PHyCLIP: $\ell_1$-Product of Hyperbolic Factors Unifies Hierarchy and Compositionality in Vision-Language Representation Learning
Authors: Daiki Yoshikawa
Takashi Matsubara
Venue: ICLR 2026 Poster (2026)
Keywords: Vision-language representation learning
compositionality
Boolean algebra
hyperbolic embedding
Primary area: unsupervised, self-supervised, semi-supervised, and supervised representation learning
TL;DR: For multi-modal representation learning, we propose to use an l1-product metric space of hyperbolic spaces, which simultaneously captures hierarchy and compositionality.

Abstract: Vision-language models have achieved remarkable success in multi-modal representation learning from large-scale pairs of visual scenes and linguistic descriptions. However, they still struggle to simultaneously express two distinct types of semantic structures: the hierarchy within a concept family (e.g., *dog* $\preceq$ *mammal* $\preceq$ *animal*) and the compositionality across different concept families (e.g., "a dog in a car" $\preceq$ *dog*, *car*). Recent works have addressed this challenge by employing hyperbolic space, which efficiently captures tree-like hierarchy, yet its suitability for representing compositionality remains unclear. To resolve this dilemma, we propose *PHyCLIP*, which employs an $\ell_1$-*P*roduct metric on a Cartesian product of *Hy*perbolic factors. With our design, intra-family hierarchies emerge within individual hyperbolic factors, and cross-family composition is captured by the $\ell_1$-product metric, analogous to a Boolean algebra. Experiments on zero-shot classification, retrieval, hierarchical classification, and compositional understanding tasks demonstrate that PHyCLIP outperforms existing single-space approaches and offers more interpretable structures in the embedding space.

Published as a conference paper at ICLR 2026
PHYCLIP: (--PRODUCT OF HYPERBOLIC FACTORS UNIFIES HIERARCHY AND COMPOSITIONALITY IN VISION-LANGUAGE REPRESENTATION LEARNING
Daiki Yoshikawa' & Takashi Matsubara 1,2 Hokkaido University "CyberAgent
ABSTRACT
Vision-language models have achieved remarkable success in multi-modal repre- sentation learning from large-scale pairs of visual scenes and linguistic descriptions. However, they still struggle to simultaneously express distinct types of mantic structures: the hierarchy within a concept family (c.g., dog 3 mammal = animal) and the compositionality across different concept families (e.g., "a dog in a car" 3 dog, car). Recent works have addressed this challenge by employing hyperbolic space, which efficiently captures tree-like hierarchy, yet its suitability for representing compositionality remains unclear. To resolve this dilemma, we propose PHyCLIP, which employs an Pi-Product metric on a Cartesian product of Hypeebollic factors. With our design, intra-family hierarchies emerge within individual hyperbolic factors, and cross-family composition is captured by the ( -r-oodduct metric, amalogous to Boolean algebra. Experiments zero-shot clas- sification, retrieval, hierarchical classification, and compositional understanding tasks demonstrate that PHyCLIP CII outperforms existing single-space approaches and offers more interpretable structures in the embedding space.
INTRODUCTION
Vision language models have become a central paradigm for learning transferable representations across visual and textual modalities. As exemplified by CLIP (Radford et al., 2021), contrastive pre- training maps images and texts to embeddings and enables strong zero-shot transfer on classification, retrieval, and related tasks. However, compressing the semantics of an instance into single point makes it challenging to faithfully encode two semantic structures at once: hierarchy (is-a relations in concept family) and compositionality (conjunction across distinct concept families).
Visual and linguistic concepts linked by is-a relations form tree-like taxonomic hierurchies. For example, dog is mammal, which in turn is animal, shown in the upper part of Fig. 1. Because the number of nodes grows exponentially with depth, Euclidean geometry struggles to faithfully represent such trees, whereas hyperbolic geometry aligns well with this growth (Bridson & Haefliger, 1999; Sarkar, 2011 1). These observations have motivated the development of hyperbolic embeddings (Nickel & Kiela, 2017) and hyperbolic entailment cones, which encode partial orders via inclusion (Ganea et al., 2018a). Within vasion-language representation learning, MERU (Desai et al., 2023) and HyCoCLIP (Pal et al., 2025) leverage these approaches capture image-text relations; for instance, an image of a dog is an instance of the linguistic concept dog (see the lower part of Fig. 1).
Beyond taxonomic structure, images and texts exhibit compositionality. For example, the description "a dog in car" binds concepts dog and car from distinct concept families (animals and transporta- tion), as shown the middle part of Fig. Classical approaches express composition via logical conjunction or additive operations (e.g., Boolean algebra, bag-of-words. and vector addition in word2vec) (Hinton et al., 1986; Mikolov et al., 2013; Vendrov et al., 2016), but these struggle to encode semantic hierarchy efficiently. Conversely, while hyperbolic geometry captures hierarchy, it lacks a canonical operation for composition. Möbius addition hyperbolic spaces (Ungar, 2008) is not aligned with standard vector addition Boolean structures (Higgins et al., 2018). Intersections of regions (such as hyperbolic entailment cones) can approximate conjunction but offer no general guarantees of representational efficiency for arbitrary co-oceurrences.

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Published as a conference paper at ICLR 2026
transpori
food
hierarchy family:
vehicle
aircraft
metric tree embedded inlo hyperbollic
bike
composiiioalliy acroes concept families:
Hoolean algehra
"a "a dog car"
"a and bike"
Ta dog riding bike"
embedded into an Ey-product metrie space
images as instances of captions
Figure 1: Conceptual diagram of hierarchical and compositional structures. While all arrows represent entailments (=), they differ in nature. (upper) Linguistic concepts organize tree-like taxonomic hierarchies of concept families, each of which be embedded into hyperbolic space (Sarkar, 2011) (middle) Images and texts exhibit compositionality across distinct concept families, which can be captured by Boolean algebra &--product metric. (lower) Images instances of their corresponding captions.
Toxt
Product Space Marperr
Imagz
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Text Iiseddrr
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Product Space Mapper
Imsage Fhooder
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Product Space Masper
Iirage Phooder
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Figure 2: Overview of PLYC Images and texts are encoded as points X in an é 11-product metric space of hyperbolic factors, (Hd)&, that is, as tuples of points (x()) hyperbolic spaces T.ę where their distance is defined by the sum of hyperbolic distances. The entailment relations X Y encoded using entailment cones (x) (z) within hyperbolic factors H!!
To resolve this dilemma, we propose PHyCLIP, which leverages an f--Product metric on a Cartesian product of Hyperbolic factors, depicted in Fig. 2 Our design follows two classical correspon- dences: (i) metric trees admit low-distortion embeddings into hyperbolic spaces, so hyperbolic factors embed intra-family taxonomies (Sarkar, 2011; Sala al., 2018; Spengler & Mettes, 2025); and (ii) finite Boolean algebras with the Hamming distance embed isometrically into an l space, so an f1 21-product metric naturally supports cross pross-family Boolean-likk composition (Deza & Laurent, 1997). Intuitively, each bit for an atomic concept (e.g., dog, car, tomato) in the Boolean algebra is replaced with hyperbolic factor for a concept family (e.g., animals, transportation, food), and the activation of multiple factors expresses composition (e.g.. "dog and car"). Unlike previous mixed-curvature models (Gu et al., 2019; Wang et al., 2024; Gao et al., 2025), our space uses an é -prruucct metric rather than a Riemannian (62) product metric and constrains each factor to have negative curvature. Our contributions are summarized as follows.
Balancing Hierarchy and Compositionality. We introduce PHyCLIP, a vision-language model that leverages an & -proruuct metric space of hyperbolic factors to jointly capture hierarchy (within factors) and compositionality (across factors).
Theoretical Support. We formally link Boolean lattices li-product metrics and metric trees hyperbolic factors, explaining that an ep-rruuuct metric space of hyperbolic factors aligns better with the dual semantic structures than standard metric spaces (c.g., Euclidean or hyperbolic).
Superior Performance and Interpretability. Experiments on zero-shot classification, image- text retrieval, hierarchical classification, and compositional understanding demonstrate that PHyCLIP achieves consistent improvements over baselines that use standard metric spaces. Visualizations find that intra-family taxonomies emerge within individual factors, and composing concepts leads the simultaneous activation of the corresponding factors, analogous to a Boolean algebra.
Codes available https: //gs teiituub.... www.n//ksmatttto ra PHyCLIIP.
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2 THEORETICAL BACKGROUND AND MOTIVATION
Geometry and Embedding of Hierarchies. Concepts in natural language linked by is-a (hy- entailment) relations form a partially ordered
pernymy/hyponymy, generalization/ppciaiiizzz set (poset) and typically exhibit deep hierarchical structure. A poset (P. is set equipped with an order relation N (which is reflexive, antisymmetrie, and transitive). A typical example dog N mammal N animal, where a dog is a type of mammal; equivalently if an entity is dog, then this entails that the entity is mammal. Large lexical resources such as Word- Net provide such relations in the form of directed acyelic graph with multiple inheritance (e.g.. dog < domestic animal) (Miller, 1995). For modeling or computational convenience, many studies approximate this hierarchy with a taxonomic tree (Morin & Bengio, 2005; Mnih & Hinton, 2008). The distance between two nodes (i.e., words) in tree is often defined the length of their shortest path, inducing a type of metric tree. The distance of a node from the root is a natural measure of specialization of the concept that the node represents. The root node specifies the most general concept (e.g., entity) and effectively means nothing (i.e., the absence). See technical details in Appendix A.
Theorem (Hyperbolic embedding of trees (Sarkar, 2011)). Let Hd be a d-dimensional hyperbolic space with the hyperbolic distance dga. For every finite metric tree T (and every infinite metric tree T with known bounds for maximum degree and minimum edge length), and for every > 0, there exist scale >> 0 and embedding f : rT > IH22 such that the distortion is at most 1 + E; that is, there exists a (1 + E, 0) quasi--sommentt embedding f up to scaling.
See Theorem 5 in Sarkar (2011) for the proof. This explains the empirical success of hyperbolic embeddings for hierarchical data (Nickel & Kiela, 2017; 2018; Ganea et al., 201 8a; Sala al., 2018; Tifrea et al., 2019; van Spengler & Mettes, 2025). In practice, Hi with d > 2 is common for achieving better performance.
Geometry and Embedding of Compositionaliiy. Beyond taxonomic structure, images and texts often exhibit compositionality: they mention multiple concepts to indicate the co-occurrence or conjunction of those concepts. For example, the description "a dog in a car" mentions concepts dog and car. Such data suggest another type of entailment relation, as an image of "a dog in a car" can be regarded as an image of dog as well as an image of car. The resulting structure is no longer tree but rather a more general poset. While hyperbolic embeddings remain an option, natural to explore alternatives that more directly capture compositionality.
Order embeddings (IR", 3) (Vendrov et al., 2016) assign each concept a point x E R" and declare = iff x4 Y/1 for all coordinates i. This is equivalent to the inclusion rela- tion between associated upper orthants U(x) == {z E R" 2; 2 2; Vi), i.e., x = y iff (xx) S Uz). Then, the coordinate- soordinate-wise max (i.e., the union of orthants) expresses conjunction (e.g., max(dog, car) includes "a dog in a car"), ) and the coordinate- wise min yields shared concepts (e.g., min("a dog in car", "a dog hyperrectangles in R. (Vilnis et al., 2018; Li et al., 2019; Dasgupta et al., 2020). In hyperbolic space, hyperbolic entailment cones use geodesic conical regions (Ganea et al., 2018a), and disk embeddings use hyperballs (Suzuki et al., 2019). Compared with hyperbolic embeddings for pure hierarchies, there has been less theoretical analysis of these region-based embeddings for compositionality. Our work extends this line to capture hierarchy and compositionality simultaneously.
sofa") Ri dog). Similarly, box embeddings use axis-aligned
Boolean Lattice and Its Relation to Order Embedding. In an is-a taxonomy, any two nodes have at least one common generalization, whereas they need not share common specialization. A lattice is a poset in which any two nodes have both a common generalization (join) and a common specialization (meet). Consider n atomic concepts C = (c.....,6, Go } (e.g.. {dog, car, tomato... ... }). A subset C C denotes the conjunction of the concepts specified in S, and the inclusion relation S 2 T (c.g.. {dog, car) 2 {dog), (car)) induces the order relation S < T (c.g.. {dog, car) < {dog]. (car)). In this way, the Boolean lattice (2 C) over all such subsets naturally represents the compositionality non-taxonomic poset. When focusing also referred to as a Boolean algebra. At the same time, using an indicator X : 2 + f0, 1}", the Boolean lattice can be regarded as metric space (40, 1)"", dHam) with the Hamming distance. See Appendix A and Ganter & Wille (1999); Davey & Priestley (2002) for more details.
of atomic concepts
operations rather than order, it is
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Definition 1 (ly f1-pprdduc metric space), Let (xx, be non-trivial metric spaces. An Ep- product metric space of {(X, dxx) is a Cartesian product space II. X, equipped with the ep-product metric dy (()(()))) .....,
(y.
= = ( (
Proposition (Embedding of Boolean Lattice). A Boolean lattice (2C, 3) for n. atomic concepts can be embedded into the poset (IR", 3) used by order embeddings while preserving the order relations. As a metric space ({0, 13", dHam), is isometrically embedded into an Ex-product metric space I X,, d1) for any k 2 n after appropriate per-factor scaling. However, it admits no isometric embedding into a hyperbolic space IH for any d 2 2 and n 2 2.
See Appendix B for the proof. The first part of the proposition explains that order embeddings can be regarded as a continuous relaxation of Boolean lattices. A pure Boolean lattice has remarkable expressivity for compositionality, but it is often too coarse as it considers combinations of all atomic concepts. Order embeddings enrich it by replacing each bit {0, } with R. The remaining part explains the affinity between Boolean lattices and the & ---pruduct metric, as well as the incompatibility between Boolean lattices and hyperbolic spaces.
Note that we can also consider a weighted version of an é--product metric, where the distances on different factors can be weighted differently. However, in the context of isometric embeddings, these weights and per-factor scalings are essentially equivalent, as one can be absorbed into the other.
PHYCLIP AND ITS LOSS FUNCTIONS
Embedding into an & Pr-Prru Metric Space of Hyperbolic Factors. Here, we model the semantics encoded in images and texts as conjunction of multiple atomic concepts each of which taken from a distinct concept family (e.g., animals, transportation, food). This model is realized by replacing each bit {0, }} of a Boolean lattice with metric tree T; for a concept family. For example, the description "a dog in a car" is represented by a pair of nodes in metric trees Ti and animal) and transportation
T3 that encode is-a taxonomies of animals (c.g., dog A mammal (e.g., car < vehicle V transport), respectively. Notably, a single hyperbolic space cannot capture this product geometry (see Proposition in Appendix B), whereas an l -1-product metric space of hyperbolic factors can.
Theorem (Embedding into an é -porruucct metric space of hyperbolic factors). Let T1........ .,,, Tk be finite metric trees (or infinite metric trees with known bounds for maximum degree and minimum edge length) with metrics dT....., dT,,. For every 8 >> 0, there exists a (1+s, 0) (u)--uas-iisometre embedding from the & fi-product metric space of these metric trees, (II= Tr, d1), into an & L1-product metric space of l hwo-dimensions hyperbolic factors, ((H2)*, dd)), after appropriate per-factor scaling.
Given the above, we propose embeddings into an iprouuet metric space of k copies of d-dimensional hyperbolic factors H. ((H4) dz). The total dimension of ( is kd. Each hyperbolic factor Hi is intended to represent the taxonomy of a concept family, as well as aspects of inter-object relations (e.g., "a dog riding something", "a car loading something") ".) An instance is embedded as tuple X = (1) ..... (k)) for (+)) HH Within each factor HH we use standard hyperbolic embeddings (Nickel & Kiela, 2017) together with hyperbolic entailment cones (Ganea et al., 201 8a) to encode intra-famity hierarchy and image-teas text entailment, while cross-family compositionality is captured by the additive geometry of the & E1-product metric space.
CII for Vision-Language Representation Learning. Here, we propose PHyCLIP for lasion-language representation learning, depicted in Fig. 2. Let / and T' denote instances of images and texts, respectively. From an instance, kd-dimensional feature vector is produced, which is then sliced into k segments (+() of dimension d for i I,... .....,K. and each segment v(() is lifted via the exponential map to its corresponding hyperbolic factor H (0), yielding the embedding X = (1) .... x (k) E ) We denote the embeddings of the image I and text T by I and T. respectively. Let B denote the index set of instances in a mini-batch; we write the mini-batch of images as {I,} [Ihee for brevity.
An image is typically more specific than its corresponding text T (I 2 T) as the text T may ignore some details of the image T. Following HyCoCLIP (Pal et al., 2025), we suppose that the
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training data enriched with box information: the image boxes 1nx are object-level crops of the original images I. and the text boxes Thx are the corresponding nouns/phrases within the text T (Ihox J T ). An image box ro and text box Thex are more general than the full image and the full text (1 7box, T 3 ) respectively, since they omit objects and words outside the boxes.
We will introduce the contrastive loss Leumt and entailment loss Lent, and the final objective is their sum weighted by a hyperparameter yy:
Coverall Leout + -entt
Contrastive Loss. To represent each hyperbolic factor H. we adopt the Lorentz model with learnable curvature -o; (Cannon et al., 1997; Nickel & Kiela, 2018; Lee, 2018). See Appendix C for implementation details. Following Definition we define the distance on the & f1-product metric space ) and its averaged version
d1 (X, Y) Σ d (xx11) ())
dang(X Y) (X,))
To pull an embedding X close its positive pair Y, while pushing it away from negatives Y, for a of b, we use the standard InfoNCE loss (Radford et al., 2021; ; Desai et al., 2023; Pal al., 2025);
com (X,,, ( (Yr) = = -Eben log EeEn exp(-den (X..Y.)/r)
(3)
where is learnable temperature parameter. We average this loss over known pairs:
from = 2 (Coum (Io); (cm)) (To))+ ((15))))) ( (T + ((T (TTx).
(1gxx) ())
Entailment Loss. We also employ hyperbolic entailment cones to capture the entailment rela- tions (Ganea et al., 2018a). See Appendix C for implementation details. For every point y each hyperbolic factor [H] we define a geodesic conical region (gy(()) with apex at y and half-aperture wyy(") (8) where all points (+)) E C(y()) are considered more specific than y (i.e., x(() 3 ()).
Then, x+ E (0) iff (xx(+), ()))) < (1)(1) for the exterior angle (xx(1), ()). To penalize the violation of the inclusion relation (x()) E (() for a pair (xx(), (y()) such that (x 3 y (0) the entailment loss Len is calculated
Leme [X, Y) = max(0, o(x(1, y (BB Lem (X, Y) E Lem, Y), (5) where hyperparameter 1) controls the margin (Pal al., 2025). We sum this loss over known pairs:
fee Σbe (Trm [T. Tb) + Te Tbbo)) + Tem (T,, I5ox) Teem (To, TbxT)))
(6)
Discussions. PHyCLIP requires h evaluations of cosh and sinh for the exponential maps and k evaluations of arcosh for factor-wise distances. These operations are fully parallelized across factors and are needed only once per embedding distance. In practice, the wall-clock time is dominated by the Vision Transformer and text encoder, and the additional cost from these operations negligible at the scale of our backbones. PHyCLIP introduces only k = 64 additional parameters learn curvatures, which is negligible compared to the 86M parameters for the base Vision Transformer.
When the negative curvature a; of hyperbolic factor H: is multiplied by 1/c², the Riemannian metric is scaled by c², and the distance between any two points becomes e times larger. Hence, learning the negative curvature effectively corresponds to learning the combination weight of weighted é uprrduut metric. Since the distance of each instance's embedding from the origin can be adjusted independently for each instance, a small curvature does not necessarily mean that the corresponding factor is emphasized. If all embedding points lie near the origin, that factor is effectively down- weighted.
4 EXPERIMENTS
4.1 TRAINING DETAILS
Datasets. We trained all models on the Grounded Image-Text Pairs (GRIT) dataset (Peng et al., 2023), which consists of automatically annotated image-text pairs with bounding boxes and corre- sponding nouns/phrases. Although the dataset is documented to contain 20.5 million pairs with 35,9 million box annotations, tions due to outdated public links. This scale remains considerably larger than manually annotated resources such Flickr30K Entities (Plummer et al., 2015).
were able obtain 14.0 million pairs with 26.6 million box annota-
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Published as a conference paper at ICLR 2026
Table 1: Zero-shot image classification.
General datasets
Fine-grained datasets
Specialized datnsets
boves
ImngeeNet
CIFAR
CIFAR- 100
SUN397
Callech- 101
SIL-10
Food. 101
Aiterftt
Flowers
EuroSAT
RESISCAS
Counryy211
CUB
Cars
Pets
DTD
CLIP
38.87
76.26
48.19
50.70
73.62
93
51.19
12,90
7.82
3.01
45.89
26
22.02
35.73
42.03
5.13
CLIP
38.81
76.53
48.59
50.80
74.29
93.34
51.05
12.70
8.40
2.89
46.19
21.32
21.74
37.49
41.78
5.10
MERU
37.96
77.63
46.37
49.39
72.10
93.14
51.67
11,09
7,80
3.53
43.36
19.98
22.18
38.81
41.77
4.86
MERU
38.08
78.14
46.80
49.59
72.69
93.28
5192
10.70
7.77
3.53
43.22
18.31
22.07
37.31
41.73
5.01
HyCoCLIP
43,80
89.00
58.59
54.49
76.14
94.96
52.64
14.90
10.24
3.57
53.33
19.41
25.90
36.36
46.97
5.64
PHyCLIP 44.31
89.33
59.05
55.32
76.35
94,84
57.26
15.90
10.89
3.24
54.18
19.98
25
36.29
48.22
5.56
The best and second best performances are emphasized by bold fonts and undertines, respectively.
Table 2: Zero-shot retrieval and hierarchical classification.
Text
Image
Image
Text
Hierarchical Classification
COCO
Flickr
COCO
Flickr
WordNet
R 5
R 10
RØ5
R 10
R
R
10
R
R 10
T(b(
LCA(3)
J(T)
Pv((1)
Ry(()
CLIP
56.29
67.53
83.15
89.58
70.32
80.09
91.60
95.60
3.750
2.276
0.7774
0.8471
0.8483
CLIP
56.20
67.50
82.75
89.42
70.35
80.19
91.10
95.63
3.736
2.279
0.7784
0.8473
0.8501
MERU
55.73
67.02
82.15
89.05
69.57
79.33
90.77
95.83
3.815
2.294
0.7733
0.8454
0.8450
MERU
55.87
67.21
81.96
88.89
69.70
79.69
91.20
95.83
3,802
2,289
0.7740
0.8457
0.8455
HyCaCLIP
57.11
68.32
83.06
89.63
69.51
79.73
91.47
95.63
3.319
2,092
0,8043
0.8676
0.8661
PHyCLIP
58.03
69.05
83.39
89.93
70.94
80.86
91.20
95.53
3.294
2,083
0.8059
0,8684
0.8672
Baselines. We compare PHyCLIP with CLIP (Radford et al., 2021). MERU (Desai et al., 2023), and HyCoCLIP (Pal et al., 2025). CLIP is seminal rision-language model trained with contrastive learning based on the cosine similarity. MERU extends CLIP by lifting embeddings to hyperbolic space and using hyperbolic entailment cones represent hierarchy. HyCoCLIP further leverages box annotations to better capture intra-modal hierarchy. All models were trained from scratch on GRIT for fair comparison, and we report the average scores over three runs with fixed random seeds. For PHyCLIP, we set the number of factors to k = 64 and the dimension of each factor to d = 8, resulting a total dimension of 512. We followed the training protocols and hyperparameters used in the official implementations of HyCoCLIP (Pal et al., 2025); see Appendix C for details. We report results obtained with the base Vision Transformer as an image encoder (Dosovitskiy et al., 2021). Supplementary results are provided in Appendix D.1.
4.2 EXPERIMENTAL RESULTS
Zero-shot Image Classification. We evaluated the geometry of embedding space via zero-shot image classification, following the protocol standardized by CLIP (Radford et al., 2021). Images are classified using the similarity to the averaged embedding of template text queries for classes across 16 datasets, grouped into General, Fine- grained, and Specialized. General datasets cover broad, heterogeneous concept families (e.g., animals, transportation, household objects). Fine-grained datasets focus on visually similar subclasses within a single concept family (e.g., specific food, dog breeds). Specialized datasets are Nomain-specifie (e.g., texture images, satellite imagery).
Table 1 summarizes top-1 accuracies. PHyCLIP obtained consistent performance gains, particularly on General datasets, which we attribute to assigning concept families to hyperbolic factors that naturally support coarse-graing grained classifications. Within Fine-grained datasets, PHyCLIP achieved remarkable improvements
Food-101 (Bossard et al., 2014) and Oxford-IIIT Pets (Parkhi et al., 2012), implying that it also learned intra- fntra-family taxonomies without confusion with other families. Although not best every dataset, the performance gap on FGVC-Aircraft (Maji al., 2013) is small, and the specialized datasets are duttof-distribution relative to GRIT. Consistent with prior findings (Pal et al., 2025), CLIP and MERU do not yield clear improvements with box annotations. Overall, PHYCLIP is the strongest zero-shot classifier among the comparison models.
Zero-shot Image and Text Retrieval. We evaluate cross- modal alignment via zero-shot retrieval in the shared embedding space: given text query, retrieve the nearest images, and vice versa. This is also a standard benchmark for vision-language models (Radford et al., 2021). We used the COCO validation set (Lin et al., 2014) and the Flickr30K test set (Young al., 2014; Karpathy & Li, 2015).

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Published as a conference paper at ICLR 2026
Table 3: Compositional understanding through hard-negative classification.
of boxes
VI.-Checkl jist Object
SugarCrepe
Location
Size
Replace
Swap
Add
Center
Mid
Margin
Lurge
Medium
Small
Ohg
Alt
Rel
Obj
All
Ohj
All
Overall
CLIP
61.90
60.30
60:40
63.87
60.17
58.23
89.37
79.95
69.54
60.54
66.02
80.39
73.36
77.72
CLIP
61.93
59.33
60.83
63,70
60.80
58,07
89.69
80.33
69.49
61.63
66.47
80.62
73.55
77.97
MERU
61.27
59.03
58.97
63.97
57.70
56.07
89.10
50.50
69
60.82
65.32
80.47
74.90
77.81
MERU
61.03
58.47
58.73
62.63
58.70
89.39
79.95
69.68
60.41
80.41
75.34
77.93
HyCoCLIP
70.43
69.50
67.80
72:57
66.13
67.20
91.38
79.74
67.24
54.69
63.66
82.57
74.23
77.99
PHyCLIP
71.20
70.30
70.37
73.73
68.10
67.83
91.06
81.05
66.36
57.41
65.87
83.24
73.80
78.32
We report Recall at & (R@k), the fraction of queries for which the paired instance appears in the top-k retrieved results.
Results are summarized in the left half of Table 2. PHyCLIP achieves the best performance across almost combinations of performance measures and datasets, which supports our choice of the 11- product metric in Eq. (2). This metric sums distances over hyperbolic factors; when an object specified in the text is absent from a candidate image, or an unspecified object is present, the corresponding factor incurs a large penalty. By contrast, single hyperbolic space implicitly encodes the presence or absence of objects as hicrarchical relations, which may weaken penalties for such mismatches and hinder separability of hard negatives. All models perform competitively the text retrieval for Flickr, suggesting the performance saturation.
Hierarchical Classification. We evaluate the expressivity for the is-a taxonomy via hierarchical classification (Kosmopoulos et al., 2015; Pal et al., 2025) on ImageNet (Russakovsky et al., 2015), where class labels are enriched by WordNet (Miller, 1995) and errors between predicted and ground- truth classes are measured on the WordNet graph with unit-length edges: Tree Induced Error (TIE) is their graph distance; Lowest Common Ancestor (LCA) error is the maximum of the distances to their LCA; Jaceard similarity J. hierarchical precision PH. and hierarchical recall R are similarities between the sets of ancestors.
Results are summarized in the right half of Table 2. PHyCLIP achieves superior scores across all metries, indicating not only higher classification accuracy but also that misclassifications tend to be close to the ground-truth class in the taxonomy. By handling cross-family compositionality via the (-prrducct metric, each hyperbolic factor can devote capacity to a cleaner intra-family is-a taxonomy, thereby yielding disentangled, hierarchy-aligned representations.
Compositional Understanding. We assess the expressivity of compositionality via hard negative classification using VL-CheckL.. (Zhao et al., 2022) and SugarCrepe (Hsieh et al., 2023). Both benchmarks provide a kind of image-to-tex retrieval task that require models to distinguish ground- truth captions from hard negatives created by altering objects, attributes, relations in the ground- truth captions. Following Pal et al. (2025), we evaluate the Object subset of VL-CheckList, in which noun for single object in each caption is randomly replaced. The results are summarized by the replaced object's s location (center/mid/margin) and size (small/medium/lrge in the image. We also evaluate all seven subsets in SugarCrepe, in which objects, attributes, and relations are replaced, swapped, or added in each caption.
As shown in Table 3, PHyCLIP yields substantial improvement VL-CheckList--Oije It successfully represents object presence robustly with respect to location and size. On SugarCrepe, PHyCLIP obtains the best average score, which exceeds that of the second-best model by a large margin, whereas the other models cluster within a small range. Performance on attribute replacement and swapping is robust, suggesting that our design decouples intra-family taxonomy from cross- family composition and thereby emphasizes attribute-object binding. By contrast, we observe modest drops relation replacement and object swapping, which implies that our design less sensitive to inter-object relations, potentially because of its Boolean-like nature. This suggests that even within image-to-ter to-text retrieval, models exhibit distinct strengths and weaknesses depending on the type of compositions. Consequently, the performance variation observed in image-to-ter retrieval on COCO and Flickr in Table 2 is likely attributed to which types of hard negatives, and in what proportion, are included in each benchmark.
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lengy lears
Tosts
Tosts Teok brnee
04
8.5
| gg
(a) PHyCLIP (factor-wise norms)
(b) PHY (overall norms)
(c) HyCoCLIP
Figure 3: Norm distributions. In (b) and (c), image norms consistently larger than text norms, because images are more specific than their paired texts (4, 3 Tb). However, in single hyperbolic factor shown in (a), image and text norms largely overlap, PHyCLIP may keep some factors unused for instances that do not contain the corresponding concept families.
Table 4: Ablation study.
Ablation Study. We investigate the contribu- tions of embedding space factorization and the --rrduuct metric through ablation studies, sum- marized in Table 4. We fix the total embedding dimension ld and vary the number of factors, k.
classification
retrieval
hicrarchical
Poffoctos, &
prodict metric
COCO, R
curvature
Image
Food 101
Image
When k = (equivalent to HyCoCLIP), perfor-
512
hyp.
43.80
52.64
57.11
69.51
3.319
0.8143
mance is the lowest on most measures; increas- ing k generally improves results, except for text retrieval, thereby demonstrating the benefit of factorization. Most performance measures peak at A: 64 or 128, although zero-shot clas- sification accuracy for Food-101 101 (Bossard et al.. 2014) drops substantially at & = 128, indicating that overly fine factorization may impair the representation of intra- taxonomy. Replacing the l f-prrducc metric with the Riemannian (f2) or -xe-product metric consistently degrades performance. We also investigated the mixed-curvature model of the Euclidean, hyperbolic, and spherical spaces of 256, 128, and 128 dimensions, respectively (Gu et al., 2019; Wang et al., 2024; Gao et al., 2025), and found overall inferior performance. These results support that the é -proddct metric provides more effective way to aggregate cross-family composition.
8
8
hyp-
44.38
54.61
80
70.80
3.273
0.8072
16
81
hyp
44.09
55.29
57.26
69.22
3,287
0.8066
32
16
hyp.
43.90
54.48
56.70
66.92
3.324
0,8035
81
hyp 44.3
57.26
58,03
3.294
0,8059
128
8,
hyp.
44.16
53.96
57.79
71.18
3.284
0,8064
64
82
hyp.
43.32
53.39
57.09
70.53
3.367
0.8011
64
800
hyp. 6.55
10.33
8.77
14.51
9.697
0.4247
82
mixed
39.34
49.05
56.72
70.81
3.712 0.7797
4.3 VISUALIZATIONS OF HYPERBOLIC FACTORS
Norm Distributions. Figure 3 plots the empirical distributions of embedding norms. (b) and (c) show that, in both PHyCLIP and HyCoCLII, image norms are consistently larger than text norms and are tightly concentrated. These models consider images to be more specific than their paired texts, < Th, which encourages the image embedding Is lie within the text's hyperbolic entailment cone C(T.) (i.e., I, C(T.)), yielding larger image norms. However, (a) shows that, within individual hyperbolic factors of PHyCLIP, the image and text distributions largely overlap and are broadly dispersed. This is because instances without particular concept family lie near the origin in the corresponding factor, other words, not use unrelated factors. Consequently, PHyCLIP leverages a broader portion of the embedding space and facilitates meaningful distances and taxonomic structures.
Intra-Family Hierarchies in Hyperbolic Factors. We obtain factor-wise embeddings (+) of single-concept prompts (c.g.. "a photo of a dog" and "a photo of car") and summarize their norms ||x[1) Iyy across 64 factors in Fig. 4 (a). The "dog" embedding exhibits its largest norm in factor i = while remaining near the origin factor
9. Conversely, the "car" embedding peaks in factor = 9 and is suppressed in factor 39. We also visualize the top 0.1% of GRIT images by the embedding norm for each factor. Factor i = 39 yields various mammals, whereas factor i shows vehicles and everyday-carry items.
Figure visualizes embeddings of various single-conceg prompts projected by HoroPCA (Chami et al., 2021) from d 8-dimensional hyperbolic factors onto 2D disks. Embeddings of mammal- related terms are spread over a wide area in factor = 39. Dog-related terms cluster in the left half, cat-related terms in the right, and "chihuahua," "corgi," and "puppy" are positioned farther from the
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dtog and
(b) Images retrieved by "maxx" of the single-concepl prompts.
Factor i
Images with I Largest Norms in H
Images with Largest Norms in Hig
(c) Images retrieved by the conjunctive prompt.
(a) Embedding norms of single-concept and conjunctive prompts.
Figure 4: Factor-wise embeddings and retrievals. (a) Single-concept prompts (e.g., "a dog" or car") activate distinct factors (i.e., i = 39 or = 9), and their textual composition (e.g., "a dog and car") activates the corresponding factors simultaneously. (b)-(c) (c) "max" of the single-concept prompts retrieves images similarly to the textual composition. See also Fig. 6 in Appendix D.2.
bag
semi-tratler
fire tuck trik
wallet snartphone
Kclinc cat
Apersian
cellphone
Akiny
school hus
Fox
@ bus
vehicle
cong camine
Vehicles and everyday-cary
hunting dog
dog
A poudle
chiluahux
puppy
Figure 5: Embeddings projected onto 2D disks by HoroPCA. A set of relevant concepts (hyponyms of mammals or words related to vehicles and everyday-car items) forms hierarchical structure in the corresponding factor (i 39 or 9), while the same concepts cluster near the origin in another.
origin than "dog." These patterns indicate that factor i 39 encodes the is-a taxonomy of mammals (or more specifically, Carnivora). In contrast, in factor i = 9, the same embeddings concentrate near this factor represent mammals. Embeddings related to vehicles and everyday-carry items exhibit the opposite pattern; they form a hierarchical arrangement factor
the origin, suggesting that PHyCLIP does
9 and cluster near the origin in factor 39.
Composition via - Product Metric. To examine the behavior of the l f1prroduct metric, we also obtain the embeddings of the textual composition (e.g.. "a photo of dog and car"). Figure 4 (a) shows that such a prompt produces large norms in both factors i = 39 and i = 9, meaning that composing concepts simultaneously activates the corresponding factors. Furthermore, we took the factor-wise "max" of the embeddings of two single-concepp prompts and found that it retrieves images similarly to the textual composition, shown (b)-(c) (see Appendix D.2 for details). These patterns align with the behavior of Boolean algebra, where multiple concepts are specified by the union of concept subsets (or the element-wiss wise max of binary indicators).
We observe the same pattern for "boy and bicycle" and "sunset and ocean." Consistent with The- orem 2, these observations empirically support that PHyCLIP organizes intra-family taxonomies within individual hyperbolic factors and expresses cross-family compositionality via the simultaneous activation of multiple factors, analogous to Boolean algebra. We emphasize that, while we provide hierarchy between images, texts, and their cropped versions, do not provide any explicit supervi- sion for factor assignments or hicrarchy between atomic concepts (i.e., words); the specialization of factors and hierarchical structures of atomic concepts emerge automatically through training.

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5 RELATED WORK
Vasion-Languuge Models and Representation Learning. Vision-language representation learn- ing contributes to retrieval (Mori et al., 1999), semantic segmentation (Barnard et al., 2003), and image generation (Ramesh et al., 2021; Labs, 2025). Early works learned alignments through object- level, word-based classification and detection (Karpathy & Li, 2015; He & Peng. 2017; Engilberge et al., 2018) or through text- image generation (Peng & Qi. 2019; Gu et al., 2018), but they often required complex annotation and network designs (Zhao al., 2022). A more generic approach maps an entire image or text to single vector and learns a shared embedding space with contrastive objective. Representative systems include DiVS (Frome et al., 2013), VSE++ (Faghri et al., 2018), CLIP (Radford et al., 2021), and ALIGN (Jia et al., 2021). Our model, PHyCLIP. follows this line, while implicitly extracting individual concepts through the geometry of an & -prrouuct metric space.
Hyperbolic Representations in Deep Learning. Data often exhibit hierarchical, tree-like structures. Many approaches have attempted to encode such structure (Nguyen et al., 2017; Vulic & Mrksie, 2018), and hyperbolic spaces have become influential due to their empirical performance and theoretical support (Sala et al., 2018; Sonthalia & Gilbert, 2020; van Spengler & Mettes, 2025). As discussed in Section 2, tree metries admit quasi-isometric embeddings into the two-dimensional hyperbolic plane, which enhances generalization and interpretability (Bridson & Haefliger, 1999; Sarkar, 2011). Hyperbolic embeddings have been applied to words (Nickel & Kiela, 2017: 2018; Tifrea et al., 2019), sentences (Dhingra et al., 201 8), graphs (Liu et al., 2019), and images (Khrulkov et al., 2020; Atigh et al., 2022; van Spengler et al., 2023; Qiu et al., 2024). There is also extensive work building neural networks on hyperbolic spaces (Ganea et al., 2018b; Shimizu et al., 2021; Takeuchi et al., 2022; Peng et al., 2022) and on optimization over Riemannian manifolds (Bonnabel, 2013; Béeigneul & Ganea, 2019). Within vision-language learning, MERU adapts CLIP to hyperbolic geometry (Desai et al., 2023). Our method also leverages hyperbolic geometry and embeds tree-like structures efficiently.
For non-hierarchical data, Euclidean, hyperspherical, or toroidal geometries can be effective (Ebisu & Ichise 2018), and several studies explore representations in Riemannian (f2) product of such spaces as cureaccurvature representations (Gu et al., 2019; Wang et al., 2024; Gao et al., 2025). PHyCLIP also employs a product space, but all factors are hyperbolic and the product metric is l we justified both choices theoretically in Section 2.
Region- Reaed Embeddings for Structured Representations. Hierarchical relations can be viewed as a form of inclusion relations. Order embeddings (Vendrov et al., 2016) represent an instance as an upper orthant of Euclidean space, and box embeddings (Vilnis et al., 2018) represent it as an axis-aligned hyperrectangle, where the inverse of set inclusion encodes the hierarchical relation. Euclidean variants include Gaussian embeddings (Vilnis& & McCallum, 2015), and hyperbolic variants include disk embeddings (Nickel & Kiela, 2018) and hyperbolic entailment cones (Ganea al., 2018a). These approaches have also been employed in the vision-language setting (Ren et al., 2016; Desai et al., 2023; Pal al., 2025). We summarize their theoretical connections in Appendix A.2. These region-based approaches permit composition via intersection of regions, which allows multiple parents and richer semantic composition. However, their compositional expressivity has not yet been fully characterized. We showed in Section 2 that order embeddings and PHyCLIP support compositionality at the level of a Boolean algebra, while a single hyperbolic space may not.
CONCLUSION
We introduced PHyCLIP, a vision- language model that learns representations using an Ep-product metric space of hyperbolic factors. We theoretically and empirically demonstrated that it simultane- ously captures compositionality across concept families through the -ppruuuct metric, as well as is-a taxonomies within hyperbolic spaces via hyperbolic embeddings. This design yields state-of-the-art performance across various downstream tasks and provides an interpretable embedding structure. While the main focus is on object composition, it also performs well for attribute binding because it decouples intra-family taxonomy from cross-family composition. By contrast, the relational structure remains unexplored; incorporating its algebraic structure is a promising direction for future work.
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ACKNOWLEDGMENTS
This study was partly supported by JST BOOST (JPMJBY24HO) and CREST (JPMJCR24QS), and partly achieved through the use of SQUID at D3 Center, Osaka University.
ETHICS STATEMENT
This study is purely focused on Dision-languren representation learning, and is not expected to have any direct negative impact society or individuals.
REPRODUCIBILITY STATEMENT
The environment, datasets, methods, evaluation metrics, and other experimental settings are provided in Section 4 and Appendix C. For full reproducibility, the source code is attached supplementary material.
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Max van Spengler, Erwin Berkhout, and Pascal Mettes. Poincaré ResNet. In /EEE/CVF International Conference on Computer Vision (ICCV), 2023.
Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is All you Need. In Advances in Neural Information Processing Systems (NeurIPS), 2017.
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A BACKGROUND THEORY
A.1 1 QUASI-ISOMETRIC EMBEDDINGS
We adopt standard notions from Bridson & Haefliger (1999). Let dx) be a metric space. A geodesic segment [r, y C X is an isometric image of an interval whose endpoints are mapped to and y in X. The space X is geodesic if every pair of points can be joined by geodesic segment. For 5 2 O, geodesic triangle is 8-slim if each side is contained in the 9-neighborhood of the union of the other two sides. A geodesic metric space X is 5-hyperbolic (in the sense of Gromov) if every geodesic triangle in X is 8-slim. A metric tree is geodesic metric space in which any two nodes joined by a unique geodesic, and every geodesic triangle is a tripod; hence, it is O-hyperbolic. Any tree with positive edge lengths, equipped with path length as distance, is metric tree. Euclidean spaces R" are not Gtrmov-hyyerboole for n 2 2, whereas hyperbolic spaces H" are o-hyperbolie, where 5 depends only on the curvature.
Definition 2 (Quasi-isometric embedding (Bridson & Haefliger, 1999)). Let (X, dx) and (Y, dy) be metric spaces. A map f (X, dx) + (Y, dyy is a (A, c) c)-quast-isometric embedding if it is injective and there exist a distortion X 2 and an error c. Z 0 such that
xxx (2, - -c S f(('()), s Xxx to for all x, x' e e..
(7)
If X- 1 and e 0, the embedding is isometric.
A.2 REPRESENTATION OF POSET AND LATTICE
A poser (P, 3) is set P with reflexive, antisymmetric, and transitive relation 3. Its Hasse diagram places an edge from x to y when x - y and there is no z such that æ < z - y: hence the existence of an upward path from x to y implies x: y (Ganter & Wille, 1999; Davey & Priestley, 2002). Given y E P. meet æ M (greatest lower bound) and a join æ Li y (least upper bound) may or may not exist. A meet-semilattice (oin-semilattice) is a poset in which the meet æ [Ty y (the join Ll y) exists for all pairs x, y. and a lattice has both for all pairs.
If a rooted tree is ordered by the ancestor relation with the root o at the bottom (so that = x for all x), then the meet x 7 of pair x, always exists, while joins need not exist. Hence, rooted tree is naturally a meet-semilattice in this orientation. Conversely, an is-a taxonomy often uses the entailment order x < y interpreted as "x entails y" or more roughly "æ is more specific than y." with the root o at the top. The join in Ll y always exists, while meets need not exist; the poset is then join-semilatice.
Let = (c.,.....c.. } be n atomic concepts (e.g., dog, car, tomato....). A subset S C C expresses the conjunction or co-occurrence of the concepts specified in S. We define the Boolean lattice (2 S) over the power set 2 of C, in which the order relation = is the inclusion relation C. Meet/join are given by intersection/union, respectively. T S S means that S specifies all concepts in T, so S entails T. Let x: 2 + {0, 1}" be the indicator map with x(5); = 1 iff e E S. Then, T < 5 iff xT)); S x();; for all i, and meet/join become bit- AND/OR, respectively. We summarize the correspondence between different representations in Table 5. In this lattice, each node is defined intensionally as a set of concepts.
From the dual perspective, each node can be defined extensionally as a set of instances that contain specified concepts, the context of formal concept analysis (Ganter & Wille, 1999). Let Z be universe of instances and let I C Z x C be an incidence relation (i.e., z I e means that z has concept | I e for all E S C}, which forms a Galois connection: 8 C T implies T' C S Also, subsets S' C 2 form the dual lattice of (2C, C), where SCT 3 T = T" S' e T' 3 S' If S" for any subset S C C. S C T T' S'.
c). For S C, define an operation S' = (z E Z
An īs-a taxonomy is typically realized as noin-subsemilatice of this dual lattice. Order embed- dings (Vendrov et al., 2016) can be regarded as an extension of the Boolean lattice, where each bit {0, 1) is replaced with a real number R. They declare "x entails y iff xr > y; for all i, similarly to the indicators of a Boolean lattice. Indeed, the ambient poset (R", =) of order embeddings is a lattice with meet/join given by coordinate- wise max/min, respectively. When regarding an embedding x as an orthant Uxx) C RR, the entailment is represented as (xx S Ut), similarly to the dual lattice. When treating the orthant U(y) as the set of all instances that contain the specified concepts
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Table 5: Correspondence of generalization, specialization, and entailment in different representations.
Generalization
Spocialization (hypanymy)
Space
Entailment
(hypemymy)
(ar or S entails y or T)
Tree of is-d Relations (is-a Taxonomy)
join LI
(meet (7)
# <
Order Embedding (as points)
max
m2 2 di for all
Order Embeedding (as orthunts)
orthants in I orthants
((m) C U(y)
Order Embedding entailment)
U(w)
Hyperbolic Entailment Cone
(union U)
intersection n
cones in H
E C(vl
Lattice power
intersection
union
2
Boolean Lattice (as lattice)
meet (T)
join L.
S E T
Boolean Lattice (with indicator)
AND
OR
[0.1)
x(S), 2 x(T), forall
Dual Lattice (as a set)
union U
intersection n
8' C T
Dual Lattice (as a lattice)
join
meet n
S' x T
Product of Trees
join
(meet ()) intersection
I1
I y for all
PHyCLIP
(union U)
cones in (H
x Ca(y for
y, the entailment is represented æ E Uty), which aligns with the definition of the dual lattice.
Hyperbolic entailment cones (Ganea et al., 2018 are hyperbolic extension of the last interpretation of order embeddings, where an orthant (gy) is replaced with a geodesic conical region C(y).
Also, our proposed PHyCLIP can be regarded as an extension of a Boolean lattice, where each bit {0, 1) is replaced with a metric tree T;, which embedded into a hyperbolic factor TH!..
B PROPOSITIONS, THEOREMS, AND PROOFS
B. PROOF OF PROPOSITION
Let (2 be Boolean lattice over all subsets of atomic concepts C = fci.....cc. C.). The indicator x maps subsets S, T C C to binary sequences x(S), E 10. where X(S); if c, E S and x(); = 0 otherwise. Then, S S T iff XSS; S x(T); for all i. The Hamming distance dHam is defined as dHam (x(5), (T)) > (X(); xT))... Consider a map f: {0, 1)"h - R". x(S)
= (x),- In) (1 (S)),...... I-(,,,,)S),,, and the product order æ 3 iff I; 2 Y1 for all i on R". Then, the map f o X embeds the Boolean lattice (2C, S) into the poset (R", used by order embeddings while preserving the order relations.
By definition, the Hamming distance is dHam (x(S), x(T) = (+S))-((III = = Ix(S), x-)) Hence, the metrie space (f0, 1)"h dHam) is equivalent to é 4pproduct metric space (10.1)* E'=1 I)). Consider map fa that maps O to the base point of the metric space X; and 1 to a point with a finite non-zero distance 1/T, > from the base point. The map fi is an isometric embedding from {0, }} to X; when scaled by тC. The map f = (fi,..., fn) is an isometric embedding from (10, 13", dHam) to (II__ T.X Σt= dx.) for any k 2 n.
Assume by contradiction that an isometric embedding ʃ ([0, 1)"h duam) + (H4, dgs) exists for some n 2 2 and d 2 2. Take four points
A = 00,0,0. .......... ) B = (1, O.O.O...... C= (1,10, ......................... D= (0, I...
in (0. 1)a. Let a f(A). (B). J(D), and fCC). Since f is an isometric embedding.
drpa (a. b) = (b. c) = dua (a, d) dizel (d, c) =
and
dn= b) dn (,. = dn (a, d) + due (d, c) = due (a, c) 2.
In hyperbolic space, a geodesic segment is unique, and its midpoint is unique, so both and d are placed at the midpoint in the geodesic segment (a. c]: hence b = d. See Proposition I.4 in Bridson Haefliger (1999). This contradicts the assumption that f is an isometric embedding (which injective).
B.2 PROPOSITION 2 AND ITS PROOF
Proposition 2 (1 product of trees is not hyperbolic). Let T1, T2 be infinite metric trees with known bounds for maximum degree and minimum edge length. Their E Ex-product metric space (Tx T2, di)
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is not 5-hyperbolic for any finite 8. Consequently, there is no (X, (X,c)-quasi--tsomeete embedding
(Ti x T2. d1) + H
quasi-geodesic q in X is (A, quaii-ssiii embedding y X, where / is an interval in IN or the intersection of Z with such an interval; see Definition 1.8.22 in Bridson & Haefliger (1999). In ô-hyperbolic space Y. the stability of quasi-geodesics asserts that the Hausdorfi distance between geodesic and (A, -)-uuss -guudeee with common endpoints is bounded by constant D D(A, c, 8); see Theorem I11.1.7 Bridson & Haefliger (1999).
Lemma (Stability of geodesic triangles under quasi-isometric embeddings). Let X be geodesic metric space and f X Y be a (λ, c) "D--qaaii-isometri embedding into a 8-hyperbolic space
Y. Then every geodesic triangle in X is ô-slim for some constant < A(8 + 2D + c). where D D(A, c, 8) the quasi- geodesic stability constant in Y.
Proof. Let A be geodesic triangle in X. Each side maps to (A, quuuii--geo in Y. By the stability of quasi-geodesics, each image side is contained in D-neighborhood of the corresponding geodesic. Geodesic triangles in Y are 5-slim; hence, each point image side is contained in 5 + 2D-neighborhood of the union the other two image sides. Pulling this back via the quasi-isometry inequalities yields the stated bound.
Let Ti, T2 be infinite trees with bounds for maximum degree and minimum edge length, which admit a geodesic ray of infinite length. For simplicity, we restrict the edge length be 1, but the following discussion holds for arbitrary non-zero edge lengths, by replacing N with the ordered set of the geodesic distances from the root to the nodes the geodesic ray.
Consider (N2, |1 II.. Let m be an even integer and take three points A = (0,0), B = (m, 0),
C= (0, m). . The midpoint
m ) of a monotone geodesic from C' is at m from [A, B]U[A, 2 DO as m + co. Hence, (N²2 11 I11) is not 6-hyperbolic for any finite
requiring 5 2 m//2
Choose two geodesic rays % N + T, for i 1. The map 4 N2 + Ti T2. 4(m, n) (7a (n)) is an isometric embedding from (N²; Ila)) into (Ti T2, d1). Given 8-slim geodesic triangle A in N², its image ((A) is also 8-slim geodesic triangle in T1 x Tg. Since (N2, (I1) is not 9-hyperbolic, neither is (Ti x T2, d1).
Assume by contradiction that f (Ti T2, d1) + H" is (A, co--quai--sommetii embedding, where H" 5-hyperbolic for a finite 8. By Lemma 1, every geodesic triangle in T, T2 is 8-slim, where 8 < X(5 + 2D + e) and D = D(A, c, 5) are constants. However, (T1 x T2, d1) ()) is not 8-hyperbolic for any finite 8, which contradicts the assumption. Therefore, there no (A, -uuiiississ embedding (Ti T2, d1) l
B.3 PROOF OF THEOREM 2
Lemma 2 (Product of quasi-isometric embeddings). If sa (Xi,dx.) + (Yi, dv:) are (Ni, ci)-quasi- isometric embeddings, then
-I Se: (II Xi, Eie dx.) (IIs Yi. Exendhy
(8)
is (A, c) --uusii-isometiic with X = max; A; and c = Ejc,
Proof of Lemma 2. Sum the index-wise inequalities and bound by max; dz- Theorem follows immediately from Theorem 1 and Lemma 2.
C IMPLEMENTATION DETAILS
C.1 LORENTZ MODEL OF HYPERBOLIC SPACE
Let 1 be the (d 1) dimeennsio Minkowski space, equipped with the Minkowski metric Ig..1 =
-dag + dx? +.. dx3 in coordinates à (Io, 21,.., R.)) Intuitively, vp denotes the time
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coordinate, and the others (I1, "il...,,all FZ) ) R denote the space coordinates. The inner product in RR.. given by
(Ø. D) -000 (ar, y)
For 0, define the upper sheet of the two-shected hyperboloid L {i Rd [ 2)
-a Zo 0}. Equivalently, every point satisfies 20 V || 1. The Riemannian metric
on Tg is the restriction of the Minkowski metric 9gd. TT with this metrie, the sectional curvature is the constant -a (Cannon et al., 1997; Lee, 2018). The geodesic distance is
d2ه (t. 9) a- 1/2 arccosh(-a- (á. B4.)) for L, y E L..
(10)
Then, a d-dimensional hyperbolic space H. with curvature -a is isometrically embedded into L by
1 H + La x a ---((( (Va- |
(11)
and we denote (ar, y) = (#t uhg and dng (x, y) = d g (di, D) in the main body.
When feature extractors (such as encoders) operate in the Euclidean space R4, their output cannot be treated directly as an embedding
hyperbolic space due to the mismatch in geometry. Instead, the output v = (E).... va) is treated as a tangent vector in the tangent space TyL ~ IR at the base point â = (a 1/2 0. 0) of IL and mapped to a point in LL via the exponential map
expo ToLLa La + a (x) (v) cosh(vailllleee
+
sinhl (wllee))
/allpllge V.
(12)
C.2 HYPERBOLIC ENTAILMENT CONES IN THE LORENTZ MODEL
Hyperbolic entailment cones capture the hierarchical relationships (Ganea et al., 2018a). For every point y in each hyperbolic factor Hd, we define a geodesic conical region Cly), where all points E Cly) are considered more specific than (i.e., 3 y). The size of this conical region is determined by its aalf-aperture wyy), which is inversely proportional to the norm:
2K
}).
w = sin
min
(13)
where is set to 0.1. Then, x E Cyy) iff ((x y) < w yy) for the exterior angle
zo + y y)
Φ(x, y) = cos"
(14)
||lllR callee//(aa ((( u)
C.3 MODEL ARCHITECTURE AND HYPERPARAMETERS
We introduce the details of our implementation and hyperparameters, which follow Desai et al. (2023); Pal et al. (2025) unless specified otherwise.
As image encoder, employ the Vision Transformer (Dosovitskiy al., 2021; Chen al., 2021; Touvron et al., 2021) with a patch size of 16. Each image is randomly resized by a scale from 0.5 to 1.0 and randomly cropped to 224 x 224 pixels, resulting in 196 tokens, concatenated with 2-D sine-cosine position embeddings. We employ the text encoder used by the original CLIP (Radford et al., 2021), which consists of a 12-layer Transformer architecture (Vaswani al., 2017) with embeddings of 512 dimensions.
The outputs of image and text encoders are scaled by learnable scalars Ging and Cxt. respectively,
before being mapped by the exponential map. These scalars are initialized to Gimg = Cex = 1/V512.
The negative curvature a; for factor i is initialized at 1.0 and clamped in [0.1, 10.0). For the contrastive loss Lcont in Eq. (3), the temperature 7 is initialized to 0.07 and clipped at minimum value of 0.01. For the entailment loss Lam in Eq. (5), the hyperparameter 7 is set to n 0.7 for inter-modality entailments (I < T and box = Tho and r = 1.2 for intra- entailments (T A /ox and T < boox). These scalars are leamed logarithmic scale.
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The hyperparameter for the overall loss in Eq. (1) is set = 0.2. We trained each model A100 GPUs for 500,000 iterations with a batch size of 768. For the large Vision Transformer, we used 8 A100 GPUs. We used the AdamW optimizer (Loshchilov & Hutter, 2019) with hyperparameters 81 = 0.9, 32 = 0.98. We applied weight decay of 0.2 to model parameters but not to scalar parameters. We used cosine learning rate scheduler (Loshchilov & Hutter, 2017) with maximum learning rate of 5 10-4 and a warm-up of 4,000 steps.
We found that the learned curvatures differ across factors and sometimes reach the upper bound 10.0 or the lower bound 0.1, but we did not observe a consistent pattern across factors.
C.4 BENCHMARKS
Zero-shot Image Classification. We follow the protocol in Desai et al. (2023). Each class is accompanied by a set of short text templates, such as "a photo of a fclass name)". The prediction made by selecting the class whose text templates are closest on average to the image in the embedding space. We summarize the datasets below.
ImageNet (Russakovsky et al., 2015): A large-scale dataset of diverse, everyday object categories.
Food-101 (Bossard al., 2014): fine-grained dataset of 101 different types of food dishes.
CIFAR-10 (Krizhevsky & Hinton, 2009); A dataset of low-resolution natural images across 10 general object classes.
CIFAR-100 (Krizhevsky & Hinton, 2009): Similar to CIFAR-10, but with 100 fine-grained object classes.
CUB-2011 (Wah et al., 2011): fine-grained dataset for the identification of 200 bird species.
SUN397 (Xiao et al., 2010): A large-scale scene recognition dataset with 397 scene categories. Stanford Cars (Krause et al., 2013); A fine-grained dataset of cars, annotated with make, model, and year.
FGVC-Aireraft (Maji et al., 2013): A fine-grained dataset for aircraft model recognition.
DTD (Cimpoi et al., 2014): The Describable Textures Dataset for texture recognition.
Oxford-IITT Pets (Parkhi et al., 2012): A fine-grained dataset of 37 different pet breeds.
Caltech-101 (Li al., 2004): One of the classic object recognition datasets with 101 categories.
Flowers-102 (Nilsback & Zisserman, 2008): A fine-grained dataset for the classification of 102 flower categories.
STI-10 (Coates et al., 2011): An image recognition dataset inspired by CIFAR-10, but with higher resolution.
EuroSAT (Helber al., 2019): A dataset of Sentinel-2 satellite images for land use and land cover classification.
RESISC45 (Cheng et al., 2017): A benchmark for Remote Sensing Image Scene Classification (RESISC).
Country211 (Radford et al., 2021): A dataset for predicting the country of origin from pho- tographs.
Zero-shot Image and Text Retrieval. In text-to-image retrieval, given a text query, the model retrieves the nearest images in the embedding space, and vice versa in image-to-text retrieval. Please refer to the detailed protocol in Desai et al. (2023). We summarize the datasets used as follows.
COCO (Lin et al., 2014): A large-scale dataset of complex everyday scenes with rich annotations.
Flickr30K (Young et al., 2014; Karpathy & Li, 2015): dataset of images from the Flickr website, each paired with five descriptive captions.
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Table 6: Results with different model sizes.
W/ booes
Hierarchical Classification
VI-CheokT C.hhhkk.ist-OOie Object
WordNet
Localion
Size
TIE(4) LCA(()
J(T)
(x
(1)
Center
Mid
Margin
Large
Medium
Small
CLIP
4,126
2.445
0.7530
0.8289
0.8314
61.83
62.23
60.33
63.30
61.23
59.07
VIT
MERU
4.189
2.438
0.7486
0.8278
0.8265
60.90
59.20
58.00
62.20
60.00
58.70
S/16
CLCCLP
3.669
2.231
0.7818
0,8514
0.8507
69.67
68.17
67.67
72.35
66.13
68.10
PHyCLIP
3.715
2.241
0.7778
0.8492
0.8476
73.33
71.37
72.00
75.27
70.77
CLIP
3,750
2.276
0.7774
0.8471
0.8483
61.90
60.30
60.40
63.87
60.17
58.23
CLIP
3,736
2.279
0.7784
0,8473
0.8501
61.93
59.33
60.83
63.70
60.80
58.07
VIT B/16
MERU
3.815
2.294
0.7733
0.8454
0.8450
61.27
59.03
58.97
63.97
57.70
56.07
MERU
3,802
2.289
0.7740
0.8457
0.8455
61,03
58.47
58.73
62.63
58.70
56.47
HyCoLLIP
3,319
2,092
0.8043
0.8676
U.6661
70.43
69.50
67.80
72.57
66.13
67.20
PHyCLIP
3.294
2,083
0.8684
0.8672
71.20
73.73
3,480
2.173
0.
0.8595
0.8613
64.57
VIT
MERU
3.571
2.191
0.7891
0.8565
0.85
62. 10
58.00
57.23
63.93
57.83
54.57
L/16
HyCoCLIP
3,113
2.012
0.8175
0.8769
0.8753
73.03
69.83
70.23
73.77
67.53
68.43
PHYC
3,038
1.991
0.8226
0.8797
0.8793
73.60
70.90
70.07
73.73
68.07
69.27
Among methods with the backhone, the best and second-best performances emphasized by bold fonts and underlines, respectively.
Hierarchical Classification. This task was introduced in Russakovsky et al. (2015), and we used the implementation in Pal et al. (2025). The class labels are enriched by WordNet (Miller, 1995), and the embeddings of class labels are obtained in the same way as the zero-shot image classification task. Errors between predicted and true classes are measured using the WordNet graph with unit-length edges. Tree Induced Error (TIE) is the distance between the nodes corresponding to predicted and true classes. Lowest Common Ancestor (LCA) error is the maximum of the distances from predicted and true classes to their LCA. Jaccard similarity J. hierarchical precision PHH and hierarchical recall RH are similarities between the sets of ancestors of predicted and true classes. Intuitively, hierarchical precision Pn quantifies correctness under over werrgeneralliztton: it takes value 1 if the predicted label is the ground truth or one of its ancestors in the taxonomy. Conversely, hierarchical recall RS quantifies correctness under over-specialization: it takes value if the predicted label is the ground truth one of its descendants.
Compositional Understanding. Samples in typical multi modtl datasets are diverse enough that there are few near-duplicate image-text pairs; consequently, models insensitive detailed semantics can still perform well on retrieval tasks. To assess whether a model truly understands the compositionality of words caption, hard negative captions are generated, which are almost correct but differ in a small, targeted way to evaluate whether models can select the true caption.
In VL-CheckLis--Oject, nouns in the caption are replaced. Because the difficulty varies with the replaced object's 's location (center/mid/margin) and size (smal//mcduum/large) in image, results are reported separately for each subset.
In SugarCrepe, three operations (replace, swap, and add) are applied to objects. attributes, and relations. Replace- Obj similar to VL-CheckList-Obpe Object. Swap exchanges roles or pairings. In Swap-Oby Objj the model must correctly resolve agent-action combinations. Add introduces nouns or adjectives that were absent from the original caption.
D ADDITIONAL RESULTS AND VISUALIZATIONS
D.1 ADDITIONAL EXPERIMENTAL RESULTS
We summarized the results of single runs with the small and large Vision Transformers as the image encoder (Dosovitskiy et al., 2021; Chen et al., 2021: Touvron al., 2021) in Table 6. As the model size increases, the overall performance improves in most cases. Nevertheless, PHyCLIP remains the best or at least competitive across all evaluation metrics for hierarchy and compositionality.
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bicycle
boy and bicycle
03
(b1) Images retrieved by "max" of the single-concepl prompts.
Factor
Images with Largest Norms in HG
Images with Largest Norms Hig
(cl) Images retrieved by the conjunctive prompt.
(al) Embedding norms of single-concept and conjunctive prompts.
37
ocean
surset
surset and ocean
033
(b2) Images retrieved by "max" of the single-concept prompts.
Factor Index i
Images Largest H
Images Largest Hij
(c2) Images retrieved by the conjunctive prompt.
(a2) Embedding norms of single-concept and conjunctive prompts.
Figure Factor-wise embeddings and retrievals. (al)(a2) Single-concept prompts activate distinct factors, and their textual composition activates the corresponding factors simultaneously. (b1)-(c1), (b2)-c2) "max" of the single-concept prompts retrieves images similarly to the textual compositions. See also Fig.
D.2 ADDITIONAL VISUALIZATIONS
In this section, provide additional visualizations that complement Section 4.3. We embed each word using the single-concept prompt, "a photo of a {word)" and the conjunctive prompt, which is the textual composition of words, "a photo of a {word 1} and (word 2)."
Figure 6 visualizes factor-wise embedding norms of single-concept and conjunctive prompts. In (al), the "boy" embedding activates factor i 51, which is also activated by various human images, indicating that this factor captures humans; the "bicycle" embedding activates factor = 4, associated with bicycles and wheels. The conjunctive prompt "boy and bicycle" activates both factors and = 51. In (a2), the "sunset" embedding activates factor = 14, which captures family of skies, whereas the "ocean" embedding activates factor 36, which captures family of natural landscapes. The conjunctive prompt "sunset and ocean" activates both factors i 14 and 36.
The contrastive loss (InfoNCE loss) uses the distance summed over all factors, so it allows embeddings to gather at the origin in factor if they are sufficiently distant from each other in another factor. As shown in Eq. (13), the aperture the hyperbolic entailment cone increases as the distance from the origin decreases, up maximum of 180 degrees. Consequently, near the origin, child instance be embedded anywhere within half of the space without incurring entailment loss. These mechanisms allow instances to effectively "turn off" some hyperbolic factors. We also considered modification in which the cone aperture could reach 360 degrees to completely eliminate the penalty, but since the original setting already behaved well in experiments, decided not to modify this component.
Figure (b)-(c), Figure (bl)-(c1) and (b2)-(c2) show top-10 GRIT images retrieved using the factor-wise "max" of single-concept prompts and conjunctive prompts. Specifically, embed two
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Published as a conference paper at ICLR 2026
single-concept prompts (e.g.. "a photo of dog" and "a photo of car") X, (ara) (1)
and X, = ( ,...., Th ()),, and then we construct a new embedding X max((a.0) by selecting, for each factor, the factor-wise embedding with the larger between two single-concept prompts, i.e, we take
X max (x (1) max(a.b)) maxfa..))
(k)
with a (4)
max(a,,) arg || for I...., k.
me (w x
Then, the factor-wise norms satisfy | maxx(a,6) |Ia: max [1 lay | laze) If each factor were bit {0, 1), this operation would reduce the union operation or the logical OR for Boolean algebra. If each factor were a real number R, it coincides with an element-wise max, examined in order embeddings (Vendrov et al., 2016). The retrieval results by both methods appropriate in most cases and often overlap. Concepts specified in the prompt are embedded with large norms in factors that capture their corresponding concept families, whereas unspecified concepts are represented with small norms. Consequently, by retaining only the high-norm factors, we can compose concepts without corrupting the semantics of the original prompts. These results suggest that PHyCLIP expresses cross-family composition in manner analogous to Boolean algebra and order embeddings.
THE USE OF LARGE LANGUAGE MODELS.
We used ChatGPT and GitHub Copilot as assistance tools for polishing the manuscript and imple- menting the experimental code. We did not use large language models for research ideation or for proofs.
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Reviews:
[{"review_id": "NZ9Ucf6hTX", "reviewer": "Reviewer_Rpq7", "summary": "Hyperbolic embeddings capture hierarchy and partial orders via inclusion. These approaches have been used to train vision-language models such as MERU and HyCoCLIP, which capture concept hierarchies. On the other hand, these hyperbolic embeddings lack a canonical composition operation. To solve this problem, the paper introduces PHyCLIP, which uses an $l_{1}$ product metric to unify hierarchy and composition of concepts. Results on zero-shot classification, image-text retrieval, and compositional benchmarks show that PHyCLIP outperforms methods using hyperbolic representations.", "questions": "See weaknesses.", "limitations": "", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The paper is well written. The related work section is comprehensive and up to date.\n\nThe method is simple to implement. The work builds on HyCoCLIP and shows that using an $l_{1}$ product to unify the hierarchy and composition of concepts improves results across all benchmark tasks.", "weaknesses": "**Missing Baseline.**\nThe paper does not compare PHyCLIP to CLIP fine-tuned on the GRIT dataset. Furthermore, fine-tuning CLIP could, in fact, improve performance on the compositionality benchmark tasks. In SugarCrepe [a], Table 5 shows much higher performance for fine-tuned CLIP. This is a strong reason to include fine-tuned CLIP as a baseline for fair comparison.\n\n**Compositionality.**\nTable 3 shows improvements for PHyCLIP, but not by a significant margin over CLIP (approximately 1.4 points on average), and only marginal improvements over CLIP on SugarCrepe. This is not a strong result for a method that aims to improve compositionality in vision-language models.\n\n**Sensitivity to k.**\nIn Table 4, changing the number of factors (k) can significantly reduce performance. However, there isn’t a clear guideline for choosing this hyperparameter.\n\n\n**References.**\n\n[a] SugarCrepe: Fixing Hackable Benchmarks for Vision-Language Compositionality. NeurIPS 23.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "5SgVOBMGhD", "reviewer": "Reviewer_LzC9", "summary": "The paper proposes PHyCLIP, which employs an ℓ1-product metric on a Cartesian product of hyperbolic factors to jointly capture hierarchical structures within concept families and compositionality across them. While the theoretical motivation is interesting and experiments are comprehensive, critical design choices lack sufficient justification and the improvements are marginal.", "questions": "1. Do ℓ1’s gains persist under a full ℓp sweep with ≥3 seeds?\n2. How do factorized Euclidean/order/box models with ℓ1 compare at equal total dimension?\n3. Is the ℓ2 baseline a true Riemannian product distance implementation? Please provide the formula/code.\n4. What is the quality-vs-cost curve as k increases (1, 8, 64, 128)?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "1. Clear design hypothesis. The paper explicitly assigns intra-family hierarchy to hyperbolic factors and inter-family composition to ℓ1 aggregation. The overview (Fig. 2, p.2) and factor-wise norm plots/activations (Figs. 4–6) align qualitatively with this hypothesis.\n2. The method is tested on classification, retrieval, hierarchical metrics, and composition benchmarks. PHyCLIP is generally competitive and often best, with notable gains on VL-CheckList–Object and small but consistent improvements on several other metrics (Tables 1–3).\n3. Interpretability hints. Factor-wise visualizations (pp.8, 21–22) show families (e.g., mammals vs. vehicles) specializing to different factors and conjunctions activating multiple factors—matching the intended Boolean-like behavior.\n4. Broad evaluation. The method is tested on classification, retrieval, hierarchical metrics, and composition benchmarks. PHyCLIP is generally competitive and often best, with notable gains on VL-CheckList–Object and small but consistent improvements on several other metrics (Tables 1–3).", "weaknesses": "1. Unconvincing ℓ1 justification \n- Prop. 1 shows Boolean lattices embed in ℓ1, but the model uses continuous hyperbolic factors, not discrete bits\n- Missing: Why not weighted combinations or learned aggregation functions?\n2. No statistical validation\n- Single runs without confidence intervals\n- Improvements often 1-2%, likely within noise\n3.  Implementation opacity\n- Curvature initialization/learning dynamics unexplained\n- \"ℓ2-product metric\" undefined mathematically\n4. Marginal gains don't justify complexity\n- Average improvement ~1% over HyCoCLIP\n- 64× more distance computations\n- 64× more curvature parameters\n- Memory/compute overhead unreported", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Atgx0bwowb", "reviewer": "Reviewer_iHKL", "summary": "This paper proposes PHyCLIP  to tackle a central limitation of contrastive vision–language models: representing both hierarchical “is-a” structure within a concept family and cross-family compositionality in a single embedding space. The proposed model PHyCLIP, which embeds images and texts into a Cartesian product of hyperbolic factors and endows that product with an ℓ₁-product metric. The design strives for a clean division of labor: intra-family hierarchies emerge inside individual hyperbolic factors, while cross-family composition is captured by the ℓ₁ aggregation across factors, in analogy with Boolean conjunction. Empirically, PHyCLIP is trained (from scratch) on the GRIT corpus with region-level supervision and evaluated on zero-shot classification, image–text retrieval, hierarchical classification, and compositional understanding; the model is reported to outperform single-space Euclidean or single-space hyperbolic baselines and to yield interpretable factor activations aligned with concept families.", "questions": "On the data side I am curious about if the authors observe any systematic sensitivity to noisy region phrases or ambiguous attributes, and could factor-wise uncertainty or gating mitigate this?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "To the best of my knowledge, the idea of representing vision–language concepts in a Cartesian product of hyperbolic factors with an ℓ₁-product metric is a clean and original way to reconcile hierarchical “is-a” structure (within factors) with cross-family compositionality (across factors) in the context of vision-language models. I think the theory–architecture fit of this paper is very strong: hyperbolic factors give you room for tree-like taxonomies, while ℓ₁ aggregation behaves like a sparse, Boolean-style conjunction that naturally penalizes missing parts in a composition. \n\nI also like that the learning objective stays simple—contrastive InfoNCE on the product metric (This is consistent with training recipes of common contrastive-based vision language models such as the baselines) plus a hyperbolic entailment-cone regularizers, which, in my view, makes comparisons fair. \n\nEmpirically, I also find the evaluation breadth convincing: zero-shot classification and retrieval improve over Euclidean and single-space hyperbolic baselines, hierarchical metrics on ImageNet+WordNet move in the right direction, and compositional checks (e.g., VL-Checklist, SugarCrepe) benefit in the settings the method is designed for.  \n\nOverall I think the contribution is solid and this work can benefit the overall community", "weaknesses": "I see the main gaps as scope and analysis, not methodology. \n\nFirst, the paper focuses on object/attribute composition; explicit relational composition (multi-object relations and their algebra) remains largely out of scope, yet those cases are where many Vision–Language systems struggle. It would be interesting to see if there are further experiments on thoes flavors of compositionality\n\nSecond, the text-retrieval side is “competitive” rather than consistently superior; If possible a targeted error analysis to separate caption ambiguity from potential factor misalignment would be very benefitial to the ML community. \n\nThird, the method’s behavior as the number of factors k grows deserves a bit more operational guidance: the results suggest a sweet spot, but it is not entirely clear how to pick k for new domains with different taxonomic granularity. \n\nFinally, because training relies on large-scale auto-annotated data, I would appreciate a brief characterization of how annotation noise (e.g., imperfect region phrases) interacts with the entailment-cone loss and with factor specialization—more as insight than as a requirement.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "njV2P50pAp", "reviewer": "Reviewer_QmqT", "summary": "This paper proposes a new model named PHyCLIP, which extends the HyCoCLIP method to a Cartesian product of hyperbolic spaces to allow for an understanding of both concept hierarchies and concept compositionality. First, the authors provide a theoretical motivation, explaining how this Cartesian product of hyperbolic spaces is capable of embedding these two types of information. This background is followed by an exposition of their PHyCLIP model. With a comprehensive suite of experiments the authors then show that their proposed model leads to significant improvements with respect to baselines such as CLIP, MERU and HyCoCLIP. Moreover, in an interesting analysis they show that the individual hyperbolic factors indeed appear to learn different concept hierarchies.", "questions": "Aside from the questions posed in the weaknesses, I have some more small questions:\n\n1. In lines 188-195 it is mentioned that \"we use standard hyperbolic embeddings (Nickel & Kiela, 2017) together with hyperbolic entailment cones (Ganea et al., 2018a). It is not completely clear to me what is meant here. Would this be the proposed approach for embedding actual trees or does this reference to the PHyCLIP method's loss presented after?\n2. Theorems 1 and 2 both use $\\mathbb{H^2}$, but I think this should be $\\mathbb{H}^d$, right?", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 4, "strengths": "- The paper proposes a solution to an important problem of hierarchical vision-language models such as MERU and HyCoCLIP, which struggle to model both compositionality and hierarchical understanding. The proposed use of Cartesian products of hyperbolic spaces seems elegant and effective, providing an important step in the right direction.\n- The theoretical background is clearly presented, well-motivated and forms a nice basis for the rest of the paper. \n- The method and its presentation are simple, yet effective.\n- The results in the extensive suite of experiments are quite convincing in my opinion.\n- To me, the analysis in Subsection 4.3 felt particularly compelling.", "weaknesses": "It feels to me that there is a slight disconnect between the theory and the implementation, due to a few minor points:\n1. The background (Theorem 2) assumes the existence of some product of trees that is to be embedded into a product of hyperbolic spaces. Here, if my understanding is correct, each tree is considered the replacement of some bit that would normally represent the presence of some concept. However, it is not completely clear from the text how this tree represents the presence or absence of a concept within the composition. Is the distance of the node to the root of the tree some form of the \"degree of presence\" of this concept? Would absence then be indicated by picking the root node? If so, that might be useful to discuss more explicitly in lines 174-181. \n2. After discussing the embedding of this product of trees, the method section dives straight into a presentation of the method. However, it seems to me that we do not actually have any known hierarchies and that we assume the model to learn these implicitly (as discussed shortly at the end of section 4). A short discussion on this point somewhere in or directly after lines 188-195 might be beneficial for the structure of Section 3 as I was not quite sure about the connection of Theorem 2 to the actual proposed method upon reading this section.\n3. The proposed loss seems to assign the same type of loss to each hyperbolic factor. However, it seems to me that this would sometimes hamper the learning of hierarchies in specific factors. For example, given an image of a dog next to a car, how should the image box containing just the car be represented in the hyperbolic factor that contains the mammal hierarchy? I would assume that this should be put close to the origin of this factor, but the loss would punish the model for such a representation. This seems like a difficult problem to address through any modifications of the loss and the analysis in 4.3 already shows that the model acts as expected despite this potential problem, but I think a discussion on this limitation should still be included somewhere. \n\nI do want to add that the theory, method and experiments individually are all convincing and that, overall, I consider these weaknesses minor compared to the strengths of the paper. \n\nLastly, a small remark: in lines 88-89 the authors mention (Sarkar, 2011; Nickel & Kiela, 2017; Ganea et al., 2018a), but given the requirement for low distortion, I think a more appropriate list would be (Sarkar, 2011; Sala et al., 2018; [1]), as the other two were not designed for low distortion and generally do not achieve it.\n\n[1] Max van Spengler, Pascal Mettes. Low-distortion and GPU-compatible Tree Embeddings in Hyperbolic Space. In _International Conference on Machine Learning (ICML)_, 2025.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)