We present EmBolic - a novel fully hyperbolic deep learning architecture for fine-grained emotion analysis from textual messages. The underlying idea is that hyperbolic geometry efficiently captures hierarchies between both words and emotions. In our context, these hierarchical relationships arise from semantic ambiguities. EmBolic aims to infer the curvature on the continuous space of emotions, rather than treating them as a categorical set without any metric structure. In the heart of our architecture is the attention mechanism in the hyperbolic disc. The model is trained to generate queries (points in the hyperbolic disc) from textual messages, while keys (points at the boundary) emerge automatically from the generated queries. Predictions are based on the Busemann energy between queries and keys, evaluating how well a certain textual message aligns with the class directions representing emotions. Our experiments demonstrate strong generalization properties and reasonably good prediction accuracy even for small dimensions of the representation space. Overall, this study supports our claim that affective computing is one of the application domains where hyperbolic representations are particularly advantageous.
Figures & tables
Figure 1: Circumplex model of affect by Russell [11] . Permissions obtained.
Figure 2: Visualization of the Poincaré disc. Two parallel geodesic lines (the shortest paths connecting points) in the left panel and visualization of distances in the right panel. The distance between points O and A is equal to the distance between B and C.
Figure 3: Embolic architecture.
Figure 4: Hyperbolic GloVe word embeddings in the Poincaré disc.
Figure 5: Randomly selected words with their representations in the Poincaré disc. This illustrates emotion-related semantic ambiguity of certain words.
Figure 6: Learned representations of textual messages for Desire, Love, and Nervousness (top row) and the corresponding representations after isometric (Möbius) correction (bottom row).
Figure 7: Visualization of the validation results. Representations of test messages for Desire, Love and Nervousness and the radii encoding these emotions. Correct predictions correspond to the points which are placed on the radii. The confidence levels are encoded by the moduli (distances from the center).
Simulation
Total samples
Top-1
Top-3
Top-5
Simulation 1
140
0.150
0.286
0.414
Simulation 2
140
0.143
0.343
0.457
Simulation 3
140
0.186
0.307
0.429
Table 1: Accuracy results for individual embeddings
Total samples
Top-1
Top-3
Top-5
140
0.264
0.464
0.579
Table 2: Cumulative accuracy results
Category
Count
Percentage (%)
Secure correct (Top-1 True, ≥ 20%)
17
12.14
Secure wrong (Top-1 False, ≥ 20%)
15
10.71
Insecure ( < 20%)
108
77.14
Total
140
100.00
Table 3: Confidence analysis of model predictions with a certainty threshold of 20%
Message
True label
Top prediction
Score (%)
True rank
I only use freedom units, sorry.
Approval
Remorse
75.41
–
I felt the same guilt when I stopped believing in [NAME].
Remorse
Caring
75.60
–
Ahhhh fond memories.
Joy
Sadness
53.94
Top-5 (4.83)
Yes, I’ve noticed it too. Or the photo of the item won’t load. This company is so shameful.
Embarrassment
Confusion
48.63
Top-2 (42.54)
Thanks for the laugh lol.
Gratitude
Amusement
33.13
Top-2 (13.04)
Table 4: High-confidence mispredictions of the emotion classification model
Message
True label
Top prediction
Score (%)
True rank
What a beautiful message—it’s all about love, and I hope it sinks in to some of the COB people who might see it.
Optimism
Admiration
21.05
Top-5 (4.52)
Sorry my teams gave you your only two losses this season ;)
Remorse
Embarrassment
25.85
Top-2 (22.78)
Here for [NAME], love this.
Love
Excitement
36.10
Top-2 (34.47)
Table 5: Examples where the true label appears in the top-5 predictions
Message
True label
Top prediction
Score (%)
True rank
[NAME] will be gone. Although I think end of the first is where he should go, he will go super early.
Optimism
Admiration
4.69
–
This is so scary. Keep safe! I would put fencing up and become a licensed carrier at that point. Sounds like crazy druggies!
Nervousness
Nervousness
5.99
Top-1 (5.99)
Table 6: Examples of uncertain model predictions
Figure 8: Emotion correlation matrix learned by EmBolic. Each cell (i,j) show how frequently emotion i appears together with j in top 5. Darker cells indicate stronger pairwise correlations.
Figure 9: Map of emotions in the Poincaré disc learned from the co-occurrence matrix of emotions depicted in Figure 8 .
Appendix figures & tables3 assets
Supplementary material from the paper’s appendix.
Appendix
Figure B.1: Continuation of Figure 4 Hyperbolic GloVe word embeddings in the Poincaré disc for two additional simulations.
Figure B.2: Learned representations of textual messages for all 28 emotion categories. Each panel shows representations of messages annotated by the same emotion. The Möbius correction is further applied to these representations, in order to align fingers with radii/angles.
Figure B.3: Validation on the test set. Representations of test messages for 28 emotions and the radii encoding these emotions. Correct predictions correspond to the points lying on the radii. Representations shown for one embedding only. The final prediction is made based on scores obtained for three embeddings.
Emotion recognition uses language, voice, face, body movement, and physiological signals to infer emotional categories or dimensions such as valence and arousal.
Emotional expression underpins natural communication and effective human-computer interaction. We present Emotion Collider (EC-Net), a hyperbolic hypergraph framework for multimodal emotion and sentiment modeling. EC-Net represents modality hierarchies using Poincare-ball embeddings and performs fusion through a hypergraph mechanism that passes messages bidirectionally between nodes and hyperedges. To sharpen class separation, contrastive learning is formulated in hyperbolic space with decoupled radial and angular objectives. High-order semantic relations across time steps and modalities are preserved via adaptive hyperedge construction. Empirical results on standard multimodal emotion benchmarks show that EC-Net produces robust, semantically coherent representations and consistently improves accuracy, particularly when modalities are partially available or contaminated by noise. These findings indicate that explicit hierarchical geometry combined with hypergraph fusion is effective for resilient multimodal affect understanding.
Multimodal Emotion Recognition (MER) aims to understand human emotions by integrating information from multiple modalities such as speech and text. Psychological studies suggest that emotions exhibit latent hierarchical structured relationships in their semantic representations. However, most MER approaches rely on Euclidean representations, which capture fine-grained cross-modal interactions but do not explicitly provide a geometric prior for organizing hierarchical emotional structures. To mitigate this gap, we propose GeoMER, a joint Euclidean-Hyperbolic modeling framework that introduces hyperbolic emotion-core routing to capture higher-level, abstract emotional representations for MER. The proposed framework jointly models token-level cross-modal semantic interactions and latent emotion representations in two complementary geometric spaces. Specifically, GeoMER consists of a Euclidean Multimodal Fusion Module (EuclFusion) for multimodal representation fusion and a Hyperbolic Geometric Core Routing (HypGeoRoute) module that performs geometry-aware core assignment and aggregation in a hyperbolic manifold to capture more abstract feature representations. The HypGeoRoute module complements the Euclidean token-level modeling. We experiment on the IEMOCAP multimodal MER benchmark and demonstrate that the proposed method achieves the best performance, with 83.54% weighted accuracy and 83.35% unweighted accuracy, outperforming Euclidean baselines, including MemoCMT.
Ting Zhu, Huizhi Liang, Boguslaw Obara +2
Newcastle University, Newcastle upon Tyne, United Kingdom · Taiyuan University of Technology, Shanxi, China
Hyperbolic spaces provide a natural geometry for representing hierarchical and tree-structured data due to their exponential volume growth. To leverage these benefits, neural networks require intrinsic and efficient components that operate directly in hyperbolic space. In this work, we lift two core components of neural networks, Multinomial Logistic Regression (MLR) and Fully Connected (FC) layers, into hyperbolic space via Busemann functions, resulting in Busemann MLR (BMLR) and Busemann FC (BFC) layers with a unified mathematical interpretation. BMLR provides compact parameters, a point-to-horosphere distance interpretation, batch-efficient computation, and a Euclidean limit, while BFC generalizes FC and activation layers with comparable complexity. Experiments on image classification, genome sequence learning, node classification, and link prediction demonstrate improvements in effectiveness and efficiency over prior hyperbolic layers. The code is available at https://github.com/GitZH-Chen/HBNN.