Two weeks ago, I did a presentation for my colleagues of the paper from Yurochkin et al. (2019), from NeurIPS 2019. It contains an interesting approach to document classification leading to strong performance, and, most importantly, excellent interpretability.

This paper seems interesting to me because of it uses two methods with strong theoretical guarantees: optimal transport and topic modelling. Optimal transport looks very promising to me in NLP, and has seen a lot of interest in recent years due to advances in approximation algorithms, such as entropy regularisation. It is also quite refreshing to see approaches using solid results in optimisation, compared to purely experimental deep learning methods.

Introduction and motivation

The problem of the paper is to measure similarity (i.e. a distance) between pairs of documents, by incorporating semantic similarities (and not only syntactic artefacts), without encountering scalability issues.

They propose a “meta-distance” between documents, called the hierarchical optimal topic transport (HOTT), providing a scalable metric incorporating topic information between documents. As such, they try to combine two different levels of analysis:

• word embeddings data, to embed language knowledge (via pre-trained embeddings for instance),
• topic modelling methods (e.g. Latent Dirichlet Allocation), to represent semantically-meaningful groups of words.

Background: optimal transport

The essential backbone of the method is the Wasserstein distance, derived from optimal transport theory. Optimal transport is a fascinating and deep subject, so I won’t enter into the details here. For an introduction to the theory and its applications, check out the excellent book from Peyré and Cuturi (2019), (available on ArXiv as well). There are also very nice posts by Gabriel Peyré on the CNRS maths blog (in French). Many more resources (including slides for presentations) are available at https://optimaltransport.github.io. For a more complete theoretical treatment of the subject, check out Santambrogio (2015), or, if you’re feeling particularly adventurous, Villani (2009).

For this paper, only a superficial understanding of how the Wasserstein distance works is necessary. Optimal transport is an optimisation technique to lift a distance between points in a given metric space, to a distance between probability distributions over this metric space. The historical example is to move piles of dirt around: you know the distance between any two points, and you have piles of dirt lying around¹. Now, if you want to move these piles to another configuration (fewer piles, say, or a different repartition of dirt a few metres away), you need to find the most efficient way to move them. The total cost you obtain will define a distance between the two configurations of dirt, and is usually called the earth mover’s distance, which is just an instance of the general Wasserstein metric.

More formally, if we have to sets of points \(x = (x_1, x_2, \ldots, x_n)\), and \(y = (y_1, y_2, \ldots, y_n)\), along with probability distributions \(p \in \Delta^n\), \(q \in \Delta^m\) over \(x\) and \(y\) (\(\Delta^n\) is the probability simplex of dimension \(n\), i.e. the set of vectors of size \(n\) summing to 1), we can define the Wasserstein distance as \[ W_1(p, q) = \min_{P \in \mathbb{R}_+^{n\times m}} \sum_{i,j} C_{i,j} P_{i,j} \] \[ \text{\small subject to } \sum_j P_{i,j} = p_i \text{ \small and } \sum_i P_{i,j} = q_j, \] where \(C_{i,j} = d(x_i, x_j)\) are the costs computed from the original distance between points, and \(P_{i,j}\) represent the amount we are moving from pile \(i\) to pile \(j\).

Now, how can this be applied to a natural language setting? Once we have word embeddings, we can consider that the vocabulary forms a metric space (we can compute a distance, for instance the euclidean or the cosine distance, between two word embeddings). The key is to define documents as distributions over words.

Given a vocabulary \(V \subset \mathbb{R}^n\) and a corpus \(D = (d^1, d^2, \ldots, d^{\lvert D \rvert})\), we represent a document as \(d^i \in \Delta^{l_i}\) where \(l_i\) is the number of unique words in \(d^i\), and \(d^i_j\) is the proportion of word \(v_j\) in the document \(d^i\). The word mover’s distance (WMD) is then defined simply as \[ \operatorname{WMD}(d^1, d^2) = W_1(d^1, d^2). \]

If you didn’t follow all of this, don’t worry! The gist is: if you have a distance between points, you can solve an optimisation problem to obtain a distance between distributions over these points! This is especially useful when you consider that each word embedding is a point, and a document is just a set of words, along with the number of times they appear.

Hierarchical optimal transport

Using optimal transport, we can use the word mover’s distance to define a metric between documents. However, this suffers from two drawbacks:

• Documents represented as distributions over words are not easily interpretable. For long documents, the vocabulary is huge and word frequencies are not easily understandable for humans.
• Large vocabularies mean that the space on which we have to find an optimal matching is huge. The Hungarian algorithm used to compute the optimal transport distance runs in \(O(l^3 \log l)\), where \(l\) is the maximum number of unique words in each documents. This quickly becomes intractable as the size of documents become larger, or if you have to compute all pairwise distances between a large number of documents (e.g. for clustering purposes).

To escape these issues, we will add an intermediary step using topic modelling. Once we have topics \(T = (t_1, t_2, \ldots, t_{\lvert T \rvert}) \subset \Delta^{\lvert V \rvert}\), we get two kinds of representations:

• representations of topics as distributions over words,
• representations of documents as distributions over topics \(\bar{d^i} \in \Delta^{\lvert T \rvert}\).

Since they are distributions over words, the word mover’s distance defines a metric over topics. As such, the topics with the WMD become a metric space.

We can now define the hierarchical optimal topic transport (HOTT), as the optimal transport distance between documents, represented as distributions over topics. For two documents \(d^1\), \(d^2\), \[ \operatorname{HOTT}(d^1, d^2) = W_1\left( \sum_{k=1}^{\lvert T \rvert} \bar{d^1_k} \delta_{t_k}, \sum_{k=1}^{\lvert T \rvert} \bar{d^2_k} \delta_{t_k} \right). \] where \(\delta_{t_k}\) is a distribution supported on topic \(t_k\).

Note that in this case, we used optimal transport twice:

• once to find distances between topics (WMD),
• once to find distances between documents, where the distance between topics became the costs in the new optimal transport problem.

The first one can be precomputed once for all subsequent distances, so it is invariable in the number of documents we have to process. The second one only operates on \(\lvert T \rvert\) topics instead of the full vocabulary: the resulting optimisation problem is much smaller! This is great for performance, as it should be easy now to compute all pairwise distances in a large set of documents.

Another interesting insight is that topics are represented as collections of words (we can keep the top 20 as a visual representations), and documents as collections of topics with weights. Both of these representations are highly interpretable for a human being who wants to understand what’s going on. I think this is one of the strongest aspects of these approaches: both the various representations and the algorithms are fully interpretable. Compared to a deep learning approach, we can make sense of every intermediate step, from the representations of topics to the weights in the optimisation algorithm to compute higher-level distances.

References