## Skipgram isn't Matrix Factorisation

The paper Neural Word Embeddings as Implicit Matrix Factorization of Levy and Goldberg was published in the proceedings of NIPS 2014 (pdf).  It claims to demonstrate that Mikolov’s Skipgram model with negative sampling is implicitly factorising the matrix of pointwise mutual information (PMI) of the word/context pairs, shifted by a global constant.  Although the paper is interesting and worth reading, it greatly overstates what is actually established, which can be summarised as follows:

Suppose that the dimension of the Skipgram word embedding is at least as large as the vocabulary.  Then if the matrices of parameters $(W, C)$ minimise the Skipgram objective, and the rows of $W$ or the columns of $C$ are linearly independent, then the matrix product $WC$ is the PMI matrix shifted by a global constant.

This is a really nice result, but it certainly doesn’t show that Skipgram is performing (even implicitly) matrix factorisation.  Rather it shows that the two learning tasks have the same global optimum  – and even this is only shown when the dimension is larger than the vocabulary, which is precisely the case where Skipgram is uninteresting.

### The linear independence assumption

The authors (perhaps unknowingly) implicitly assume that the word vectors on one of the two layers of the Skipgram model are linearly independent.  This is a stronger assumption than what the authors explicitly assume, which is that the dimension of the hidden layer is at least as large as the vocabulary.  It is also not a very natural assumption, since Skipgram is interesting to us precisely because it captures word analogies in word vector arithmetic, which are linear dependencies between the word vectors!  This is not a deal breaker, however, since these linear dependencies are only ever approximate.

In order to see where the assumption arises, first recall some notation of the paper:

The authors consider the case where the negative samples for Skipgram are drawn from the uniform distribution $P_D$ over the contexts $V_C$, and write

for the log likelihood.  The log likelihood is then rewritten as another double summation, in which each summand (as a function of the model parameters) depends only upon the dot product of one word vector with one context vector:

The authors then suppose that the values of the parameters $W, C$ are such that Skipgram is at equilibrium, i.e. that the partial derivatives of $l$ with respect to each word- and content-vector component vanish.  They then assume that this implies that the partial derivatives of $l$ with respect to the dot products vanish also.  To see that this doesn’t necessarily follow, apply the chain rule to the partial derivatives:

This yields systems of linear equations relating the partial derivatives with respect to the word- and content- vector components (which are zero by supposition) to the partial derivatives with respect to the dot products, which we want to show are zero.  But this only follows if one of the two systems of linear equations has a unique solution, which is precisely when its matrix of coefficients (which are just word- or context- vector components) has linearly independent rows or columns.  So either the family of word vectors or the family of context vectors must be linearly independent in order for the authors to proceed to their conclusion.

Word vectors that are of dimension the size of the vocabulary and linearly independent sound to me more akin to a one-hot or bag of words representations than to Skipgram word vectors.

### Skipgram isn’t Matrix Factorisation (yet)

If Skipgram is matrix factorisation, then it isn’t shown in this paper.  What has been shown is that the optima of the two methods coincide when the dimension is larger that the size of the vocabulary. Unfortunately, this tells us nothing about the lower dimensional case where Skipgram is actually interesting.  In the lower dimensional case, the argument of the authors can’t be applied, since it is then impossible for the word- or context- vectors to be linearly independent.  It is only in the lower dimensional case that the Skipgram and Matrix Factorisation are forced to compress the word co-occurrence information and thereby learn anything at all.  This compression is necessarily lossy (since there are insufficient parameters) and there is nothing in the paper to suggest that the two methods will retain the same information (which is what it means to say that the two methods are the same).

### Appendix: Comparing the objectives

To compare Skipgram with negative sampling to MF, we might compare the two objective functions.  Skipgram maximises the log likelihood $l$ (above). MF, on the other hand, typically minimises the squared error between the matrix and its reconstruction:

The partial derivatives of $E$, needed for a gradient update, are easy to compute:

Compare these with the partial derivatives of the Skipgram log-likehood $l$, which can be computed as follows:

## Does vector direction encode word frequency?

In a paper with Adriaan Schakel, we presented controlled experiments for word embeddings using pseudo-words. Performing these experiments in the case of word2vec CBOW showed that, in particular, the vector direction of any particular word changed only moderately when the frequency of the word was varied. Shortly before we released the paper, Schnabel et al presented an interesting paper at EMNLP, where (amongst other things), they showed that it was possible to distinguish rare from frequent words using logistic regression on the normalised word vectors, i.e. they showed that vector direction does approximately encode coarse (i.e. binary, rare vs. frequent) frequency information.  Here, I wanted to quickly report that the result of Schnabel et al. holds for the vectors obtained from our experiments, as they should. Below, I’ll walk through exactly what I checked.

I took the word vectors that we trained during our experiments. You can check our paper for a detailed account. In brief, we trained a word2vec CBOW model on popular Wikipedia pages with a hidden layer of size 100, negative sampling with 5 negative samples, a window size of 10, a minimum frequency of 128, and 10 passes through the corpus. Sub-sampling was not used so that the influence of word frequency could be more clearly discerned. There were 81k unigrams in the vocabulary. Then:

1. the word vectors were normalised so that their (Euclidean-) length was 1.
2. the frequency threshold of 5000 was chosen (somewhat arbitrarily) to define the boundary between rare and frequent words. This gave 8428 “frequent” words. A random sample of the same size of the remaining “rare” words was then chosen, so that the two classes, “rare” and “frequent”, were balanced. This yielded approximately 17k data points, where a data point is a normalised word vector labelled with either “frequent” (1) or “rare” (0).
3. the data points were split into training- and test- sets, with 70% of the data points in the training set.
4. a logistic regression model was fit on the training set. An intercept was fit, but this boosted the performance only slightly. No regularisation was used since the number of training examples wass high compared to the number of parameters.
5. The performance on the test set was assessed by calculating the ROC curve on the training and test sets and the accuracy on the test set.

Model performance
Consider the ROC curve below. We see from that fact that the test curve approximately tracks the training curve that the model generalises reasonably to unseen data. We see also from the closeness of the curves to the axes at the beginning and the end that the model is very accurate in detecting frequent words when it gives a high probability (bottom left of the curve) and at detecting infrequent words when it gives a low probability (top right).

The accuracy of the model on the test set was 82%, which agrees very nicely with what was reported in Schnabel et al., summarised in the following image:

The training corpus and parameters of Schnabel, though not reported in full detail (they had a lot of other things to report), seem similar. We know that their CBOW model was 50 dimensional, had a vocabulary of 103k words, and was trained on the 2008 Wikipedia.

## Musings on "adjectives as matrices"

The advantage of considering (e.g.) adjectives as transformations rather than points in space is that these transformations can be applied in unseen combinations. This counters one of Chomsky’s objections to statistical modelling of language, that is, that language is effectively infinite, whereas language models are trained on only a finite amount of data (so are humans, but humans are supposed to be born with a universal grammar). The case, considered by Baroni et al., of adjective as linear transform has a couple of disadvantages, however. The first that there are a large number of parameters to be learnt for each adjective, the second being that it doesn’t capture the near commutativity of adjectives, i.e. in most cases adjectives can be applied to a noun in different orders without significantly changing the meaning.

I can think of several approaches for enforcing the commutativity of adjective matrices:

1. simply using diagonal matrices (this reduces to one of the approaches already considered), or
2. penalising the off-diagonal elements via regularisation, or
3. interleaving existing parameter updates with updates that penalise (co-occurring?) adjective matrices for not commuting with one another, e.g. using the gradient of the matrix commutator $AB – BA$

## (Linear) Maps of the Impossible: Capturing semantic anomalies in distributional space

Eva Maria Vecchi, Marco Baroni and Roberto Zamparelli.

Presented at the workshop “Distributional Semantics and Compositionality” (2011) PDF

The authors attempt to use distributional models to distinguish between acceptable and “semantically deviant” adjective-noun combinations (an example of this distinction is given by “blue rose” vs “residential steak”). They hypothesise in particular that the length of the vector representation of the adjective-noun combination is an indication of its acceptability. Their reasoning for this hypothesis assumes that directions and in particular axes are interpretable in distributional models (this does not apply in the case of word2vec, at least). They further hypothesise that the combination will be spatially isolated with respect to the cosine similarity.

The distributional representation is derived from a POS-tagged and lemmatised corpus by considering sentence-internal co-occurrence between the vocabulary as a whole and the 10k most frequent nouns, verbs and adjectives, transformed via the “local mutual information” measure and reduced to rank 300 using PCA.

Different methods of transforming the noun representation using the adjective to obtain the adjective-noun combination are studied and the results are evaluated against human judgements of semantic deviance.

## Polyglot: Distributed Word Representations for Multilingual NLP

Rami Al-Rfou, Bryan Perozzi, Steven Skiena (all at Stony Brook University)

Published in the proceedings of CoNLL 2013 (PDF).

The authors train word embeddings for 117 different languages using Wikipedia. The embeddings are trained using an architecture similar to that of SENNA of Collobert et al. This architecture computes a score representing the likelihood that the words given as input occurred together in order. A short window is scanned over a stream of text, and the score of the phrase in the window is compared to the score of a corrupted version of the same phrase where the middle word is substituted randomly. The model is penalised (using hinge loss, i.e. one-way error) according to whether the uncorrupted or corrupted phrase was more highly scored.

The score of a phrase is computed as per the following:

1. Each of the words is transformed from a one-hot to a distributed representation via the application of a shared matrix $C$, and these representations are concatenated;
2. The hyperbolic tan of an affine transformation of this concatenation is calculated component-wise, yielding a “hidden” vector.
3. The components of this vector are combined via an affine transformation to yield the score.

So this neural network has three layers and the parameters are the shared matrix $C$ together with the two affine transformations.

The word embedding is given by the rows of the shared matrix $C$.

The models are trained using Theano for extensive periods of time (the authors mention “weeks”). The window size is taken to be radial length 2, the word embedding rank is 64 and the hidden layer size is 32.

To demonstrate the utility of the word representations, the authors the representations as initialisation for a model performing parts of speech tagging.

The paper was published at about the same time as word2vec (it does not refer to word2vec at all). The approach, the notation and the terminology, however, demonstrate that certain things that I had thought particular to word2vec were in fact already accepted practice, including:

• the use of discriminative tasks for training word embeddings
• sampling contexts by scanning a short window over text
• the use of the middle word in a context for the discriminative task
• dividing through by the “fan out” for initialisation (page 187, TBC)
• the symbols <S> and </S> for delimiting sentences

## A Unified Model for Word Sense Representation and Disambiguation

Chen, Liu, Sun, published in the conference proceedings of EMNLP 2014 (PDF).

The authors leverage the word2vec skipgram model and WordNet glosses (i.e. word sense definitions) for word sense disambiguation. This is achieved as follows:

1. A skipgram model is trained.
2. For each sense of a word according to WordNet, a vector is derived by taking the average of the content words in the WordNet definition (“gloss”) of that sense (“gloss vectors”)
3. The gloss vectors are used to identify the sense of a word occurrence by considering its dot product with the context of that occurrence. The sense whose gloss vector has the highest dot product with the context vector is chosen, as long as it is wins by a sufficient margin.

The authors are then able to train word sense vectors (distinct from the gloss vectors) by modifying the skip-gram objective. These word sense vectors are then used for similarity tasks and not for word sense disambiguation. It seems to me that it would have been simpler to annotate word occurrences in the corpus with the senses than to modify the objective.

Evaluation is performed for coarse-grained WSD (i.e. disambiguating homographs).

Independence assumptions in iterative word sense disambiguation
The authors disambiguate the senses of a words one word at a time, based upon the disambiguation that has already taken place. Two different strategies are considered for choosing the order in which to disambiguate the words in a context. These strategic approaches make a problematic independence assumption – that the sense of the word to be disambiguated is independent of the senses of the words not yet disambiguated. I haven’t read many WSD papers – I suspect these independence assumptions aren’t particular to the approach of the authors.

## GloVe: Global Vectors for Word Representations

Pennington, Socher, Manning, 2014.
PDF

GloVe trains word embeddings by performing a weighted factorisation of the log of the word co-occurrence matrix. The model scales to very large corpora (Common Crawl 840B tokens) and performs well on word analogy tasks.

Model
The cost function is given by:

$\displaystyle \sum_{i, j = 1}^V f(X_{i,j}) (u_i^T v_j + b_i + c_j – \text{log} X_{i,j})^2$

where:

• $V$ is the size of the vocabulary,
• $X$ denotes the word co-occurrence matrix (so $X_{i,j}$ is the number of times that word $j$ occurs in the context of word $i$)
• the weighting $f$ is given by $f(x) = (x / x_{\text{max}})^\alpha$ if $x < x_{\text{max}}$ and $1$ otherwise,
• $x_{\text{max}} = 100$ and $\alpha = 0.75$ (determined empirically),
• $u_i, v_j$ are the two layers of word vectors,
• $b_i, c_j$ are bias terms.

Note that the product is only over pairs $i, j$ for which $X_{i,j}$ is non-zero. This means that GloVe (in contrast to word2vec with negative sampling) trains only “positive samples” and also that we don’t have to worry about the logarithm of zero.

This is essentially just weighted matrix factorisation with bias terms:

Note that in the implementation (see below), the $X_{i,j}$ are not raw co-occurrence counts, but rather the accumulated inverse distance between the two words, i.e.

$\displaystyle X_{w, w’} := \sum_{\text{windows containing\ } w, w’} (\text{distance between\ } w, w’)^{-1}.$

I am fairly sure that the implementation of Adagrad is incorrect. See my post to the forum.

The factor weighting f

The authors go to some trouble to motivate the definition of this cost function (section 3).  The authors note that many different functions could be used in place of their particular choice of $f$, and further that their $\alpha$ coincides with that used by word2vec for negative sampling. I can’t see the relevance of the latter, however (in word2vec, the $0.75$th power it is used to define the noise distribution; moreover powering a value in the range $[0, 1]$ has a very different effect to powering a value in the range $[0, 100]$).

Graphing the function (see above) hints that it might have been specified more simply, since the non-linear region is in fact almost linear.

Word vectors
The resulting word embeddings ($u_i$ and $v_j$) are unified via a direct sum of their vector spaces.

The cosine similarity is used to find the missing word in word similarity tasks. It is not stated if the word vectors were normalised before forming the arithmetic combination of word vectors.

Source code
The authors take the exemplary step of making the source code available.

Evaluation and comparison with word2vec
The authors do a good job of demonstrating their approach, but do a scandalously bad job of comparing their approach to word2vec. This seems to reflect a profound misunderstanding on the part of the authors as to how word2vec works. While it has to be admitted that the word2vec papers were not well written, it is apparent that the authors made very little effort at all.

The greatest injustice is the comparison of the performance of GloVe with an increasing number of iterations to word2vec with an increasing number of negative samples:

The most important remaining variable to control
for is training time. For GloVe, the relevant
parameter is the number of training iterations.
For word2vec, the obvious choice would be the
number of training epochs. Unfortunately, the
code is currently designed for only a single epoch:
it specifies a learning schedule specific to a single
pass through the data, making a modification for
multiple passes a non-trivial task. Another choice
is to vary the number of negative samples. Adding
negative samples effectively increases the number
of training words seen by the model, so in some
ways it is analogous to extra epochs.

Firstly, it is simply impossible that it didn’t occur to the authors to simulate extra iterations through the training corpus for word2vec by simply concatenating the training corpus with itself multiple times. Moreover, the authors themselves are capable programmers (as demonstrated by their own implementation). The modification to word2vec that they avoided is the work of ten minutes.

Secondly, the notion that increasing the exposure of word2vec to noise is comparable to increasing the exposure of GloVe to training data is ridiculous. The authors clearly didn’t take the time to understand the model they were at pains to criticise.

While some objections were raised about the evaluation performed in this article and subsequent revisions have been made, the GloVe iterations vs word2vec negative sample counts evaluation persists in the current version of the paper.

Another problem with the evaluation is that the GloVe word vectors formed as the direct sum of the word vectors resulting from each matrix factor. The authors do not do word2vec the favour of also direct summing the word vectors from the first and second layers.

## Language Understanding for Text-based Games using Deep Reinforcement Learning

Appeared on the arXiv, June 2015.

The joint work of Karthik Narasimhan, Tejas Kulkarni and Regina Barzilay.

The aim of the paper is to create an autonomous agent that solves quests in text-based adventure games. The agent has no knowledge of the underlying game state, and must decide upon what action to take based only upon the representation of the game state that is afforded by the game. In this sense it seeks to solve a similar problem to that of the now famous Atari deep learning paper. This is also an interesting model for how humans communicate with one another.

There are similarities in approach, moreover, in that both employ reinforcement learning. In contrast, this paper employs a Long-Short Term Memory network.

They use Evennia, a Python framework for building multiplayer online text games (used here in a single player context).

Adriaan S.: Q-learning does not scale well. (This could account for the small vocabulary used.)

## Word2vec weight initialisation

The initialisation of the weights in word2vec is not what I expected.

• syn1 The weights connecting the hidden- to the output-layer are initialised to zero (in both the hierarchical softmax and the negative sampling cases)
• syn0 The initial values for the weights connecting the input- to the hidden-layer are drawn uniformly and independently from the interval $[\frac{-1}{2n}, \frac{1}{2n} ]$, where $n$ is the rank of the hidden layer (i.e. number of hidden units.)

The range of interval from which the syn0 weights are sampled was chosen depends on the rank. I had presumed that this was to account for the dependency of the distribution of the dot product (and in particular the L2-norm) on the rank. However, estimating these distributions empirically, this doesn’t seem to be the case:

According to Mikolov (in a helpful response in the word2vec google group), the initialisation of the weights was chosen empirically, since it seemed to work well.

Questions:

1. I was unable to derive an expression for the distribution of L2-norms mathematically. Can someone help with that?

## Document Classification by Inversion of Distributed Language Representations

This is a note on the arxiv by Matt Taddy from April 2015. It reads very clearly and has a simple point to make: language modelling techniques can be used in classification tasks by training a separate language model for each class; documents are assigned to the class of the model where the document has the highest likelihood (hence “inversion”). In our discussion, we assume a uniform prior over the classes.

Taddy considers the particular case of predicting the sentiment of Yelp reviews at different levels of granularity. Different approaches are considered:

• word2vec inversion is inversion in the sense described above where document vectors are taken as the average of the word vectors of the constituent words;
• phrase regression, where separate logistic regression models are trained for each output class, taking as input phrase count vectors;
• doc2vec regression, is as per phrase regression, but taking as input one of:
• doc2vec DBOW
• doc2vec DM
• doc2vec DBOW and DM combined, i.e. in direct sum
• MNIR, the authors own Multinomial Inverse Regression

Three separate classification tasks are considered, labelled “a”, “b” and “c” in the diagram below, representing two-, three- and five-class sentiment classification.

As illustrated in the following figure, only the word2vec inversion technique would do a decent job when the gravity of a misclassification is considered (so penalising less if, e.g. predicted star rating is off by only one star):

Missing from Taddy’s comparison is inversion using the document vectors, though this is certainly the sort of thing his paper suggests might work well. Also missing is regression using the document vectors obtained as aggregates of word vectors.

## Document Embedding with Paragraph Vectors

Presented at NIPS 2014 (PDF) by Dai, Olah, Le and Corrado.

## Model

The authors consider a modified version of the PV-DBOW paragraph vector model. In previous work, PV-DBOW had distinguished words appearing in the context window from non-appearing words given only the paragraph vector as input. In this modified version, the word vectors and the paragraph vectors take turns playing the role of the input, and word vectors and paragraph vectors are trained together. That is, a gradient update is performed for the paragraph vector in the manner of regular PV-DBOW, then a gradient update is made to the word vectors in the manner of Skipgram, and so on. This is unfortunately less than clear from the paper. The authors were good enough to confirm this via correspondence, however (thanks to Adriaan Schakel for communicating this). For the purposes of the paper, this is the paragraph vector model.

The representations obtained from paragraph vector (using cosine similarity) are compared to those obtained using:

• an average of word embeddings
• LDA, using Hellinger distance (which is proportional to the L2 distance between the component-wise square roots)
• paragraph vector with static, pre-trained word vectors

In the case of the average of word embeddings, the word vectors were not normalised prior to taking the average (confirmed by correspondence).

## Corpora

Two corpora are considered, the arXiv and Wikipedia:

• 4.5M articles from Wikipedia, with a vocabulary of size 915k
• 886k articles from the arXiv, full texts extracted from the PDFs, with a vocabulary of 970k words.

Only unigrams are used. The authors observed that bigrams did not improve the quality of the paragraph vectors. (p3)

## Quantitative Evaluation

Performance was measured against collections of triples, where each triple consisted of a test article, an article relevant to the test article, and an article less relevant to the test article. While not explicitly stated, it is reasonable to assume that the accuracy is then taken to be the rate at which similarity according to the model coincides with relevance, i.e. the rate at which the model says that the relevant article is more similar than the less relevant article to the test article. Different sets of triples were considered, the graph below shows performance of the different methods relative to a set of 172 Wikipedia triples that the authors built by hand (these remain unreleased at the time of writing).

It is curious that, with the exception of the averaged word embeddings, the accuracy does not seem to saturate as the dimension increases for any of the methods. However, as each data point is the accuracy of a single training (confirmed by correspondence), this is likely nothing more than the variability inherent to each method. It might suggest, for example, that the paragraph vectors method has a tendency to get stuck in local minima. This instability in paragraph vector is not apparent, however, when tested on the triples that are automatically generated from Wikipedia (Figure 5). In this latter case, there are many more triples.

Performance on the arXiv is even more curious: accuracy decreases markedly as the dimension increases!

## Implementations

I am not sure there are any publicly available implementations of this modified paragraph vectors method. According to Dai, the implementation of the authors uses Google proprietary code and is unlikely to be released. However it should be simple to modify the word2vec code to train the paragraph vectors, though some extra code will need to be written to infer paragraph vectors after training has finished.

I believe that the gensim implementation provides only the unmodified version of PV-DBOW, not the one considered in this paper.

It is interesting that the paragraph vector is chosen so as to best predict the constituent words, i.e. it is inferred. This is a much better approach from the point of view of word sense disambiguation than obtaining the paragraph vector as a linear image of an average of the word vectors (NMF vs PCA, in their dimension reductions on bag of words, is another example of this difference).

Questions

1. Is there is an implementation available in GenSim? (see e.g. this tutorial).
2. (Tangent) What is the motivation (probabilistic meaning) for the Hellinger distance?

## Literary treasure hunting with the Lateral API

A good friend, Sam Matthews, recently gave a talk in December 2014 at a conference of the Australian Modernist Studies Network on “Transnational Modernisms”. Sam spoke about his discovery of a reference to a print-shop from Balzac’s “Two Poets” in Christina Stead‘s novel Seven Poor Men of Sydney. Sam later suggested that I check if we couldn’t use Lateral’s text matching service (the “Recommender (BYO documents!)” API) to confirm this reference to Balzac and potentially uncover other ones. As you’ll see below, the preliminary results are very encouraging. This is hardly a conclusive experiment, but

In case you would like to search for references to Balzac’s works yourself, you can do so by reusing the API key I created: b4de9b9183df4cbf8d70cde15609800a .

This is how I proceeded:

1. I downloaded the Complete works of Balzac from Project Gutenberg. This gives one HTML file for each of Balzac’s works.
2. I split each work into paragraphs, labelling the paragraphs by their work and position within the work. Balzac wrote many paragraphs, it turns out!
3. I subscribed to the API at Lateral, obtaining an API key.
4. I installed Francis Tzeng’s python package for accessing the Lateral API
5. Using the python package, I added the paragraphs of Balzac to the Lateral recommender. Short paragraphs containing not enough meaningful words were rejected; in total, the number of meaningful paragraphs of Balzac indexed was over 21,000.
6. Again using the python package, I searched for the closest paragraphs of Balzac to the passage of Stead that Sam had indicated to me (see below).

The passage of Stead’s novel that evokes the print-shop appears below (from Chapter 3):

devil’s kitchen where  the word is made bread … triangular park … A wide old doorway opened beside the tobacconist’s shop, and over it was a name, white on blue, “Tank Steam Press, Ground Floor.” The tobacconist owned the old single-storey building and rented out to several establishments the mouldy apartments of the ground and first floor. In the attic was the man who did heliogravure. The building had once been a private house. Its court was now a cart-dock and opened into the other street. Its first-floor bathroom at the head of the stairs contained the old water-closet, used by all the workers in the house, a gas-ring to make tea, and the usual broken chairs and out of-date telephone directories. The distinctive smell of the building came from this closet and from the printing-ink.Joseph walked through the old doorway, went by a staircase and entered the large airy double room occupied by the Press. He opened the glass back-door and moved about among the presses, curiously inspecting the jobs in their various stages, picking up a paper, looking through the bills on a bill-hook, putting his finger in the dust in the little glassed-in office of Chamberlain, the owner, and shutting off the stove, lighted by the cleaner, because the day was warm enough.

Below are the paragraphs of Balzac that are semantically closest to the text above, according to Lateral. As you can see, the 1st and the 9th closest paragraphs (of over 21,000!) indeed come from “Two Poets”, and inspection reveals that they indeed concern the printshop! You can click the links to fetch the corresponding paragraphs using the API. The intermediately ranked results seem to be architectural descriptions.

 [ { "distance": 0.034905, "document_id": "TWO POETS-00019" }, { "distance": 0.035945, "document_id": "THE COLLECTION OF ANTIQUITIES-00557" }, { "distance": 0.037409, "document_id": "SONS OF THE SOIL-01139" }, { "distance": 0.038067, "document_id": "A MAN OF BUSINESS-00034" }, { "distance": 0.038168, "document_id": "URSULA-01020" }, { "distance": 0.038216, "document_id": "COUSIN PONS-01938" }, { "distance": 0.03837, "document_id": "COLONEL CHABERT-00023" }, { "distance": 0.038545, "document_id": "COUSIN BETTY-01508" }, { "distance": 0.038823, "document_id": "TWO POETS-00018" }, { "distance": 0.038891, "document_id": "RISE AND FALL OF CESAR BIROTTEAU-01382" }, { "distance": 0.039151, "document_id": "THE RED INN and others-00045" }, { "distance": 0.039195, "document_id": "THE LESSER BOURGEOISIE(The Middle Classes)-00635" }, { "distance": 0.039369, "document_id": "SCENES FROM A COURTESAN'S LIFE-00761" }, { "distance": 0.039377, "document_id": "THE TWO BROTHERS-00663" }, { "distance": 0.039471, "document_id": "HONORINE-00036" }, { "distance": 0.039808, "document_id": "Z. MARCAS-00043" }, { "distance": 0.039896, "document_id": "RISE AND FALL OF CESAR BIROTTEAU-00623" }, { "distance": 0.040041, "document_id": "THE VILLAGE RECTOR-00313" }, { "distance": 0.040253, "document_id": "A WOMAN OF THIRTY-00700" }, { "distance": 0.04031, "document_id": "CATHERINE DE' MEDICI-01059" } ]