efficiency for stronger accuracy on several datasets.Source: Table 2 · p. 7, §4.2 · p. 7
  • HFS is invariant by construction because it depends on pairwise distances and heat-field Hessian eigenvalues.The rotation comparison evaluates the same trained models on original and independently SO(3)-rotated test clouds; HFS+DGCNN changes by at most 1.1 points across four datasets.Source: Table 3 · p. 7, §4.2 · pp. 7–8
  • HFS-full reaches 99.8% on DisksAnnuli versus 86.9% for GraphCode, while HFS-desc reaches 98.3% on Processes versus 83.4%.These density-aware benchmarks compare HFS with multiparameter-persistence methods.Source: Table 4 · p. 8, §4.2 · p. 8
  • No single descriptor block matches the full descriptor on any dataset, and removing all four local blocks costs up to 10.7 points.The log-Hessian block is especially important on SCOP and NeuroMorpho, where its removal costs 8.4 and 6.2 points, respectively.Source: §4.2 · p. 8, Table 5 · p. 9

4.3 DISCUSSION

  • The experiments support computing intrinsic multiscale geometry when class identity carries that information, while rigid-global-shape or localized-appearance tasks can favor coordinate networks.Much of HFS’s advantage is already present in the closed-form descriptor; learned pooling and backbone integration improve it further.Source: §4.3 · p. 9

5 CONCLUSION

HFS lifts irregular point clouds into smooth ambient heat fields across scale, giving geometric analysis a direct interface to finite samples. The appendix interprets the field through global measurements and scale-dependent heat landscapes.

5 CONCLUSION

  • HFS lifts an irregular point cloud to smooth ambient heat fields, making geometric analysis directly accessible to point-cloud learning.The conclusion frames this as an interface from discrete data to a multiscale geometric object.Source: §5 · p. 9, A HEAT FIELD SIGNATURES: DETAILS AND… · p. 13

A.1 FROM POINT CLOUDS TO HEAT FIELDS

  • Small scales resolve individual points, intermediate scales reveal branches, sheets, clusters, and junctions, and large scales retain coarse organization.The evolution is measured without explicitly constructing a mesh, graph Laplacian, or surface.Source: §A.1 · p. 13

A.2 GLOBAL HEAT-FIELD SIGNATURES

  • Global functionals track heat concentration and its scale response, while roughness measures describe the heat landscape and its energy-normalized form.The global block records how concentration and roughness evolve as neighboring structures interact.Source: §A.2 · p. 13
  • Heat bodies visualize smooth neighborhoods at suitable scales, but their grid-evaluated volume and boundary geometry keep them outside the main closed-form HFS representation.Filaments can appear tube-like, surfaces slab-like, and disconnected structures can merge as scale increases.Source: §A.2 · p. 14

A.3 LOCAL HEAT GEOMETRY

Local heat-field derivatives characterize dimension, orientation, and scale transitions around points; HDS aggregates local dimensional composition across a shape and scale. The appendix also describes scale selection and the benchmark scope.

A.3 LOCAL HEAT GEOMETRY

  • At smooth manifold points, heat dimension approximates intrinsic dimension between sampling resolution and geometric-variation scales, while excessively small scales can yield values near zero.The finite-sample approximation has geometry/density bias and sampling noise, with a smaller self-point correction.Source: §A.3 · p. 14
  • Local signatures distinguish effective dimension, tangent-normal structure, and geometric change with scale, capturing anisotropy, branching, boundaries, and transitions.The heat dimension counts effective directions, the log-Hessian spectrum describes orientation, and the transition response tracks changing local dimension.Source: §A.3 · p. 16

A.4 HEAT DIMENSION SPECTRUM

  • HDS summarizes the distribution of local heat dimensions across the cloud and scale, preserving dimensional composition that a mean dimension can lose.A half-filament/half-volume cloud and a surface-like cloud can both have mean dimension 2 but different spectra.Source: §A.4 · p. 17

A.5 SCALE CHOICE AND COMPUTATION

  • The scale scheme uses logarithmically spaced scales, and truncated pairwise computation can reduce cost from O(m^2T) to O(mkT) when only k neighbors contribute appreciably.The upper endpoint captures coarse shape organization; the lower endpoint tracks sampling resolution.Source: §A.5 · p. 17

B DATASET DETAILS

  • The seven main benchmarks span synthetic density, scale, and topology tasks in R2 and real neuronal, subcellular, tree, and protein morphology in R3.The benchmark summary identifies these domains and their dimensional settings.Source: B DATASET DETAILS · p. 18

B.1 SYNTHETIC BENCHMARKS

The synthetic benchmarks probe distinct spatial properties, including random interaction patterns, fine-scale density structure, density-based localization, and topology.

B.1 SYNTHETIC BENCHMARKS

  • Random Point Processes classifies Poisson, Matérn, Strauss, and Baddeley–Silverman clouds as random, clustered, repulsive, or multiscale processes, following Kerber & Russold (2024).It contains 1,000 clouds per class, with point counts from 105 to 321, and uses mean accuracy over 20 random 80/20 splits.Source: §B.1 · p. 18
  • Orbit5k tests five dynamical-system classes whose hardest pairs share coarse geometry but differ in fine-scale density, following Carrière & Blumberg (2020).It uses 1,000 clouds of 1,000 points per class and mean accuracy over 20 random 70/30 splits.Source: §B.1 · p. 18
  • DisksAnnuli jointly tests density-based localization and disk-versus-annulus topology, with object densities 2.4–4.8× the background density.Clouds contain 874–2,717 points; evaluation uses mean accuracy over 20 random 80/20 splits.Source: §B.1 · p. 18

B.2 REAL-WORLD MORPHOLOGY BENCHMARKS

The real-world benchmarks cover neuronal, subcellular, tree, and protein point clouds with different sampling and evaluation choices. SCOP uses backbone coordinates alone and a reduced point budget to avoid duplicate-induced scale collapse.

B.2 REAL-WORLD MORPHOLOGY BENCHMARKS

  • NeuroMorpho is a balanced nine-class benchmark of 6,198 neuronal reconstructions sampled to 1,024 points by arc length and evaluated with stratified 5-fold cross-validation.The benchmark uses native micron coordinates; its classes span nine neuron types.Source: §B.2 · p. 18
  • Allen uses 8,400 balanced interphase-cell clouds across seven subcellular classes, randomly subsampled to 1,024 points and evaluated with stratified 5-fold cross-validation.The task primarily distinguishes multiscale spatial distributions and clustering of intracellular structures.Source: §B.2 · p. 18
  • FOR-species20K uses 3,435 trees from 15 species, each randomly subsampled to 1,024 points, with occlusion- and viewpoint-driven LiDAR density variation as a robustness test.Evaluation uses stratified 5-fold cross-validation.Source: §B.2 · pp. 18–19
  • SCOP contains 1,455 proteins from 15 balanced folds and uses only Cα backbone coordinates, excluding sequence, residue identity, side-chain, and chemical features.The point budget is 128 to avoid duplicate-induced collapse of the nearest-neighbor scale; clouds are centered but not orientation-normalized, and results are not directly comparable to models using additional protein information.Source: §B.2 · p. 19

C.1 IMPLEMENTATION DETAILS

HFS implementation uses a shared, untuned multiscale schedule and offers fixed-statistical and learned representations. Comparisons use specified classifier, baseline, sampling, and evaluation protocols.

C.1 IMPLEMENTATION DETAILS

  • The HFS-desc descriptor concatenates global, HDS, and pooled-local features into 336 dimensions; HFS-simple omits the log-Hessian block and has 216 dimensions.The shared schedule uses T = 8 logarithmically spaced times between cmin = 0.05 and cmax = 0.25, with no per-dataset tuning.Source: §C.1 · p. 19
  • HFS-full replaces fixed statistical pooling with a learned shared MLP, while HFS+DGCNN adds 48 per-point HFS features to xyz and leaves the DGCNN otherwise unmodified.HFS-full has about 0.2M parameters and is trained for 250 epochs; its blocks are standardized using training-split statistics.Source: §C.1 · p. 19
  • HKS uses a symmetric 10-NN graph Laplacian and FPFH uses normal and feature radii; on SCOP, FPFH falls back to 30- and 100-neighbor radii for about a third of clouds.The SCOP adjustment addresses clouds where the fixed radii enclose no neighbors.Source: §C.1 · p. 19
  • Descriptor rows use the same standardized two-layer MLP classifier, making their comparison one of representations rather than downstream models.The classifier has hidden widths (128, 64) and is trained to convergence, with specified iteration limits for some rows.Source: §C.1 · p. 19
  • Deep baselines follow published training recipes, and the rotation comparison adds full-SO(3)-augmented DGCNN and rotation-equivariant VN-DGCNN.The listed networks use standard augmentation without full SO(3), except DGCNN-SO(3); VN-DGCNN’s invariance was numerically verified at 6 × 10^-8 relative logit change.Source: §C.1 · pp. 19–20
  • Real-benchmark methods receive identical clouds, while synthetic descriptor methods use native-resolution clouds and deep baselines use fixed budgets; SCOP uses m = 128 to avoid duplicated residues.PointNet++ accuracy falls at the lower SCOP budget because its hierarchy repeatedly subsamples input points; other k-NN baselines and descriptor rows improve when duplication is removed. Evaluation uses shared splits or folds, with timings collected on one H200 node.Source: §C.1 · p. 20

C.2 BOUNDARY DATASETS: WHERE HFS IS NEUTRAL

HFS is designed for labels carried by intrinsic multiscale geometry, not localized or rigid-shape cues. On the boundary datasets, coordinate networks are at least as strong, and the authors recommend choosing the representation to match the label.

C.2 BOUNDARY DATASETS: WHERE HFS IS NEUTRAL

  • 2.8 points behind DGCNN on IntrA (best HFS, second by mACC) and 1.7 behind Point Transformer on ATLAS-1; coordinate networks are at least as strong.The datasets label a localized aneurysm bulge or overall cell shape and size, rather than multiscale organization of the whole cloud.Source: §C.2 · pp. 20–21, Table 8 · p. 21
  • The practical guidance is to use HFS for intrinsic morphology and a coordinate encoder for rigid shape or localized appearance.HFS remains competitive on these boundary tasks, but the morphology-benchmark margins disappear.Source: §C.2 · pp. 20–21

C.3 SCALE-SCHEDULE ABLATION

The scale ablation supports using multiple diffusion times, with diminishing gains as scale sampling becomes finer. The authors use T = 8 as an accuracy/size compromise, not a tuned optimum.

C.3 SCALE-SCHEDULE ABLATION

  • 58.8 → 69.5 → 72.3 → 74.2 on NeuroMorpho for T = 2, 4, 8, 16; moving from 8 to 16 adds 1.9 points for twice the descriptor length.The accuracy increase is monotonic and saturates; T = 16 improves every reported HFS number.Source: Table 9 · p. 21, §C.3 · p. 21
  • The ablation varies T with the range fixed, then varies range multipliers at T = 8, using Processes OA over 20 splits and NeuroMorpho mACC over five folds.The defaults throughout are T = 8 and (0.05, 0.25).Source: Table 9 · p. 21, §C.3 · p. 21
  • Accuracy spans 98.2–98.6 on Processes and 72.2–73.0 on NeuroMorpho across five range settings, comparable to fold-to-fold standard deviation.The defaults are cmin = 0.05 and cmax = 0.25; no per-dataset tuning is performed.Source: §C.3 · p. 21, Table 9 · p. 21

C.4 ROTATION-INVARIANCE PROTOCOL

The rotation protocol compares each model on original and independently rotated test clouds without retraining, and directly checks HFS feature invariance. The accompanying persistence overview contrasts single-scale barcodes with density–distance bifiltrations and their vectorized representations.

C.4 ROTATION-INVARIANCE PROTOCOL

  • The test protocol applies independently sampled Haar-uniform SO(3) rotations to each test cloud, with no training on rotated test data or retraining between evaluations.The rotation comparison therefore isolates sensitivity to a change of frame under each method’s training recipe.Source: §C.4 · pp. 21–22
  • HFS feature deviations under rotation are at floating-point round-off, 10^-7 to 4 × 10^-6, supporting exact invariance by construction rather than empirical robustness alone.The protocol’s Δ is original-test balanced accuracy minus rotated-test balanced accuracy; values within roughly ±2 points are treated as run-to-run noise.Source: §C.4 · pp. 21–22
  • Principal-axis alignment is 0.94 for FOR-species and 0.44 for SCOP; arbitrary SCOP reference frames make rotation nearly in-distribution for baselines.The alignment score approaches 1 for a concentrated canonical orientation and 0 for axes spread over the sphere.Source: §C.4 · p. 22

D.1 ONE-PARAMETER PERSISTENCE

  • Single-parameter persistence records feature birth and death scales in stable barcodes, but one scale can conflate density with geometry and confuse sparse structure with outliers.For example, connecting a sparse filament can collapse a dense cluster at the same scale.Source: §D.1 · p. 22

D.2 MULTIPARAMETER PERSISTENCE

  • Multiparameter modules index homology over a parameter grid, but lack a complete discrete barcode analogue, so practical pipelines map them to vectors or graphs for classification.The appendix frames this as an account of multiparameter persistence and the density–distance bifiltration used by the topological baselines.Source: §D.2 · pp. 22–23, D MULTIPARAMETER PERSISTENCE FOR POINT… · p. 22

D.3 THE DENSITY–DISTANCE BIFILTRATION

  • The density–distance bifiltration first selects points by codensity threshold, then builds a Rips complex by distance; the benchmarks probe joint density–scale signatures that HFS targets through heat-field channels.On DisksAnnuli, the annulus loop persists only within bands on both axes; on Processes, interaction types imprint distinct joint signatures.Source: §D.3 · p. 23
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