Data Skeleton: Essential Structural Proxy

• Data skeletons are compact structural proxies that distill key topological, geometrical, and kinematic features from complex datasets.
• They enable efficient analysis in diverse applications such as topological data analysis, active clustering, and human motion recognition.
• Various forms—from sparse graphs and simplicial complexes to learned skeletal points—offer tailored preservation of data invariants with computational efficiency.

Data skeleton denotes a family of compact structural representations that preserve salient organization while discarding ambient redundancy. In the literature, the term ranges from homologically persistent subcomplexes for finite metric spaces, to sparse directed graphs and graph-forests for active clustering, to anatomical medial structures and articulated human-joint sequences used in recognition, re-identification, and biomechanics (Verovsek et al., 2017, Xie et al., 10 Sep 2025, Chen et al., 28 Sep 2025, Khargonkar et al., 2023, Yang et al., 6 Mar 2026). This breadth suggests that “skeleton” is not a single object but a recurring abstraction principle: represent the essential topology, geometry, density, or kinematics of data with a lower-complexity surrogate.

1. Conceptual scope and representational forms

Across the cited work, a data skeleton may be a simplicial subcomplex, a sparse weighted directed graph, a granular-ball forest, a set of skeletal points with associated radii and spoke vectors, or a tensorized sequence of human joint coordinates. The formal object depends on what is being preserved. In topological data analysis, the goal is to preserve persistent homology across all scales. In active clustering, it is to retain the few edges most relevant to clustering uncertainty. In anatomical shape analysis, it is to summarize both the interior and exterior of a 3D object. In human-motion analysis, it is to encode body structure and dynamics in a form suitable for temporal, graph-based, or vision-pretrained models (Verovsek et al., 2017, Xie et al., 10 Sep 2025, Chen et al., 28 Sep 2025, Khargonkar et al., 2023, Rao et al., 2022, Yang et al., 6 Mar 2026).

A common source of ambiguity is that the same term denotes structurally different objects. For Kališnik Verovšek–Kurlin–Lešnik, a skeleton is a minimal-weight subcomplex that remains $d$-fitting at every filtration scale. In Data Skeleton Learning, it is a sparse, weighted directed graph $G_s=(V,E_s)$ with at most $n-1$ edges over data indices. In GBSK, it is a forest $S=(V,E)$ whose nodes are density-bearing granular-balls. In skeletal point representations for anatomy, it is an s-rep-like collection of predicted skeletal points $S=\{s_j\}$, radii $R=\{r(s_j)\}$, and spoke directions $U=\{u_j\}$ derived directly from a point cloud (Verovsek et al., 2017, Xie et al., 10 Sep 2025, Chen et al., 28 Sep 2025, Khargonkar et al., 2023).

This suggests that the unifying property of a data skeleton is not its datatype but its role as a parsimonious structural proxy. The proxy is judged by different invariants in different domains: Betti numbers and persistence intervals, clustering accuracy and constraint efficiency, Chamfer and Hausdorff distances, or action-recognition and re-identification performance.

2. Topological skeletonization and persistent structure

“The Higher-Dimensional Skeletonization Problem” formulates skeletonization for a finite point cloud $C$ in a metric space as an optimization over weighted simplicial complexes (Verovsek et al., 2017). A monotone weighting $w:P(C)\to[0,\infty)$ induces a filtration

$$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$

and the target is a subcomplex $G_s=(V,E_s)$0 minimizing total weight while ensuring that, for every scale $G_s=(V,E_s)$1, the reduced complex

$G_s=(V,E_s)$2

is $G_s=(V,E_s)$3-fitting in $G_s=(V,E_s)$4. The paper defines $G_s=(V,E_s)$5-spanning, $G_s=(V,E_s)$6-forest, $G_s=(V,E_s)$7-tree, and $G_s=(V,E_s)$8-fitting, and generalizes the classical minimal spanning tree to a minimal spanning $G_s=(V,E_s)$9-tree.

The construction has three stages. First, a minimal spanning $n-1$0-tree $n-1$1 is built greedily by adding $n-1$2-simplices in increasing weight order only when they do not create a $n-1$3-cycle. Second, the critical $n-1$4-simplices $n-1$5 are identified, and death times are assigned through relative-homology kernels together with the elder rule. Third, the homologically persistent $n-1$6-skeleton is assembled as

$n-1$7

Its reduced form at scale $n-1$8 keeps exactly those simplices whose birth–death interval contains $n-1$9 (Verovsek et al., 2017).

The principal claims are optimality and fidelity. For every $S=(V,E)$0, the reduced $S=(V,E)$1 is $S=(V,E)$2-fitting in $S=(V,E)$3, and among all $S=(V,E)$4-fitting subcomplexes it has minimal total weight. The labels on critical $S=(V,E)$5-simplices agree with their birth–death coordinates in the persistence diagram. The paper illustrates the $S=(V,E)$6 case with four points forming a square, where the minimal spanning tree is extended by a single critical edge realizing the persistent $S=(V,E)$7-cycle, and the $S=(V,E)$8 case with a tetrahedral configuration whose $S=(V,E)$9-cycle is captured by a critical $S=\{s_j\}$0-face (Verovsek et al., 2017).

Within the broader data-skeleton literature, this work supplies the most explicit topological criterion for what it means to preserve “essential structure”: not merely connectivity, but all homology up to dimension $S=\{s_j\}$1 at all filtration scales.

3. Geometric skeletal point representations for 3D anatomy

“Skeletal Point Representations with Geometric Deep Learning” addresses skeletonization as a learned, one-shot replacement for slow template-based s-rep fitting on anatomical shapes (Khargonkar et al., 2023). The input is a uniformly sampled point cloud $S=\{s_j\}$2 with $S=\{s_j\}$3, normalized to zero mean and unit radius. A PointNet++ encoder extracts per-point features $S=\{s_j\}$4, and a shared MLP predicts a weight matrix $S=\{s_j\}$5 with $S=\{s_j\}$6. Each column lies on the probability simplex, so each skeletal point is a convex combination of boundary points,

$S=\{s_j\}$7

Radii are then predicted through

$S=\{s_j\}$8

and spoke directions are extracted toward the highest-weight contributing boundary point (Khargonkar et al., 2023).

The training objective combines four terms,

$S=\{s_j\}$9

Three of these are presented as novel geometric losses. The weak-supervision term $R=\{r(s_j)\}$0 is a symmetric Chamfer distance between predicted skeletal points and template-fitted s-rep points from SlicerSALT. The medial-enforcement term

$R=\{r(s_j)\}$1

forces each skeletal sphere to touch the three closest boundary points, reflecting the definition that true medial-axis points are equidistant to at least three boundary points. The spread regularizer

$R=\{r(s_j)\}$2

prevents collapse and encourages the sheet to span the interior. The final term $R=\{r(s_j)\}$3 is the standard point-to-sphere distance from Point2Skeleton (Khargonkar et al., 2023).

Training uses PyTorch 1.12 on a single NVIDIA RTX A5000 (24 GB), Adam with learning rate $R=\{r(s_j)\}$4, zero weight decay, and batch size $R=\{r(s_j)\}$5. Pre-training is performed on 5000 synthetic deformed ellipsoids with known s-reps, generated by axis scaling, bending, and twisting; fine-tuning is then performed separately on hippocampi and tricuspid valve leaflets. On held-out synthetic ellipsoids, the full model with spread and medial losses yields Skeleton C.D. $R=\{r(s_j)\}$6 and H.D. $R=\{r(s_j)\}$7, compared with $R=\{r(s_j)\}$8 and $R=\{r(s_j)\}$9 for supervision only. On held-out clinical data, hippocampi achieve Skeleton C.D. $U=\{u_j\}$0, H.D. $U=\{u_j\}$1, Surf-recon C.D. $U=\{u_j\}$2, H.D. $U=\{u_j\}$3; leaflets achieve Skeleton C.D. $U=\{u_j\}$4, H.D. $U=\{u_j\}$5, Surf-recon C.D. $U=\{u_j\}$6, H.D. $U=\{u_j\}$7. Typical runtime is approximately $U=\{u_j\}$8 s per shape, versus $U=\{u_j\}$9–$C$0 min, and up to $C$1 h in difficult cases, for SlicerSALT’s s-rep optimization (Khargonkar et al., 2023).

The significance of this formulation is that the skeleton is not regressed as an unconstrained latent code. By forcing convex-combination geometry, medial contact, and interior coverage, the model produces a smooth, sheet-like distribution of skeletal points that closely matches classical s-reps while reducing computation time by two orders of magnitude.

4. Sparse graph skeletons for scalable clustering

In active clustering, a data skeleton is an explicitly sparse graph that concentrates uncertainty into a small number of critical edges. Data Skeleton Learning defines

$C$2

as a sparse, weighted directed graph over the dataset $C$3, with edge weights $C$4 and at most $C$5 edges (Xie et al., 10 Sep 2025). DSInit repeatedly links each current representative to its nearest neighbor, detects reciprocal nearest-neighbor pairs, and retains the higher in-degree node as the next representative, yielding a directed tree of depth $C$6 with exactly $C$7 edges. Recons then couples this skeleton with a second sparse constraint graph $C$8, repeatedly querying the heaviest skeleton edge, applying shortest-path deduction for must-link and cannot-link inference, and either confirming, severing, or reattaching the edge. Current labels are given by the connected components of the zero-weight subgraph. The reported properties are $C$9 initialization, $w:P(C)\to[0,\infty)$0 query complexity, $w:P(C)\to[0,\infty)$1 memory, and response-time scaling approximately $w:P(C)\to[0,\infty)$2 with $w:P(C)\to[0,\infty)$3. On 17 of 18 UCI datasets, DSL reaches ARI $w:P(C)\to[0,\infty)$4 with fewer constraints than any competitor, and its mean AUIC@n is approximately $w:P(C)\to[0,\infty)$5 versus approximately $w:P(C)\to[0,\infty)$6 for the next best baseline (Xie et al., 10 Sep 2025).

GBSK and AGBSK use a different abstraction: a statistical skeleton formed by a forest of granular-balls rather than a nearest-neighbor tree (Chen et al., 28 Sep 2025). Each node is a quintuple

$w:P(C)\to[0,\infty)$7

where $w:P(C)\to[0,\infty)$8 is the represented subset, $w:P(C)\to[0,\infty)$9 the center, $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$0 the radius, $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$1 the density, and $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$2 a smoothing term. Parent links are defined by the nearest higher-density ball,

$$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$3

with roots at local density peaks. Multi-sampling draws $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$4 random subsets of size $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$5, builds multi-grained granular-balls through a two-means splitting rule governed by $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$6 versus $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$7, aggregates representative centers, and constructs a final key-ball forest. AGBSK fixes $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$8, $$Q(\alpha)=\{\sigma\subseteq C\mid w(\sigma)\le\alpha\},$$9, and $G_s=(V,E_s)$00, leaving only $G_s=(V,E_s)$01 as user input. The analysis gives $G_s=(V,E_s)$02 and $G_s=(V,E_s)$03, and experiments include 3M2D5 with ACC $G_s=(V,E_s)$04 in $G_s=(V,E_s)$05 s, MNIST8M with ACC approximately $G_s=(V,E_s)$06 in approximately $G_s=(V,E_s)$07 s, and AGC100M with 100 million instances across 256 dimensions, where GBSK attains ACC approximately $G_s=(V,E_s)$08 in approximately $G_s=(V,E_s)$09 s and AGBSK ACC approximately $G_s=(V,E_s)$10 in approximately $G_s=(V,E_s)$11 s (Chen et al., 28 Sep 2025).

Method	Skeleton object	Reported properties
DSL	Sparse weighted directed graph $G_s=(V,E_s)$12 with exactly $G_s=(V,E_s)$13 edges	$G_s=(V,E_s)$14 initialization, $G_s=(V,E_s)$15 memory, ARI $G_s=(V,E_s)$16 on 17/18 datasets with fewer constraints
GBSK / AGBSK	Graph-forest of key balls $G_s=(V,E_s)$17	$G_s=(V,E_s)$18 time and space, demonstrated on datasets up to 100 million instances

These two lines of work use different primitives, but both treat the skeleton as an operational sparsifier. Instead of approximating all pairwise structure, they keep only the relations that most strongly determine cluster formation.

5. Human skeleton sequences as learnable structured data

For human-motion applications, the skeleton is typically a time-indexed collection of 3D joint coordinates. In depression detection from gait, each frame is represented by a 75-dimensional vector from 25 Kinect V2 joints,

$G_s=(V,E_s)$19

followed by coordinate alignment, root-centering at SpineBase, temporal Gaussian smoothing with kernel $G_s=(V,E_s)$20, and joint reduction to 17 effective joints for several augmentations (Yang et al., 2022). The paper studies five augmentation operators: rotation, shear, Gaussian-noise injection, joint masking, and channel masking. Mutual information with raw data is used to distinguish “non-noise” augmentations from “noise” augmentations. Rotation has $G_s=(V,E_s)$21 bits and channel mask $G_s=(V,E_s)$22 bits, compared with $G_s=(V,E_s)$23 for Gaussian noise, $G_s=(V,E_s)$24 for shear, and $G_s=(V,E_s)$25 for joint mask. The corresponding depression-detection results peak at $G_s=(V,E_s)$26 for rotation and $G_s=(V,E_s)$27 for channel mask, while low-MI augmentations have unstable or smaller gains (Yang et al., 2022).

Low-data action recognition emphasizes cross-sample structure rather than augmentation alone. SkeletonX targets one-shot and limited-scale regimes by constructing two pair types for each sample $G_s=(V,E_s)$28: DASP, which keeps the same performer and changes the action, and SADP, which keeps the action and changes the performer (Zhang et al., 16 Apr 2025). A GCN backbone outputs a spatio-temporal tensor $G_s=(V,E_s)$29, which is disentangled into spatial and temporal features, aggregated across a sample pair, and optimized only with cross-entropy terms: $G_s=(V,E_s)$30 The module adds at most approximately $G_s=(V,E_s)$31 M parameters to a $G_s=(V,E_s)$32–$G_s=(V,E_s)$33 M backbone. On NTU in the 10-samples-per-class setting, CTR-GCN improves from $G_s=(V,E_s)$34 to $G_s=(V,E_s)$35. In one-shot NTU-120, SkeletonX with ProtoNet reaches $G_s=(V,E_s)$36 with $G_s=(V,E_s)$37 M parameters and $G_s=(V,E_s)$38 GFLOPs, compared with $G_s=(V,E_s)$39 for “M-scale” at $G_s=(V,E_s)$40 M parameters and $G_s=(V,E_s)$41 GFLOPs (Zhang et al., 16 Apr 2025).

Unsupervised person re-identification introduces another representational layer: structured graphs over body parts. SPC-MGR constructs three graph levels per frame—part-level with $G_s=(V,E_s)$42 nodes, body-level with $G_s=(V,E_s)$43, and hyper-body-level with $G_s=(V,E_s)$44—then applies a multi-head structural relation layer to capture physically connected relations and a full-level collaborative relation layer to infer same-level and cross-level collaboration (Rao et al., 2022). Sequence features are clustered by DBSCAN into prototypes, and a contrastive loss pulls instances toward their cluster prototype and away from others. The model reports $G_s=(V,E_s)$45 top-1 on KS20 versus $G_s=(V,E_s)$46 for SM-SGE and $G_s=(V,E_s)$47 for PoseGait, while using $G_s=(V,E_s)$48 M parameters and $G_s=(V,E_s)$49 GFLOPs (Rao et al., 2022).

A still more radical reformulation is Skeleton-to-Image Encoding. S2I partitions joints into five semantic parts—Spine, Left Arm, Right Arm, Left Leg, Right Leg—orders them along each kinematic chain, normalizes channels by training-set mean and standard deviation, and resizes the resulting $G_s=(V,E_s)$50 grid into a pseudo-RGB image $G_s=(V,E_s)$51, typically $G_s=(V,E_s)$52 (Yang et al., 6 Mar 2026). This permits direct use of ImageNet-pretrained ViT-B/16 models with MAE or DiffMAE objectives. On NTU-60 linear evaluation, S2I with DiffMAE attains $G_s=(V,E_s)$53 / $G_s=(V,E_s)$54 on C-Sub / C-View, and 3-stream fusion reaches $G_s=(V,E_s)$55 / $G_s=(V,E_s)$56. In full fine-tuning, 3s-S2I reaches $G_s=(V,E_s)$57 / $G_s=(V,E_s)$58 on NTU-60 and $G_s=(V,E_s)$59 / $G_s=(V,E_s)$60 on NTU-120. Cross-format transfer from NTU-60 to Toyota CV1 improves from $G_s=(V,E_s)$61 to $G_s=(V,E_s)$62 under 3-stream fusion, without manual down-sampling or zero-padding (Yang et al., 6 Mar 2026).

Taken together, these studies show that human skeleton data can be modeled as temporally smoothed coordinate series, paired samples, multi-level graphs, or image-like tensors. The choice of representation determines which inductive biases are retained: viewpoint invariance, performer/action disentanglement, part relations, or compatibility with large-scale vision pretraining.

6. Generative augmentation and adversarial robustness

Augmentation of skeleton data increasingly shifts from handcrafted transforms to learned generative models. Attack-AAIRS studies person identification from LiDAR-based skeleton videos, where each sample forms a tensor of size $G_s=(V,E_s)$63, with preprocessing that centers the root joint at the origin, rotates the subject to face the LiDAR sensor, and normalizes coordinates into $G_s=(V,E_s)$64 (Zalameda et al., 25 Mar 2026). The baseline recognizer is HCN-ID, which fuses a raw stream and a temporal-difference stream $G_s=(V,E_s)$65. AAIRS augments training through an AC-GAN; Attack-AAIRS extends this by freezing a pretrained generator $G_s=(V,E_s)$66 and HCN-ID, cloning $G_s=(V,E_s)$67 into an attack generator $G_s=(V,E_s)$68, and optimizing

$G_s=(V,E_s)$69

This learns a distribution of synthetic attack samples rather than perturbing only the limited real training set. Over 10-fold cross-validation on 9 identities with 3-frame gait segments and $G_s=(V,E_s)$70 joints, HCN-ID trained on real data alone yields $G_s=(V,E_s)$71, AAIRS with 50% real / 50% synthetic yields $G_s=(V,E_s)$72, and Attack-AAIRS inoculation yields $G_s=(V,E_s)$73 for the real-only model and $G_s=(V,E_s)$74 for the AAIRS model. Average robustness gains after inoculation include $G_s=(V,E_s)$75 and $G_s=(V,E_s)$76 for FGSM, $G_s=(V,E_s)$77 and $G_s=(V,E_s)$78 for PGD, and $G_s=(V,E_s)$79 and $G_s=(V,E_s)$80 for BIM on the two model settings. The HCN-ID Score is $G_s=(V,E_s)$81 for real data, $G_s=(V,E_s)$82 for AAIRS synthetic data, and $G_s=(V,E_s)$83 for Attack-AAIRS adversarial synthetic data (Zalameda et al., 25 Mar 2026).

For action recognition, “Generative Data Augmentation for Skeleton Action Recognition” proposes a conditional diffusion model over sequences $G_s=(V,E_s)$84, with per-frame features expanded to $G_s=(V,E_s)$85 (Dong et al., 16 Apr 2026). A Transformer-based encoder–decoder receives skeleton embeddings, sinusoidal temporal encodings, a label embedding, and a timestep embedding; a Generative Refinement Module filters low-fidelity samples by an $G_s=(V,E_s)$86 threshold, and dropout remains active at sampling time to balance fidelity and diversity. The training loss is

$G_s=(V,E_s)$87

optimized with Adam for 600 epochs at initial learning rate $G_s=(V,E_s)$88, batch size $G_s=(V,E_s)$89, on an NVIDIA RTX 3090. In few-shot and full-data settings, augmenting each real sample with 5 synthetic motions improves multiple backbones. On HumanAct12 with STGCN++, $G_s=(V,E_s)$90 real data improves from $G_s=(V,E_s)$91 to $G_s=(V,E_s)$92, and $G_s=(V,E_s)$93 real data improves from $G_s=(V,E_s)$94 to $G_s=(V,E_s)$95. On NTU-VIBE with STGCN++, $G_s=(V,E_s)$96 real data improves from $G_s=(V,E_s)$97 to $G_s=(V,E_s)$98. Reported generation-quality metrics on HumanAct12 are FID $G_s=(V,E_s)$99, Diversity $n-1$00, and Precision/Recall $n-1$01, compared with MDM at $n-1$02 FID and T2M-GPT at $n-1$03 FID (Dong et al., 16 Apr 2026).

These results indicate two complementary functions of generative skeleton models. One is defensive: synthesize attacks that expose failure modes and inoculate the classifier. The other is data-expansive: sample class-consistent, diverse motions that improve downstream generalization under limited supervision.

7. Benchmarks, biomechanics, and anatomically grounded skeletons

Reliable evaluation requires datasets with exact geometric and kinematic ground truth. The “Synthetic Human Model Dataset for Skeleton Driven Non-rigid Motion Tracking and 3D Reconstruction” provides seven motion sequences of a single human model, each with per-frame ground-truth mesh geometry, ground-truth skeleton, and skinning weights (Elanattil et al., 2019). The rig has 15 joints and 14 bones organized in a pelvis-rooted hierarchy, with per-frame joint positions $n-1$04 stored in camera coordinates. Pose is represented by homogeneous transforms $n-1$05, and mesh deformation is generated by linear-blend skinning over $n-1$06 vertices and 14 bone weights summing to 1. The dataset contains $n-1$07 faces and RGB-D resolution $n-1$08 with intrinsics $n-1$09. Suggested evaluation metrics include MPJPE,

$n-1$10

surface reconstruction error, and joint-angle error (Elanattil et al., 2019).

“From Skin to Skeleton: Towards Biomechanically Accurate 3D Digital Humans” pushes the notion of data skeleton toward anatomical realism by re-rigging SMPL with a biomechanics skeleton to create SKEL (Keller et al., 8 Sep 2025). The authors construct BioAMASS from over 9 hours of AMASS sequences spanning 113 subjects, 2,198 motion clips, and approximately 935,000 frames. Each frame provides SMPL shape $n-1$11, pose $n-1$12, and vertices $n-1$13. An OpenSim-based biomechanical skeleton model with $n-1$14 rigid bone groups and $n-1$15 anatomical degrees of freedom is then optimized inside SMPL meshes using synthetic markers and a scale prior, yielding joint locations $n-1$16 and bone rotations $n-1$17. Joint locations are regressed from SMPL vertices via non-negative least squares, and per-bone base rotations are learned by minimizing a Frobenius-norm rotation discrepancy. SKEL is then posed with biomechanically realistic kinematics while remaining animatable like SMPL, and can “upgrade” existing SMPL-based datasets by fitting SKEL pose $n-1$18 to a mesh generated from $n-1$19 (Keller et al., 8 Sep 2025).

The biomechanical claim is precise: SKEL has more biomechanically accurate joint locations than SMPL, and the bones fit inside the body surface better than previous methods. This marks an important boundary condition for the broader literature. Some skeletons are sparse graphs or latent abstractions; others are intended to coincide with real joint locations, joint limits, and bone placements. The distinction matters whenever the downstream task is biomechanics, rehabilitation, clinical modeling, or anatomically constrained animation rather than purely discriminative recognition.