Hyperbolic Structured Classification for Robust Single Positive Multi-label Learning

TL;DR

This work tackles SPMLL by replacing traditional point-based label representations with hyperbolic balls in the Poincaré model, enabling natural encoding of hierarchical, co-occurrence, and independent relationships without explicit graphs. It introduces a temperature-adaptive hyperbolic ball classifier and a physics-inspired double-well regularizer, jointly optimizing CLIP-derived features projected into hyperbolic space to discover structured label correlations under incomplete supervision. Empirical results on MS-COCO, PASCAL VOC, NUS-WIDE, and CUB-200-2011 show competitive mAP and improved interpretability, with learned embeddings correlating strongly with real-world co-occurrence patterns. The approach demonstrates that hyperbolic geometry provides a robust and transparent framework for structured classification in weakly supervised settings, offering practical benefits for multi-label recognition tasks with incomplete annotations.

Abstract

Single Positive Multi-Label Learning (SPMLL) addresses the challenging scenario where each training sample is annotated with only one positive label despite potentially belonging to multiple categories, making it difficult to capture complex label relationships and hierarchical structures. While existing methods implicitly model label relationships through distance-based similarity, lacking explicit geometric definitions for different relationship types. To address these limitations, we propose the first hyperbolic classification framework for SPMLL that represents each label as a hyperbolic ball rather than a point or vector, enabling rich inter-label relationship modeling through geometric ball interactions. Our ball-based approach naturally captures multiple relationship types simultaneously: inclusion for hierarchical structures, overlap for co-occurrence patterns, and separation for semantic independence. Further, we introduce two key component innovations: a temperature-adaptive hyperbolic ball classifier and a physics-inspired double-well regularization that guides balls toward meaningful configurations. To validate our approach, extensive experiments on four benchmark datasets (MS-COCO, PASCAL VOC, NUS-WIDE, CUB-200-2011) demonstrate competitive performance with superior interpretability compared to existing methods. Furthermore, statistical analysis reveals strong correlation between learned embeddings and real-world co-occurrence patterns, establishing hyperbolic geometry as a more robust paradigm for structured classification under incomplete supervision.

Figures (5)

• Figure 1: Label correlation encoding through hyperbolic ball interactions. Left: Hierarchical correlations with nested response patterns discovered from data. Center: Co-occurrence correlations through intersecting decision regions. Right: Statistical independence via separated geometric regions. Heat maps show response intensities, white contours indicate decision boundaries (0.5), and red stars mark label centers. These patterns emerge automatically from learning statistical correlation structures in the training data.
• Figure 2: Comprehensive temperature scaling analysis combining baseline and temperature-adaptive methods. Top row displays 2D response maps for baseline and three temperature settings ($\tau = 0.1, 1.0, 5.0$) applied to mid-position label ($\|c\| = 0.5$). Bottom row shows 1D response curves: (left) baseline response across different class positions, (right three) direct comparisons between baseline and temperature-scaled methods. This unified analysis demonstrates the consistent superiority of temperature scaling across all geometric positions in hyperbolic space.
• Figure 3: Double-well regularizer parameter analysis. (a) Platform width control via $\beta_1$ balances correlation pattern stability. (b) Boundary steepness control via $\beta_2$ enforces geometric constraints. (c) Asymmetric platform configuration adapts to correlation complexity distributions.
• Figure 4: Temperature scaling effects in hyperbolic classification. The figure shows comprehensive analysis across three class positions ($\|c\| = 0.07, 0.5, 0.8$): (left column) temperature-coordinate heatmaps, (middle column) response curves for different temperatures, (right column) 2D response maps with $\tau = 1.0$. Temperature scaling enables adaptive decision boundaries that respect hyperbolic geometry properties.
• Figure 5: Statistical validation of co-occurrence pattern learning in COCO dataset. The scatter plot shows the relationship between real-world co-occurrence probability and learned hyperbolic embedding center distances across 2,574 category pairs. Strong negative correlation (Pearson $r = -0.629$, $p < 10^{-280}$) demonstrates that our model automatically captures meaningful statistical patterns from the data. Color coding distinguishes frequency groups: red (high co-occurrence $\geq 2\%$), orange (medium $0.5$--$2\%$), and blue (low $< 0.5\%$). The systematic organization validates that our ball-based framework learns meaningful label relationships rather than arbitrary spatial arrangements.