Prediction-Driven Top-K Jaccard Coefficient

• The paper introduces a dynamic refinement of Jaccard similarity by predicting optimal Top-K neighbors using neural models and a Sparse Differential Transformer.
• It employs a supervised prediction task to adaptively determine neighborhood sizes, which significantly enhances robustness and discriminative power in clustering.
• Empirical results on large-scale datasets demonstrate improved face clustering performance and generalization across multiple domains.

The prediction-driven Top-K Jaccard similarity coefficient is a dynamic, data-adaptive refinement of the traditional Jaccard approach for measuring pairwise relationships in face clustering graphs. Central to this methodology is the replacement of a static, globally fixed neighbor count with an individually predicted optimal Top-K for each node, determined through a supervised prediction task powered by neural models and further stabilized via a Sparse Differential Transformer (SDT). The resulting framework achieves increased discriminative power, improved robustness to noise, and state-of-the-art clustering performance on multiple large-scale datasets (Zhang et al., 27 Dec 2025).

1. Mathematical Foundations of the Top-K Jaccard Similarity

The classical Jaccard similarity for two sets $A$ and $B$ is given by:

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In the context of face clustering, the sets $N_i$ and $N_j$ typically correspond to the K nearest neighbors of nodes $i$ and $j$ in the embedding space, measured, for instance, by cosine similarity:

$$J(N_i, N_j) = \frac{|N_i \cap N_j|}{|N_i \cup N_j|}.$$

In the prediction-driven Top-K extension, the neighbor count $K$ for each node is not fixed but predicted. If $\mathrm{Top}\text{-}K_i$ is the predicted number for node $i$ (rounded to $\hat{k}_i$), then

$$N_i^{\hat k_i} = \{\text{the first } \hat k_i \text{ neighbors of } i\}$$

$$N_j^{\hat k_i} = \{\text{the first } \hat k_i \text{ neighbors of } j\}$$

with the intersection $$\mathcal M_{ij}^{\hat k_i} = N_i^{\hat k_i} \cap N_j^{\hat k_i}$$.

The prediction-driven Top-K Jaccard edge probability is then:

$$\widetilde p_{ij} = \frac{\sum_{h\in \mathcal M_{ij}^{\hat k_i}} (\hat p_{ih} + \hat p_{hj})}{\sum_{h\in N_i^{\hat k_i}} \hat p_{ih} + \sum_{h\in N_j^{\hat k_i}} \hat p_{hj}}$$

where $\hat p_{uv}$ is a normalized pairwise similarity, obtained via distance-to-probability transformation:

$$a_{ij} = \frac{f_i \cdot f_j}{\|f_i\| \|f_j\|}, \quad d_{ij} = 2 - 2a_{ij}$$

$$p_{ij} = \frac{1}{1 + \exp(\delta d_{ij} + \epsilon)} \quad (\delta = 7.5, \epsilon = -5)$$

$$\hat p_{ij} = \frac{p_{ij}}{\sum_{k\in N_i} p_{ik}}$$

This approach increases the reliability of similarity measurements by focusing on a purified, node-specific neighborhood (Zhang et al., 27 Dec 2025).

2. Data-Driven Prediction of Optimal Top-K

Rather than applying a fixed neighbor threshold, the optimal neighborhood size for each node is formalized as a supervised prediction problem. For node $i$, the model considers the top-$K$ candidate neighbors (ranked by cosine similarity) and predicts a score $s_{i \to j} \in [0,1]$ for each candidate $j$, approximating the likelihood that $j$ shares the same identity as $i$.

During inference, the predicted Top-K is obtained by thresholding:

$$\mathrm{Top}\text{-}K_i = \max \{j \mid s_{i\to j} \geq \eta\}$$

where $\eta$ is typically set to 0.90.

The input to the predictor consists of the embedding sequence $X_i = [f_i; f_{i,1}; f_{i,2}; \cdots; f_{i,K}]$ and corresponding positional or relative distance encodings. These are projected into queries, keys, and values for Transformer-based processing.

Training employs a binary cross-entropy loss:

$$\mathcal{L} = -\frac{1}{N}\sum_{i}\sum_{j=1}^K \left[ y_{ij}\log s_{i \to j} + (1 - y_{ij})\log(1 - s_{i \to j}) \right]$$

with $y_{ij}$ denoting the ground-truth same-identity indicator. Training samples primarily target a window around the true decision boundary to maintain class balance (Zhang et al., 27 Dec 2025).

3. Transformer-Based Predictors and the Sparse Differential Transformer (SDT)

The initial prediction framework employs a vanilla Transformer architecture (three layers, eight self-attention heads, hidden size 1024) with standard self-attention. However, the vanilla Transformer tends to overvalue irrelevant relationships, introducing noise into the neighborhood prediction.

Differential Self-Attention

The attention computation is refined by splitting queries and keys, and applying a subtractive update:

$$F_{\rm DIFF} = \left[ S(Q_1 K_1^\top/\sqrt{d}) - \lambda S(Q_2 K_2^\top/\sqrt{d}) \right] V$$

where $S(\cdot)$ denotes softmax and $\lambda$ is learnable.

Sparse Differential Transformer (SDT)

Sparsity is imposed by masking all but the top-K entries per row:

$$F_{\rm SDT} = \left[ S(M(Q_1 K_1^\top/\sqrt{d})) - \lambda S(M(Q_2 K_2^\top/\sqrt{d})) \right] V$$

with $M(\cdot)$ denoting the Top-K mask. This structure masks weak key–query pairs, focusing the model on the most informative local relationships and improving resilience to noise.

Mixture-of-Experts SDT

MoE-SDT introduces a mixture over masks $M_{K-u}, M_K, M_{K+u}$ (with $u = 5$) and combines their outputs using learnable weights $(\alpha, \beta, \gamma)$:

$$\begin{aligned} F_{\rm MoE\text{-}SDT} =\;& \alpha [ S(M_{K-u}(Q_1 K_1^\top)) - \lambda S(M_{K-u}(Q_2 K_2^\top)) ] V \ +& \beta [ S(M_K(Q_1 K_1^\top)) - \lambda S(M_K(Q_2 K_2^\top)) ] V \ +& \gamma [ S(M_{K+u}(Q_1 K_1^\top)) - \lambda S(M_{K+u}(Q_2 K_2^\top)) ] V \end{aligned}$$

This construct further mitigates prediction errors near the Top-K boundary (Zhang et al., 27 Dec 2025).

4. Integration Into Clustering Workflow

The integration of the prediction-driven Top-K Jaccard coefficient into face clustering follows these algorithmic steps:

1. Extract face embeddings $\{f_i\}$.
2. For each $i$, determine top-$K$ candidates via cosine similarity.
3. Process $[f_i; f_{i,1}; \ldots; f_{i,K}]$ using the SDT predictor, yielding scores $s_{i\to j}$.
4. Compute $$\mathrm{Top}\text{-}K_i = \max\{j \mid s_{i\to j} \geq \eta\}$$.
5. Round to $\hat{k}_i$ (e.g., nearest multiple of 10 up to $K$).
6. Build refined neighborhoods $N_i^{\hat k_i}$ and compute $\widetilde p_{ij}$.
7. Threshold $\widetilde p_{ij}$ to construct a sparse adjacency matrix.
8. Apply the Map-Equation codec for community detection.

Pseudocode for the procedure is:

for i in 1…N: neighbors = topK_by_cosine(f_i, all_f, K) scores = SDT_predictor([f_i; neighbors]) TopK = max{ j | scores[j] ≥ η } ĥ = round_to_10(TopK) if TopK<K else K refined = neighbors[:ĥ] for j in refined: compute p̂_ij via (9),(7) and add to numerator/denominator of (11) build edges where ṗ_ij>τ_edge cluster = MapEquation(adjacency)

(Zhang et al., 27 Dec 2025)

5. Empirical Results and Robustness

Evaluation on several benchmarks demonstrates the efficacy of the prediction-driven Top-K Jaccard approach.

MS-Celeb-1M Clustering:

Method	$F_P$ (584K)	$F_B$ (584K)	$F_P$ (5.21M)	$F_B$ (5.21M)
FC-ESER	95.28	93.85	89.40	88.80
Diff-Cluster	95.46	94.14	90.08	89.14

Sigmoid distance-transform yields +0.18% $F_P$, +0.16% $F_B$ versus exponential; adding Top-K filtering yields +0.35% $F_P$, +0.46% $F_B$.

Transformer Variant Ablations:

Architecture	$F_P$	$F_B$
Vanilla Transformer	94.25	92.73
Vanilla+Top-K Mask	94.78	93.25
Differential (no mask)	95.05	93.59
Differential+Top-K (SDT)	95.34	93.93
MoE-SDT	95.46	94.14

Noisy similarity matrix experiments simulate up to 40% noise: SDT maintains pairwise F-score, while vanilla Transformer degrades significantly.

Generalization evaluations confirm improvements across MSMT17 (person re-ID) and DeepFashion, as well as gains from substituting SDT into contemporary clustering pipelines (Zhang et al., 27 Dec 2025).

6. Impact and Significance

The prediction-driven Top-K Jaccard similarity coefficient constitutes an advancement in clustering methodology by dynamically adapting local connectivity based on learned data relationships rather than static heuristics. By defining neighborhood boundaries as a supervised prediction problem and employing SDT for robust selection, the approach produces neighborhood sets with enhanced purity, resulting in more reliable graph-based similarity estimation.

Empirical evidence establishes increases in overall clustering accuracy, robustness to random noise in similarity calculations, and improved performance generalization into domains beyond face clustering. A plausible implication is that prediction-driven Top-K filtering can be fruitfully adapted to related graph construction problems in metric learning and instance-level retrieval beyond the specific setting investigated.

In summary, this method advances the state of the art in graph-based face clustering with potential for transferability across adjacent domains (Zhang et al., 27 Dec 2025).