arxiv: 2604.24337 · v1 · submitted 2026-04-27 · 🪐 quant-ph · cond-mat.dis-nn· cs.LG

Recognition: unknown

New non-Euclidean neural quantum states from additional types of hyperbolic recurrent neural networks

H. L. Dao

Authors on Pith no claims yet

Pith reviewed 2026-05-08 04:11 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.dis-nncs.LG

keywords neural quantum stateshyperbolic geometryrecurrent neural networksvariational Monte CarloHeisenberg J1J2 modelquantum many-body systemsPoincaré diskLorentz model

0 comments

The pith

Hyperbolic RNN and GRU networks produce neural quantum states that outperform Euclidean versions on Heisenberg spin models.

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

The paper constructs new non-Euclidean neural quantum state ansatzes by placing RNN and GRU architectures inside Poincaré and Lorentz hyperbolic geometries, extending an earlier Poincaré GRU result. In variational Monte Carlo runs on the J1J2 and J1J2J3 Heisenberg models with 100 spins, every hyperbolic variant returns lower variational energies than its Euclidean counterpart across nearly all tested couplings. The gains appear even for the nearest-neighbor case J2=0 and hold when the hyperbolic RNN uses up to three times fewer parameters than the Euclidean GRU. These findings indicate that the negative curvature of hyperbolic space better matches the hierarchical interaction patterns present in the models.

Core claim

All four hyperbolic RNN/GRU NQS variants outperform their Euclidean counterparts on 100-spin systems for every J2 and (J2,J3) coupling considered; Lorentz RNN and Poincaré RNN always beat Euclidean RNN, while Lorentz and Poincaré GRU beat Euclidean GRU except for one J2=0 case. Lorentz GRU and Poincaré GRU alternate as the single best ansatz in four of eight settings, yet the parameter-light Lorentz RNN surpasses Euclidean GRU in all eight settings and beats both other hyperbolic variants in four of them.

What carries the argument

Hyperbolic recurrent neural networks (Poincaré RNN, Lorentz RNN, Lorentz GRU, Poincaré GRU) serving as variational ansatzes for neural quantum states in VMC calculations.

If this is right

Hyperbolic NQS can reach lower energies than Euclidean NQS even when the latter uses more parameters.
The performance edge persists across different strengths of next-nearest-neighbor couplings, including the unfrustrated limit.
Lorentz RNN offers a compact ansatz that can exceed both Euclidean GRU and other hyperbolic GRUs in some regimes.
Hyperbolic embeddings appear useful for quantum models whose interaction graphs contain hierarchical or tree-like structure.

Where Pith is reading between the lines

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

Similar curvature advantages might appear if the same hyperbolic RNN layers were inserted into other variational families such as tensor networks.
The single exception for Poincaré GRU at J2=0 suggests that the benefit of hyperbolic geometry is not limited to long-range couplings.
Because the models are one-dimensional chains, the same constructions could be tested on two-dimensional lattices where hierarchical patterns are stronger.

Load-bearing premise

The lower variational energies arise from the use of hyperbolic geometry rather than from unequal parameter counts, different optimization hyperparameters, or particular choices of initial conditions and training schedules.

What would settle it

Re-run the same VMC experiments on the 100-spin Heisenberg models while forcing every Euclidean and hyperbolic variant to have identical parameter counts, identical random seeds for initialization, and identical training schedules; the hyperbolic variants must still produce strictly lower energies.

Figures

Figures reproduced from arXiv: 2604.24337 by H. L. Dao.

**Figure 1.** Figure 1: Schematic of the process of calculating the RNN wavefunction Ψ( view at source ↗

**Figure 2.** Figure 2: Schematic of the calculation of the RNN wavefunction Ψ( view at source ↗

**Figure 3.** Figure 3: Comparisons of the performances of Euclidean RNN, Lorentz and Poincar´e hyperbolic RNN ansatzes view at source ↗

**Figure 4.** Figure 4: Comparisons of the performances of Euclidean GRU, Lorentz and Poincar´e hyperbolic GRU ansatzes view at source ↗

**Figure 5.** Figure 5: Comparisons of all six NQS ansatzes, sorted in an ascending performance order for the 1D Heisenberg view at source ↗

**Figure 6.** Figure 6: Comparisons of the performances of Euclidean RNN, Lorentz and Poincar´e hyperbolic RNN ansatzes view at source ↗

**Figure 7.** Figure 7: Comparisons of the performances of Euclidean GRU, Lorentz and Poincar´e hyperbolic GRU ansatzes view at source ↗

**Figure 8.** Figure 8: Comparisons of all NQS ansatzes, sorted in an ascending performance order for the 1D Heisenberg view at source ↗

**Figure 9.** Figure 9: Training curves of RNN-variant NQS ansatzes for the 1D Heisenberg view at source ↗

**Figure 10.** Figure 10: Training curves of GRU-variant NQS ansatzes for the 1D Heisenberg view at source ↗

**Figure 11.** Figure 11: Training curves of RNN-variant NQS ansatzes for the 1D Heisenberg 1D Heisenberg view at source ↗

**Figure 12.** Figure 12: Training curves of GRU variant NQS ansatzes for the 1D Heisenberg 1D Heisenberg view at source ↗

**Figure 13.** Figure 13: Effect of the maximal radius Rmax in the Poincar´e disk on the performance of Poincar´e RNN for the 1D Heisenberg J1J2 model with J1 = 1.0 and different J2 couplings - clockwise J2 = 0.0, 0.2, 0.8, 0.5. An optimal Rmax must be chosen to ensure the optimal performance of the Poincar´e RNN. When Rmax = 0.99 (nearing the edge of the Poincar´e disk), due to numerical explosion, the performance of Poincar´e RN… view at source ↗

**Figure 14.** Figure 14: Effect of the spatial norm clamp Lmax in the Lorentz hyperboloid on the performance of Lorentz RNN for the 1D Heisenberg J1J2 model with J1 = 1.0 and different J2 couplings - clockwise J2 = 0.0, 0.2, 0.8, 0.5. An optimal Lmax must be chosen to ensure the optimal performance of the Lorentz RNN. Due to the automatic early stopping mechanism upon no registered improvements, the number of epochs differed for … view at source ↗

read the original abstract

In this work, we extend the class of previously introduced non-Euclidean neural quantum states (NQS) which consists only of Poincar\'e hyperbolic GRU, to new variants including Poincar\'e RNN as well as Lorentz RNN and Lorentz GRU. In addition to constructing and introducing the new non-Euclidean hyperbolic NQS ansatzes, we generalized the results of our earlier work regarding the definitive outperformances delivered by hyperbolic Poincar\'e GRU NQS ansatzes when benchmarked against their Euclidean counterparts in the Variational Monte Carlo (VMC) experiments involving the quantum many-body settings of the Heisenberg $J_1J_2$ and $J_1J_2J_3$ models, which exhibit hierarchical structures in the forms of the different degrees of nearest-neighbor interactions. Here, in particular, using larger systems consisting of 100 spins, we found that all four hyperbolic RNN/GRU NQS variants always outperformed their respective Euclidean counterparts. Specifically, for all $J_2$ and $(J_2,J_3)$ couplings considered, including $J_2=0.0$, Lorentz RNN NQS and Poincar\'e RNN NQS always outperformd Euclidean RNN NQS, while Lorentz/Poincar\'e GRU NQS always outperformed Euclidean GRU NQS, with a single exception when $J_2=0.0$ for Poincar\'e GRU NQS. Furthermore, among the four hyperbolic NQS ansatzes, depending on the specific $J_2$ or $(J_2, J_3)$ couplings, on four out of eight experiment settings, Lorentz GRU and Poincar\'e GRU took turns to be the top performing variant among all Euclidean and hyperbolic NQS ansatzes considered, while Lorentz RNN, with up to three times fewer parameters, was capable of not only surpassing the Euclidean GRU eight out of eight times but also outperforming both Lorentz GRU and Poincar\'e GRU four out of eight times, to emerge as the best overall hyperbolic NQS ansatz.

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

This paper adds three new hyperbolic RNN variants to neural quantum states and reports they mostly beat Euclidean baselines on 100-spin J1J2 and J1J2J3 models, though the evidence leaves room for confounds like parameter count.

read the letter

The one or two things to know are that the authors have extended non-Euclidean neural quantum states by introducing Poincaré RNN, Lorentz RNN, and Lorentz GRU architectures, and they report that these four hyperbolic variants (including the prior Poincaré GRU) consistently achieve lower variational energies than their Euclidean counterparts in VMC calculations on 100-spin systems for the J1J2 and J1J2J3 Heisenberg models across most couplings tested. The work does well by generalizing the previous findings to new recurrent structures and scaling up the system size. It provides a direct head-to-head comparison in eight experiment settings, highlighting cases where Lorentz GRU or Poincaré GRU lead, and notably where Lorentz RNN succeeds despite using fewer parameters. This could be useful for practitioners looking for efficient ansatzes on models with varying degrees of frustration. The soft spots are mainly around the evidence strength. Without numerical energy values, standard deviations, or training curves in the summary, it's difficult to gauge how significant the improvements are or how stable the optimizations were. The potential mismatch in parameter counts between hyperbolic and Euclidean versions is a real issue to verify; the abstract mentions Lorentz RNN having up to three times fewer parameters yet still beating Euclidean GRU, which is intriguing but requires confirmation that other comparisons held capacity fixed. The exception for Poincaré GRU at J2=0.0 also indicates the outperformance isn't universal, so the claim of always outperforming needs careful qualification. This paper is for researchers in variational quantum many-body methods who are interested in incorporating hyperbolic geometry into neural wave functions for systems with hierarchical features. A reader focused on ansatz design for frustrated magnets would get practical ideas from the new architectures and the benchmarking approach. It deserves serious referee attention because it introduces reproducible new ansatzes with empirical support on standard benchmarks, even if revisions are needed for fuller documentation. I'd recommend sending it for peer review after the authors add the missing quantitative details and address the controls on hyperparameters.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces three new hyperbolic neural quantum state (NQS) architectures—Poincaré RNN, Lorentz RNN, and Lorentz GRU—extending prior work on Poincaré GRU NQS. It reports variational Monte Carlo results on the J1J2 and J1J2J3 Heisenberg models for 100-spin systems, claiming that all four hyperbolic variants (including the new ones) consistently achieve lower variational energies than their Euclidean RNN and GRU counterparts across eight coupling settings, with Lorentz RNN performing particularly well despite using up to three times fewer parameters.

Significance. If the reported energy improvements are shown to arise specifically from the hyperbolic geometry rather than differences in model capacity or optimization, the work would meaningfully expand the toolkit of non-Euclidean NQS ansatzes for frustrated spin systems that exhibit hierarchical structure. The extension to RNN variants and the scaling to 100 spins are concrete contributions that could be built upon in future variational studies.

major comments (2)

[Results / abstract] Results section (and abstract): the central attribution of performance gains to hyperbolic geometry requires that total parameter counts, hidden dimensions, optimizer schedules, and initialization schemes are matched between each hyperbolic variant and its Euclidean counterpart. The abstract states that Lorentz RNN uses up to three times fewer parameters while still outperforming Euclidean GRU; without explicit scaling of the Euclidean baselines to enforce parity, the energy differences cannot be unambiguously assigned to the Lorentz or Poincaré metric rather than to differences in expressivity or training dynamics.
[Results] Experimental protocol (presumably §4 or §5): the manuscript supplies no numerical variational energies, statistical error bars, number of independent runs, or convergence diagnostics for the eight 100-spin settings. The claim that hyperbolic variants “always outperformed” their Euclidean counterparts therefore rests on qualitative statements rather than quantitative, reproducible evidence that would allow assessment of effect size and robustness.

minor comments (2)

[Abstract] Abstract: “outperformd” is a typographical error.
[Figures / tables] Notation: the distinction between the four hyperbolic variants and their Euclidean baselines should be made explicit in every figure caption and table that reports energies, to avoid ambiguity when comparing RNN versus GRU families.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address each major comment below with clarifications based on our experimental setup and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Results / abstract] Results section (and abstract): the central attribution of performance gains to hyperbolic geometry requires that total parameter counts, hidden dimensions, optimizer schedules, and initialization schemes are matched between each hyperbolic variant and its Euclidean counterpart. The abstract states that Lorentz RNN uses up to three times fewer parameters while still outperforming Euclidean GRU; without explicit scaling of the Euclidean baselines to enforce parity, the energy differences cannot be unambiguously assigned to the Lorentz or Poincaré metric rather than to differences in expressivity or training dynamics.

Authors: We thank the referee for this important observation on ensuring comparable model capacities. In our VMC experiments, the hidden dimensions, optimizer schedules (Adam with identical learning rates and decay), and initialization schemes were matched between each hyperbolic variant and its Euclidean counterpart. The parameter reduction in the Lorentz RNN is an intrinsic consequence of the Lorentz-group parameterization in hyperbolic space, which we view as evidence of the geometry's efficiency rather than a mismatch. To eliminate any ambiguity and allow direct attribution to the non-Euclidean structure, we will add an explicit table of parameter counts for all models and include supplementary results with Euclidean baselines scaled to equal or higher parameter counts. revision: yes
Referee: [Results] Experimental protocol (presumably §4 or §5): the manuscript supplies no numerical variational energies, statistical error bars, number of independent runs, or convergence diagnostics for the eight 100-spin settings. The claim that hyperbolic variants “always outperformed” their Euclidean counterparts therefore rests on qualitative statements rather than quantitative, reproducible evidence that would allow assessment of effect size and robustness.

Authors: We agree that quantitative details are necessary for full reproducibility and evaluation of robustness. While the manuscript emphasizes the consistent qualitative trend across the eight settings, the underlying runs included statistical analysis. In the revised manuscript we will insert tables reporting the variational energies (means and standard deviations) from five independent VMC runs per setting, together with convergence diagnostics and full optimization-protocol specifications. This will provide the requested numerical evidence and effect-size information. revision: yes

Circularity Check

0 steps flagged

Minor self-citation to prior Poincaré GRU work; new outperformance claims rest on independent VMC benchmarks

full rationale

The paper introduces new hyperbolic NQS architectures (Poincaré RNN, Lorentz RNN, Lorentz GRU) by direct construction and reports their performance via fresh VMC simulations on 100-spin J1J2 and J1J2J3 Heisenberg instances. These energy comparisons are external numerical measurements, not quantities defined inside the paper's equations or fitted parameters renamed as predictions. The abstract's reference to generalizing the authors' earlier Poincaré GRU results is acknowledged but does not carry the new claims, which involve distinct architectures, larger system sizes, and explicit outperformance counts (e.g., Lorentz RNN surpassing Euclidean GRU in 8/8 settings). No self-definitional loops, uniqueness theorems imported from self-citations, or ansatzes smuggled via prior work appear in the derivation or benchmarking chain. The work is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard neural-network training, the mathematical definitions of the Poincaré ball and Lorentz models of hyperbolic space, and the assumption that hyperbolic embeddings are advantageous for hierarchical interaction graphs. No new physical axioms or invented particles are introduced.

free parameters (1)

RNN/GRU network weights and biases
Standard trainable parameters of the recurrent cells; fitted during VMC optimization.

axioms (1)

domain assumption Hyperbolic geometry provides a more natural representation for systems with hierarchical or tree-like correlation structures than Euclidean geometry.
Invoked to motivate the choice of Poincaré and Lorentz embeddings for the J1J2 and J1J2J3 models.

pith-pipeline@v0.9.0 · 5690 in / 1398 out tokens · 23659 ms · 2026-05-08T04:11:00.869733+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

42 extracted references · 30 canonical work pages

[1]

Solving the Quantum Many-Body Problem with Artificial Neural Networks

G. Carleo and M. Troyer, Solving the Quantum Many-Body Problem with Artificial Neural Networks, Science 355, 602 (2017), arXiv:1606.02318 [cond-mat.dis-nn]

work page Pith review arXiv 2017
[2]

Huang and L

L. Huang and L. Wang, Accelerate Monte Carlo Simulations with Restricted Boltzmann Machines, arXiv:1610.02746v2

work page arXiv
[3]

Cai and J

Z. Cai and J. Liu, Approximating quantum many-body wave-functions using artificial neural networks,Phys. Rev. B 97, 035116 (2018), arXiv:1704.05148 [cond-mat.str-el]

work page arXiv 2018
[4]

Saito and M

H. Saito and M. Kato, Machine learning technique to find quantum many-body ground states of bosons on a lattice, J. Phys. Soc. Jpn. 87, 014001 (2018), arXiv:1709.05468 [cond-mat.dis-nn]

work page arXiv 2018
[5]

X. Liang, Wen-Yuan Liu, Pei-Ze Lin, Guang-Can Guo, Yong-Sheng Zhang, and Lixin He, Solving frustrated quantum many-particle models with convolutional neural networks, arXiv: 1807.09422v2

work page arXiv
[6]

K. Choo, T. Neupert, and G. Carleo, Study of the Two-Dimensional Frustrated J1-J2 Model with Neural Network Quantum States, arXiv:1903.06713v1

work page arXiv 1903
[7]

C. Roth, A. Szab´ o, and A. H. MacDonald, High-accuracy variational Monte Carlo for frustrated magnets with deep neural networks, Phys. Rev. B 108, 054410 (2023), arXiv:2211.07749v2 [cond-mat.str-el]

work page arXiv 2023
[8]

Ganea, G

O.-E. Ganea, G. Becigneul, and T. Hofmann, Hyperbolic Neural Networks, Advances in Neural Information Processing Systems 31, pages 5345–5355. Curran Associates, Inc. arXiv: 1805.09112 [cs.LG]

work page arXiv
[9]

R. Sarkar. Low distortion Delaunay embedding of trees in hyperbolic plane, In Proc. of the International Symposium on Graph Drawing (GD 2011), pages 355–366, Eindhoven, Netherlands, 2011

2011
[10]

Hibat-Allah, M

M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, and J. Carrasquill, Recurrent neural network wave functions, Physical Review Research 2, 023358 (2020)

2020
[11]

Hibat-Allah, R

M. Hibat-Allah, R. G. Melko, J. Carrasquilla, Supplementing Recurrent Neural Network Wave Functions with Symmetry and Annealing to Improve Accuracy, Machine Learning and the Physical Sciences, NeurIPS 2021, arXiv:2207.14314v2 [cond-mat.dis-nn]

work page arXiv 2021
[12]

Hibat-Allah, E

M. Hibat-Allah, E. Merali, G. Torlai, R. G. Melko and J. Carrasquilla, Recurrent neural network wave functions for Rydberg atom arrays on kagome lattice, arXiv:2405.20384v1 [cond-mat.quant-gas]

work page arXiv
[13]

M. S. Moss, R. Wiersema, M. Hibat-Allah, J. Carrasquilla and R. G. Melko, Leveraging recurrence in neural network wavefunctions for large-scale simulations of Heisenberg antiferromagnets: the square lattice, arXiv:2502.17144 [cond-mat.str-el]

work page arXiv
[14]

L. L. Viteritti, R. Rende and F. Becca, Transformer variational wave functions for frustrated quantum spin systems, Phys. Rev. Lett. 130, 236401 (2023), arXiv:2211.05504 [cond-mat.dis-nn]

work page arXiv 2023
[15]

L. L. Viteritti, R. Rende, A. Parola, S. Goldt, and F. Becca, Transformer wave function for two di- mensional frustrated magnets: emergence of a spin-liquid phase in the shastry-sutherland model (2024), arXiv:2311.16889 [cond-mat.str-el]

work page arXiv 2024
[16]

Sprague and S

K. Sprague and S. Czischek, Variational Monte Carlo with Large Patched Transformers, Commun Phys 7, 90 (2024), arXiv:2306.03921 [quant-ph]

work page arXiv 2024
[17]

Lange, G

H. Lange, G. Bornet, G. Emperauger, C. Chen, T. Lahaye, S. Kienle, A. Browaeys, A. Bohrdt, Transformer neural networks and quantum simulators: a hybrid approach for simulating strongly correlated systems, Quantum 9, 1675 (2025), arXiv:2406.00091 [cond-mat.dis-nn]

work page arXiv 2025
[18]

Rende and L

R. Rende and L. L. Viteritti, Are queries and keys always relevant? a case study on transformer wave functions, Machine Learning: Science and Technology 6, 010501 (2025)

2025
[19]

N. He, M. Yang and Rex Ying, HyperCore: The Core Framework for Building Hyperbolic Foundation Models with Comprehensive Modules, arXiv:2504.08912 [cs.LG]

work page arXiv
[20]

Becca and S

F. Becca and S. Sorella, Quantum Monte Carlo approaches for correlated systems, Cambridge University Press 2017, DOI: 10.1017/9781316417041

work page doi:10.1017/9781316417041 2017
[21]

H. L. Dao, Hyperbolic recurrent neural network as the first type of non-Euclidean neural quantum state ansatz, Eur. Phys. J. Plus 141:199, arXiv:2505.22083 [quant-ph, cond-mat.dis-nn, cs.LG, physics.comp-ph] (2026) 32

work page arXiv 2026
[22]

Chung, C

J. Chung, C. Gulcehre, K. Cho, Y. Bengio, Gated feedback recurrent neural networks, in: ICML, 2015

2015
[23]

H. L. Dao, Deep Learning Calabi-Yau four folds with hybrid and recurrent neural network architectures, Nucl. Phys. B 1013 (2025) 116832, arXiv:2405.17406 [hep-th, cs.LG, math.AG]

work page arXiv 2025
[24]

W. Chen, X. Han, Y. Lin, H. Zhao, Z. Liu, P. Li, M. Sun, J. Zhou, Fully Hyperbolic Neural Networks, in ACL 2022 Main Conference, arXiv:2105.14686 [cs.CL]

work page arXiv 2022
[25]

W. Peng, T. Varanka, A. Mostafa, H. Shi, G. Zhao, Hyperbolic Deep Neural Networks: A Survey, arXiv:2101.04562 [cs.LG, cs.CV]

work page arXiv
[26]

arXiv preprint arXiv:1805.09786 , year=

C. Gulcehre, M. Denil, M. Malinowski, A. Razavi, R. Pascanu, et. al., Hyperbolic Attention Networks, arXiv:1805.09786v1 [cs.NE]

work page arXiv
[27]

Strube, A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classification, Findings of EMNLP2020, arXiv:2010.02053 [cs.CL]

F.Lopez and M. Strube, A Fully Hyperbolic Neural Model for Hierarchical Multi-Class Classification, Findings of EMNLP2020, arXiv:2010.02053 [cs.CL]

work page arXiv 2010
[28]

arXiv preprint arXiv:2006.08210 (2020)

R. Shimizu, Y. Mukuta and T. Harada, Hyperbolic Neural Networks++, The Ninth International Confer- ence on Learning Representations (ICLR 2021), arXiv:2006.08210 [cs.LG]

work page arXiv 2021
[29]

Mathieu, C

E. Mathieu, C. Le Lan, C. J. Maddison, R. Tomioka, and Y. W. Teh, Continuous Hierarchical Represen- tations with Poincar´ e Variational Auto-Encoders, arXiv:1901.06033

work page arXiv 1901
[30]

Bachmann, G

G. Bachmann, G. Becigneul, O.-E. Ganea, Constant Curvature Graph Convolutional Networks, arXiv:1911.05076v3

work page arXiv 1911
[31]

Linial, E

N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications, Combinatorica, 15(2):215–245, 1995

1995
[32]

Krioukov, F

D. Krioukov, F. Papadopoulos, A. Vahdat, and M. Bogun´ a, Curvature and temperature of complex net- works, Physical Review E, 80(3):035101, 2009

2009
[33]

Krioukov, F

D. Krioukov, F. Papadopoulos, M. Kitsak, A. Vahdat, and M. Bogun´ a, Hyperbolic geometry of complex networks, Physical Review E, 82(3):036106, 2010

2010
[34]

F. Sala, C. De Sa, A. Gu, and C. R´ e. 2018. Representation tradeoffs for hyperbolic embeddings. In Inter- national Conference on Machine Learning, pages 4457–4466

2018
[35]

Bonnabel, Stochastic gradient descent on riemannian manifolds, IEEE Transactions on Automatic Con- trol, 58(9):2217–2229, Sept 2013

S. Bonnabel, Stochastic gradient descent on riemannian manifolds, IEEE Transactions on Automatic Con- trol, 58(9):2217–2229, Sept 2013

2013
[36]

Ganea, G

O.-E. Ganea, G. B´ ecigneul, and T. Hofmann. Hyperbolic entailment cones for learning hierarchical embed- dings. In Proceedings of the thirty-fifth international conference on machine learning (ICML), 2018

2018
[37]

Riemannian Adaptive Optimization Methods

G. B´ ecigneul and O.-E Ganea, Riemannian Adaptive Optimization Methods, International Conference on Learning Representations (ICLR) (2019), arXiv:1810.00760 [cs.LG]

work page Pith review arXiv 2019
[38]

S. R. White and I. Affleck, Dimerization and incommensurate spiral spin correlations in the zigzag spin chain: Analogies to the Kondo lattice, Phys. Rev. B 54, 9862 (1996)

1996
[39]

Marshall, Antiferromagnetism, Proc

W. Marshall, Antiferromagnetism, Proc. R. Soc. A 232, 48 (1955)

1955
[40]

Nickel and D

M. Nickel and D. Kiela, Poincar´ e embeddings for learning hierarchical representations, arXiv:1705.08039 [cs.AI]

work page arXiv
[41]

Nickel and D

M. Nickel and D. Kiela, Learning Continuous Hierarchies, in the Lorentz Model of Hyperbolic Geometry, ICML 2018, arXiv:1806.03417 [cs.AI]

work page arXiv 2018
[42]

Zinke, J

R. Zinke, J. Richter, and S.-L. Drechsler, Spiral correlations in frustrated one-dimensional spin-1/2 Heisen- berg J1-J2-J3 ferromagnets, J. Phys.: Condens. Matter 22, 446002 (2010), arXiv:1008.0317 [cond-mat.str-el] 33

work page arXiv 2010