1 Introduction & Motivation
Word embeddings are ubiquitous nowadays as first layers in neural network and deep learning models for natural language processing. They are essential in order to move from the discrete word space to the continuous space where differentiable loss functions can be optimized. The popular models of Glove(Pennington et al., 2014), Word2Vec (Mikolov et al., 2013b) or FastText (Bojanowski et al., 2016), provide efficient ways to learn word vectors fully unsupervised from raw text corpora, solely based on word co-occurrence statistics. These models are then successfully applied to word similarity and other downstream tasks and, surprisingly (or not (Arora et al., 2016)), exhibit a linear algebraic structure that is also useful to solve word analogy.
However, unsupervised word embeddings still largely suffer from revealing asymmetric word relations including the latent hierarchical structure of words. This is currently one of the key limitations in automatic text understanding, e.g. for tasks such as textual entailment (Bowman et al., 2015). To address this issue, (Vilnis & McCallum, 2015; Muzellec & Cuturi, 2018)
propose to move from point embeddings to probability density functions, the simplest being Gaussian or Elliptical distributions. Their intuition is that the variance of such a distribution should encode the generality/specificity of the respective word. However, this method results in losing the arithmetic properties of point embeddings (e.g. for analogy reasoning) and becomes unclear how to properly use them in downstream tasks. To this end, we propose to take the best from both worlds: we embed words as points in a Cartesian product of hyperbolic spaces and, additionally, explain how they are bijectively mapped to Gaussian embeddings with diagonal covariance matrices, where the hyperbolic distance between two points becomes the Fisher distance between the corresponding probability distribution functions (PDFs). This allows us to derive a novel principled is-a score on top of word embeddings that can be leveraged for hypernymy detection. We learn these word embeddings unsupervised from raw text by generalizing the Glove method. Moreover, the linear arithmetic property used for solving word analogy has a mathematical grounded correspondence in this new space based on the established notion of parallel transport in Riemannian manifolds. In addition, these hyperbolic embeddings outperform Euclidean Glove on word similarity benchmarks. We thus describe, to our knowledge, the first word embedding model that competitively addresses the above three tasks simultaneously. Finally, these word vectors can also be used in downstream tasks as explained byGanea et al. (2018b).
We provide additional reasons for choosing the hyperbolic geometry to embed words. We explain the notion of average -hyperbolicity of a graph, a geometric quantity that measures its ”democracy” (Borassi et al., 2015). A small hyperbolicity constant implies ”aristocracy”, namely the existence of a small set of nodes that ”influence” most of the paths in the graph. It is known that real-world graphs are mainly complex networks (e.g. scale-free exhibiting power-law node degree distributions) which in turn are better embedded in a tree-like space, i.e. hyperbolic (Krioukov et al., 2010). Since, intuitively, words form an ”aristocratic” community (few generic ones from different topics and many more specific ones) and since a significant subset of them exhibits a hierarchical structure (e.g. WordNet (Miller et al., 1990)), it is naturally to learn hyperbolic word embeddings. Moreover, we empirically measure very low average -hyperbolicity constants of some variants of the word log-co-occurrence graph (used by the Glove method), providing additional quantitative reasons for why spaces of negative curvature (i.e. hyperbolic) are better suited for word representations.
2 Related work
Recent supervised methods can be applied to embed any tree or directed acyclic graph in a low dimensional space with the aim of improving link prediction either by imposing a partial order in the embedding space (Vendrov et al., 2015; Vilnis et al., 2018; Athiwaratkun & Wilson, 2018), by using hyperbolic geometry (Nickel & Kiela, 2017, 2018), or both (Ganea et al., 2018a).
To learn word embeddings that exhibit hypernymy or hierarchical information, supervised methods (Vulić & Mrkšić, 2018; Nguyen et al., 2017) leverage external information (e.g. WordNet) together with raw text corpora. However, the same goal is also targeted by more ambitious fully unsupervised models which move away from the ”point” assumption and learn various probability densities for each word (Vilnis & McCallum, 2015; Muzellec & Cuturi, 2018; Athiwaratkun & Wilson, 2017; Singh et al., 2018).
There have been two very recent attempts at learning unsupervised word embeddings in the hyperbolic space (Leimeister & Wilson, 2018; Dhingra et al., 2018). However, they suffer from either not being competitive on standard tasks in high dimensions, not showing the benefit of using hyperbolic spaces to model asymmetric relations, or not being trained on realistically large corpora. We address these problems and, moreover, the connection with density based methods is made explicit and leveraged to improve hypernymy detection.
3 Hyperbolic spaces and their cartesian product
In order to work in the hyperbolic space, we have to choose one model, among the five isometric models that exist. We choose to embed words in the Poincaré ball . This is illustrated in Figure (a)a for , where dark lines represent geodesics. The distance function in this space is given by , being the conformal factor. We will also embed words in products of hyperbolic spaces, and explain why later on. A product of balls , with the induced product geometry, is known to have distance function . Finally, another model of interest for us is the Poincaré half-plane illustrated in Figure (d)d, with distance function . Figure 1 shows an isometry from to mapping the vertical segment to and fixing , sending the radius to .
4 Adapting GloVe
The Glove (Pennington et al., 2014) algorithm is an unsupervised method for learning word representations in the Euclidean space from statistics of word co-occurrences in a text corpus, with the aim to geometrically capture the words’ meaning and relations.
We use the notations: is the number of times word occurs in the same window context as word ; ; is the probability that word appears in the context of word . An embedding of a (target) word is written , while an embedding of a context word is written .
The initial formulation of the Glove model suggests to learn embeddings as to satisfy the equation . Since is symmetric in but is not, (Pennington et al., 2014) propose to restore the symmetry by introducing biases for each word, absorbing into ’s bias:
Finally, the authors suggest to enforce this equality by optimizing a weighted least-square loss:
where is the size of the vocabulary and down-weights the signal coming from frequent words (it is typically chosen to be , with and ).
Glove in metric spaces.
Note that there is no clear correspondence of the Euclidean inner-product in a hyperbolic space. However, we are provided with a distance function. Further notice that one could rewrite Eq. (1) with the Euclidean distance as , where we absorbed the squared norms of the embeddings into the biases. We thus replace the Glove loss by:
is a function to be chosen as a hyperparameter of the model, andcan be any differentiable distance function. Although the most direct correspondence with Glove would suggest , we sometimes obtained better results with other functions, such as (see sections 8 & 9). Note that De Sa et al. (2018) also apply to their distance matrix for hyperbolic MDS before applying PCA. Understanding why is a good choice would be interesting future work.
5 Connecting Gaussian embeddings & hyperbolic embeddings
In order to endow Euclidean word embeddings with richer information, Vilnis & McCallum (2015) proposed to represent words as Gaussians, i.e. with a mean vector and a covariance matrix111diagonal or even spherical, for simplicity., expecting the variance parameters to capture how generic/specific a word is, and, hopefully, entailment relations. On the other hand, Nickel & Kiela (2017) proposed to embed words of the WordNet hierarchy (Miller et al., 1990) in hyperbolic space, because this space is mathematically known to be better suited to embed tree-like graphs. It is hence natural to wonder: is there a connection between the two?
The Fisher geometry of Gaussians is hyperbolic.
It turns out that there exists a striking connection (Costa et al., 2015). Note that a 1D Gaussian can be represented as a point in . Then, the Fisher distance between two distributions relates to the hyperbolic distance in :
For -dimensional Gaussians with diagonal covariance matrices written , it becomes:
Hence there is a direct correspondence between diagonal Gaussians and the product space .
Fisher distance, KL & Gaussian embeddings.
The above paragraph lets us relate the word2gauss algorithm (Vilnis & McCallum, 2015) to hyperbolic word embeddings. Although one could object that word2gauss
is trained using a KL divergence, while hyperbolic embeddings relate to Gaussian distributions via the Fisher distance, let us remind that KL and define the same local geometry. Indeed, the KL is known to be related to , as its local approximation (Jeffreys, 1946). In short, if and denote two closeby probability distributions for a small , then , where is the Fisher information metric, inducing .
A benefit of representing words in (products of) hyperbolic spaces, as opposed to (diagonal) Gaussian distributions, is that one can use recent Riemannian adaptive optimization tools such as Radagrad (Bécigneul & Ganea, 2018). Note that without this connection, it would be unclear how to define a variant of Adagrad (Duchi et al., 2011) intrinsic to the statistical manifold of Gaussians. Empirically, we indeed noticed better results using Radagrad, rather than simply Riemannian sgd (Bonnabel, 2013). Similarly, note that Glove trains with Adagrad.
6 Analogies for hyperbolic/Gaussian embeddings
The connection exposed in section 5 allows us to provide mathematically grounded (i) analogy computations for Gaussian embeddings using hyperbolic geometry, and (ii) hypernymy detection for hyperbolic embeddings using Gaussian distributions.
A common task used to evaluate word embeddings, called analogy, consists in finding which word is to the word , what the word is to the word . For instance, queen is to woman what king is to man. In the Euclidean embedding space, the solution to this problem is usually taken geometrically as . Note that the same is also to , what is to .
How should one intrinsically define “analogy parallelograms” in a space of Gaussian distributions? Note that Vilnis & McCallum (2015) do not evaluate their Gaussian embeddings on the analogy task, and that it would be unclear how to do so. However, since we can go back and forth between (diagonal) Gaussians and (products of) hyperbolic spaces as explained in section 5, we can use the fact that parallelograms are naturally defined in the Poincaré ball, by the notion of gyro-translation (Ungar, 2012, section 4). In the Poincaré ball, the two solutions and are respectively generalized to
The formulas for these operations are described in closed-forms in appendix C, and are easy to implement. The fact that and differ is due to the curvature of the space. For evaluation, we chose a point located on the geodesic between and for some ; if , this is called the gyro-midpoint and then , which is at equal hyperbolic distance from as from . We explain in appendix A.2 how to select , and that continuously deforming the Poincaré ball to the Euclidean space (by sending its radius to infinity) lets these analogy computations recover their Euclidean counterparts, which is a nice sanity check.
7 Towards a principled score for entailment/hypernymy
We now use the connection explained in section 5 to introduce a novel principled score that can be applied on top of our unsupervised learned Poincaré Glove embeddings to address the task of hypernymy detection, i.e. to predict relations of type is-a(v,w) such as is-a(dog, animal). For this purpose, we first explain how learned hyperbolic word embeddings are mapped to Gaussian embeddings, and subsequently we define our hypernymy score.
Invariance of distance-based embeddings to isometric transformations.
The method of Nickel & Kiela (2017)
uses a heuristic entailment score in order to predict whetheris-a , defined for as is-a, based on the intuition that the Euclidean norm should encode generality/specificity of a concept/word. However, such a choice is not intrinsic to the hyperbolic space when the training loss involves only the distance function. We say that training is intrinsic to , i.e. invariant to applying any isometric transformation to all word embeddings (such as hyperbolic translation). But their “is-a” score is not intrinsic, i.e. depends on the parametrization. For this reason, we argue that an isometry has to be found and fixed before using the trained word embeddings in any non-intrinsic manner, e.g. to define hypernymy scores. To discover it, we leverage the connection between hyperbolic and Gaussian embeddings as follows.
Mapping hyperbolic embeddings to Gaussian embeddings via an isometry.
For a 1D Gaussian representing a concept, generality should be naturally encoded in the magnitude of . As shown in section 5, the space of Gaussians endorsed with the Fisher distance is naturally mapped to the hyperbolic upper half-plane , where the variance corresponds to the (positive) second coordinate in . Moreover, as shown in section 3, can be isometrically mapped to , where the second coordinate corresponds to the open vertical segment in . However, in , any (hyperbolic) translation or any rotation w.r.t. the origin is an isometry222See http://bulatov.org/math/1001 for intuitive animations describing hyperbolic isometries.. Hence, in order to map a word to a Gaussian via , we first have to find the correct isometry. This transformation should align with whichever geodesic in encodes generality. For simplicity, we assume it is composed of a centering and a rotation operations in . Thus, we start by identifying two sets and of potentially generic and specific words, respectively. For the re-centering, we then compute the means and of and respectively, and , and Möbius translate all words by the global mean with the operation . For the rotation, we set , and rotate all words so that is mapped to . Figure 2 and Algorithm 1 illustrate these steps.
In order to identify the two sets and , we propose the following two methods.
Unsupervised 5K+5K: a fully unsupervised method. We first define a restricted vocabulary of the 50k most frequent words among the unrestricted one of 190k words, and rank them by frequency; we then define as the 5k most frequent ones, and as the 5k least frequent ones of the 50k vocabulary (to avoid extremely rare words which might have received less signal during training).
Weakly-supervised WN +: a weakly-supervised method that uses words from the WordNet hierarchy. We define as the top words from the top levels of the WordNet hierarchy, and as of the bottom words from the bottom levels, randomly sampled in case of ties.
Vilnis & McCallum (2015) propose using is-a for distributions , the argument being that a low indicates that we can encode easily as , implying that entails . However, we would like to mitigate this statement. Indeed, if and are two 1D Gaussian distributions with same mean, then where , which is not a monotonic function of . This breaks the idea that the magnitude of the variance should encode the generality/specificity of the concept.
Another entailment score for Gaussian embeddings.
What would constitute a good number for the variance’s magnitude of a -dimensional Gaussian distribution ? It is known that 95% of its mass is contained within a hyper-ellipsoid of volume , where denotes the volume of a ball of radius in . For simplicity, we propose dropping the dependence in and define a simple score is-a. Note that using difference of logarithms has the benefit of removing the scaling constant , and makes the entailment score invariant to a rescaling of the covariance matrices: is-a is-a.
8 Embedding symbolic data in a continuous space with matching hyperbolicity
Why would we embed words in a hyperbolic space? Given some symbolic data, such as a vocabulary along with similarity measures between words in our case, co-occurrence counts can we understand in a principled manner which geometry would represent it best? Choosing the right metric space to embed words can be understood as selecting the right inductive bias an essential step.
A particular quantity of interest describing qualitative aspects of metric spaces is the -hyperbolicity which we formally define in appendix B. This metric introduced by Gromov (1987) quantifies the tree-likeliness of a space. However, for various reasons discussed in appendix B, we used the averaged -hyperbolicity, denoted , defined by Albert et al. (2014). Intuitively, a low of a finite metric space characterizes that this space has an underlying hyperbolic geometry, i.e. an approximate tree-like structure, and that the hyperbolic space would be well suited to isometrically embed it. We also report the ratio (invariant to metric scaling), where is the average distance in the finite space, as suggested by Borassi et al. (2015), whose low value also characterizes the “hyperbolicness” of the space.
Since our methods are trained on a weighted graph of co-occurrences, it makes sense to look for the corresponding hyperbolicity of this symbolic data. The lower this value, the more hyperbolic is the underlying nature of the graph, thus indicating that the hyperbolic space should be preferred over the Euclidean space for embedding words. However, in order to do so, one needs to be provided with a distance for each pair of words , while our symbolic data is only made of similarity measures. Note that one cannot simply associate the value to , as this quantity is not symmetric. Instead, inspired from Eq. (3), we associate to words the distance333One can replace with to avoid computing the logarithm of zero. with the choice , i.e.
Table 1 shows values for different choices of . The discrete metric spaces we obtained for our symbolic data of co-occurrences appear to have a very low hyperbolicity, i.e. to be very much “hyperbolic”, which suggests to embed words in (products of) hyperbolic spaces. We report in section 9 empirical results for and .
9 Experiments: similarity, analogy, entailment
We trained all models on a corpus provided by Levy & Goldberg (2014); Levy et al. (2015) used in other word embeddings related work. Corpus preprocessing is explained in the above references. The dataset has been obtained from an English Wikipedia dump and contains 1.4 billion tokens. Words appearing less than one hundred times in the corpus have been discarded, leaving unique tokens. The co-occurrence matrix contains approximately millions non-zero entries, for a symmetric window size of
. All models were trained for 50 epochs, and unless stated otherwise, on the full corpus of 189,533 word types. For certain experiments, we also trained the model on a restricted vocabulary of themost frequent words, which we specify by appending either “(190k)” or “(50k)” to the experiment’s name in the table of results.
Poincaré models, Euclidean baselines.
We report results for both 100D embeddings trained in a 100D Poincaré ball, and for 50x2D embeddings, which were trained in the Cartesian product of 50 2D Poincaré balls. Note that in the case of both models, one word will be represented by exactly 100 parameters. For the Poincaré models we employ both and . All hyperbolic models were optimized with Radagrad (Bécigneul & Ganea, 2018) as explained in Sec. 5. On the tasks of similarity and analogy, we compare against a 100D Euclidean GloVe model which was trained using the hyperparameters suggested in the original GloVe paper (Pennington et al., 2014). The vanilla GloVe model was optimized using Adagrad (Duchi et al., 2011). For the Euclidean baseline as well as for models with we used a learning rate of 0.05. For Poincaré models with we used a learning rate of 0.01.
The initialization trick.
We obtained improvement in the majority of the metrics when initializing our Poincaré model with pretrained parameters. These were obtained by first training the same model on the restricted (50k) vocabulary, and then using this model as an initialization for the full (190K) vocabulary. This will be referred to as the “initialization trick”. For fairness, we also trained the vanilla (Euclidean) GloVe model in the same fashion.
Word similarity is assessed using a number of well established benchmarks shown in Table 2. We summarize here our main results, but more extensive experiments (including in lower dimensions) are in Appendix A.1. We note that, with a single exception, our 100D and 50x2D models outperform the vanilla Glove baselines in all settings.
|sixties||seventies, eighties, nineties, 60s, 70s, 1960s, 80s, 90s, 1980s, 1970s|
|dance||dancing, dances, music, singing, musical, performing, hip-hop, pop, folk, dancers|
|daughter||wife, married, mother, cousin, son, niece, granddaughter, husband, sister, eldest|
|vapor||vapour, refrigerant, liquid, condenses, supercooled, fluid, gaseous, gases, droplet|
|ronaldo||cristiano, ronaldinho, rivaldo, messi, zidane, romario, pele, zinedine, xavi, robinho|
|mechanic||electrician, fireman, machinist, welder, technician, builder, janitor, trainer, brakeman|
|algebra||algebras, homological, heyting, geometry, subalgebra, quaternion, calculus, mathematics, unital, algebraic|
For word analogy, we evaluate on the Google benchmark (Mikolov et al., 2013a) and its two splits that contain semantic and syntactic analogy queries. We also use a benchmark by MSR that is also commonly employed in other word embedding works. For the Euclidean baselines we use 3COSADD (Levy et al., 2015). For our models, the solution to the problem “which is to , what is to ” is selected as , as described in section 6. In order to select the best without overfitting on the benchmark dataset, we used the same 2-fold cross-validation method used by (Levy et al., 2015, section 5.1) (see our Table 15) which resulted in selecting . We report our main results in Table 4, and more extensive experiments in various settings (including in lower dimensions) in appendix A.2. We note that the vast majority of our models outperform the vanilla Glove baselines, with the 100D hyperbolic embeddings being the absolute best.
datasets. We classify all the methods in three categories depending on the supervision used for word embedding learning and for the hypernymy score, respectively. For Hyperlex we report results in Tab.6 and use the baseline scores reported in (Nickel & Kiela, 2017; Vulić et al., 2017). For WBLess we report results in Tab. 7 and use the baseline scores reported in (Nguyen et al., 2017).
|reptile||amphibians, carnivore, crocodilian, fish-like, dinosaur, alligator, triceratops|
|algebra||mathematics, geometry, topology, relational, invertible, endomorphisms, quaternions|
|music||performance, composition, contemporary, rock, jazz, electroacoustic, trio|
|feeling||sense, perception, thoughts, impression, emotion, fear, shame, sorrow, joy|
Hypernymy results discussion.
We first note that our fully unsupervised 50x2D, model outperforms all its corresponding baselines setting a new state-of-the-art on unsupervised WBLESS accuracy and matching the previous state-of-the-art on unsupervised HyperLex Spearman correlation.
Second, once a small amount of weakly supervision is used for the hypernymy score, we obtain significant improvements as shown in the same tables and also in Fig. 4. We note that this weak supervision is only as a post-processing step (after word embeddings are trained) for identifying the best direction encoding the variance of the Gaussian distributions as described in Sec. 7. Moreover, it does not consist of hypernymy pairs, but only of 400 or 800 generic and specific sets of words from WordNet. Even so, our unsupervised learned embeddings are remarkably able to outperform all (except WN-Poincaré) supervised embedding learning baselines on HyperLex which have the great advantage of using the hypernymy pairs to train the word embeddings.
Which model to choose?
While there is no single model that outperforms all the baselines on all presented tasks, one can remark that the model 50x2D, , with the initialization trick obtains state-of-the-art results on hypernymy detection and is close to the best models for similarity and analogy (also Poincaré Glove models), but almost constantly outperforming the vanilla Glove baseline on these. This is the first model that can achieve competitive results on all these three tasks, additionally offering interpretability via the connection to Gaussian word embeddings.
|Supervised embedding learning & Unsupervised hypernymy score||OrderEmb||0.191|
|PARAGRAM + CF||0.320|
|WN-Eucl from (Nickel & Kiela, 2017)||0.389|
|WN-Poincaré from (Nickel & Kiela, 2017)||0.512|
|Unsupervised embedding learning & Weakly-supervised hypernymy score||50x2D Poincaré GloVe, , init trick (190k)|
|50x2D Poincaré GloVe, , init trick (190k)|
|Unsupervised embedding learning & Unsupervised hypernymy score||Word2Gauss-DistPos||0.206|
|DIVE-WS (Chang et al., 2018)||0.333|
|SBOW-PPMI-CS from (Chang et al., 2018)||0.345|
|50x2D Poincaré GloVe, , init trick (190k) Unsupervised 5k+5k||0.284|
|50x2D Poincaré GloVe, , init trick (190k) Unsupervised 5k+5k||0.341|
|Supervised embedding learning & Unsupervised hypernymy score||(Weeds et al., 2014)||0.75|
|WN-Poincaré from (Nickel & Kiela, 2017)||0.86|
|(Nguyen et al., 2017)||0.87|
|Unsupervised embedding learning & Weakly-supervised hypernymy score||50x2D Poincaré GloVe, , init trick (190k)|
|50x2D Poincaré GloVe, , init trick (190k)|
|Unsupervised embedding learning & Unsupervised hypernymy score||SGNS from (Nguyen et al., 2017)||0.48|
|(Weeds et al., 2014)||0.58|
|50x2D Poincaré GloVe, , init trick (190k) Unsupervised 5k+5k||0.575|
|50x2D Poincaré GloVe, , init trick (190k) Unsupervised 5k+5k||0.652|
We propose to adapt the GloVe algorithm to hyperbolic spaces and to leverage a connection between statistical manifolds of Gaussian distributions and hyperbolic geometry, in order to better interpret entailment relations between hyperbolic embeddings. We justify the choice of products of hyperbolic spaces via this connection to Gaussian distributions and via computations of the hyperbolicity of the symbolic data upon which GloVe is based. Empirically we present the first model that can simultaneously obtain state-of-the-art results or close on the three tasks of word similarity, analogy and hypernymy detection.
- Albert et al. (2014) Réka Albert, Bhaskar DasGupta, and Nasim Mobasheri. Topological implications of negative curvature for biological and social networks. Physical Review E, 89(3):032811, 2014.
- Arora et al. (2016) Sanjeev Arora, Yuanzhi Li, Yingyu Liang, Tengyu Ma, and Andrej Risteski. A latent variable model approach to pmi-based word embeddings. Transactions of the Association for Computational Linguistics, 4:385–399, 2016.
- Athiwaratkun & Wilson (2017) Ben Athiwaratkun and Andrew Wilson. Multimodal word distributions. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), volume 1, pp. 1645–1656, 2017.
- Athiwaratkun & Wilson (2018) Ben Athiwaratkun and Andrew Gordon Wilson. Hierarchical density order embeddings. arXiv preprint arXiv:1804.09843, 2018.
- Baroni & Lenci (2011) Marco Baroni and Alessandro Lenci. How we blessed distributional semantic evaluation. In Proceedings of the GEMS 2011 Workshop on GEometrical Models of Natural Language Semantics, pp. 1–10. Association for Computational Linguistics, 2011.
- Bécigneul & Ganea (2018) Gary Bécigneul and Octavian-Eugen Ganea. Riemannian adaptive optimization methods. arXiv preprint arxiv:1810.00760, 2018.
- Bojanowski et al. (2016) Piotr Bojanowski, Edouard Grave, Armand Joulin, and Tomas Mikolov. Enriching word vectors with subword information. arXiv preprint arXiv:1607.04606, 2016.
- Bonnabel (2013) Silvere Bonnabel. Stochastic gradient descent on riemannian manifolds. IEEE Transactions on Automatic Control, 58(9):2217–2229, 2013.
- Borassi et al. (2015) Michele Borassi, Alessandro Chessa, and Guido Caldarelli. Hyperbolicity measures democracy in real-world networks. Physical Review E, 92(3):032812, 2015.
- Bowman et al. (2015) Samuel R Bowman, Gabor Angeli, Christopher Potts, and Christopher D Manning. A large annotated corpus for learning natural language inference. In Proceedings of the 2015 Conference on Empirical Methods in Natural Language Processing, pp. 632–642, 2015.
- Chang et al. (2018) Haw-Shiuan Chang, Ziyun Wang, Luke Vilnis, and Andrew McCallum. Distributional inclusion vector embedding for unsupervised hypernymy detection. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), volume 1, pp. 485–495, 2018.
- Chen et al. (2013) Wei Chen, Wenjie Fang, Guangda Hu, and Michael W Mahoney. On the hyperbolicity of small-world and treelike random graphs. Internet Mathematics, 9(4):434–491, 2013.
- Costa et al. (2015) Sueli IR Costa, Sandra A Santos, and João E Strapasson. Fisher information distance: a geometrical reading. Discrete Applied Mathematics, 197:59–69, 2015. URL https://arxiv.org/pdf/1210.2354.pdf.
De Sa et al. (2018)
Christopher De Sa, Albert Gu, Christopher Ré, and Frederic Sala.
Representation tradeoffs for hyperbolic embeddings.
International Conference on Machine Learning, 2018.
- Dhingra et al. (2018) Bhuwan Dhingra, Christopher Shallue, Mohammad Norouzi, Andrew Dai, and George Dahl. Embedding text in hyperbolic spaces. In Proceedings of the Twelfth Workshop on Graph-Based Methods for Natural Language Processing (TextGraphs-12), pp. 59–69, 2018.
- Duchi et al. (2011) John Duchi, Elad Hazan, and Yoram Singer. Adaptive subgradient methods for online learning and stochastic optimization. Journal of Machine Learning Research, 12(Jul):2121–2159, 2011.
- Ganea et al. (2018a) Octavian-Eugen Ganea, Gary Bécigneul, and Thomas Hofmann. Hyperbolic entailment cones for learning hierarchical embeddings. In International Conference on Machine Learning, 2018a.
- Ganea et al. (2018b) Octavian-Eugen Ganea, Gary Bécigneul, and Thomas Hofmann. Hyperbolic neural networks. In Advances in Neural Information Processing Systems, 2018b.
- Gromov (1987) Mikhael Gromov. Hyperbolic groups. In Essays in group theory, pp. 75–263. Springer, 1987.
- Jeffreys (1946) Harold Jeffreys. Proc. R. Soc. Lond. A, 186(1007):453–461, 1946.
- Kiela et al. (2015) Douwe Kiela, Laura Rimell, Ivan Vulić, and Stephen Clark. Exploiting image generality for lexical entailment detection. In Proceedings of the 53rd Annual Meeting of the Association for Computational Linguistics and the 7th International Joint Conference on Natural Language Processing (Volume 2: Short Papers), volume 2, pp. 119–124, 2015.
- Krioukov et al. (2010) Dmitri Krioukov, Fragkiskos Papadopoulos, Maksim Kitsak, Amin Vahdat, and Marián Boguná. Hyperbolic geometry of complex networks. Physical Review E, 82(3):036106, 2010.
- Leimeister & Wilson (2018) Matthias Leimeister and Benjamin J Wilson. Skip-gram word embeddings in hyperbolic space. arXiv preprint arXiv:1809.01498, 2018.
- Levy & Goldberg (2014) Omer Levy and Yoav Goldberg. Linguistic regularities in sparse and explicit word representations. In Proceedings of the Eighteenth Conference on Computational Natural Language Learning, pp. 171–180. Association for Computational Linguistics, 2014. doi: 10.3115/v1/W14-1618. URL http://www.aclweb.org/anthology/W14-1618.
- Levy et al. (2015) Omer Levy, Yoav Goldberg, and Ido Dagan. Improving distributional similarity with lessons learned from word embeddings. Transactions of the Association for Computational Linguistics, 3:211–225, 2015. ISSN 2307-387X. URL https://transacl.org/ojs/index.php/tacl/article/view/570.
- Mikolov et al. (2013a) Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781, 2013a.
- Mikolov et al. (2013b) Tomas Mikolov, Ilya Sutskever, Kai Chen, Greg S Corrado, and Jeff Dean. Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems, pp. 3111–3119, 2013b.
- Miller et al. (1990) George A Miller, Richard Beckwith, Christiane Fellbaum, Derek Gross, and Katherine J Miller. Introduction to wordnet: An on-line lexical database. International journal of lexicography, 3(4):235–244, 1990.
- Muzellec & Cuturi (2018) Boris Muzellec and Marco Cuturi. Generalizing point embeddings using the wasserstein space of elliptical distributions. arXiv preprint arXiv:1805.07594, 2018.
- Nguyen et al. (2017) Kim Anh Nguyen, Maximilian Köper, Sabine Schulte im Walde, and Ngoc Thang Vu. Hierarchical embeddings for hypernymy detection and directionality. In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pp. 233–243, 2017.
- Nickel & Kiela (2018) Maximilian Nickel and Douwe Kiela. Learning continuous hierarchies in the lorentz model of hyperbolic geometry. In International Conference on Machine Learning, 2018.
- Nickel & Kiela (2017) Maximillian Nickel and Douwe Kiela. Poincaré embeddings for learning hierarchical representations. In Advances in Neural Information Processing Systems, pp. 6341–6350, 2017.
- Pennington et al. (2014) Jeffrey Pennington, Richard Socher, and Christopher D Manning. Glove: Global vectors for word representation. In EMNLP, volume 14, pp. 1532–43, 2014.
- Singh et al. (2018) Sidak Pal Singh, Andreas Hug, Aymeric Dieuleveut, and Martin Jaggi. Wasserstein is all you need. arXiv preprint arXiv:1808.09663, 2018.
- Ungar (2012) Abraham A Ungar. Beyond the Einstein addition law and its gyroscopic Thomas precession: The theory of gyrogroups and gyrovector spaces, volume 117. Springer Science & Business Media, 2012.
- Ungar (2008) Abraham Albert Ungar. A gyrovector space approach to hyperbolic geometry. Synthesis Lectures on Mathematics and Statistics, 1(1):1–194, 2008.
- Vendrov et al. (2015) Ivan Vendrov, Ryan Kiros, Sanja Fidler, and Raquel Urtasun. Order-embeddings of images and language. arXiv preprint arXiv:1511.06361, 2015.
- Vilnis & McCallum (2015) Luke Vilnis and Andrew McCallum. Word representations via gaussian embedding. ICLR, 2015.
- Vilnis et al. (2018) Luke Vilnis, Xiang Li, Shikhar Murty, and Andrew McCallum. Probabilistic embedding of knowledge graphs with box lattice measures. arXiv preprint arXiv:1805.06627, 2018.
- Vulić & Mrkšić (2018) Ivan Vulić and Nikola Mrkšić. Specialising word vectors for lexical entailment. In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), volume 1, pp. 1134–1145, 2018.
- Vulić et al. (2017) Ivan Vulić, Daniela Gerz, Douwe Kiela, Felix Hill, and Anna Korhonen. Hyperlex: A large-scale evaluation of graded lexical entailment. Computational Linguistics, 43(4):781–835, 2017.
- Weeds et al. (2014) Julie Weeds, Daoud Clarke, Jeremy Reffin, David Weir, and Bill Keller. Learning to distinguish hypernyms and co-hyponyms. In Proceedings of COLING 2014, the 25th International Conference on Computational Linguistics: Technical Papers, pp. 2249–2259. Dublin City University and Association for Computational Linguistics, 2014.
Appendix A More experiments
We show here extensive empirical results in various settings, including lower dimensions, different product structures, changing the vocabulary and using different functions.
In the experiment’s name, we first indicate which dimension was used: “” denotes while “” denotes . “Vanilla” designates the baseline, i.e. the standard Euclidean GloVe from Eq. (1), while “Poincaré” designates our hyperbolic GloVe from Eq. (3). For Poincaré models, we then append to the experiment’s name which function was applied to distances during training. Every model was trained for epochs. Vanilla models were optimized with Adagrad (Duchi et al., 2011) while Poincaré models were optimized with Radagrad (Bécigneul & Ganea, 2018). For each experiment we tried using learning rates in , and found that the best were for and for and for Vanilla models accordingly, we only report the best results. For similarity, we only considered the “target word vector” and always ignored the “context word vector”. We also tried using the Euclidean/Möbius average444A Möbius average is a gyro-midpoint, as explained in section 6. of these, but obtained (almost) consistently worse results for all experiments (including baselines) and do not report them.
Reported scores are Spearman’s correlations on the ranks for each benchmark dataset, as usual in the literature. We used (minus) the Poincaré distance as a similarity measure to rank neighbors.
|Experiment’s name||Rare Word||WordSim||SimLex||SimVerb||MC||RG|
|100D Vanilla (init)||0.3787||0.5668||0.2964||0.1639||0.6562||0.6757|
|100D Poincaré, (init)||0.4187||0.6209||0.3208||0.1915||0.7833||0.7578|
|Experiment’s name||Rare Word||WordSim||SimLex||SimVerb||MC||RG|
|100D Vanilla (190k)||0.4443||0.5986||0.3071||0.1705||0.7245||0.7114|
Note that restricting the vocabulary incurs a loss of certain pairs of words from the benchmark similarity datasets, hence similarity results on the restricted (50k) vocabulary from Table 9 should be analyzed with caution, and in the light of Tables 10 and 11 (especially for Rare Word).
Details and notations.
In the column “method”, “3.c.a” denotes using 3COSADD to solve analogies, which was used for all Euclidean baselines; for Poincaré models, as explained in section 9, the solution to the analogy problem is computed as with , and then the nearest neighbor in the vocabulary is selected either with the Poincaré distance on the corresponding space, which we denote as “
”, or with cosine similarity on the full vector, which we denote as”. Finally, note that each cell contains two numbers, designated by and respectively: denotes ignoring the context vectors, while denotes considering as our embeddings the Euclidean/Möbius average between the target vector and the context vector . In each dimension, we bold best results for .
|Experiment’s name||Method||Semantic Google analogy accuracy using||Syntactic Google analogy accuracy using||Total Google analogy accuracy using||MSR analogy accuracy using|
|100D Vanilla||3.c.a||0.6005 / 0.6374||0.5869 / 0.5540||0.5931 / 0.5918||0.4868 / 0.4427|
|100D Vanilla (init)||3.c.a||0.6427 / 0.6878||0.5950 / 0.5672||0.6167 / 0.6219||0.4826 / 0.4509|
|100D Poincaré,||0.4289 / 0.4444||0.5892 / 0.5484||0.5165 / 0.5012||0.4625 / 0.4186|
|0.4834 / 0.4908||0.5736 / 0.5514||0.5326 / 0.5239||0.4833 / 0.4395|
|100D Poincaré,||0.6010 / 0.6308||0.6121 / 0.5659||0.6070 / 0.5954||0.4793 / 0.4375|
|(init)||0.6641 / 0.6776||0.6088 / 0.5740||0.6339 / 0.6210||0.4971 / 0.4600|
|100D Poincaré,||0.1013 / 0.5110||0.2388 / 0.4865||0.1764 / 0.4976||0.1461 / 0.3235|
|0.4329 / 0.7152||0.2507 / 0.4596||0.3334 / 0.5756||0.2042 / 0.3628|
|50x2D Poincaré,||0.4511 / 0.4745||0.5766 / 0.5365||0.5196 / 0.5083||0.4763 / 0.4268|
|0.3274 / 0.3553||0.4326 / 0.3924||0.3849 / 0.3756||0.3329 / 0.2914|
|50x2D Poincaré,||0.6426 / 0.6709||0.5940 / 0.5560||0.6160 / 0.6081||0.4576 / 0.4166|
|0.4754 / 0.5255||0.4544 / 0.4271||0.4639 / 0.4718||0.3425 / 0.2980|
|48D Vanilla||3.c.a||0.3642 / 0.3650||0.451 / 0.4156||0.4115 / 0.3927||0.3467 / 0.3139|
|48D Poincaré,||0.2368 / 0.2403||0.4693 / 0.4242||0.3638 / 0.3407||0.3755 / 0.3255|
|0.2479 / 0.2449||0.4704 / 0.4264||0.3694 / 0.3440||0.3919 / 0.3405|
|48D Poincaré,||0.2108 / 0.4575||0.2752 / 0.4452||0.2460 / 0.4508||0.1842 / 0.2790|
|0.4513 / 0.5848||0.3137 / 0.4334||0.3762 / 0.5021||0.2386 / 0.3232|
|24x2D Poincaré,||0.2338 / 0.2412||0.4509 / 0.4116||0.3524 / 0.3343||0.3426 / 0.3039|
|0.1294 / 0.1445||0.2240 / 0.1971||0.1811 / 0.1733||0.1619 / 0.1427|
|24x2D Poincaré,||0.4663 / 0.4851||0.4834 / 0.4482||0.4756 / 0.4650||0.3456 / 0.3124|
|0.2479 / 0.2477||0.2626 / 0.2445||0.2559 / 0.2460||0.1670 / 0.1388|
|20D Vanilla||3.c.a||0.1234 / 0.1202||0.2133 / 0.2004||0.1724 / 0.1640||0.1481 / 0.1281|
|20D Poincaré,||0.1043 / 0.1020||0.2159 / 0.1946||0.1653 / 0.1526||0.1751 / 0.1527|
|0.1027 / 0.0993||0.2184 / 0.1955||0.1659 / 0.1519||0.1781 / 0.1505|
|20D Poincaré,||0.1728 / 0.1840||0.2717 / 0.2646||0.2268 / 0.2280||0.1580 / 0.1451|
|0.2133 / 0.2018||0.2950 / 0.2762||0.2579 / 0.2424||0.1821 / 0.1611|
|10x2D Poincaré,||0.1005 / 0.1015||0.2102 / 0.1915||0.1604 / 0.1506||0.1570 / 0.1365|
|0.0424 / 0.0392||0.0773 / 0.0686||0.0615 / 0.0553||0.0520 / 0.0446|
|10x2D Poincaré,||0.1635 / 0.1618||0.2530 / 0.2263||0.2124 / 0.1970||0.1580 / 0.1446|
|0.0599 / 0.0548||0.0992 / 0.0861||0.0814 / 0.0719||0.0501 / 0.0408|
|4D Vanilla||3.c.a||0.0036 / 0.0045||0.0012 / 0.0015||0.0023 / 0.0028||0.0011 / 0.0012|
|4D Poincaré,||0.0089 / 0.0092||0.0043 / 0.0041||0.0064 / 0.0064||0.0046 / 0.0054|
|0.0036 / 0.0039||0.0020 / 0.0026||0.0027 / 0.0032||0.0015 / 0.0016|
|4D Poincaré,||0.0135 / 0.0133||0.0058 / 0.0061||0.0093 / 0.0094||0.0051 / 0.0056|
|0.0045 / 0.0050||0.0024 / 0.0029||0.0034 / 0.0038||0.0015 / 0.0011|
|Experiment’s name||Method||Semantic Google analogy accuracy using||Syntactic Google analogy accuracy using||Total Google analogy accuracy using||MSR analogy accuracy using|
|100D Vanilla (190k)||3.c.a||0.4789 / 0.4966||0.5684 / 0.5450||0.5278 / 0.5230||0.4382 / 0.3990|
|100D Vanilla||3.c.a||0.2848 / 0.3043||0.5003 / 0.5103||0.4025 / 0.4168||0.3545 / 0.3655|
|100D Poincaré,||0.3684 / 0.3803||0.5820 / 0.5545||0.4851 / 0.4754||0.4394 / 0.3970|
|0.3982 / 0.4014||0.5786 / 0.5504||0.4968 / 0.4828||0.4494 / 0.4016|
|100D Poincaré,||0.1265 / 0.4005||0.2209 / 0.4693||0.1781 / 0.4381||0.1384 / 0.3066|
|0.2634 / 0.5179||0.2521 / 0.4460||0.2572 / 0.4786||0.1933 / 0.3354|
|50x2D Poincaré,||0.3956 / 0.4012||0.5799 / 0.5451||0.4963 / 0.4798||0.4482 / 0.3957|
|0.2809 / 0.2789||0.4146 / 0.4067||0.3539 / 0.3488||0.3464 / 0.2880|
|50x2D Poincaré,||0.5204 / 0.5275||0.5819 / 0.5518||0.5540 / 0.5407||0.4404 / 0.3980|
|0.3873 / 0.4172||0.4517 / 0.4411||0.4225 / 0.4303||0.3335 / 0.2933|
|48D Vanilla||3.c.a||0.3212 / 0.3299||0.4727 / 0.4303||0.4039 / 0.3847||0.3550 / 0.3156|
|48D Poincaré,||0.2127 / 0.2163||0.4680 / 0.4239||0.3521 / 0.3297||0.3581 / 0.3078|
|0.2180 / 0.2220||0.4690 / 0.4228||0.3551 / 0.3317||0.3708 / 0.3134|
|48D Poincaré,||0.2035 / 0.3676||0.2572 / 0.4129||0.2329 / 0.3923||0.1787 / 0.2652|
|0.3063 / 0.4212||0.3090 / 0.4174||0.3078 / 0.4192||0.2243 / 0.2951|
|24x2D Poincaré,||0.2307 / 0.2308||0.4506 / 0.4090||0.3508 / 0.3281||0.3289 / 0.2979|
|0.1328 / 0.1334||0.2475 / 0.2153||0.1955 / 0.1781||0.1850 / 0.1544|
|24x2D Poincaré,||0.3649 / 0.3788||0.4738 / 0.4343||0.4244 / 0.4091||0.3424 / 0.2985|
|0.2041 / 0.2080||0.2680 / 0.2469||0.2390 / 0.2293||0.1805 / 0.1611|
|20D Vanilla||3.c.a||0.1223 / 0.1164||0.2472 / 0.2120||0.1905 / 0.1686||0.1550 / 0.1289|
|20D Poincaré,||0.0925 / 0.0903||0.2292 / 0.1967||0.1672 / 0.1484||0.1601 / 0.1286|
|0.0917 / 0.0890||0.2355 / 0.1964||0.1702 / 0.1477||0.1629 / 0.1271|
|20D Poincaré,||0.1583 / 0.1661||0.2619 / 0.2479||0.2149 / 0.2108||0.1624 / 0.1419|
|0.1757 / 0.1777||0.2970 / 0.2613||0.2408 / 0.2220||0.1804 / 0.1554|
|10x2D Poincaré,||0.0962 / 0.0945||0.2177 / 0.1919||0.1626 / 0.1477||0.1546 / 0.1279|
|0.0403 / 0.0387||0.0850 / 0.0704||0.0647 / 0.0560||0.0483 / 0.0432|
|10x2D Poincaré,||0.1440 / 0.1467||0.2533 / 0.2231||0.2037 / 0.1884||0.1580 / 0.1417|
|0.0614 / 0.0583||0.1014 / 0.0880||0.0832 / 0.0745||0.0473 / 0.0435|
|4D Vanilla||3.c.a||0.0050 / 0.0053||0.0011 / 0.0012||0.0029 / 0.0031||0.0011 / 0.0011|
|4D Poincaré,||0.0054 / 0.0056||0.0037 / 0.0037||0.0045 / 0.0046||0.0041 / 0.0044|
|0.0038 / 0.0036||0.0023 / 0.0025||0.0030 / 0.0030||0.0006 / 0.0006|
|4D Poincaré,||0.0127 / 0.0127||0.0072 / 0.0077||0.0097 / 0.0100||0.0061 / 0.0057|
|0.0060 / 0.0061||0.0030 / 0.0037||0.0043 / 0.0048||0.0024 / 0.0025|
Note that restricting the vocabulary to the most frequent 190k or 50k words will remove some of the test instances in the benchmark analogy datasets. These are described in Table 14.
||Semantic Google||Syntactic Google||Total Google||MSR|
||Validation accuracy||Test accuracy|
About analogy computations.
Note that one can rewrite Eq. (6) with tools from differential geometry as
where denotes the parallel transport along the unique geodesic from to . The and maps of Riemannian geometry were related to the theory of gyrovector spaces (Ungar, 2008) by Ganea et al. (2018b), who also mention that when continuously deforming the hyperbolic space into the Euclidean space , sending the curvature from to (i.e. the radius of from to ), the Möbius operations recover their respective Euclidean counterparts . Hence, the analogy solutions of Eq. (6) would then all recover , which seems a nice sanity check.
We show here more plots illustrating the method (described in section 7) that we use to map points from a (product of) Poincaré disk(s) to a (diagonal) Gaussian. Colors indicate WordNet levels: low levels are closer to the root. Figures 5,6,7,8 show the three steps (centering, rotation, isometric mapping to half-plane) for 20D embeddings in , i.e. each of these steps in each of the corresponding 2D spaces. In these figures, centering and rotation were determined with our proposed semi-supervised method, i.e. selecting 400+400 top and bottom words from the WordNet hierarchy. We show these plots for two models in : one trained with and one with .
It is easily noticeable that words trained with are embedded much closer to each other than those trained with . This is expected: is applied to the distance function, and according to Eq. (3), should match , which is smaller for than for .
Appendix B -hyperbolicity
Let us start by defining the -hyperbolicity, introduced by Gromov (1987). The hyperbolicity of a -tuple is defined as half the difference between the biggest two of the following sums: , , . The -hyperbolicity of a metric space is defined as the supremum of these numbers over all -tuples. Following Albert et al. (2014), we will denote this number by , and by the average of these over all -tuples, when the space is a finite set. An equivalent and more intuitive definition holds for geodesic spaces, i.e. when we can define triangles: its -hyperbolicity is the smallest such that for any triangle , there exists a point at distance at most from each side of the triangle. Chen et al. (2013) and Borassi et al. (2015) analyzed and for specific graphs, respectively. A low hyperbolicity of a graph indicates that it has an underlying hyperbolic geometry, i.e. that it is approximately tree-like, or at least that there exists a taxonomy of nodes. Conversely, a high hyperbolicity of a graph suggests that it possesses long cycles, or could not be embedded in a low dimensional hyperbolic space without distortion. For instance, the Euclidean space is not -hyperbolic for any , and is hence described as -hyperbolic, while the Poincaré disk is known to have a -hyperbolicity of . On the other-hand, a product is -hyperbolic, because a plane could be isometrically embedded in it using for instance the first coordinates of each . However, if would constitute a good choice to embed some given symbolic data, then most likely would as well. This stems from the fact that -hyperbolicity () is a worst case measure which does not reflect what one could call the “hyperbolic capacity” of the space. Furthermore, note that computing requires for a graph of size , while can be approximated via sampling. Finally, is robust to adding/removing a node from the graph, while is not. For all these reasons, we choose as a measure of hyperbolicity.
As explained in section 8, we computed hyperbolicities of the metric space induced by different functions, on the matrix of co-occurrence counts, as reported in Table 1. We also conducted similarity experiments, reported in Table 17. Apart from WordSim, results improved for higher powers of , corresponding to more hyperbolic spaces. However, also note that higher powers will tend to result in words embedded much closer to each other, i.e. with smaller distances, as explained in appendix A.3. In order to know whether this benefit comes from contracting distances or making the space more “hyperbolic”, it would be interesting to learn (or cross-validate) the curvature of the Poincaré ball (or equivalently, its radius) jointly with the function. Finally, it order to explain why WordSim behaved differently compared to other benchmarks, we investigated different properties of these, as reported in Table 16. The geometry of the words appearing in WordSim do not seem to have a different hyperbolicity compared to other benchmarks; however, WordSim seems to contain much more frequent words. Since hyperbolicities are computed with the assumption that (see Eq. (7)), it would be interesting to explore whether learned biases indeed take these values. We left this as future work.
|# of test instances||353||999||3,500|
|# of different words||419||1,027||822|
|min. index (frequency)||57||38||21|
|max. index (frequency)||58,286||128,143||180,417|
|median of indexes||2,723||4,463||9,338|
|Experiment’s name||Rare Word||WordSim||SimLex||SimVerb|
Appendix C Closed-form formulas of Möbius operations
The Möbius addition of and in is defined as
We define .
Möbius scalar multiplication.
The Möbius scalar multiplication of by is defined as
Exponential and logarithmic maps.
For any point , the exponential map and the logarithmic map are given for and by:
Gyro operator and parallel transport.
Parallel transport is given for