Topology Understanding Net (TUN)

• Topology Understanding Net (TUN) is a multi-modal deep learning framework that distinguishes genuine topological signals from noise in one-dimensional persistence diagrams.
• It employs enhanced 4D descriptors, self-attention encoding, and PointNet-style point cloud processing to effectively integrate global geometric context.
• Empirical evaluations across diverse datasets show near-perfect F1 scores, underscoring TUN's precision and robustness in automated topological analysis.

Topology Understanding Net (TUN) refers to two distinct architectures in the literature: one for significance detection in persistence diagrams for topological data analysis (Chen et al., 16 Dec 2025), and another for unified lane topology relationship reasoning in autonomous driving (Li et al., 23 Jul 2025). Both frameworks address challenges in extracting reliable topological structure from complex geometric or visual data but are applied to fundamentally different domains. The following entry details the Topology Understanding Net as proposed by (Chen et al., 16 Dec 2025) for the identification of significant homological features in persistence diagrams.

A Topology Understanding Net (TUN) is a multi-modal deep learning framework designed to identify which points in one-dimensional persistence diagrams (1-PDs), computed from point clouds, represent true topological signal (e.g., genuine loops) versus topological noise. It addressed core limitations of classical persistence-based heuristics by fusing geometric, topological, and auxiliary statistical cues. TUN achieves state-of-the-art accuracy for automated PD significance annotation, facilitating robust downstream decision-making in applied topological data analysis.

1. Motivation and Problem Definition

TUN is motivated by the inability of traditional persistence-based thresholds to reliably distinguish signal from noise in persistence diagrams, particularly when global PD structure and underlying geometry are nontrivial. Given a 1-PD $P = \{(b_i, d_i)\}_{i=1}^N$, typically computed using an alpha filtration on a point cloud, the task is to predict, for each point, the probability that it encodes a "true" one-dimensional homology generator (i.e., a persistent 1-cycle) of the original shape. Classic methods rely on heuristics such as ordering by persistence ($d_i-b_i$) or thresholding the point's distance from the diagonal, but these approaches are insufficient in settings with non-homogeneous noise, variable feature scale, or complex point distributions. TUN is designed as a data-driven alternative: it learns to combine per-point PD descriptors, global PD and point cloud context, and additional geometric statistics to resolve ambiguities inherent to classical topological analysis (Chen et al., 16 Dec 2025).

2. Persistence Diagrams and Enhanced Descriptors

A one-dimensional persistence diagram is the multiset $$P = \{p_i = (b_i, d_i) \in \mathbb{R}^2\mid 1 \leq i \leq N\}$$, each $p_i$ representing appearance (birth) and disappearance (death) filtration values for a homological feature. The principal scalar, persistence, is $p_i = d_i - b_i$.

TUN expands upon raw $(b_i, d_i)$ representation by mapping each diagram point to a 4-dimensional feature vector: $$v_i = [b_i, d_i, p_i=d_i-b_i, \delta_i = \log\frac{d_i}{b_i}] \in \mathbb{R}^4$$ with stabilized computation of $\delta_i = \log(d_i/\max\{b_i, \varepsilon\})$ for $\varepsilon > 0$. These descriptors, encoding both absolute and relative persistence lifetimes, provide a richer per-point feature space for learning per-point significance.

3. Architecture and Information Fusion

TUN's architecture is organized in four principal modules: (1) a persistence diagram encoder with self-attention, (2) a PointNet-style point cloud encoder, (3) a learned multi-modal fusion of global context, and (4) a per-point classifier.

3.1 Persistence Diagram Encoder

The enhanced PD descriptors $v_i$ are transformed via a shared MLP ($\mathrm{MLP}_{pd}$) to yield embeddings $u_i$. These embeddings are then contextualized with a self-attention mechanism: $Q = UW^Q,\quad K = UW^K,\quad V = UW^V$

$A = \mathrm{softmax}(QK^T / \sqrt{H})$

$\widetilde U = AV$

The output $F_i$ encompasses per-point features with global context via normalization and residual connections. Pooling over $F_i$ yields a global PD feature $g_{pd}$.

3.2 Point Cloud Encoder

Input point clouds $C = \{c_j \in \mathbb{R}^3\}_{j=1}^M$ are processed with shared $1 \times 1$ convolutions (analogous to PointNet), followed by batch normalization and ReLU, and aggregated with max-pooling to yield a global cloud feature $g_{pc} \in \mathbb{R}^H$.

3.3 Multi-Modal Fusion

The global features $g_{pd}$ and $g_{pc}$ are projected into a shared space; auxiliary global statistics $a$ (PD and PC counts, persistence statistics, geometric spread, bounding-box extents, noise/uniformity estimators) are also projected and concatenated. The entire fused vector $z$ is passed through an MLP to produce the unified global context $f$.

3.4 Per-Point Significance Classification

For each diagram point, its contextualized representation $F_i$ is concatenated with the global feature $f$, then processed by a point-wise MLP. The resulting logits $s_i$ yield significance scores $p_i = \mathrm{softmax}(s_i)$, where $p_i[1]$ reflects the probability of genuine topological signal.

4. Stable Preprocessing and Auxiliary Features

TUN employs strict preprocessing for numerical stability and batch uniformity. PDs are truncated or zero-padded to $N_{pd}=100$ points; point clouds are subsampled or padded to $N_{pc}=50,000$ points. The $\delta_i$ ratio is numerically stabilized using small $\varepsilon \approx 10^{-8}$. Auxiliary global features encompass: PD statistics (count, mean/std/max persistence, mean birth), point cloud statistics (counts, spread, mean norm), bounding box dimensions, and nonuniformity (KNN distance variance, PCA ratios), yielding $a \in \mathbb{R}^{14}$.

5. Imbalance-Aware Training

Significant diagram points are rare. TUN employs a weighted focal loss per point: $$\ell_i = w_{y_i} \alpha (1 - p_{t, i})^\gamma (-\log p_{t, i})$$ with $y_i \in \{0,1\}$ as the ground-truth label for significance, $w_1 > w_0$ to upweight positives, $\alpha=1.0$, $\gamma=2.0$, and masking $M_i$ for valid points. The aggregated loss is

$$\mathcal{L} = \frac{1}{\sum_i M_i}\sum_i M_i \ell_i$$

This class-imbalance focus is critical to maintain precision and recall on minority significant points.

6. Empirical Results and Ablations

TUN was evaluated on 1,500 test samples from four domains: planar shapes, CAD models, triply periodic minimal surfaces (TPMS), and zeolites, each with up to 100 hand-labeled PD points. Against 2-means clustering on persistence and Confidence Set thresholding, TUN demonstrated near-perfect performance:

Dataset	TUN F1	2-means F1	CS (Recall)
Planar	1.0000	0.45–0.65	0
CAD	1.0000	0.45–0.65	0
Zeolite	1.0000	0.45–0.65	0
TPMS	0.9997	0.45–0.65	0

Ablation studies highlighted that the PD encoder alone achieves high recall but allows more false positives; integrating the point cloud encoder and auxiliary features restores precision to 1.0 and boosts recall. Removal of any auxiliary block modestly degrades recall while preserving precision.

7. Practical Guidelines and Hyperparameters

Default hyperparameters include: embedding size $H=256$, fusion dimension $F=256$, 8 attention heads, dropout rate 0.3–0.4, batch size 16, AdamW optimizer, cosine-annealing LR, gradient clipping at norm 1.0, and early stopping after 10 validation epochs. PD and PC caps are respected as in preprocessing. Higher point cloud sampling density may be required for detecting small loops but increases memory cost (GPU: $\sim$24 GB VRAM per batch; per epoch training time: minutes).

In summary, TUN integrates enhanced descriptor learning, attention-based global context, geometric encoding, auxiliary statistics, and focal loss into a unified pipeline, yielding robust and automated significance detection in persistence diagrams—substantially surpassing baseline and classical methods in empirical studies (Chen et al., 16 Dec 2025).