Hamming Sentence Embeddings for Information Retrieval

TL;DR

This paper introduces a neural encoder-decoder method to compress sentence embeddings into Hamming space, enabling efficient retrieval with minimal performance loss at high compression ratios.

Contribution

It presents a novel neural compression technique that produces binary Hamming embeddings for sentences, maintaining semantic similarity performance at high compression ratios.

Findings

Comparable performance to uncompressed embeddings at 256:1 compression ratio

Strong feature decorrelation and good generalization from Wikipedia training

Source code and experimental results publicly available

Abstract

In retrieval applications, binary hashes are known to offer significant improvements in terms of both memory and speed. We investigate the compression of sentence embeddings using a neural encoder-decoder architecture, which is trained by minimizing reconstruction error. Instead of employing the original real-valued embeddings, we use latent representations in Hamming space produced by the encoder for similarity calculations. In quantitative experiments on several benchmarks for semantic similarity tasks, we show that our compressed hamming embeddings yield a comparable performance to uncompressed embeddings (Sent2Vec, InferSent, Glove-BoW), at compression ratios of up to 256:1. We further demonstrate that our model strongly decorrelates input features, and that the compressor generalizes well when pre-trained on Wikipedia sentences. We publish the source code on Github and all experimental results.

Tables

Table 1. Table 1. The average absolute correlation of sentence embedding’s dimensions (Equation ( 1 )) on the Wikipedia-1mio dataset. The compressed embeddings’ correlation is about 3-4 lower than the original embeddings’.

	avg. correlation (%)
embedding type	original	compressed
infersent	10.17	2.26
mbow	9.31	2.73
sent2vec	5.37	1.83

Equations

$$f_1(\mathbf{x}):=\text{tanh}(A\cdot \mathbf{x}+\mathbf{b})$$

$$z_i^k:=\frac{\exp (l(y_i^k))}{\exp (l(y_1^k))+\exp (l(y_2^k))}\text{with}l(y)=\frac{1}{\tau }\cdot (y+g).$$

$$\underset{A,\mathbf{b},A^{\prime },{\mathbf{b}}^{\prime },C}{\mathrm{argmin}}\sum _{i=1}^n||{\mathbf{e}}_i-{\mathbf{e}}_i^{\prime }|{|}_{{}_2}^{{}^2}$$

$$\frac{1}{d^2}\sum _{i=1}^d\sum _{j=1}^d|\rho (i,j)|.$$

Full text

In quantitative experiments on several benchmarks for semantic similarity tasks, we show that our compressed hamming embeddings yield a comparable performance to uncompressed embeddings (Sent2Vec, InferSent, Glove-BoW), at compression ratios of up to 256:1. We further demonstrate that our model strongly decorrelates input features, and that the compressor generalizes well when pre-trained on Wikipedia sentences. We publish the source code on Github111https://github.com/ungol-nlp and all experimental results222http://bit.ly/2H8cP7Q.

hashing, semantic similarity, autoencoder, sentence embeddings

††copyright: none††conference: ; ; ††price: ††isbn:

1. Introduction

Dense, real-valued embeddings play a fundamental role as representations for words (Mikolov et al., 2013; Pennington et al., 2014; Bojanowski et al., 2016) or sentences and documents (Pagliardini et al., 2017; Conneau et al., 2017; Arora et al., 2016) in neural NLP models. They are known to convey semantics – for example, the semantic similarity of two embedded entities is commonly expressed by their embeddings’ distance.

Since embeddings are high-dimensional vectors (often of 100-1000 features), they require memory of up to multiple GB of memory for large vocabularies or corpora. To allow low-capacity devices such as mobile phones or embedded systems to facilitate machine learning models that process word embeddings, recent work implements encoder-decoder models for compressing word embeddings (Tissier et al., 2018; Shu and Nakayama, 2017), which are trained to reconstruct the original embeddings. The resulting (binary) latent representation of the encoder can then be used for downstream tasks like classification or retrieval, or can be used in conjunction with an instance of the decoder to create a reproduction of the original embedding. This approach not only reduces the required memory significantly, but also speeds up the distance calculation (either in a brute-force comparison or in combination with index structures (Norouzi et al., 2014)).

While the above approaches have focused on word-level compression (Tissier et al., 2018; Shu and Nakayama, 2017), we study the compression of sentence-level embeddings. This is of practical relevance, since unlike words – which come with a fixed vocabulary, such that similarities or clusters can be cached – the space of sentences is virtually infinite. We apply an encoder-decoder model similar to Shu’s and Nakayama’s (Shu and Nakayama, 2017) to several state-of-the-art sentence embeddings (Sent2Vec, InferSent, GloVe-BoW) and use the resulting binary embeddings in textual similarity and retrieval tasks. Our contributions are:

• •

We show that the spatial properties of sentence embeddings are retained well by the produced hash codes, which holds regardless of the upstream sentence embedding model. Thereby, accuracy depends on the task at hand: While compressed embeddings yield competitive results for semantic similarity (STS 2012-16), they are outperformed in topic-oriented categorization tasks.

• •

We show that a compressor trained on relatively few (100K) Wikipedia sentences in a few minutes generalizes well.

• •

An explanation for the good performance of our model is that it decorrelates redundant dimensions in the input data, as we demonstrate quantitatively.

2. Related Work

One of the earliest methods to produce hash codes based on neural architectures was semantic hashing (Salakhutdinov and Hinton, 2009), which produces bit-codes by training a stack of RBMs learning a latent representation of word frequency. In recent work either the combination of existing word embeddings (Wieting et al., 2015; Arora et al., 2016) or independent machine learning models (Logeswaran and Lee, 2018; Conneau et al., 2017; Le and Mikolov, 2014; Pagliardini et al., 2017; Zhu et al., 2015) transform phrases, sentences or documents to embeddings. To our knowledge, no attempt to create hash codes for such sentence embeddings has been evaluated so far.

On word level, hashing has been studied by introducing a virtual quantization function to the CBOW approach (Lam, 2018). Shu and Nakayama (2017) introduce an auto-encoder for compressing word embeddings but did not evaluate a configuration which produces binary codes. Recently (Tissier et al., 2018) proposed a neural auto-encoder that produces hash codes by thresholding the latent representation. To preserve the spatial information of the input space, a regularization method is added to the loss function.

3. Approach

We explore the usefulness of hashes for fast retrieval using a model similar to Shu’s and Nakayama’s (Shu and Nakayama, 2017). Figure 1 illustrates our approach: Sentence embeddings (gained by a pre-trained model) are compressed using an encoder-decoder architecture (in the following referred to as the compressor) which is trained to minimize reconstruction error. The compressor consists of an encoder $\phi$ and a decoder $\psi$. The encoder’s output is mapped to a binary vector which is then used for a low-resource comparison in retrieval.

The encoder transforms a real-valued embedding $\mathbf{e}$ to $b$ one-hot encoded representations, each of the same dimension $p$. We use a fixed value of $p=2$ (which we found to work best) in this paper, such that the bottleneck is a $(2\cdot b)$-dimensional binary feature. More precisely, the encoder consists of two fully connected layers: $\phi(\mathbf{e}):=\sigma(f_{2}(f_{1}(\mathbf{e})))$ where

[TABLE]

with $A\in\mathbb{R}^{b\times d}$ and $A^{\prime}\in\mathbb{R}^{2b\times b}$. The first layer $f_{1}$ has $b$ output neurons, the second layer produces $2b$ logit values $(y_{1}^{1},y_{2}^{1}),(y_{1}^{2},y_{2}^{2}),$ $\dots,(y_{1}^{b},y_{2}^{b})$ viewed in pairs, $\mathbf{y}^{k}:=(y_{1}^{k},y_{2}^{k})$. The function $\sigma$ applies a softmax to each pair, i.e. $\sigma(\mathbf{y}^{1},...,\mathbf{y}^{b})=\mathbf{z}^{1},...,\mathbf{z}^{b}$ with

[TABLE]

With the function $l(\cdot)$, we apply the Gumbel Softmax trick (Jang et al., 2016; Maddison et al., 2016) to bias the model towards a categorical distribution, whereas the free parameter $\tau$ controls the degree of discreteness and $g$ denotes random noise sampled from the Gumbel extreme value distribution (Gumbel, 1954). When applying the trained model, we set $g=0$ and obtain a $b$-dimensional binary representation by thresholding $(z_{1}^{1},z_{1}^{2},...,z_{1}^{b})$ at 0.5. This representation is referred to as the compressed embedding in the following, and is used for similarity evaluation.

The decoder transforms the bottleneck back to a reconstruction $\mathbf{e}^{\prime}$ of the input embedding $\mathbf{e}$ . It is based on the idea of additive vector quantization for compression (Babenko and Lempitsky, 2014; Jegou et al., 2011), i.e. the reconstruction is a linear combination of basis vectors organised in codebooks. There are $2b$ such basis vectors stored in a matrix $C\in\mathbb{R}^{d\times 2b}$. Given an encoder output $\mathbf{x}=(\mathbf{z}^{1},...,\mathbf{z}^{b})^{T}$ of size $2b$, the decoder output $\mathbf{e}^{\prime}$ is defined as $\psi(\mathbf{x}):=C\cdot\mathbf{x}$.

Training: Given a set of training embeddings $\mathbf{e}_{1},\mathbf{e}_{2},...,\mathbf{e}_{n}$, the model’s parameters $\{A,\mathbf{b},A^{\prime},\mathbf{b}^{\prime},C\}$ are fitted such that the average Euclidean distance of all training samples to their respective reconstruction is minimized:

[TABLE]

Optimization is carried out using Stochastic Gradient Descent with the Adam Optimizer (initial learning rate $10^{-4}$). The decoder’s codebook’s vectors $c_{i}$ are initialised by randomly sampling from the training embeddings $\mathbf{e}_{i}$: $c_{i}=\frac{1}{xb}\mathbf{e}_{i}$ with $x\in[1,2]$ to obtain about the same norm for the reconstruction when training starts. Training is stopped when no significant change ($\Delta<10^{-5}$ to $10^{-4}$) of the loss value can be observed over a fixed period of $100$ epochs. The parameter $\tau$ is set to one for the majority of models333We found that very few models only trained by annealing from $\tau=1$ to $0.75$..

4. Experiments

To examine how well the spatial information of sentence embeddings is retained when compressed with our encoder, we use two similarity-based tasks, namely semantic textural similarity (STS, Section 4.1) and k-NN document classification (Section 4.4). We compare three types of embeddings:

• •

The original real-valued sentence embeddings, using InferSent (Conneau et al., 2017), Sent2Vec (Pagliardini et al., 2017), or bag-of-words with averaged Glove vectors (Pennington et al., 2014). All models were used in their pre-trained versions, no fine-tuning was applied. We tested several common distance measures but report the cosine similarity (which we found to work best).

• •

The compressed sentence embeddings from our encoder. We trained compressors for $128$, $256$ and $512$ bits, either randomly sampled sentences from Wikipedia or on sentences collected from the respective task’s training set (e.g., $10518$ sentences from the STS datasets). Any binary embeddings are compared using the Hamming distance.

• •

a simple baseline binarization, which transforms a real-valued input embedding $\mathbf{e}\in\mathbb{R}^{d}$ into a binary embedding $\mathbf{e}^{\prime}\in\{0,1\}^{d}$ by thresholding each dimension at the median: $\mathbf{e}^{\prime}_{i}:=\mathbf{1}_{\mathbf{e}_{i}\geq\,\text{median}(E_{i,*})}$ (where $E=[\mathbf{e}_{1},...,\mathbf{e}_{n}]\in\mathbb{R}^{d\times n}$ denotes the embedding matrix).

4.1. Semantic Similarity

We use the SentEval (Conneau and Kiela, 2018) implementation of the SemEval Semantic Textual Similarity (STS) tasks 2012-2016. Figure 2 compares our compressor model (trained on 1 mio Wikipedia sentences) with the other methods. We report Spearman’s $\rho$ between ground truth similarity annotations and similarity scores (higher is better).

Our model achieves competitive results for all three sentence embeddings, even outperforming the original real-valued embeddings. This improvement is strongest for averaged word vectors ( mbow). Also, Figure 1 shows that our model generalizes well from Wikipedia to other text domains without the need of any refinement, and that training on a relatively small dataset (100,000 sentences) – which takes in the order of minutes – already yields competitive performance. For the remainder of the paper, models trained on 1 mio. Wikipedia sentences will be used.

4.2. Memory Footprint

The biggest advantage of the binary representation is the compactness of its encoding. For example, storing 10 mio. 700-dimensional sent2vec embeddings takes 28GB of disk space ($700\times 4$ bytes per embedding at single float precision), while the corresponding (512-bit) hash codes require only $640$ MB while reaching the same accuracy. For Infersent ($4096$ dimensions), memory is even reduced from $163$GB to $640$MB (which corresponds to a reduction factor of 256:1).

4.3. Correlation Inspection

The competitive performance of our compressor could be due to the fact that the original float embeddings contain redundant features (i.e. certain dimensions in the embeddings carry the same information), which may have an impact on semantic similarity calculations. In contrast to this, the compressor – forced to remove redundancy in the embeddings – aggregates these dimensions.

To test this hypothesis, we inspect the scale of correlations between dimensions in the embeddings: Let $\mathbf{e}_{1},...,\mathbf{e}_{n}$ denote d-dimensional embeddings, and let $\rho(i,j)$ (for $i,j=1,...,d$) denote the correlation between two dimensions in these embeddings. We measure the overall correlation within the embeddings as

[TABLE]

Table 1 illustrates the overall correlation on the Wikipedia-1m dataset, using (a) the original real-valued embeddings, and (b) the 512-dimensional hamming embeddings. We observe a strong reduction of correlation between the embeddings by a factor of about 3-4, which illustrates the decorrelating effect of our compressor.

4.4. k-NN Document Classification

Finally, we assess how well the Hamming sentence embeddings can be used in a more heterogeneous, topic-oriented information retrieval setting. We employ three datasets: (1) amazon (Blitzer et al., 2007) (reviews from four different product categories; 6400/1600 documents), (2) classic (Tunali, 2010) (short citations from academic papers of four different categories; 5600/1400 documents) and (3) dbpedia (self-crawled Wikipedia pages sampled from 13 classes; 47000/11400 documents; inspired by (Zhang and LeCun, 2015)). All documents are pruned to the first 1000 tokens. Each test document is classified by a voting over its $10$ nearest training documents (computed by their similarity scores on our different embeddings), whereas each vote is weighted such that for the $n$-th neighbour the weight for the vote is defined as $1/\sqrt{n}$.

The results of this evaluation are mixed (Figure 4): Working with the original embeddings and determining the cosine similarity works best in general. The median baseline works surprisingly well, in some cases even outperforming all other methods, but failing for Infersent vectors on classic and amazon. Also, the picture with respect to the training domain is mixed, as sometimes training on the target domain and sometimes training on Wikipedia performs better. Overall, we found results for this topical evaluation to be more instable than for sentence similarity. Figure 5 matches these quantitative results with a visualization of embeddings: For the classes Film (top), Athlete (center) and Plant (bottom) from the dbpedia dataset, we picked 300 random samples and visualized the binary hash codes ($b=512$, trained on $1$ mio. Wikipedia sentences). The figure illustrates that the binary hashes show some, but not a strong correlation to the document category, an issue that will require further investigation.

5. Conclusion

In this paper, we have studied a neural encoder-decoder to produce binary hash codes for an efficient similarity matching on sentence- and paragraph level. We have shown that the spatial information is retained well on a simple sentence-level similarity task. Also, we found our model to decorrelate the input embeddings’ dimensions, and training on a limited number of Wikipedia sentences generalizes well (at least for the STS task). When applied to topic-oriented k-NN classification task, the model yields mixed results. We publish a spreadsheet with all experimental results and the source code.