SPINE: SParse Interpretable Neural Embeddings

Introduction

Distributed representations map words to vectors of real numbers in a continuous space. These word vectors have been exploited to obtain state-of-the-art results in NLP tasks, such as parsing (?), named entity recognition (?), and sentiment analysis (?). However, word vectors have dense representations that humans find difficult to interpret. For instance, we are often clueless as to what a “high” value along a given dimension of a vector signifies when compared to a “low” value. To demonstrate this, we analyze embeddings of few randomly selected words (see Table 1). For these randomly picked words, we examine top participating dimensions (Top participating dimensions are the dimensions that have highest absolute values for that word). For each of these selected top dimensions, we note the words that have the highest absolute values in that dimension. We observe that for embeddings from state-of-the-art word models like GloVe (?) and word2vec (?) are not ‘interpretable’, i.e. the top participating words do not form a semantically coherent group. This notion of interpretability — one that requires each dimension to denote a semantic concept — resonates with post-hoc interpretability, introduced and discussed in (?).

We argue that this notion of interpretability can help in gaining better understanding of neural representations and models. Interpretability in a general neural network pipeline would not just help us reason about the outcomes that they predict, but would also provide us cues to make them more efficient and robust. In various feature norming studies (?; ?; ?), where participants were asked to list the properties of several words and concepts, it was observed that they typically used few sparse characteristic properties to describe the words, with limited overlap between different words. For instance, to describe the city of Pittsburgh, one might talk about phenomena typical of the city, like erratic weather and large bridges. It is redundant and inefficient to list negative properties, like the absence of the Statue of Liberty. Thus, sparsity and non-negativity are desirable characteristics of representations, that make them interpretable. Many recent studies back this hypothesis (?; ?; ?; ?; ?; ?). This raises the following question:

How does one transform word representations to a new space where they are more interpretable?

To address the question, in this paper, we make following contributions:

We employ a denoising $k$-sparse autoencoder to obtain SParse Interpretable Neural Embeddings (SPINE), a transformation of input word embeddings.

We train the autoencoder using a novel learning objective and activation function to attain interpretable and efficient representations.

We evaluate SPINE using a large scale, crowdsourced, intrusion detection test, along with a battery of downstream tasks. We note that SPINE is more interpretable and efficient than existing state-of-the-art baseline embeddings.

The outline of the rest of the paper is as follows. First, we describe prior work that is closely related to our approach, and highlight the key differences between our approach and existing methods. Next, we provide a mathematical formulation of our proposed method. Thereafter, we describe model training and tuning, and our choice of hyperparameters. Further, we discuss the performance of the embeddings generated by our method on interpretability tests and on various downstream tasks. We conclude by discussing future work.

Related Work

We first discuss previous efforts to attain interpretability in word representations. Then, we discuss prior work related to $k$-sparse autoencoders.

Murphy et al. (?) proposed NNSE (Non-Negative Sparse Embeddings) to learn interpretable word embeddings. They proposed methods to learn sparse representations of words using non-negative matrix factorization on the co-occurrence matrix of words. Faruqui et al. (?, a) consider linguistically inspired dimensions as a means to induce sparsity and interpretability in word embeddings. However, since their dimensions are binary valued, there is no notion of the extent to which a word participates in a particular dimension. Park et al. (?) apply rotations to the word vectors to improve the interpretability of the vectors.

Our method is different from these approaches in two ways. Firstly, our method is based on neural models, and is hence more expressive than linear matrix factorization or simple transformations like rotation. Secondly, we allow for different words to participate at varying levels in different dimensions, and these dimensions are discovered naturally during the course of training the network.

Their work is similar to ours in spirit and motivation, and is the closest match to our goal of producing interpretable and efficient representations. Thus, we compare our approach with SPOWV in both performance and interpretability tests.

k𝑘k-sparse autoencoders

A $k$-sparse autoencoder (?) is an autoencoder for which, with high probability, at most $k$ hidden units are active for any given input. Ng (?) introduced a mechanism to train $k$-sparse autoencoders. The underlying idea is to achieve an expected activation value for a hidden unit that is equivalent to $k$ completely activated hidden units. The training algorithm does so by augmenting a standard input reconstruction loss with a term that measures the deviation between the observed and the desired mean activation values. However, as (?) note, equating the expected activation values does not necessarily produce exactly (or less than) $k$-sparse representations. Our proposed novel objective function and choice of activation function mitigate this issue.

Methodology

In contrast to the sparse coding (matrix factorization) approach of SPOWV, we obtain sparse, interpretable embeddings using a neural model. Specifically, we make use of a denoising $k$-sparse autoencoder that is trained to minimize a loss function that concisely captures the required sparsity constraints. The sparse activations corresponding to the given input embeddings are the interpretable vectors generated by our model. Figure 1 depicts a $k$-sparse autoencoder that produces sparse and interpretable activations for the input word ‘internet’.

Let $\widetilde{\mathbf{X}_{i}}$ be the predicted output for an input embedding $\mathbf{X}_{i}\in\mathcal{D}$. That is,

In this setting, given $\mathcal{D}$, our $k$-sparse autoencoder is trained to minimize the following loss function.

where $RL(\mathcal{D})$ is the reconstruction loss over the data set, $ASL(\mathcal{D})$ is the average sparsity loss over the data set, and $PSL(\mathcal{D})$ is the partial sparsity loss over the data set. The coefficients $\lambda_{1}$ and $\lambda_{2}$ determine the relative importance of the two penalty terms. We define these loss terms below.

In the denoising autoencoder setting, we add isotropic Gaussian noise (with mean $\mathbf{0}$, and variance $\sigma^{2}\mathbf{I}$) to enable the autoencoder to learn more robust intermediate representations of the input.

Average Sparsity Loss (ASL)

In order to enforce $k$-sparse activations in the hidden layer, (?) describe a modification to the basic autoencoder loss function that penalizes any deviation of the observed average activation value from the desired average activation value of a given hidden unit, over a given data set. We formulate this loss as follows.

Please note that in addition to the original formulation in (?), we also allow for the observed average activation value to be less than the desired average activation value, using a max operator.

Partial Sparsity Loss (PSL)

It is possible to obtain an ASL value of without actually having $k$-sparse representations. For example, to obtain an average activation value of $0.5$ for a given hidden unit across 4 examples, one feasible solution is to have an activation value of $0.5$ for all the four examples ((?) too note this problem).

To obtain activation values that are truly $k$-sparse, we introduce a novel partial sparsity loss term that penalizes values that are neither close to nor $1$, pushing them close to either or $1$. We use the following formulation of PSL to do so.

This key addition to the loss term facilitates the generation of sparse embeddings with activations close to 0 and 1.

Choice of activation function

As motivated earlier, non-negativity in the output embeddings is a useful property in the context of interpretability. This drives us to use activation functions that produce non-negative values for all possible inputs values. The activations produced by Rectified Linear Units (ReLU) and Sigmoid units are necessarily positive, making them promising candidates for our use case. Since we wish to allow for strict sparsity (the possibility of exact values), we rule out the Sigmoid activation function, due to its asymptotic nature with respect to activation.

Note that the ReLU activation function produces values in the range $[0,\infty)$, which makes it difficult to argue about the degree of activation of a given hidden unit. Moreover, PSL is not well defined over this range of values. We resolve these issues by using a capped ReLU (cap-ReLU) activation function, that produces activation values in the $$ range. Mathematically,

Experimental Setup

In this section, we discuss model training, hyperparameter tuning and the baseline embeddings that we compare our method against.

Using the formulation described earlier, we train autoencoder models on pre-trained GloVe and word2vec embeddings. The GloVe vectors were trained on 6 billion tokens from a 2014 dump of Wikipedia and Gigaword5, while the word2vec vectors were trained on around 100 billion words from a part of the Google News dataset. Both the GloVe and word2vec embeddings are 300 dimensions long, and we select the 17k most frequently occurring words according to the Leipzig corpus (?). We use 15k of these words for training, and use the remaining 2k for hyperparameter tuning.

Note that through this exercise we evaluate the utility of our additional loss formulation of Partial Sparsity Loss ($PSL$), and we observe that $\lambda_{2}=1$ outperforms $\lambda_{2}=0$ (i.e without the loss) by attaining 22% more sparsity on GloVe and 67% more sparsity on word2vec embeddings at comparable reconstruction loss values.

Baseline embeddings

We compare the embeddings generated by our model (SPINE) against their corresponding starting embeddings (i.e GloVe and word2vec). We also compare our word vectors against Sparse Overcomplete Word Vectors (SPOWV) from (?) , which we believe is a more meaningful comparison, as their approach is also tailored to generate sparse, effective, and interpretable embeddings. In order to perform a fair comparison, we use their method to generate 1000-dimensional output embeddings, the same as ours. All other hyperparameters were used as per the authors’ recommendations.

Interpretability

We evaluate the interpretability of the resulting representations against the ones obtained from baseline models. We estimate the interpretability of the dimensions in two ways. First, we conduct word intrusion detection tests to quantitatively estimate the interpretability of dimensions in the output embedding space. Second, we qualitatively examine the top participating words from a few randomly sampled dimensions and see if they possess observable similarities.

We adopt the evaluation mechanism from reading tree leaves (?), that was first used to interpret probabilistic topic models. Since then, intrusion detection tests have been widely used in estimating interpretability (?), (?), (?).

For a given dimension (column) $Z_{h}$ of the generated embeddings matrix $Z$, we sort the column in the decreasing order of values, and then select the top 4 most active words in that dimension. These four words, if coherent, should be easily identifiable when mixed with a random intruder word. Following the strategies in (?; ?), we select a random intruder word that is both present in the bottom half of the dimension $h$ in question, and in the top 10 percentile in at least one other dimension $h^{\prime}\in\mathcal{H}\setminus\{h\}$. A sample intrusion detection question can be found in Figure 2.

We used the Amazon Mechanical Turk (MTurk) platform to conduct these intrusion detection tests on a large scale. For the two representations SPOWV and SPINE, and for two different initializations (GloVe & word2vec), we randomly sample 300 of the 1000 dimensions, and pose an intrusion detection test for each of these 300 dimensions. Each Human Intelligence Task (HIT) on MTurk consisted of six such questions. Further, every single HIT was independently solved by three different workers. We also conducted similar intrusion detection tests for each of the original GloVe and word2vec embedding dimensions. Accounting for the various settings, a total of 5400 questions were annotated. From the three different annotations we receive for every question, we take the majority vote, and choose that as the answer to that question. In cases where all three annotators mark a different intruder, we randomly select one of the three choices marked by annotators.

Benchmark Downstream Tasks

To test the quality of the embeddings generated by our model, we use them in the following benchmark downstream classification tasks: sentiment analysis, news classification, noun phrase chunking, and question classification. We also test them using the word similarity task discussed in this section (Table 6). Like in (?), we use the average of the word vectors of the words in a sentence as features for text classification tasks. We experiment with SVMs, Logistic Regression and Random forests, which are tuned on the development set. Accuracy is reported on the test set.

Sentiment Analysis: This task tests the semantic properties of word embeddings. It is a sentence-level binary classification task on the Stanford Sentiment Treebank dataset (?). We used the provided train, dev. and test splits with only the non-neutral labels, of sizes 8337, 1081 and 2166 sentences respectively.

Noun Phrase Bracketing: We evaluate the word vectors on NP bracketing task (?), wherein a noun phrase of 3 words is classified as left bracketed or right bracketed. The NP bracketing dataset contains 2,227 noun phrases split into 10 folds. We append the word vectors of three words to get feature representation (?). For words not present in the subset of 17K words we have chosen, we use all zero vectors. We tune on the first fold and report cross-validation accuracy on the remaining nine folds.

Question Classification (TREC): To facilitate research in question answering, (?) propose a dataset of categorizing questions into six different types, e.g., whether the question is about a location, about a person, or about some numeric information. The TREC dataset comprises of 5,452 labeled training questions, and 500 test questions. By isolating 10% of the training questions for validation, we use train/validation/test splits of 4906/546/500 questions respectively.

News Classification: Following (?), we consider three binary news classification tasks from the 20 Newsgroups datasethttp://qwone.com/~jason/20Newsgroups/. Each task involves categorizing a document according to two related categories (1) Sports: baseball vs. hockey (958/239/796) (2) Computers: IBM vs. Mac (929/239/777) (3) Religion: atheism vs. christian (870/209/717).

Word Similarity Task: We use the WS-353 dataset (?), which contains 353 pairs of English words. Each pair of words has been assigned similarity ratings by multiple human annotators. We use the cosine similarity between the embeddings of each pair of words, and report the Spearman’s rank correlation coefficient $\rho$ between the human scored list and the predicted similarity list. We consider only those pairs of words where both words are present in the vocabulary. This leads to the removal of $59$ of the $353$ pairs ($17.3\%$).

Results and Discussion

In this section, we report the results of aforementioned experiments and discuss the implications.

Table 4 lists the precision scores of word intrusion detection task for each model with different starting vectors. We observe that our precision scores are notably higher than those of the original vectors, and those of the Sparse Overcomplete Word Vectors. This implies that annotators could select the intruder much more accurately from our dimensions with higher agreement (Table 5), showing that the resulting representations are highly coherent and interpretable. This forms the key result of our effort to produce more interpretable representations.

Performance on downstream tasks

From the results in Table 6, it is clear that the embeddings generated by our method perform competitively well on all benchmark tasks, and do significantly better on a majority of them.

Qualitative assessment

For a few sampled words, we investigate the top words from dimensions where the given word is active. Table 1 lists the results of this exercise for three particular words (mathematics, internet and remote), for different models and different starting vectors. From Table 1, we observe that the top dimensions of the word embeddings generated by our model (SPINE) are both coherent, and relevant to the word under examination. Often, our representations are able to capture different interpretations of a given word. For instance, the word ‘remote’ can be used in various settings: remote areas (like remote villages, huts), the electronic remote (like buttons, click), and conditions in remote areas (like poverty). From these examples, we get anecdotal evidence about the higher interpretability achieved by our model on the resulting representations.

Retroffiting vs joint learning

We follow a retrofitting approach to obtain sparse and interpretable embeddings. An alternate to this is to add various sparsity and non-negativity inducing regularizers when training Word2Vec or GloVe. One practical advantage of our retrofitting approach is that it does not need access to a giant corpus. Moreover, our proposed method is agnostic to, and abstracted from, the underlying method used to create the word embeddings, making it more widely applicable.

Interpretability and downstream performance

Embeddings from our method outperform original word embeddings for some downstream tasks. This is counter-intuitive, as an increase in interpretability might have come at the expense of downstream performance. We believe that this increased performance is because sparse embeddings directly capture the salient characteristics of concepts, which allow downstream task models to efficiently converge to optimum weight values. For example, if a dimension in the sparse embedding signifies a location name, downstream chunking task models would find it useful.

Further, on choosing 2000 or more hidden dimensions, we found both topic coherence and interpretability to improve, though at a severe cost of performance on downstream tasks. On choosing 500 dimensions, topic coherence and interpretability deteriorated. Theoretically, in the extreme case, one can obtain one-hot vectors by setting the dimension size to be equal to the vocabulary size. These representations would be highly interpretable, but would perform significantly worse on downstream tasks.

General discussion

We attribute the success of our method to the expressiveness of a neural autoencoder framework, that facilitates non linear transformations in contrast to existing linear matrix factorization based approaches. We further strengthen the hypothesis that non-negativity and sparsity lead to semantically coherent (interpretable) dimensions. As per this notion of interpretability, GloVe and word2vec embeddings are highly uninterpretable, whose individual dimensions, by themselves, do not represent any concrete concept or topic. Please note that we do not imply that GloVe and word2vec representations fail to capture the underlying semantic structure. In these representations, similar words are close in the embeddings space, and neighbouring words form a semantically coherent group. In fact, it is due to this characteristic that these representations achieve good scores in word similarity tasks (Table 6).

However, we argue that our notion of post-hoc interpretability – one that requires each dimension to capture a semantic concept – is a more pragmatic one. In many prediction settings, a softmax layer precedes the class probabilities. Weights from these softmax layers bind to the final layer representation, and large positive and negative weights sway the output class probabilities. In order to explain a prediction, one necessarily has to understand the semantic concepts that each of the dimensions corresponding to these large weights represent. Hence, this notion of post-hoc interpretability is more useful in explaining predictions.

Conclusion

We have presented a novel mechanism to generate interpretable word embeddings using denoising $k$-sparse autoencoders. Large scale crowd-sourced experiments show that our word embeddings are more interpretable than the embeddings generated by state-of-the-art sparse coding approaches. Also, our embeddings outperform popular baseline representations on a diverse set of downstream tasks. Our approach uses sub-differentiable loss functions and is trained through back propagation, potentially allowing for seamless integration into neural models, and end-to-end training capabilities. As a part of future work, we are investigating the effect of inducing varying amounts of sparsity at multiple hidden layers in more sophisticated networks, and studying the properties of the resultant sparse activations.

Acknowledgments

We thank Dr. Partha Talukdar, Dr. Jason Eisner, Pradeep Dasigi, Dr. Graham Neubig, Dr. Matthew R. Gormley and anonymous AAAI reviewers for their valuable suggestions. The authors also acknowledge Siddharth Dalmia and Mansi Gupta for carefully reviewing the paper draft. DP is supported in part by a Siebel Scholarship and by DARPA Big Mechanism program under ARO contract W911NF-14-1-0436.

References