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Title: Don't Settle for Average, Go for the Max: Fuzzy Sets and Max-Pooled Word Vectors

Paper content:
Recent literature suggests that averaged word vectors followed by simple post-processing outperform many deep learning methods on semantic textual similarity tasks.
Furthermore, when averaged word vectors are trained supervised on large corpora of paraphrases, they achieve state-of-the-art results on standard STS benchmarks.
Inspired by these insights, we push the limits of word embeddings even further.
We propose a novel fuzzy bag-of-words (FBoW) representation for text that contains all the words in the vocabulary simultaneously but with different degrees of membership, which are derived from similarities between word vectors.
We show that max-pooled word vectors are only a special case of fuzzy BoW and should be compared via fuzzy Jaccard index rather than cosine similarity.
Finally, we propose DynaMax, a completely unsupervised and non-parametric similarity measure that dynamically extracts and max-pools good features depending on the sentence pair.
This method is both efficient and easy to implement, yet outperforms current baselines on STS tasks by a large margin and is even competitive with supervised word vectors trained to directly optimise cosine similarity.
Natural languages are able to encode sentences with similar meanings using very different vocabulary and grammatical constructs, which makes determining the semantic similarity between pieces of text a challenge.
It is common to cast semantic similarity between sentences as the proximity of their vector representations.
More than half a century since it was first proposed, the Bag-of-Words (BoW) representation (Harris, 1954; BID47 BID37 remains a popular baseline across machine learning (ML), natural language processing (NLP), and information retrieval (IR) communities.
In recent years, however, BoW was largely eclipsed by representations learned through neural networks, ranging from shallow BID36 BID21 to recurrent BID12 BID53 , recursive BID51 BID55 , convolutional BID30 BID32 , self-attentive BID57 BID9 and hybrid architectures BID19 BID56 BID66 .Interestingly
, BID5 showed that averaged word vectors BID38 BID44 BID6 BID29 weighted with the Smooth Inverse Frequency (SIF) scheme and followed by a Principal Component Analysis (PCA) post-processing procedure were a formidable baseline for Semantic Textual Similarity (STS) tasks, outperforming deep representations. Furthermore,
BID59 and BID58 showed that averaged word vectors trained supervised on large corpora of paraphrases achieve state-of-the-art results, outperforming even the supervised systems trained directly on STS.Inspired by these insights, we push the boundaries of word vectors even further. We propose a
novel fuzzy bag-of-words (FBoW) representation for text. Unlike classical
BoW, fuzzy BoW contains all the words in the vocabulary simultaneously but with different degrees of membership, which are derived from similarities between word vectors.Next, we show that max-pooled word vectors are a special case of fuzzy BoW. Max-pooling significantly
outperforms averaging on standard benchmarks when word vectors are trained unsupervised.Since max-pooled vectors are just a special case of fuzzy BoW, we show that the fuzzy Jaccard index is a more suitable alternative to cosine similarity for comparing these representations. By contrast, the fuzzy Jaccard
index completely fails for averaged word vectors as there is no connection between the two. The max-pooling operation is commonplace
throughout NLP and has been successfully used to extract features in supervised systems BID10 BID32 BID31 BID13 BID12 BID15 ; however, to the best of our knowledge, the present work is the first to study max-pooling of pre-trained word embeddings in isolation and to suggest theoretical underpinnings behind this operation.Finally, we propose DynaMax, a completely unsupervised and non-parametric similarity measure that dynamically extracts and max-pools good features depending on the sentence pair. DynaMax outperforms averaged word vector
with cosine similarity on every benchmark STS task when word vectors are trained unsupervised. It even performs comparably to BID58 's
vectors under cosine similarity, which is a striking result as the latter are in fact trained supervised to directly optimise cosine similarity between paraphrases, while our approach is completely unrelated to that objective. We believe this makes DynaMax a strong
baseline that future algorithms should aim to beat in order to justify more complicated approaches to semantic similarity.As an additional contribution, we conduct significance analysis of our results. We found that recent literature on STS
tends to apply unspecified or inappropriate parametric tests, or leave out significance analysis altogether in the majority of cases. By contrast, we rely on nonparametric
approaches with much milder assumptions on the test statistic; specifically, we construct bias-corrected and accelerated (BCa) bootstrap confidence intervals BID17 for the delta in performance between two systems. We are not aware of any prior works that
apply such methodology to STS benchmarks and hope the community finds our analysis to be a good starting point for conducting thorough significance testing on these types of experiments.
In this work we combine word embeddings with classic BoW representations using fuzzy set theory.
We show that max-pooled word vectors are a special case of FBoW, which implies that they should be compared via the fuzzy Jaccard index rather than the more standard cosine similarity.
We also present a simple and novel algorithm, DynaMax, which corresponds to projecting word vectors onto a subspace dynamically generated by the given sentences before max-pooling over the features.
DynaMax outperforms averaged word vectors compared with cosine similarity on every benchmark STS task when word vectors are trained unsupervised.
It even performs comparably to supervised vectors that directly optimise cosine similarity between paraphrases, despite being completely unrelated to that objective.Both max-pooled vectors and DynaMax constitute strong baselines for further studies in the area of sentence representations.
Yet, these methods are not limited to NLP and word embeddings, but can in fact be used in any setting where one needs to compute similarity between sets of elements that have rich vector representations.
We hope to have demonstrated the benefits of experimenting more with similarity metrics based on the building blocks of meaning such as words, rather than complex representations of the final objects such as sentences.
In the word fuzzification step the membership values for a word w are obtained through a similarity function sim (w, u (j) ) between the word embedding w and the rows of the universe matrix U , i.e. DISPLAYFORM0 In Section 2.2, sim(w, u (j) ) was the dot product w · u (j) and we could simply write µ = wU T .
There are several reasons why we chose a similarity function that takes values in R as opposed to DISPLAYFORM1 First, we can always map the membership values from R to (0, 1) and vice versa using, e.g. the logistic function σ(x) = 1 1+e −ax with an appropriate scaling factor a > 0.
Intuitively, large negative membership values would imply the element is really not in the set and large positive values mean it is really in the set.
Of course, here both 'large' and 'really' depend on the scaling factor a.
In any case, we see that the choice of R vs. [0, 1] is not very important mathematically.
Interestingly, since we always max-pool with a zero vector, fuzzy BoW will not contain any negative membership values.
This was not our intention, just a by-product of the model.
For completeness, let us insist on the range [0, 1] and choose sim (w, u (j) ) to be the clipped cosine similarity max (0, cos(w, u (j) )).
This is in fact equivalent to simply normalising the word vectors.
Indeed, the dot product and cosine similarity become the same after normalisation, and max-pooling with the zero vector removes all the negative values, so the resulting representation is guaranteed to be a [0, 1]-fuzzy set.
Our results for normalised word vectors are presented in TAB3 .After
comparing TAB0 we can draw two conclusions. Namely
, DynaMax still outperforms avg-cos by a large margin even when word vectors are normalised. However
, normalisation hurts both approaches and should generally be avoided. This is
not surprising since the length of word vectors is correlated with word importance, so normalisation essentially makes all words equally important BID48 .

assistant: text: Max-pooled word vectors with fuzzy Jaccard set similarity are an extremely competitive baseline for semantic similarity; we propose a simple dynamic variant that performs even better.