RegD: Hierarchical Embeddings via Dissimilarity between Arbitrary Euclidean Regions

Hui Yang; Jiaoyan Chen

arxiv: 2501.17518 · v3 · submitted 2025-01-29 · 💻 cs.LG · cs.AI

RegD: Hierarchical Embeddings via Dissimilarity between Arbitrary Euclidean Regions

Hui Yang , Jiaoyan Chen This is my paper

Pith reviewed 2026-05-23 04:36 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords hierarchical embeddingsEuclidean embeddingsregion-based representationsdepth-based dissimilarityhyperbolic geometry emulationontology embeddingsexponential growtharbitrary geometric regions

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The pith

RegD embeds hierarchies using arbitrary Euclidean regions such as boxes or balls by defining a depth-based dissimilarity that reproduces hyperbolic exponential growth.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents RegD as a flexible framework for embedding hierarchical data entirely in Euclidean space while supporting arbitrary geometric regions as representations. It formally proves that incorporating depth-based dissimilarity between these regions allows the method to emulate key hyperbolic geometry properties including exponential growth and hierarchy preservation. This addresses limitations in prior hyperbolic approaches that rely on specific geometric constructs and struggle to integrate with broader semantic modeling techniques. A sympathetic reader would care because the approach promises greater generalizability for applications in life sciences and e-commerce without sacrificing theoretical expressiveness. The work also demonstrates consistent empirical gains on real-world datasets and extension to ontology embedding tasks.

Core claim

RegD is a Euclidean framework that supports the use of arbitrary geometric regions such as boxes and balls as embedding representations and incorporates a depth-based dissimilarity between regions to achieve hyperbolic-like expressiveness, including the ability to emulate exponential growth, while remaining entirely in Euclidean space.

What carries the argument

Depth-based dissimilarity between arbitrary Euclidean regions, which measures separation in a way that incorporates hierarchy depth to produce hyperbolic-like volume growth and structure preservation.

If this is right

Hierarchical data can be represented with flexible region shapes like boxes or balls instead of being locked to specific hyperbolic constructs.
Embeddings produced by RegD can be combined more readily with techniques that model semantic relationships beyond pure hierarchies.
The framework delivers performance improvements over existing methods across multiple real-world hierarchical datasets.
Ontology embedding tasks that extend beyond strict hierarchies become feasible within the same Euclidean region-based setup.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same dissimilarity construction might be adapted to preserve other geometric properties such as metric distortion bounds in non-hierarchical data.
Computational cost of region dissimilarity calculations could become a bottleneck when regions have complex boundaries, suggesting a need for approximation schemes.
Because the method stays in Euclidean space it may inherit existing optimization tools and hardware accelerations developed for standard vector embeddings.

Load-bearing premise

The depth-based dissimilarity between arbitrary Euclidean regions can be defined and computed so that it reproduces the exponential growth and hierarchy-preserving properties of hyperbolic geometry for any choice of region shape.

What would settle it

A dataset or synthetic hierarchy where RegD embeddings using the depth-based dissimilarity fail to preserve parent-child relations or exhibit sub-exponential volume growth compared to standard hyperbolic baselines.

Figures

Figures reproduced from arXiv: 2501.17518 by Hui Yang, Jiaoyan Chen.

**Figure 2.** Figure 2: Illustration of dbd(reg1 ,reg2 ) (red) for reg2 ⊆ reg1 (left) or reg2 ̸⊆ reg1 (right). Green lines shows the inverse: dbd(reg2 ,reg1 ). reg2 reg1 reg1 reg2 reg3 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 4.** Figure 4: Mapping from balls to cones in the case of dimension 1. [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗

**Figure 5.** Figure 5: All F1 scores of 10 random runs on Mammal, WordNet noun, MCG, and Hearst. [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

read the original abstract

Hierarchical data is common in many domains like life sciences and e-commerce, and its embeddings often play a critical role. While hyperbolic embeddings offer a theoretically grounded approach to representing hierarchies in low-dimensional spaces, current methods often rely on specific geometric constructs as embedding candidates. This reliance limits their generalizability and makes it difficult to integrate with techniques that model semantic relationships beyond pure hierarchies, such as ontology embeddings. In this paper, we present RegD, a flexible Euclidean framework that supports the use of arbitrary geometric regions -- such as boxes and balls -- as embedding representations. Although RegD operates entirely in Euclidean space, we formally prove that it achieves hyperbolic-like expressiveness by incorporating a depth-based dissimilarity between regions, enabling it to emulate key properties of hyperbolic geometry, including exponential growth. Our empirical evaluation on diverse real-world datasets shows consistent performance gains over state-of-the-art methods and demonstrates RegD's potential for broader applications such as the ontology embedding task that goes beyond hierarchy.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

RegD gives a Euclidean hierarchy embedding using arbitrary regions plus a depth-based dissimilarity that is claimed to be formally proved to match hyperbolic exponential growth.

read the letter

The main thing here is that RegD keeps everything in Euclidean space but uses arbitrary regions as embeddings and a depth-based dissimilarity that the authors say they prove emulates the exponential growth and hierarchy properties of hyperbolic geometry. They also test it on ontology embedding tasks beyond pure hierarchies and report consistent gains over prior methods on real datasets. The new element is the specific pairing of flexible region shapes with this dissimilarity construction and the formal proof that it delivers hyperbolic-like expressiveness without leaving Euclidean space. That combination looks distinct from the hyperbolic and box-embedding lines cited in the abstract, and the empirical results plus the ontology extension are useful additions. The central soft spot is the proof itself. The claim that the dissimilarity reproduces the key properties for any choice of region shape is load-bearing, and the abstract gives no derivation details, so it is impossible to check for hidden assumptions or cases where the emulation breaks. The computational cost of the dissimilarity for complex regions is another practical question that is not addressed in the provided text. Nothing in the abstract or stress-test note shows an internal contradiction or unsupported step, but the proof needs to be examined directly. This paper is aimed at people working on hierarchical embeddings who want Euclidean options that can mix with other semantic models. Readers focused on alternatives to hyperbolic geometry or on ontology tasks would find the construction and results relevant. It deserves a serious referee because the formal claim plus the experimental evaluation on relevant tasks are substantial enough to warrant detailed review.

Referee Report

2 major / 2 minor

Summary. The paper introduces RegD, a Euclidean embedding method for hierarchical data that represents entities as arbitrary geometric regions (e.g., boxes, balls) and defines a depth-based dissimilarity between them. It claims a formal proof that this construction achieves hyperbolic-like expressiveness, including exponential volume growth and hierarchy preservation, while remaining in Euclidean space. Experiments on real-world datasets report consistent gains over SOTA hyperbolic and Euclidean baselines, with extension to ontology embedding tasks.

Significance. If the formal proof holds, the result would allow hierarchical embeddings to be combined with arbitrary Euclidean semantic models without switching geometries, expanding applicability to mixed hierarchy-plus-relation tasks.

major comments (2)

[§3] §3 (Depth-based dissimilarity definition and proof): The central claim that the depth-based dissimilarity reproduces exponential growth and hierarchy preservation for arbitrary region shapes rests on an unexamined construction; the manuscript must supply the full derivation (including how depth is computed for non-nested or overlapping regions) to verify that the exponential property emerges without shape-specific assumptions.
[Theorem 1] Theorem 1 (hyperbolic-like expressiveness): The proof sketch in the abstract asserts parameter-free emulation of hyperbolic volume growth, but the definition of depth appears to require a choice of reference point or ordering; this must be shown to be independent of such choices for the claim to hold for arbitrary regions.

minor comments (2)

[Table 1, Figure 2] Table 1 and Figure 2: axis labels and legend entries use inconsistent notation for the dissimilarity measure; standardize to match the definition in §3.1.
[§5.3] §5.3 (ontology embedding experiment): clarify whether the reported gains are statistically significant across the five random seeds; add standard deviations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major point below and will revise the paper to improve the clarity and completeness of the proof sections.

read point-by-point responses

Referee: [§3] §3 (Depth-based dissimilarity definition and proof): The central claim that the depth-based dissimilarity reproduces exponential growth and hierarchy preservation for arbitrary region shapes rests on an unexamined construction; the manuscript must supply the full derivation (including how depth is computed for non-nested or overlapping regions) to verify that the exponential property emerges without shape-specific assumptions.

Authors: We agree that additional detail is warranted. In the revised manuscript we will expand Section 3 with the complete, step-by-step derivation of the depth-based dissimilarity. The expanded text will explicitly define depth computation for non-nested and overlapping regions and will show that the exponential-volume property follows from the construction without requiring shape-specific assumptions. revision: yes
Referee: [Theorem 1] Theorem 1 (hyperbolic-like expressiveness): The proof sketch in the abstract asserts parameter-free emulation of hyperbolic volume growth, but the definition of depth appears to require a choice of reference point or ordering; this must be shown to be independent of such choices for the claim to hold for arbitrary regions.

Authors: We will revise the presentation of Theorem 1 to include a formal argument establishing that the depth measure is invariant under choice of reference point or ordering. The added material will demonstrate this independence directly from the definition, thereby confirming that the hyperbolic-like expressiveness result holds for arbitrary regions. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents RegD as a new construction in Euclidean space that defines a depth-based dissimilarity on arbitrary regions and formally proves it emulates hyperbolic properties such as exponential growth. No equations or claims reduce by construction to fitted parameters, prior self-citations, or renamed known results; the central expressiveness result is derived directly from the introduced dissimilarity measure and its proof, rendering the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the introduction of a depth-based dissimilarity measure whose properties are asserted to match hyperbolic geometry; no explicit free parameters are named in the abstract.

axioms (2)

domain assumption Hierarchical data can be faithfully represented by containment or overlap relations among arbitrary Euclidean regions
Implicit in the choice of boxes and balls as embedding candidates
ad hoc to paper A depth-based dissimilarity defined on Euclidean regions can reproduce exponential volume growth and hierarchy preservation
This is the load-bearing mathematical step claimed in the abstract

invented entities (1)

depth-based dissimilarity between regions no independent evidence
purpose: To measure dissimilarity while encoding hierarchy depth so that Euclidean regions emulate hyperbolic geometry
New measure introduced by the paper; no independent evidence supplied in abstract

pith-pipeline@v0.9.0 · 5690 in / 1319 out tokens · 26434 ms · 2026-05-23T04:36:41.797627+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

d_dep(reg1,reg2)=g(‖P(reg1)−P(reg2)‖_p / (f(reg1)f(reg2))) … lim reg→∅ f(reg)=0 … emulates … hyperbolic space, where the dissimilarity … grow[s] rapidly as they approach the boundary

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

[1]

URLhttp://proceedings.mlr.press/v80/ganea18a.html

PMLR, 2018b. URLhttp://proceedings.mlr.press/v80/ganea18a.html. Yuan He, Jiaoyan Chen, Hang Dong, Ian Horrocks, Carlo Allocca, Taehun Kim, and Brahmananda Sapkota. Deeponto: A python package for ontology engineering with deep learning.Semantic Web, 15(5):1991–2004, 2024a. doi: 10.3233/SW-243568. URL https://doi.org/10.3233/ SW-243568. Yuan He, Zhangdie Yu...

work page doi:10.3233/sw-243568 1991
[2]

URLhttp://proceedings.mlr.press/v80/sala18a.html

PMLR, 2018. URLhttp://proceedings.mlr.press/v80/sala18a.html. Jiaming Shen and Jiawei Han.Automated Taxonomy Discovery and Exploration. Synthesis Lectures on Data Mining and Knowledge Discovery. Springer, 2022. ISBN 978-3-031-11404-5. doi: 10. 1007/978-3-031-11405-2. URLhttps://doi.org/10.1007/978-3-031-11405-2. Jingchuan Shi, Hang Dong, Jiaoyan Chen, Zhe...

work page doi:10.1007/978-3-031-11405-2 2018
[3]

Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun

URLhttp://proceedings.mlr.press/v97/suzuki19a.html. Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. In Yoshua Bengio and Yann LeCun (eds.),4th International Conference on Learning Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings,

work page 2016
[4]

Order-Embeddings of Images and Language

URLhttp://arxiv.org/abs/1511.06361. Luke Vilnis, Xiang Li, Shikhar Murty, and Andrew McCallum. Probabilistic embedding of knowl- edge graphs with box lattice measures. In Iryna Gurevych and Yusuke Miyao (eds.),Pro- ceedings of the 56th Annual Meeting of the Association for Computational Linguistics, ACL 2018, Melbourne, Australia, July 15-20, 2018, Volume...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.18653/v1/p18-1025 2018
[5]

For any ball ball(c′, r′)⊆ball(c 2, r2) withr ′ ≤0.5·r 1, we must have: ||c1 −c ′||p p +|r 1 −r ′|p >(0.5·r 1)p

Let reg1 =ball(c 1, r1) and reg2 =ball(c 2, r2). For any ball ball(c′, r′)⊆ball(c 2, r2) withr ′ ≤0.5·r 1, we must have: ||c1 −c ′||p p +|r 1 −r ′|p >(0.5·r 1)p. Therefore, we have: ddep(reg1,ball(c ′, r′)) = ||c1 −c ′||p p +|r 1 −r ′|p f(reg1)f(ball(c ′, r′)) > (0.5·r 1)p f(reg1)f(ball(c ′, r′)) . Thus, when: f(ball(c ′, r′))< ϵ:= (0.5·r 1)p f(reg1)[ddep...

work page
[6]

For any n, M >0 , we can select n distinct vectors c1,

Let reg=ball(c, r) . For any n, M >0 , we can select n distinct vectors c1, . . . ,cn ∈ ball(c,0.5·r)such that: ∥ci −c j∥p p > δ >0for someδ >0. Similarly to item 1, we can choose ri small enough such that for any ball ball(ci, ri), we have: f(ball(c i, ri))< δ M 0.5 . Thus, we obtain: ddep(ball(ci, ri),ball(c j, rj))> δ f(ball(c i, ri))f(ball(c j, rj)) ≥...

work page
[7]

Next, we focus on the cased bd(reg1,reg 2) = 0

By definition, we have dbd(reg1,reg 2)≤0 if and only if reg1 ⊆reg 2. Next, we focus on the cased bd(reg1,reg 2) = 0. Note that ifd bd(reg1,reg 2) = 0, we must havereg 1 ⊆reg 2. Otherwise, we have dbd(reg1,reg 2) = max x2∈reg2\reg1 {d(reg1,x 2)}= 0 Therefore, for any x2 ∈reg 2, we have d(reg1,x 2) = 0, therefore x2 ⊆reg 1 (assuming reg1 is a closed set). C...

work page
[8]

Similarly, since x2 ⊆reg c 1 ⊆reg c 2 and x2 ∈reg 2, we also havex 2 ∈∂(reg 2)

Since we have x2 ⊆reg 2 ⊆reg 1, therefore, x2 ∈∂(reg 1). Similarly, since x2 ⊆reg c 1 ⊆reg c 2 and x2 ∈reg 2, we also havex 2 ∈∂(reg 2). This finishes the proof of the first case

work page
[9]

By assumption, we have dbd(reg1,reg 2) = max x2∈reg2 {−d(regc 1,x 2)}, d bd(reg1,reg ′

work page
[10]

Sincereg ′ 2 ⊆reg 2, of course we have dbd(reg1,reg ′ 2)≤d bd(reg1,reg 2)

= max x2∈reg′ 2 {−d(regc 1,x 2)}. Sincereg ′ 2 ⊆reg 2, of course we have dbd(reg1,reg ′ 2)≤d bd(reg1,reg 2). This finishes the proof of the second case. 14 Preprint A.5 GENERALPARAMETERIZEDREGIONS A set ofparameterized regionsover Euclidean Space Rn can be defined as all regions reg of the following form: reg={x∈R n :f i(x, θ)≤0,fori= 1, . . . , n}, where...

work page
[11]

For any reg1,reg 2 ∈ R and any ∆>0 ,for any subset reg′ ⊆reg 2 reg1 sufficiently small, we have ddep(reg1,reg ′)> d dep(reg1,reg 2) + ∆

work page
[12]

,regn ⊆reg such that for any distincti, j∈ {1,

For any reg∈ R , any integer n, and any M >0 , there exist subsets reg1, . . . ,regn ⊆reg such that for any distincti, j∈ {1, . . . , n}, ddep(regi,reg j)> M. Outline of proof. We sketch the main ideas, which closely follow the argument in the proof of Theorem 1. For the first statement, by selecting a sufficiently small subsetreg′ ⊆reg 2, we can ensure t...

work page arXiv 2024
[13]

In certain cases—such as using balls on the Mammal dataset with d= 2 and on the Noun dataset with d= 5 —performance is slightly lower, though still superior to that of other baseline methods

work page
[14]

SubclassOf

In some settings, performance is even improved on average, for example, when using balls on the Noun dataset withd= 10. These results indicate that the proposed method is robust across different random seeds, with consistent performance across most settings. Impact of λ In Table 5, we evaluate the performance of λ= 0 , 0.5, and 1, and report results for b...

work page 2024
[15]

The energy function is given by: E(u, v) =∥max{|c u −c v|+o v −o u,0}∥, where cu and cv denote the center vectors of the boxes, and ou and ov denote the offsets of the boxes

ELBE:Nodes u and v are embedded as boxes boxu and boxv, respectively. The energy function is given by: E(u, v) =∥max{|c u −c v|+o v −o u,0}∥, where cu and cv denote the center vectors of the boxes, and ou and ov denote the offsets of the boxes

work page
[16]

The energy function is defined as: E(u, v) = max{∥cu −c v∥+r v −r u,0}, wherec u andc v represent the centers of the balls, andr u andr v represent their radii

ELEM:Nodes u and v are embedded as balls ballu and ballv, respectively. The energy function is defined as: E(u, v) = max{∥cu −c v∥+r v −r u,0}, wherec u andc v represent the centers of the balls, andr u andr v represent their radii. Note that the original ELEM method includes a regularization term that enforces the norms ∥cu∥ and ∥cv∥ to be close to 1. Ho...

work page 2013
[17]

ELEM includes a regularization term, similar to TransE, to constrain the norm of the center of each ball to be close to 1

work page
[18]

This allows it to define box(A⊓B) as box(A)∩box(B)

ELBE handles conjunctions more effectively, as the intersection of two boxes is still a box. This allows it to define box(A⊓B) as box(A)∩box(B) . In contrast, ELEM, which uses balls, cannot handle intersections directly and must use specialized mechanisms to approximate satisfaction of the axiomA⊓B⊑B ′. To integrate RegD and τBox into ELBE or ELEM, it suf...

work page

[1] [1]

URLhttp://proceedings.mlr.press/v80/ganea18a.html

PMLR, 2018b. URLhttp://proceedings.mlr.press/v80/ganea18a.html. Yuan He, Jiaoyan Chen, Hang Dong, Ian Horrocks, Carlo Allocca, Taehun Kim, and Brahmananda Sapkota. Deeponto: A python package for ontology engineering with deep learning.Semantic Web, 15(5):1991–2004, 2024a. doi: 10.3233/SW-243568. URL https://doi.org/10.3233/ SW-243568. Yuan He, Zhangdie Yu...

work page doi:10.3233/sw-243568 1991

[2] [2]

URLhttp://proceedings.mlr.press/v80/sala18a.html

PMLR, 2018. URLhttp://proceedings.mlr.press/v80/sala18a.html. Jiaming Shen and Jiawei Han.Automated Taxonomy Discovery and Exploration. Synthesis Lectures on Data Mining and Knowledge Discovery. Springer, 2022. ISBN 978-3-031-11404-5. doi: 10. 1007/978-3-031-11405-2. URLhttps://doi.org/10.1007/978-3-031-11405-2. Jingchuan Shi, Hang Dong, Jiaoyan Chen, Zhe...

work page doi:10.1007/978-3-031-11405-2 2018

[3] [3]

Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun

URLhttp://proceedings.mlr.press/v97/suzuki19a.html. Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. In Yoshua Bengio and Yann LeCun (eds.),4th International Conference on Learning Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings,

work page 2016

[4] [4]

Order-Embeddings of Images and Language

URLhttp://arxiv.org/abs/1511.06361. Luke Vilnis, Xiang Li, Shikhar Murty, and Andrew McCallum. Probabilistic embedding of knowl- edge graphs with box lattice measures. In Iryna Gurevych and Yusuke Miyao (eds.),Pro- ceedings of the 56th Annual Meeting of the Association for Computational Linguistics, ACL 2018, Melbourne, Australia, July 15-20, 2018, Volume...

work page internal anchor Pith review Pith/arXiv arXiv doi:10.18653/v1/p18-1025 2018

[5] [5]

For any ball ball(c′, r′)⊆ball(c 2, r2) withr ′ ≤0.5·r 1, we must have: ||c1 −c ′||p p +|r 1 −r ′|p >(0.5·r 1)p

Let reg1 =ball(c 1, r1) and reg2 =ball(c 2, r2). For any ball ball(c′, r′)⊆ball(c 2, r2) withr ′ ≤0.5·r 1, we must have: ||c1 −c ′||p p +|r 1 −r ′|p >(0.5·r 1)p. Therefore, we have: ddep(reg1,ball(c ′, r′)) = ||c1 −c ′||p p +|r 1 −r ′|p f(reg1)f(ball(c ′, r′)) > (0.5·r 1)p f(reg1)f(ball(c ′, r′)) . Thus, when: f(ball(c ′, r′))< ϵ:= (0.5·r 1)p f(reg1)[ddep...

work page

[6] [6]

For any n, M >0 , we can select n distinct vectors c1,

Let reg=ball(c, r) . For any n, M >0 , we can select n distinct vectors c1, . . . ,cn ∈ ball(c,0.5·r)such that: ∥ci −c j∥p p > δ >0for someδ >0. Similarly to item 1, we can choose ri small enough such that for any ball ball(ci, ri), we have: f(ball(c i, ri))< δ M 0.5 . Thus, we obtain: ddep(ball(ci, ri),ball(c j, rj))> δ f(ball(c i, ri))f(ball(c j, rj)) ≥...

work page

[7] [7]

Next, we focus on the cased bd(reg1,reg 2) = 0

By definition, we have dbd(reg1,reg 2)≤0 if and only if reg1 ⊆reg 2. Next, we focus on the cased bd(reg1,reg 2) = 0. Note that ifd bd(reg1,reg 2) = 0, we must havereg 1 ⊆reg 2. Otherwise, we have dbd(reg1,reg 2) = max x2∈reg2\reg1 {d(reg1,x 2)}= 0 Therefore, for any x2 ∈reg 2, we have d(reg1,x 2) = 0, therefore x2 ⊆reg 1 (assuming reg1 is a closed set). C...

work page

[8] [8]

Similarly, since x2 ⊆reg c 1 ⊆reg c 2 and x2 ∈reg 2, we also havex 2 ∈∂(reg 2)

Since we have x2 ⊆reg 2 ⊆reg 1, therefore, x2 ∈∂(reg 1). Similarly, since x2 ⊆reg c 1 ⊆reg c 2 and x2 ∈reg 2, we also havex 2 ∈∂(reg 2). This finishes the proof of the first case

work page

[9] [9]

By assumption, we have dbd(reg1,reg 2) = max x2∈reg2 {−d(regc 1,x 2)}, d bd(reg1,reg ′

work page

[10] [10]

Sincereg ′ 2 ⊆reg 2, of course we have dbd(reg1,reg ′ 2)≤d bd(reg1,reg 2)

= max x2∈reg′ 2 {−d(regc 1,x 2)}. Sincereg ′ 2 ⊆reg 2, of course we have dbd(reg1,reg ′ 2)≤d bd(reg1,reg 2). This finishes the proof of the second case. 14 Preprint A.5 GENERALPARAMETERIZEDREGIONS A set ofparameterized regionsover Euclidean Space Rn can be defined as all regions reg of the following form: reg={x∈R n :f i(x, θ)≤0,fori= 1, . . . , n}, where...

work page

[11] [11]

For any reg1,reg 2 ∈ R and any ∆>0 ,for any subset reg′ ⊆reg 2 reg1 sufficiently small, we have ddep(reg1,reg ′)> d dep(reg1,reg 2) + ∆

work page

[12] [12]

,regn ⊆reg such that for any distincti, j∈ {1,

For any reg∈ R , any integer n, and any M >0 , there exist subsets reg1, . . . ,regn ⊆reg such that for any distincti, j∈ {1, . . . , n}, ddep(regi,reg j)> M. Outline of proof. We sketch the main ideas, which closely follow the argument in the proof of Theorem 1. For the first statement, by selecting a sufficiently small subsetreg′ ⊆reg 2, we can ensure t...

work page arXiv 2024

[13] [13]

In certain cases—such as using balls on the Mammal dataset with d= 2 and on the Noun dataset with d= 5 —performance is slightly lower, though still superior to that of other baseline methods

work page

[14] [14]

SubclassOf

In some settings, performance is even improved on average, for example, when using balls on the Noun dataset withd= 10. These results indicate that the proposed method is robust across different random seeds, with consistent performance across most settings. Impact of λ In Table 5, we evaluate the performance of λ= 0 , 0.5, and 1, and report results for b...

work page 2024

[15] [15]

The energy function is given by: E(u, v) =∥max{|c u −c v|+o v −o u,0}∥, where cu and cv denote the center vectors of the boxes, and ou and ov denote the offsets of the boxes

ELBE:Nodes u and v are embedded as boxes boxu and boxv, respectively. The energy function is given by: E(u, v) =∥max{|c u −c v|+o v −o u,0}∥, where cu and cv denote the center vectors of the boxes, and ou and ov denote the offsets of the boxes

work page

[16] [16]

The energy function is defined as: E(u, v) = max{∥cu −c v∥+r v −r u,0}, wherec u andc v represent the centers of the balls, andr u andr v represent their radii

ELEM:Nodes u and v are embedded as balls ballu and ballv, respectively. The energy function is defined as: E(u, v) = max{∥cu −c v∥+r v −r u,0}, wherec u andc v represent the centers of the balls, andr u andr v represent their radii. Note that the original ELEM method includes a regularization term that enforces the norms ∥cu∥ and ∥cv∥ to be close to 1. Ho...

work page 2013

[17] [17]

ELEM includes a regularization term, similar to TransE, to constrain the norm of the center of each ball to be close to 1

work page

[18] [18]

This allows it to define box(A⊓B) as box(A)∩box(B)

ELBE handles conjunctions more effectively, as the intersection of two boxes is still a box. This allows it to define box(A⊓B) as box(A)∩box(B) . In contrast, ELEM, which uses balls, cannot handle intersections directly and must use specialized mechanisms to approximate satisfaction of the axiomA⊓B⊑B ′. To integrate RegD and τBox into ELBE or ELEM, it suf...

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