Aligning two clouds of embeddings, or high dimensional real vectors, is a fundamental problem in machine learning with applications in natural language processing such as unsupervised word and sentence translation(Rapp, 1995; Fung, 1995) or in computer vision such as point set registration (Cootes et al., 1995) and structure-from-motion (Tomasi and Kanade, 1992). Most of the successes of these approaches were made in domains where either the dimension of the vectors were small or some geometrical constraints on the point clouds were known (Fischler and Bolles, 1987; Leordeanu and Hebert, 2005; Liu et al., 2008).
When dealing with unstructured sets of high dimensional vectors, it is quite common to use a few anchor points to learn the matching (Mikolov et al., 2013; Xing et al., 2015). The supervision for this problem can be limited or noisy, for example using exact string matches in the context of word vectors alignment (Artetxe et al., 2017). Recently, several unsupervised approaches have obtained compelling results, by framing this problem as some form of distance minimization between distributions, using either the Wasserstein distance or adversarial training (Cao et al., 2016; Zhang et al., 2017a; Conneau et al., 2017). The methods typically require a relatively sophisticated framework that leads to a hard, and sometimes unstable, optimization problem. Moreover, both weakly supervised and unsupervised methods greatly benefit from a refinement procedure (Artetxe et al., 2017; Conneau et al., 2017), often based on some variants of iterative closest points (ICP, Besl and McKay, 1992). It is thus not surprising that a more direct approach, only relying on ICP, was able to achieve similar performance (Hoshen and Wolf, 2018)
. However, this method is still very sensitive to the initialization, and requires a large number of random restarts, based on randomized principal component analysis to converge.
In this work, we propose a simple approach based on jointly learning the alignment and the linear transformation between the two point clouds. Our objective function is similar to the one ofZhang et al. (2017b) and Hoshen and Wolf (2018) but our algorithm largely differs. In particular, our algorithm shares similarities with the work of Bojanowski and Joulin (2017) where a non-linear transformation and an alignment between two point clouds are jointly learned. While their goal is completely different, we take a similar optimization scheme to efficiently scale our approach to very large sets of points. Our formulation is not convex and we propose a convex relaxation for the initialization based on a standard relaxation of the quadratic alignment problem for graph matching (Gold and Rangarajan, 1996). Overall our approach makes little assumption about the clouds or their distributions, is flexible and converges in a few minutes.
An interesting by-product of our formulation is that it draws some similarities between graph matching and aligning embeddings that could be used in either way. We validate this observation on several toy examples designed to give some insights on the strengths and weaknesses of our approach. We also show that our approach is competitive with the state-of-the-art among unsupervised approaches on word translation while running in a few minutes. More precisely, we make the following contributions: starting from a simple formulation, based on the joint estimation of an orthogonal matrix and a permutation matrix, we introduce a stochastic algorithm to minimize our cost function, which can scale to large datasets ; we show that this algorithm can be initialized using the solution of a convex problem, leading to better convergence ; we evaluate our method on various toy experiments, as well as the task of bilingual lexicon induction.
In this section, we first describe Procrustes, which is a standard approach to align points when correspondences are known, and then the Wasserstein distance which is used to measure the distance between sets of points. We then present our method which is derived from these two techniques.
Procrustes analysis learns a linear transformation between two sets of matched points and . If the correspondences between the two sets are known (i.e., which point of corresponds to which point of ), then the linear transformation can be recovered by solving the least square problem:
This technique has been successfully applied in many different fields, from analyzing sets of 2D shapes (Goodall, 1991) to learning a linear mapping between word vectors in two different languages with the help of a bilingual lexicon (Mikolov et al., 2013). Constraints on the mapping can be further imposed to suit the geometry of the problem. For example, Xing et al. (2015) have shown empirically that orthogonal transformations are well suited to the mapping of word vectors. The corresponding orthogonal Procrustes corresponds to the following optimization problem:
where is the set of orthogonal matrices. This orthogonality constraint is particularly interesting since it ensures that the distances between points are unchanged by the transformation. As shown by Schönemann (1966), the orthogonal Procrustes problem has a closed form solution equal to , where
is the singular value decomposition of.
On the other hand, if we assume that the transformation between the two sets of points is known, finding the correspondences between these sets can be formulated as the optimization problem:
where is the set of permutation matrices, i.e. the set of binary matrices that enforces a -to- mapping: Enforcing a -to-
mapping is not always realistic but it has the advantage to be related to a set of orthogonal matrices. Indeed, this makes the previous optimization problem equivalent to the following linear program:
This is known as the optimal assignment problem or maximum weight matching problem (Kuhn, 1955). It can be solved using the Hungarian algorithm, which has a complexity of . For large number of points, the Hungarian algorithm is impractical and an approximation is required. For example, in the context of word translation, Zhang et al. (2017b) uses the fact that this problem is equivalent to minimizing the squared Wasserstein distance between the two sets of points and :
to use an approximate Earth Mover Distance solver based on the Sinkhorn algorithm (Cuturi, 2013).
Procrustes in Wasserstein distance.
In our case, we do not know the correspondence between the two sets, nor the linear transformation. Formally, our goal is thus to learn an orthogonal matrix , such that the set of point is close to the set of point and -to- correspondences can be inferred. We use the Wasserstein distance defined in Eq (2), as the measure of distance between our two sets of points and we combine it with the orthogonal Procrustes defined in Eq. (1), leading to the problem of Procrustes in Wasserstein distance:
The problem is not jointly convex in and but, as shown in the previous sections, there are exact solutions for each optimization problem if the other variable is fixed. Naively alternating the minimization in each variable does not scale and empirical results show that it quickly converges to bad local minima even on small problems (Zhang et al., 2017b).
In this section, we describe a scalable optimization scheme for the problem defined in Eq. (2). The objective of the problem in Eq. (2) can be intepreted as the Wasserstein distance , between and , two empirical distributions of size of the vectors of and . One possible approach would be to alternate full minimization of in and a gradient-based update in . One difficulty with this method, and our formulation, is that the dimension of the permutation matrix scales quadratically with the number of points . Finding the optimal matching for a given orthogonal matrix has a complexity of , or of up to logarithmic terms for an approximate matching via Sinkhorn (Cuturi, 2013; Altschuler et al., 2017). As a consequence, we use instead at each step a new batch of two subsamples of size , denoted by and . We compute at each step the optimal matching and the value . We then perform a gradient-guided step in for . This surrogate objective, at each step, can be seen as a subsampled version of size of our objective.
As in Genevay et al. (2018), we use samples from distributions for stochastic minimization of an objective involving a Wasserstein distance. Similarly, our algorithm can be intepreted as the stochastic optimization of a population version objective , seeing as i.i.d. samples of size from latent unknown distributions and , and as samples of size . At a high level, this approach can be motivated by the convergence of to , where and are the empirical distributions of i.i.d. samples of size from latent distributions and , via the analysis of and the triangle inequality. Results of these type, and analysis of the rates of convergence go back to Dudley (1969) for the distance, and have been extended to all distances, including (Weed and Bach, 2017), the case of discrete distributions (Tameling et al., 2017; Sommerfeld et al., 2018), and the use of these methods in variational problems (Bernton et al., 2017). While these results do not give formal guarantees for our approach, they provide some motivation for the success of this method.
Optimizing a general function over the manifold of the orthogonal matrices, the Stiefel manifold, as been thoroughly studied (Absil et al., 2009). A simple solution is to take a step in a descending direction (given the full gradient or a stochastic approximation) while pulling back the update on the manifold. Most of the pull back operators are computationally expensive but our matrix is relatively small. In particular, we use the projection even though it requires a singular value decomposition as shown in the previous section. Overall, our optimization in
is a stochastic gradient descent (SGD) with a projection on the Stiefel manifold.
The overall optimization scheme is summarized in Algorithm 1. At the iteration , we sample a mini-batch and of points of size from the matrices and . We then compute the optimal permutation for this batch by solving the same problem as in Eq. (2). Given this permutation, we compute the gradient in to update the matrix. While this procedure is efficient and can scale to very large sets of points, it is not convex and the quality of its solution depends on its initialization. In particular, the initial point is very important for the quality of the final solution. In the next section, we discuss a strategy to initialize this optimization problem.
In this section, we propose a convex relaxation of the problem defined in Eq. (2). This relaxation comes from the observation that our problem is equivalent to
Solving a linear program over is equivalent to solving it as over its convex hull, i.e., the set of matrices with a spectral norm lower than . This value at this maximum is thus equal to the dual norm of the spectral norm, i.e., the trace norm (or nuclear norm) of . Thus, the problem is equivalent to:
where is the trace norm. The trace norm requires to compute the singular values of the matrix , which is computationally expensive. Another possible formulation is to replace the trace norm by the Frobenius norm, leading to the following:
where and . This formulation has the advantage of leading to a quadratic assignment program, i.e.,
since the permutation matrices are orthogonal, and the Frobenius norm is invariant by multiplication with an orthogonal matrix. This problem is a standard formulation for the graph isomorphism, or graph matching problem and it is known to be NP-hard in general (Garey and Johnson, 2002). However, many convex relaxations have been proposed for this problem. Of particular interest, Gold and Rangarajan (1996) replace the set of permutation matrices by its convex hull, the set of doubly stochastic matrices, namely the Birkhoff polytope, leading to the following convex relaxation:
where is the Birkhoff polytope. It would be interesting to see what would be the equivalent relaxation for the trace norm but this goes beyond the scope of this paper.
We use the Frank-Wolfe algorithm to minimize this problem (Frank and Wolfe, 1956). Once the global minimizer has been attained, we compute a corresponding orthogonal matrix by solving:
This corresponds to taking the singular value of , and setting . Note that the matrix is not necessarily a permutation matrix, but only doubly stochastic.
Improving the nearest-neighbor search.
Once the source embeddings are mapped to the target space, they are not perfectly aligned with target embeddings and a retrieval procedure is required. In high dimensional vector space, some points, called hubs, tend to be close to disproportionally many vectors and a direct nearest-neighbor (NN) search favors these hubs (Aucouturier and Pachet, 2008; Radovanović et al., 2010). This problem has been observed in many retrieval situations (Doddington et al., 1998; Pachet and Aucouturier, 2004; Dinu et al., 2014).
Several strategies have been proposed to diminish the effects of hubs, most are based on some definition of a local distance (Jegou et al., 2010; Conneau et al., 2017) or by reversing the direction of the retrieval (Dinu et al., 2014; Smith et al., 2017). In this paper, we consider the Inverted Softmax (ISF) proposed by Smith et al. (2017) and the Cross-Domain Similarity Local Scaling (CSLS) of Conneau et al. (2017). The ISF is defined for normalized vectors as:
where is a temperature parameter. CSLS is a similarity measure between the vectors and from 2 different sets and , defined as where
is the cosine similarity betweenand , and is the average of the cosine similarity between and its nearest neighbors among the vectors in . It is an extension of the contextual dissimilarity measure of Jegou et al. (2010) to two sets of points. An advantage of the CSLS measure over ISF is that its free parameter (i.e., the number of nearest neighbors) is much simpler to set than the temperature of the ISF.
3 Related work
Embedding alignment with no bilingual dictionary.
In the case where the permutation matrix is given by a bilingual dictionary, the problem stated in Eq. (2) has been studied in the context of word translation by Mikolov et al. (2013) with as an unconstrained linear transformation. Xing et al. (2015) have shown that constraining to be an orthonormal matrix yields better results if used with normalized embeddings, and Artetxe et al. (2016) have improved further the quality of supervised alignment by using centered normalized embeddings. Hoshen and Wolf (2018) is the closest attempt to learn unsupervised word translation with a formulation similar to Eq. (2). They use an iterative closest point approach initialized with a randomized PCA. Others have looked into unsupervised aligning word embedding distributions instead of an explicit alignment: Cao et al. (2016)2017a; Conneau et al., 2017) have preferred the popular Generative Adversarial Networks (GANs) framework of Goodfellow et al. (2014). Recently, Zhang et al. (2017b) have proposed to use the earth mover distance to refine the alignment of word embeddings obtained from a generative adversarial network.
The graph matching formulation of Eq. (5) is a quadratic assignment problem, that is NP-Hard (Garey and Johnson, 2002). Polynomial algorithms exist for restricted classes of graphs, like planar graphs (Hopcroft and Wong, 1974) or trees (Aho and Hopcroft, 1974)
, and for the general case, many heuristics have been proposed, some can be found in the review papers ofConte et al. (2004) and Foggia et al. (2014). Closer to our approach is the convex relaxation to a linear program proposed in Almohamad and Duffuaa (1993) and even more, the convex relaxation proposed by Gold and Rangarajan (1996) of the set of permutation matrices to its convex hull, i.e., the set of doubly-stochastic matrices. Under certain conditions on the graph structures, the solution of this relaxation is the same as the exact isomorphism (Aflalo et al., 2014), but in most cases, there are no guarantees to converge to a corner of the convex hull, and can even provably converge to poor solutions for some graph families (Lyzinski et al., 2016). However, Lyzinski et al. (2016) have suggested that they can be used as an initialization for an inexact or non-convex algorithm like the concave minimization problem of Zaslavskiy et al. (2009) or the problem formulated in Eq. (2).
As in Bojanowski and Joulin (2017), our formulation in Eq. (2) minimizes the Wasserstein distance (also known as Earth Mover’s distance) to estimate the minimal transformation between two distributions of points (Rubner et al., 1998). We refer the reader to Peyré and Cuturi (2017) for an exhaustive survey on the subject. Kusner et al. (2015) use an EMD between word embeddings to compute the distance between documents, and later Rolet et al. (2016) use a smooth Wasserstein distance to align word embeddings distribution and discover cross-language topics. Flamary et al. (2016) optimize a different Wasserstein objective over the Stiefel manifold, for the task of discriminant analysis. The Wasserstein distance has also been used in the field of statistics to give meaningful notions of means for misaligned signals (Panaretos et al., 2016; Zemel and Panaretos, 2017). Finally, our approach bears some similarities with the Gromov-Wasserstein that has been successfully used to align shapes in Mémoli (2007); Solomon et al. (2016).
In this section, we evaluate our method in two settings. First, we perform toy experiments to gain some insight regarding our approach. We then compare it to various state of the art algorithms on the task of unsupervised word translation.
4.1 Toy experiments
Instead of generating purely random datasets, we train various word embeddings, using the same corpus but introducing some noise in the training process. More specifically, we train skipgram and cbow models using the fasttext tool (Bojanowski et al., 2016) on 100M English tokens from the 2007 News Crawl corpus.111http://statmt.org/wmt14/translation-task.html We consider the following settings to generate our toy datasets:
Seed: we learn two models on the same data, with the same hyper-parameters. The only difference between the two models is the seed used to initialize the parameters of the models. Please note that during the training of skipgram models, frequent words are randomly subsampled, as well as negative examples. There is thus two sources of randomness between the two models: the initial parameters of the model and the sampling during training.
Data: in that instance, we learn two embeddings using different split of the data. More precisely, we use the first 100M tokens for the first model and the next 100M tokens for the second model. Since both training splits come from the same domain, the models are closed.
Window: in that instance, we learn two skipgram models on the same data, but with different window size. The first model uses a window of size , while the second model uses a window of size . We also use different seeds, similarly to the first instance.
Algorithm: for the last setting, we consider two word embeddings from different models: skipgram and cbow.
Because all models are trained on English data, we have the ground truth matching between vectors from the two sets. We can thus estimate an orthogonal matrix using Procrustes, and measure the “distance” between the two sets of points. We report these for the different settings in Table 1. For these experiments, we compare the different approaches on the first 10,000 points of our models.
First, we compare our stochastic algorithm with random initialization with the same algorithm initialized with the solution of the convex relaxation. We observe that while the random initialization sometimes converges for the easy problems, its rate of success rapidly decreases. On the other hand, when initialized with the solution of the convex formulation, our stochastic algorithm always converges to a good solution. Interestingly, even when the solution of the relaxed problem is not a good solution, it still provides a good initialization point for the stochastic gradient method.
4.2 Unsupervised word translation
In our second set of experiments, we evaluate our method on the task of unsupervised word translation. Given word embeddings trained on two monolingual corpora, the goal is to infer a bilingual dictionary by aligning the corresponding word vectors. We use the same exact setting as Conneau et al. (2017). In particular, we use the same evaluation datasets and code, as well as the same word vectors.
For CSLS, we use nearest neighbors and for ISF, we use a temperature of . Following Xing et al. (2015) and Artetxe et al. (2016), we normalize and center the embeddings for both Procrustes and our approach. The refinement approach is taken from Conneau et al. (2017) and consists in alternating, for epochs, between building a dictionary of mutual nearest neighbors (using CSLS) and running Procrustes on this dictionary. This procedure is thus ICP with the CSLS criterion. Our convex relaxation is too demanding to run on all the vectors, and thus, we use the 2.5K most frequent words for our initialization. As noted in section 2, the choice of the batch size is important: a larger batch size will lead to a better approximation of our cost function, but is also more computationally intensive. Thus, we start with a small batch size of , and double the batch size after a third and two third of the optimization. Finally, we use the Sinkhorn solver of Cuturi (2013) to compute approximate solutions of optimal transport problems.
In order to quantitatively assess the quality of each approach, we consider the problem of bilingual lexicon induction. This can be formulated as a retrieval problem, and following standard practice, we report the precision at one. As observed by Artetxe et al. (2017) and Conneau et al. (2017), refining the alignment significantly improves the performance of unsupervised approaches. We thus report the performance of different methods after refinement in Table 2. It is interesting to see that the refinement also improves supervised approaches. Several reasons can explain this: first the lexicons for supervision are not clean, and the
-loss is sensible to outliers. Second the lexicons are small (aroundK) and the mapping is not regularized, besides the orthogonality constraint.
|Proc. - NN||78.3||80.9||77.0||80.0||70.8||71.8||49.5||64.6|
|Proc. - CSLS||81.1||84.8||81.6||84.0||74.7||74.4||52.0||67.8|
|Proc. - ISF||81.4||83.4||81.2||82.9||71.8||72.4||50.4||67.8|
|Adv. - NN||69.8||71.3||70.4||61.9||63.1||59.6||29.1||41.5|
|Adv. - CSLS||75.7||79.7||77.8||71.2||70.1||66.4||37.2||48.1|
|Ours - NN||77.2||75.6||75.0||72.1||66.0||62.9||32.6||48.6|
|Ours - CSLS||79.8||81.8||79.8||78.0||69.4||66.4||37.5||50.3|
|Ours - ISF||80.2||80.3||79.6||77.2||66.9||64.2||36.9||50.3|
Overall our performance is on par with ICP and significantly better than adversarial training. The fact that ICP and our approach achieve comparable results is not surprising as we are considering a similar loss function. However, ICP requires a lot of random initializations to converge to a good solution, while initializing our approach with the result of the convex relaxation guarantees the same performance with a single run.
Results without refinement.
The refinement step used in Table 2 is an ICP with a CSLS criterion. This procedure requires a good initialization to converge to a satisfying solution. Hoshen and Wolf (2018) uses hundreds of initializations to find a good starting point, while the adversarial training often gives a decent initialization with a single run. We compare the quality of this intialization with the solution of our relaxed method in Table 3. For both NN and CSLS, our approach performs better than the GAN based method of Conneau et al. (2017) on out of pairs of languages while being simpler. However, we are still significantly worse than the supervised approach, by often more than . It is interesting to notice that for both Procrustes and our approach, ISF and CSLS obtain similar performance. Empirically, we found that ISF is more sensitive to its free parameter. In particular, the temperature in ISF should vary with the number of points, and the fact that a single value works across all these datasets is potentially an indirect consequence of the evaluation, that is to restrict the size of the sets to words. We thus propose to use CSLS in most of our experiments.
Impact of the batch size on the performance.
The size of the batch size plays an important role on our surrogate loss as it trades off speed for distance to the original formulation. Table 4 shows its influence on the performance as well as running time. As expected, our model converges faster for small since we run the Sinkhorn algorithm on matrices, thus leading to a complexity of up to logarithmic terms (Altschuler et al., 2017). The larger is, the better the approximation of the squared Wasserstein distance is, and the closer we are to the real loss function. Despite saying nothing about the quality of the local minima reached by the surrogate, it is interesting to see that in practice this converts into better performance as well.
This paper presents a general approach to align embeddings in high dimensional space. While the overall problem is non-convex and computationally expensive, we present an efficient stochastic algorithm to solve the problem. We also develop a convex relaxation that can be used to initialize our approach. We validate our method on a few toy examples and a real application, namely unsupervised word translation, where we achieve performances that are on par with the state-of-the-art. A few questions remain, in particular the link between graph matching and point cloud alignments can be further investigated. It should be possible to identify the type of problems where our approach is guaranteed to work and where it will provably fail. Finally, we think that it is possible to improve the relaxation procedure.
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