2021 [1]Elnaz Shafaei-Bajestan 1]Eberhard Karls Universität Tübingen, Germany
2]Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany

# Semantic properties of English nominal pluralization

Insights from word embeddings

<EMAIL_ADDRESS>Masoumeh Moradipour-Tari
<EMAIL_ADDRESS>Peter Uhrig
<EMAIL_ADDRESS>R. Harald Baayen<EMAIL_ADDRESS>[  [

###### Abstract

Semantic differentiation of nominal pluralization is grammaticalized in many
languages. For example, plural markers may only be relevant for human nouns.
English does not appear to make such distinctions. Using distributional
semantics, we show that English nominal pluralization exhibits semantic
clusters. For instance, pluralization of fruit words is more similar to one
another and less similar to pluralization of other semantic classes.
Therefore, reduction of the meaning shift in plural formation to the addition
of an abstract plural meaning is too simplistic. A semantically informed
method, called CosClassAvg, is introduced that outperforms pluralization
methods in distributional semantics which assume plural formation amounts to
the addition of a fixed plural vector. In comparison with our approach, a
method from compositional distributional semantics, called FRACSS, predicted
plural vectors that were more similar to the corpus-extracted plural vectors
in terms of direction but not vector length. A modeling study reveals that the
observed difference between the two predicted semantic spaces by CosClassAvg
and FRACSS carries over to how well a computational model of the listener can
understand previously unencountered plural forms. Mappings from word forms,
represented with triphone vectors, to predicted semantic vectors are more
productive when CosClassAvg-generated semantic vectors are employed as gold
standard vectors instead of FRACSS-generated vectors.

###### keywords:

Pluralization, Plural semantics, Distributional semantics, Proportional
analogies, Vector averaging, Compositional distributional semantics

This article is under review at Morphology.

##  1 Introduction

According to Corbett (2000), grammatical number in English can be
conceptualized as a ‘feature’ (see also Jakobson ., 1928/2012) with two
opposing values: singular and plural. For English, the formal realization of
the nominal plural is the addition of the suffix -s, a marker whose absence
signals the nominal singular. Textbooks such as Lieber (2010) and Aronoff
Fudeman (2011) take the semantics of plurality for granted and call attention
to the way in which the plural is realized. According to Aronoff Fudeman
(2011, p. 156), number is generally not marked in the English lexicon, with
the exception of pluralia tantum such as pants.

Two major approaches to morphology, compositional and realizational
morphology, analyze English pluralization in subtly different ways. In
compositional models, pluralization is argued to be a process that operates on
the less complex word (the singular) and results in the more complex word (the
plural). The morpheme -s is concatenated with the stem, and in parallel, the
abstract meaning ‘plural’ that is registered in the lexical entry of the
morpheme -s overrides the singular meaning associated with the stem.

In realizational models, the semantic part of the word-schema for plural nouns
contains the description ‘plurality of $x$s’ to capture the semantic
similarity of all things plural while abstracting away from their differences;
hence the use of the variable ‘$x$’. The schema for plural nouns stands in
correspondence to a schema for singular words with the same semantic symbol
‘$x$’ (Haspelmath Sims, 2010; Booij, 2010). The operator ‘plurality of’ is
assumed to implement the same semantic change for all instances ‘$x$’ to which
it is applied.

While compositional models and realizational models are rather different with
respect to how they view the relationship between form and meaning at the
level of sub-word units, they are in remarkable agreement when it comes to how
the semantics of number for nouns is understood. Both models assume that
English plural formation amounts to a change in form that goes hand in hand
with the addition of an abstract semantic feature of plurality to a singular
word, shifting its denotation from one entity to more than one entity.

As a matter of fact, formal semantics also has assumed a simple theory of
plural meaning. Lasersohn (1995) opens with “Plurality is a simple notion – it
just means ‘more than one’.” Similarly, Link (1983/2012) in his study of the
logical analysis of plurals restricts the domain of entities from which
singular nominals take values to atoms (or, ordinary individuals), and plural
nominals to non-atomic summations of more than one atom. This interpretation
of the plural nominal is often called exclusive since it excludes atoms from
its denotation (de Swart Farkas, 2010). It applies to the example 1 below.
Although an exclusive interpretation of plural meaning is often fine, there
are cases where a plural can refer inclusively to one or more entities, as in
1. Here, children in 1 is number neutral – it can mean ‘one’ child or ‘more
than one’ children. This is an inclusive interpretation that includes atoms
and non-atomic sums in its denotation.

. . You’re welcome to bring your two children. .̱ You’re welcome to bring your
children. $\mbox{\rm\bf z}_{.}$ (adapted from Sauerland ., 2005)

Following the maximize presupposition principle in pragmatics (Heim,
1991/2008), Sauerland . (2005) and Liter . (2017) argue that an exclusive
interpretation of a plural form is a consequence of pragmatic inference, which
depends on a range of contextual factors (Chemla, 2008). In a situation where
the speaker is unsure of the addressee’s number of children, 1 is appropriate
and 1 is odd. Assuming that these two sentences are in competition, the use of
the singular child in 1 is blocked because we know from experience that people
may have more than one child and the sentence in 1 with stronger
presuppositions is preferred.

. Context: People can have more than one child. . *You’re welcome to bring
your child. $\mbox{\rm\bf z}_{.}$ (adapted from Sauerland ., 2005)

In other words, a plural form can be used to denote an unspecified quantity
(one, more than one, one or more than one) (see also Mattens, 1970) and the
exact quantity has to be resolved through interpretation from context. A
principal presupposition that underlies this approach to plurality is that the
conceptualization of the number feature is orthogonal to the meaning of the
nominal phrase. Sauerland (2003) argues for an additional syntactic (zero)
head above the determiner phrase which contains the number feature and its
semantic content. In what follows, we question this presupposition by using
distributional semantics (DS).

DS represents words’ meanings with high-dimensional numeric vectors, which we
will refer to primarily as ‘semantic vectors’ and alternatively as ‘word
embeddings’—as they are known in Natural Language Processing (NLP).
Distributional semantics builds on the hypotheses that words that are similar
in meaning occur in similar contexts (Rubenstein Goodenough, 1965) and “words
that occur in the same contexts tend to have similar meaning” (Pantel, 2005).

There are many different ways in which semantic vectors for words can be
constructed. Early implementations made use of word by document contingency
tables (Landauer Dumais, 1997) or word by context-word contingency tables
(Lund Burgess, 1996; Shaoul Westbury, 2010). These tables typically yield very
high-dimensional vectors with thousands or tens of thousands of dimensions. By
means of dimensionality reduction techniques such as singular value
decomposition, the dimensionality of semantic vectors is substantially
reduced. Landauer Dumais (1997) recommended 300-dimensional vectors, as in
their experience lower-dimensional vectors performed with higher accuracy in a
range of tasks such as synonymy detection.

More recent models make use of artificial neural networks that are trained to
predict target words from the words in their immediate context (e.g., CBOW;
Mikolov, Chen ., 2013) or to predict the words in the immediate context of a
target word from that target word (e.g., Skip-gram; Mikolov, Chen ., 2013). A
simple three-layer neural network for the Skip-gram model was implemented by
Mikolov, Sutskever . (2013), using stochastic gradient descent and back-
propagation of error. The model was trained on 100 billion words from the
Google News corpus, and the resulting word2vec semantic vectors were made
available at https://code.google.com/archive/p/word2vec/.

Other word embeddings extend the word2vec methodology by incorporating
character n-grams of words (fastText; Bojanowski ., 2017) or by modifying the
objective function being optimized (GloVe; Pennington ., 2014). All these
methods extract the semantic vectors purely from textual information. Other
studies integrate visual information on top of that and create multi-modal
embeddings (e.g., Shahmohammadi ., 2021).

Word embeddings are employed advantageously in several tasks within NLP such
as named entity recognition, part of speech tagging, sentiment analysis, word
sense disambiguation (Wang ., 2019), and in many areas of psychology and
psycholinguistics (Günther, Rinaldi Marelli, 2019). Boleda (2020) discusses
their relevance for theoretical linguistics in the areas of diachronic
semantic change, polysemy, and the interface between semantics and syntax or
semantics and morphology.

The traditional demarcation of morphology and semantics in linguistics is less
prominent in DS models. Nevertheless, the statistics used in these models have
been shown to encode morphological and syntactic information besides semantic
information (Westbury Hollis, 2019). For morphologically related words,
measurements from DS models, such as vector similarity, are consistent with
human semantic similarity ratings and lexical decision latencies (Rastle .,
2000, 2004; Moscoso del Prado Martín ., 2005; Milin ., 2009). The degree of
semantic transparency in English derivation (Marelli Baroni, 2015) and Dutch
compounds (Heylen De Hertog, 2012) were explained by DS similarity measures.
Findings of Smolka . (2014) regarding the effect of semantic transparency on
morphological priming of German complex verbs were replicated with DS
similarity measures by Padó . (2015) (although Shafaei-Bajestan, 2017, could
not fully replicate the latter study) (see also Baayen Smolka, 2020). Shen
Baayen (2021) find that semantic transparency measured by DS is linked to the
productivity of adjective–noun compounds in Mandarin. DS models used in
investigating the paradigmatic relation between two Indonesian prefixes
(Denistia ., 2021) corroborated the findings of earlier corpus-based analyses.
The discriminative lexicon model of Baayen . (2019) is a computational model
of lexical processing, including morphologically complex words, that
incorporates insights from distributional semantics for the representation of
word meanings.

In what follows, we utilize word embeddings to study the meaning of English
nominal pluralization. DS models from machine learning produce semantic
vectors for both singular and plural word forms. However, in order to be
useful for the study of morphology, we need to consider additional questions:
What does the process of English pluralization, i.e., going from the singular
to the plural semantics, mean in this context? How can we model this process?
Given a singular meaning, can we conceptualize the plural, and conversely,
given the plural meaning, can we conceptualize the singular? As it is more
likely that we encounter previously unseen plurals of known singulars, than
previously unseen singulars given known plurals, we focus specifically on the
productivity of the conceptualization of plural forms and ask: How well can we
estimate the semantics of previously unseen plural words? And how does form
relate to the estimations for meaning?

In the following section, we first introduce the corpus used in the present
study. Sections 3 and 4 investigate the aforementioned questions using methods
inspired by realizational morphology and compositional morphology,
respectively. In doing so, we illustrate that the widespread assumption in
morphology about plural meaning is too simplistic, and we study alternative
approaches that stay closer to the actual complexity of noun pluralization in
English. In section 5, the semantic vectors developed by the best-performing
methods (formalizing realizational and compositional morphological theories)
are put to use in a word comprehension model (based on the discriminative
lexicon model by Baayen ., 2019) to study which kind of vectors are optimal
for the recognition of previously unseen plural words. A discussion of the
findings concludes the study.

##  2 Data

The corpus data used in this study is taken from the NewsScape English Corpus
(Uhrig, 2018, ). The corpus consists of 269 million tokens from the subtitles
of more than 35,000 hours of recordings of US-American TV news collected in
the UCLA Library Broadcast NewsScape (Steen ., 2018). After capture, the
recordings undergo compression, during which the audio channel is recoded into
a 96 kbit/sec AAC stream with the Fraunhofer FDK library. For this project,
the subtitles collected in the NewsScape text files were processed in an NLP
pipeline.

In a first step of this pipeline, sentence splitting was carried out with a
purpose-built splitter that takes into account the fact that captions are
transmitted in upper case. The resulting sentences were processed with
Stanford CoreNLP (Manning ., 2014) version 3.7.0, i.e. with PTB3 tokenization.
The caseless model included in CoreNLP1 was used to tag every word with a Penn
Treebank part-of-speech tag.111Note that the caseless mode is only available
for the left3distsim model but not for the slower but usually better
bidirectional tagger model. Then CoreNLP’s TrueCase annotator was deployed,
which overwrites the original text for further processing (preserving the
original on a separate level). Dependency Parsing, Named-Entity Recognition
and any further processing steps are then based on the case-restored text to
ensure consistent results from tools that do not offer caseless models.

After the NLP pipeline, the data was run through a modified version of the
forced alignment system Gentle (Ochshorn Hawkins, 2015), which basically runs
an automatic speech recognition process with a language model created from the
subtitles and then attempts to match the recognized words with the words in
the subtitles. The quality of the forced alignment results crucially depends
on the accurateness of the transcript it is fed. However, TV subtitles are not
exact transcripts. Not only do they often ignore disfluencies such as false
starts, but they also omit words and sometimes even change them. The
commercials included in the recordings do not systematically come with
subtitles either. Thus, Gentle only aligns between 90 and 95 percent of the
words in the subtitles successfully, and of these, 92.5% in a manual
evaluation were deemed to be aligned correctly by a human annotator listening
to them (Uhrig, ). We have to bear in mind, though that the cutoff points may
not have been exact on these words. To increase the quality of the dataset
used in the present study, only files where Gentle reported at least 97% of
successfully aligned words were used.

Words’ meanings are represented with semantic vectors of word2vec, which is
widely used within NLP and theoretical linguistics. The nearest neighbors of a
target word in this semantic space are often semantically similar (e.g., good
and great) or related (good and bad) words. The top 10 closest neighbors to
Germany are German, Europe, European, Sweden, Switzerland, Austria, France,
Spain, Poland, and Russia. Wang . (2019) show that similarities computed
between pairs of word2vec vectors are highly correlated ($r(2998)=0.72$) with
similarity ratings between word pairs obtained from human subjects in the MEN
data set (Bruni ., 2014), and that word2vec vectors are best performing on
syntactic word analogy tasks (see section 3) juxtaposed with 5 other semantic
spaces. Westbury Hollis (2019) argue that Mikolov, Chen . (2013)’s approach
for training of the word2vec vectors is closely related to the cognitively
plausible learning rule of Rescorla Wagner (1972).

We compiled a noun pluralization dataset with 14,699 singular-plural noun
pairs from the NewsScape English Corpus with a word2vec vector. Proper names,
plurals endings with anything other than an -s, plural-singular pairs with the
same word-form, and named entities were excluded from the dataset.

A second set brought together 500,000 tokens of 14,640 orthographic word types
from more than 100 television programs in the NewsScape English Corpus with no
restriction on their morphological and syntactic categories. The 13,902 words
from this set that have a vector available in word2vec compose a vocabulary
dataset.

##  3 Pluralization with realizational morphology

Realizational morphology (also known as word and paradigm morphology;
Matthews, 1991; Stump, 2001) posits whole words rather than morphemes as the
basic units. A central notion in this theory is the paradigm. In English, the
inflectional paradigm for the verb talk is {talk, talks, talked, talking} and
for the noun talk is {talk, talks}. Productivity of the lexicon as a system
emerges from proportional analogies between words within paradigms, such as
$talk:talks::walk:walks$ (talk is to talks as walk is to walks).

###  3.1 Proportional analogies with word embeddings

Analogical reasoning using word embeddings has been studied for different
types of analogical relations including semantic analogies, such as

$man:king::woman:queen,$

derivational analogies, as in

$quiet:quietly::happy:happily,$

and inflectional analogies similar to

$pen:pens::table:tables.$ (1)

Various implementations of proportional analogies with word embeddings have
been worked out, such as 3CosAdd (Mikolov, Yih Zweig, 2013), 3CosMul (Levy
Goldberg, 2014), LRCos, and 3CosAvg (Drozd ., 2016). Performance varies
extensively for the different methods and the different types of analogical
relations. Rogers . (2017) report that, for English, analogical reasoning with
embeddings is most successful for inflectional analogies across different
methods. These methods are considered below in the context of plural
formation.

Most of the aforementioned methods operate on three input vectors to estimate
a vector for the target word in a given analogy. For instance, to implement
the analogy in (1), 3CosAdd predicts a vector for tables, labeled
$\overrightarrow{tables}_{p}$, by computing

$\overrightarrow{tables}_{p}=\overrightarrow{pens}-\overrightarrow{pen}+\overrightarrow{table}.$
(2)

The word selected as the predicted plural is the word the vector of which is
closest to the composed vector, $\overrightarrow{tables}_{p}$ in (2), in terms
of cosine similarity. As a consequence, evaluation of these methods is
restricted to predefined analogy test sets such as Google’s (Mikolov, Chen .,
2013) which provide a series of analogies similar to the examples above.
Another limitation of these methods is that their prediction for the target
word tables highly depends on the prime word pair, here pen and pens, and not
on just the singular word table (Rogers ., 2017). Thus, the predicted plural
vector for tables is different when the prediction builds on another analogy
such as

$banana:bananas::table:tables.$

3CosAvg, on the other hand, operates on just one input vector, the vector of
the base word. Given the input word table, the predicted plural vector by
3CosAvg is

$\overrightarrow{tables}_{p}=\overrightarrow{table}+\overrightarrow{\textsc{avg
shift}}.$

The word selected as the plural form is again, exactly as for 3CosAdd, that
word the vector of which is closest to the assembled vector. For plural
analogies, Drozd . (2016) define the average shift vector as

$\overrightarrow{\textsc{avg
shift}}=\frac{1}{m}\sum_{i=1}^{m}\vec{p_{i}}-\frac{1}{n}\sum_{i=1}^{n}\vec{s_{i}},$
(3)

assuming there are $m$ plural word-forms with vectors $\vec{p_{i}}$ and $n$
singular word-forms with vectors $\vec{s_{i}}$. The average shift vector is
fixed given the data, and represents the semantics of pluralization, just as
the plural vector in the discriminative lexicon model (Baayen ., 2019)
provides a fixed representation for plurality.

For a dataset with $m$ plural and $m$ singular word-forms, the average shift
vector, i.e., the difference vector between the average vector of plurals and
the average vector of singulars, formulated in (4), is equal to the average
vector of the difference vectors between plurals and singulars, formulated in
(5):

$\displaystyle\overrightarrow{\textsc{avg shift}}$
$\displaystyle=\frac{1}{m}\sum_{i=1}^{m}\vec{p_{i}}-\frac{1}{m}\sum_{i=1}^{m}\vec{s_{i}}$
(4)
$\displaystyle=\frac{1}{m}(\sum_{i=1}^{m}\vec{p_{i}}-\sum_{i=1}^{m}\vec{s_{i}})$
$\displaystyle=\frac{1}{m}\sum_{i=1}^{m}(\vec{p_{i}}-\vec{s_{i}}).$ (5)

Henceforth, we refer to the vector $\vec{p_{i}}-\vec{s_{i}}$ for word $i$ as
this word’s individual shift vector. Such a shift vector is exemplified in
Figure 1. Importantly, if plural and singular forms for different lexemes are
consistently used across similar contexts, as captured by word embeddings,
then the difference between individual shift vectors and the average shift
vector is expected to be small.

Figure 1: Individual shift vector for the lexeme pen is calculated as
$\overrightarrow{\text{pens}}-\overrightarrow{\text{pen}}$.

A range of studies have adopted shift vectors to study the semantics of
various lexical relations. For instance, Roller . (2014) and Weeds . (2014)
used shift vectors for hypernymy detection. Bonami Paperno (2018) used shift
vectors to model inflectional and derivational contrasts in French, and Mickus
. (2019) made use of shift vectors for tracing contrasts in grammatical gender
of nouns and adjectives.

###  3.2 Individual and average shift vectors

How well does an average shift vector approximate the shifts between
individual singulars and their plurals? To address this question, we
investigated what the individual shift vectors look like and whether the
average shift vector is representative for the individual shift vectors. For
each noun pair in the pluralization dataset represented by word2vec semantic
vectors, we first calculated its individual shift vector by subtracting the
singular vector from the plural vector. As a next step, we calculated the
length (or magnitude), the direction, and the neighborhood structure of the
shift vectors.

We gauged the length of vectors with the $\ell_{2}$ norm, i.e., the Euclidean
distance of a vector from the origin. Fig. 2 shows notched box and whiskers
plots for the length of singular, plural, and individual shift vectors. Vector
lengths differed in the mean for singular, plural and shift vectors (Friedman
test, $\tilde{\chi}^{2}(2)=7201$, $p\ll 0.0001$). Pairwise Wilcoxon signed-
rank test between groups with Bonferroni correction revealed significant
differences in length for all pairwise comparisons (all $p\ll 0.0001$). Plural
vectors are, on average, longer than singular vectors (the difference between
the medians $\Delta\mathrm{MD}$ is 0.13). This fits well with the intuition
that English plurals are semantically more complex than their corresponding
singulars.

Shift vectors are, on average, smaller than the singular
($\Delta\mathrm{MD}=0.43$) and the plural vectors ($\Delta\mathrm{MD}=0.56$),
which is only to be expected given that the shift vectors are, by definition,
difference vectors. Although the average length of the shift vectors is
smaller than the average lengths of singular or plural vectors, shift vectors
turn out to nevertheless be surprisingly long. Their range ($1.1-6.8$,
$\mathrm{MD}=2.88$) is nearly as wide as the ranges of the singular vectors
and the plural vectors.

Figure 2: Box plots for the length of 14,699 word2vec’s singular, plural, and
individual shift vectors.

We quantified the angles of vectors in word2vec’s 300-dimensional vector space
with respect to the standard unit vector $\vec{e}_{300}$ in degrees, using
(3.2). This 300-dimensional unit vector has a 1 as the last element and zeros
elsewhere. Notched boxplots for angle are presented in Fig. 3. The range of
angles for shift vectors is even more similar to the ranges of angles of the
singular and plural vectors, compared to vector lengths.

$\displaystyle\theta(\vec{v})$
$\displaystyle=\frac{180}{\pi}(\arccos{\frac{\vec{v}\cdot\vec{e}_{300}}{\left\lVert\vec{v}\right\rVert_{2}\left\lVert\vec{e}_{300}\right\rVert_{2}})}$
$\displaystyle=\frac{180}{\pi}(\arccos{\frac{v_{300}}{{\sqrt{\sum\limits_{i=1}^{300}{v_{i}^{2}}}}}})$
(6)

Figure 3: Box plots for the angle of 14,699 word2vec’s singular, plural, and
individual shift vectors.

Fig. 4 plots the length of shift vectors against their angle. Considerable
variability is visible in the length and the angle of individual shift
vectors. The average of a set of vectors radiating from the origin that point
in various directions and have various lengths will inevitably end up close to
the origin of that vector space. The average shift vector, in red, at (89.25,
0.64) is smaller than all of the individual shift vectors, and has an
$\ell_{2}$ norm of only 0.64. When such a small vector is added to the
singular, it is hardly distinguishable from the singular vector, and at a
large distance from the actual corresponding plural vector.

Figure 4: Scatter plot visualizing the relationship between the length, on the
y-axis, and angle, on the x-axis, for the individual shift vectors. The
isolated red dot below the cloud of all other data points at (89.25, 0.64)
belongs to the average shift vector.

Upon closer inspection, it turns out that, rather than being random, the set
of individual shift vectors exhibits structure. The length of plural vectors
increases with the length of their singular vectors, and likewise, the length
of shift vectors increases with the length of the singular vectors, as
illustrated in Fig. 5. From this, we can draw the conclusion that the
semantics of shift vectors is changing in close association with the semantics
of the singular and plural words.

Figure 5: Scatterplots for the length of plural vectors (vertical axis in the
left panel) and length of shift vectors (vertical axis in the right panel)
against the length of singular vectors (horizontal axis in both plots), with
Locally WEighted Scatterplot Smoothing (LOWESS) trend lines in red. The dashed
black lines represent the identity line $y=x$.

Given that singular words that have similar semantics have closer vectors, and
singular words with less similar meanings have more diverging vectors, we now
consider the question of whether the shift vectors themselves show structuring
that goes beyond the structure provided at the level of individual lexemes. To
address this question, we made use of the t-SNE algorithm for visualizing
high-dimensional data (van der Maaten Hinton, 2008) as implemented in the
scikit-learn Python library (Pedregosa ., 2011), version 1.0.1, to plot the
300-dimensional shift vectors in a two-dimensional plane.222Following the
recommendations of van der Maaten (2021), we searched the t-SNE’s parameter
space between possible combinations of perplexity (either 10, 15, 20, 25, 30,
or 35), number of iterations (either 500, 1000, 2000, 3000, or 4000), random
state (either 1, 12, or 123), and initialization method (either random or PCA)
for the t-SNE with the lowest Kullback-Leibler divergence. The lowest KL-
divergence was obtained with the following setting: perplexity $=$ 35, number
of iterations = 4000, early exaggeration = 12, random state = 1, learning rate
= ‘auto’, metric=‘euclidean’, and initialization = ‘random’. This
visualization technique is known to have a very high chance of recovering the
clustering structure present in the input space in the reduced output space
(Linderman Steinerberger, 2019; Arora ., 2018).

Fig. 6 presents the scatter of data points in this plane, coloured with the
label of the first synset in WordNet (Fellbaum, 1998; Miller, 1995) for the
singular word form. From the 14,699 pairs in our pluralization dataset, 11,749
pairs are found in WordNet and used in the remainder of this study. The
labels, indicated in the figure’s legend, often referred to as supersenses,
include 26 broad semantic categories for nouns (Ciaramita Johnson, 2003).
Interestingly, the individual shift vectors form clusters that are reasonably
well approximated by the WordNet supersenses. Some supersenses show well-
defined clusters, such as person towards the bottom right corner of the plane
and animal towards the top right corner. This indicates that pluralization is
similar for nouns denoting animal nouns and is different for nouns denoting
persons. Importantly, the average shift vector (highlighted by a red cross) is
located near the origin of this space at $(0.4,\,-1.8)$. Interpretation of the
t-SNE dimensions is not very straightforward. Preliminary investigation
suggests that the first dimension is to a very large extent differentiating
between concrete and abstract words (see supplementary materials for details).
The second dimension is less interpretable, and rather similar to the first
dimension.

Figure 6: A projection of shift vectors onto a two-dimensional plane using
t-SNE reveals semantic clustering. Colors correspond to WordNet supersenses.
The average shift vector marked with a red cross, which is located very close
to the origin at $(0.4,\,-1.8)$, is blind to this rich structure. This figure
is available as an interactive plot in the supplementary materials.

Although some well-defined clusters are present in Fig. 6, other clusters show
considerable overlap. This is due to two problems. The first problem is that
nouns can have multiple senses. We selected the first sense listed in WordNet,
which, according to Jurafsky Martin (September 21, 2021), is the most frequent
sense and hence a strong baseline. However, inaccuracies are inevitable. For
instance, strawberry is assigned the food category while blueberry is labeled
as a plant. A related problem is that we have one embedding for all senses,
instead of sense-specific embeddings.

The second problem is that the supersenses are often too broad and too over-
populated to form semantically coherent groups. For instance, the supersense
artifact brings together musical instruments, vehicles, clothes, guns, and
buildings among others. In the t-SNE plane, this supersense is found in two
distinct regions. The fuzziness of the 26 supersenses is clearly demonstrated
by Linear Discriminant Analysis (LDA) given the task of assigning shift
vectors to supersenses. From an evaluation of the LDA on all of the data
points ($N=11749)$, accuracy and weighted average F-score were both 58.4% . To
put the multiclass classification performance of the LDA into perspective, the
weighted average F-score by the LDA is 7 times greater than the weighted
average F-score of a baseline classifier that always predicts the most
frequent superset. The LDA’s performance indicates that on the one hand there
is structure and the structure is captured by both a supervised algorithm,
i.e., LDA, and an unsupervised algorithm, i.e., t-SNE. On the other hand, it
indicates that there is also considerable uncertainty about superset
membership.

To address the first problem, one would have to make use of techniques for
word-sense disambiguation. Word sense disambiguation has a very long history
in computational linguistics and there are many supervised and unsupervised
algorithms designed for this task. One might combine WordNet and FrameNet
(Baker ., 1998) annotations as proposed by Baker Fellbaum (2009), train a
supervised model (e.g., Zhong Ng, 2010), or search for words’ nearest
neighbors in a contextual word embeddings space (Loureiro Jorge, 2019). Given
a high-accuracy word sense disambiguation pipeline, one could then apply word
sense disambiguation before calculating embeddings using word2vec. Such a
programme, if at all feasible, is outside the scope of the present study.

The second problem is more straightforward to address. Instead of using the 26
supersenses shown in Fig. 6, we can zoom in on smaller, more semantically
homogeneous sense sets. For instance, the supersense person covers 2725
lexemes in our data. By moving to semantic classes one level below this
supersense, we obtain more coherent subsets such as relative, scientist, and
lover. For our pluralization dataset, we constructed a total of 411 classes,
by moving zero steps or one step down from the supersenses. On average, a
class has 28.6 ($SD=39.8$) members. No class has fewer than 5 members. The
most populous class has 481 members. These new semantic classes are more
semantically cohesive, as can be seen in Figure 7 for a number of sub-classes
within the supersense artifact as an example. Furthermore, the performance of
LDA increased despite the substantial increase in the number of classes.
Accuracy and weighted average F-score are both 61% from an evaluation of an
LDA that predicts 411 classes given the shift vectors. In comparison, the
weighted average F-score by this model is 189 times greater than the weighted
average F-score of a baseline classifier that always predicts the most
frequent class.

Figure 7: A projection of shift vectors onto a two-dimensional plane using
t-SNE for a subset of data points within the supersense artifact that form
rudimentary sub-clusters such as clothing and musical instruments.

Although the idea of an abstract semantic representation is appealing, it
turns out that a simple average shift vector fails to do justice to the
intricate semantic structure that characterizes nominal pluralization in
English. Apparently, English pluralization is substantially more subtle, and
varies systematically with the semantic category (supersense) of a noun.

###  3.3 CosClassAvg

This new set of 411 classes, or a similarly cohesive set of classes of
semantically highly related words, makes it possible to formalize a new model
for plural semantics. We first calculated the average shift vector for each of
the 411 classes. The mean length of these average shift vectors is 1.2, and
its standard deviation was 0.3. Compared to the distribution of shift vectors
shown in Fig. 2, both mean and standard deviation are substantially reduced.
The same holds for their angles ($M=89.1$, $\mathrm{SD}=2.6$). This clarifies
that by-class shift vectors are more similar to each other than is the case
for the shift vectors in the undifferentiated set of all nouns.

We can now introduce our ‘CosClassAvg’ theory for noun plurals. Given an input
word and its semantic class, the plural vector predicted by CosClassAvg is
obtained by taking the singular vector and adding to it the average shift
vector for that class. Thus, the vector for bananas is predicted using

$\overrightarrow{bananas}_{p}=\overrightarrow{banana}+\overrightarrow{\textsc{avg-
shift}}_{\textsc{fruit}},$

while the vector for cars is predicted based on

$\overrightarrow{cars}_{p}=\overrightarrow{car}+\overrightarrow{\textsc{avg-
shift}}_{\textsc{vehicle}}.$

We can assess the quality of predicted vectors by inspecting the cosine
similarities of a predicted vector with the vectors of all words. Ideally, the
vector that is closest to the predicted vector represents the meaning of the
targeted plural.

How well does CosClassAvg perform? To address this question, we first
investigated whether predicted plurals are better differentiated from their
singular counterparts. As our baseline for comparisons, we used the Only-b
method introduced in Linzen (2016), where b represents the vector for the base
word. This method simply returns the input singular vector, without adding
anything to it, as the predicted plural vector. As a consequence, this method
will always predict the nearest neighbor in terms of cosine similarity, i.e.
the word that is most similar to the base word in the vocabulary.

We calculated the predicted plural vectors for all singular words in our
pluralization dataset ($N=11,749$) using 3CosAvg, CosClassAvg, and the
baseline method. Many implementations of proportional analogies with word
embeddings exclude the input words such as the singular word from the
vocabulary as a potential predicted word. However, in an “honest” practice, as
Rogers . (2017) put it, we do not exclude any words from the vocabulary. We
therefore compared predicted vectors with a broader set of words covering all
30,497 word-form types in our pluralization and our vocabulary datasets.

The notched boxplots in Fig. 8 summarize the distributions of cosine
similarities (left) and Euclidean distances (right), for the baseline model
(Only-B), the 3CosAvg model, and the new CosClassAvg model, of the predicted
vectors and the corresponding plural vectors provided by word2vec. The lowest
boxplots in blue produced by the baseline method, indicate that the singular
and the plural vectors in word2vec are already astonishingly similar. Both
3CosAvg and CosClassAvg improve on the baseline and generate more similar and
less distant vectors to the actual plural vector, with CosClassAvg in the
lead.

Figure 8: Comparison between the predicted plural vectors and the corpus-
extracted plural vectors using cosine similarity (left panel) and Euclidean
distance (right) for the Only-B, the 3CosAvg, and the CosClassAvg method.

For predicted vectors to well approximate the true plural vectors, they should
be less close to their corresponding singular vectors. Fig. 9 visualizes
cosine similarity to singular vectors and Euclidean distance from singular
vectors of the predicted plural vectors. Similarity decreases from one and
distance increases from zero with 3CosAvg and CosClassAvg plural vectors. The
Euclidean distance between 3CosAvg plural vectors and their singular vectors
is always equal to the length of the average shift vector. The length of this
overall average shift vector is smaller than the length of any CosClassAvg
class-specific shift vector.

Figure 9: Comparison between the predicted plural vectors and the singular
vectors using cosine similarity (left panel) and Euclidean distance (right)
for the Only-B, the 3CosAvg, and the CosClassAvg method.

When we use the stringent criterion that any word, including the singular, can
be a neighbor of the predicted plural, then performance of both 3CosAvg and
CosClassAvg is disappointing. 3CosAvg always selects the singular as closest
neighbor, and CosClassAvg only correctly selects 42 plurals (0.4%). Although
CosClassAvg yields predicted vectors that are further away from their
singulars and closer to their plurals, compared to 3CosAvg, predicted plural
vectors remain very close to their singular vectors. Table. 1 lists the
percentages of lexemes for which the targeted plural vector is among the top-n
neighbors. Of the three methods, CosClassAvg clearly outperforms the other
two, with percentages ranging from 79% to 95%. In other words, if we relax our
criterion and filter out singular vectors as candidates, the accuracy of
CosClassAvg is at 79%.

Table 1: Percentage of the lexemes ($N=11,749$) for which the plural vector is among the 2 (Top 2), 3 (Top 3), 10 (Top 10), and 20 nearest neighbors (Top 20) of the predicted plural vector. Method | Top 2 | Top 3 | Top 10 | Top 20
---|---|---|---|---
Only-B | 61 | 74 | 88 | 92
3CosAvg | 70 | 80 | 91 | 93
CosClassAvg | 79 | 86 | 93 | 95
|  |  |  |

###  3.4 Discussion

According to the 3CosAvg method proposed by Drozd . (2016), pluralization can
be formalized as a function adding an average shift vector to the singular
vector:

$\bm{v}_{\text{pl}}=f_{\text{\tiny{3CosAvg}}}(\bm{v}_{\text{sg}})=\bm{v}_{\text{sg}}+\bm{v}_{\textsc{pl}}.$

We have shown that this formalization of plurality is too simple: shift
vectors form semantically motivated clusters. CosClassAvg brings these classes
into a modified function

$\bm{v}_{\text{pl}}=g_{\text{\tiny{CosClassAvg}}}(\bm{v}_{\text{sg}})=\bm{v}_{\text{sg}}+\bm{v}_{\textsc{pl}\mid\text{semantic
class}}.$

The meaning shift in pluralization is similar for lexemes within a semantic
class and is different for lexemes from different semantic classes.
CosClassAvg capitalizes on this observation, computes several average shift
vectors, one per semantic class, which enables it to generate improved
predictions for plurals.

Pluralization with CosClassAvg requires two pieces of information to make a
prediction, namely, information on the semantic clusters (and their centroids)
and information on the semantic class membership of a given singular noun. The
current study shows that, given this information, more precise predictions for
plural vectors are obtained. Although outside the scope of this contribution,
it may well be possible to develop an end-to-end model that does class
induction and pluralization jointly. To that end, semantically cohesive
clusters within the shift space may be obtained using unsupervised clustering
algorithms. We have shown that clusters found by the unsupervised t-SNE
algorithm independently of the WordNet tags, are well-supported by the
supervised LDA classification using WordNet tags. In the present approach, we
accept the 411 classes as given, leaving it to further research to address the
question of how these classes might be grounded in unsupervised learning.

Regarding the second source of information, we gauged how straightforward it
is to classify singular nouns according to their semantics. From a 5-fold
stratified cross-validation evaluation of an LDA predicting the 411 semantic
classes from singular vectors, the mean weighted average F-score was 61%
($\mathrm{SD}=0.1$) on the training sets and 32% ($\mathrm{SD}=0.5$) on the
test sets. The weighted average F-scores by the LDA from the 5 evaluations are
on average 190.3 times ($\mathrm{SD}=0.5$) greater on the training sets and on
average 100.1 times ($\mathrm{SD}=2.3$) greater on the the test sets than the
weighted average F-scores of a baseline classifier that always predicts the
most frequent class. A straightforward LDA performs quite well under cross-
validation. Thus, the classes that we derived from WordNet are to a large
extent implicit in the word embeddings.

The CosClassAvg method may also provide enhanced predictivity for human
lexical processing, compared to the 3CosAvg method. For instance, Westbury
Hollis (2019) calculated average vectors for words belonging to different
syntactic categories, or containing different derivational affixes, and showed
that these average vectors can be leveraged to model human categorization
decisions. Following their approach, we computed the average vector of plural
nouns ($\frac{1}{m}\sum_{i=0}^{m}\vec{p_{i}}$ in equation 4) using the 14,699
plural words in our pluralization dataset introduced in section 2. Nearest
neighbors to the average plural were retrieved among our vocabulary dataset.
We replicated Westbury Hollis (2019)’s findings for the average plural vector.
Within the closest neighbors of the average plural vector, 79% are plural
nouns. However, other than being plural, these nouns are semantically highly
heterogeneous. If human category decisions are also influenced by the lexical
semantics of nouns, more precise predictions can perhaps be obtained by
further conditioning on the semantic class of the noun. We leave this issue to
further research.

The clustering of plural shift vectors by semantic class likely reflects
differences in how plural objects configure in our (culture-specific)
constructions of the world. Multiple cars occur in different configurations
which tend to share alignments, as in parking lots or traffic jams. Multiple
oranges or multiple cherries occur in very different configurations, typically
piled up in boxes or on plates, and bananas occur in hands on banana plants
and fruit stands. Apparently, the different properties of the objects that we
refer to in the plural are reflected in our language use, as captured by
distributional semantics. However, apples and oranges are more similar than
apples and bananas. As a consequence, the vectors predicted by CosClassAvg
will always be a bit off for individual words. This observation necessitates
updating the plural semantic function $g$ with an error term, as follows

$\bm{v}_{\text{pl}}=g_{\text{\tiny{CosClassAvg}}}(\bm{v}_{\text{sg}})=\bm{v}_{\text{sg}}+\bm{v}_{\textsc{pl}\mid\text{semantic
class}}+\bm{\epsilon}_{\text{lexeme}}.$

The error vector $\bm{\epsilon}_{\text{lexeme}}$ represents the lexeme-
specific semantics that cannot be captured by the semantic commonalities of
the lexeme’s semantic class. In usage-based grammar and corpus linguistics,
individual words, including inflected words, have been argued to have their
own highly specific usage profiles (see, e.g., Sinclair, 1991). ‘Error’
components such as $\bm{\epsilon}_{\text{lexeme}}$ formalize this important
insight. However, since semantic vectors themselves are measurements, and as
such subject to measurement error, we need to add a second error vector
representing measurement noise:

$\bm{v}_{\text{pl}}=g_{\text{\tiny{CosClassAvg}}}(\bm{v}_{\text{sg}})=\bm{v}_{\text{sg}}+\bm{v}_{\textsc{pl}\mid\text{semantic
class}}+\bm{\epsilon}_{\text{lexeme}}+\bm{\epsilon}.$

Since CosClassAvg decomposes semantic vectors into constituent semantic
vectors, it constitutes a ‘decompositional’ or ‘analytical’ method for
accounting for inflectional semantics. In the next section, we compare
decompositional CosClassAvg with a compositional method, FRACCS (Marelli
Baroni, 2015).

##  4 Pluralization with FRACSS

Marelli Baroni (2015), building on previous research on compositional
semantics (Mitchell Lapata, 2008; Baroni Zamparelli, 2010; Lazaridou ., 2013),
proposed to model derivational semantics with the help of a linear
transformation that takes the semantic vector of the base word as input, and
maps it onto the semantic vector of the corresponding plural using a linear
mapping $\bm{B}$:

$\bm{v}_{\text{pl}}=h_{\text{\tiny{FRACSS}}}(\bm{v}_{\text{sg}})=\bm{v}_{\text{sg}}\bm{B}.$

This model, known as the FRACSS model, has been applied to German complex
verbs (Günther, Smolka Marelli, 2019), and an extended version has been used
to study compounding in English and German (Günther Marelli, 2016, 2019;
Marelli ., 2017; Günther ., 2020). In the following, we apply FRACSS to
English plural inflection, and compare its predictions with those of
CosClassAvg.

FRACSS transforms singular vectors into plural vectors using straightforward
matrix multiplication. Let $\bm{X}$ denote a matrix with as row vectors the
word embeddings of singulars, and let $\bm{Y}$ denote a matrix with the same
number of row and column vectors representing the meanings of the
corresponding plurals:

${\displaystyle\bm{X}={\begin{pmatrix}x_{1,1}&x_{1,2}&\cdots&x_{1,n}\\\
x_{2,1}&x_{2,2}&\cdots&x_{2,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
x_{t,1}&x_{t,2}&\cdots&x_{t,n}\\\
\end{pmatrix}},\quad\bm{Y}={\begin{pmatrix}y_{1,1}&y_{1,2}&\cdots&y_{1,n}\\\
y_{2,1}&y_{2,2}&\cdots&y_{2,n}\\\ \vdots&\vdots&\ddots&\vdots\\\
y_{t,1}&y_{t,2}&\cdots&y_{t,n}\\\ \end{pmatrix}}.}$

The mapping $\bm{B}$ is a $n\times n$ dimensional matrix that satisfies

$\bm{X}\bm{B}=\bm{Y}.$

We estimate $\bm{B}$ as follows:

$\bm{B}=\bm{X}^{+}\bm{Y}=(\bm{X}^{T}\bm{X})^{-1}\bm{X}^{T}\bm{Y},$

where $\bm{X}^{+}$ is the pseudo-inverse of $\bm{X}$ and $\bm{X}^{T}$ is its
transpose, and $(.)^{-1}$ denotes a matrix inverse operation. Given $\bm{B}$
and the vector of a singular, the predicted plural vector is given by

${\displaystyle{\begin{bmatrix}x_{1}&x_{2}&\cdots&x_{n}\end{bmatrix}}{\begin{pmatrix}b_{1,1}&b_{1,2}&\cdots&b_{1,m}\\\
b_{2,1}&b_{2,2}&\cdots&b_{2,m}\\\ \vdots&\vdots&\ddots&\vdots\\\
b_{n,1}&b_{n,2}&\cdots&b_{n,m}\\\
\end{pmatrix}}={\begin{bmatrix}\hat{y}_{1}&\hat{y}_{2}&\cdots&\hat{y}_{m}\end{bmatrix}},}$

which, according to the definition of matrix multiplication, implies that

$\hat{y}_{j}=\sum_{i=1}^{n}b_{i,j}\cdot x_{i},\qquad{\small\text{for }1\leq
j\leq m}.$

In other words, the $j$-th element of the semantic vector of a given plural is
a weighted sum of the values of its singular vector.

###  4.1 Conceptualizing noun plurals with FRACSS

We estimated the mapping matrix $\bm{B}$ for 90% of the singular-plural pairs
in our pluralization dataset (10,574 pairs) using 300-dimensional word2vec
vectors. The remaining 1,175 word pairs were set aside as held-out testing
data. The resulting 300$\times$300 $\bm{B}$ matrix implements the change in
the meaning of singular words that goes hand in hand with the affixation of
the plural -s. With $\bm{B}$ in hand, we can calculate predicted plural
vectors for both the training data and the test data. The model correctly
predicts plural forms for 88% of training items and for 76% of test items.
Clearly, the mapping appears robust as a memory for seen items and it is also
productive for unseen items.

To better understand the performance of the FRACSS model, recall that in
section 3 we observed that word2vec’s singular and plural vectors are very
similar. That is to say, any model for finding a mapping between the singular
and the plural space is a-priori in an advantageous position since the
relationship between the two spaces is already a given property of the
semantic space constructed by word2vec. In other words, the mapping matrix
$\bm{B}$ must be somewhat similar to an identity matrix (i.e., a matrix with
ones on the diagonal and zeroes elsewhere). The cool-to-warm heat map in Fig.
10, that visualizes the FRACSS matrix, shows that this is indeed the case.
Input vector dimensions are on the vertical axis, indexed by $i$ from 1 to
300, and output vector dimensions, indexed by $j$ from 1 to 300, are on the
horizontal axis. The color indicates the magnitude of the value at index
$(i,j)$. The value at index $(i,j)$ of this matrix, $b_{i,j}$, shows the
association strength between the $i$-th dimension of the singular vectors and
the $j$-th dimension of the plural vectors. Association strengths are highest
on the diagonal entries of this matrix, which links every singular with its
own plural.

Figure 10: FRACSS matrix for English plural suffix -s.

The mean value of the diagonal elements is 0.57 ($SD=0.02$). Barely any
structure is evident elsewhere: the mean value of off-diagonal elements is a
mere $9.8\times 10^{-5}$ ($SD=0.017$). We can therefore approximate the effect
of multiplication with $\bm{B}$ with a much simpler operation:

$\hat{\bm{Y}}=0.57\bm{X}\bm{I}+\bm{\epsilon},\bm{\epsilon}\sim{\cal
N}_{300}({\bf-0.001},0.08\bm{I}),$ (7)

where $\bm{I}$ is the identity matrix, $\bm{\epsilon}$ is a matrix of
300-dimensional random vectors as row vectors all chosen from the same
multivariate normal distribution with mean vector $\bf{-0.001}$ (a
300-dimensional vector with -0.001 everywhere) and covariance matrix
$0.08\bm{I}$.333For almost all predicted plural and singular vector pairs, the
epsilons were normally distributed with an average mean of -0.001 and an
average standard deviation of 0.08 (D’Agostino’s $K^{2}$ departure from
normality hypothesis test; $p>0.001$ for 99.8% of 11749 tests). Note that this
approximation of $\bm{B}$ predicts that the semantic vectors predicted by
FRACSS are shorter in length than their singulars: this follows from the
multiplication factor 0.57.

How do the FRACSS predicted vectors compare to the vectors predicted by
CosClassAvg? To address this question, we first consider similarity evaluated
by means of the angle between vectors, and subsequently by means of the
Euclidean distance of the corpus-extracted vectors. The median cosine
similarity of predicted and target vectors is 0.75 for FRACSS and 0.71 for
CosClassAvg (Wilcoxon signed-rank test $W=65105609.0$, $p\ll 0.0001$ one-
tailed, $N=11749$). Furthermore, the median cosine similarity between singular
vectors and predicted vectors is 0.87 for FRACSS and 0.95 for CosClassAvg
(Wilcoxon signed-rank test $W=649018.0$, $p\ll 0.0001$ one-tailed).

When accuracy is evaluated with the cosine similarity measure, the FRACSS
plural vectors are now close enough to the target plural vectors to capture
the plural word correctly as the first nearest neighbor in 1520 cases (13%).

Similar results are obtained when we use the Euclidean distance measure. The
median Euclidean distance to corpus-extracted plural vectors is shorter from
predicted vectors for FRACSS at 2.28 in comparison with vectors for
CosClassAvg at 2.64 (Wilcoxon signed-rank test $W=1530725.0$, $p\ll 0.0001$
one-tailed). Inversely, the median Euclidean distance between singular vectors
and predicted vectors is 1.67 for FRACSS and 1.04 for CosClassAvg (Wilcoxon
signed-rank test $W=68556852.0$, $p\ll 0.0001$ one-tailed).

Thus far, we have based our evaluation on the angle and distance between
vectors. We have seen that FRACSS vectors have smaller angles and shorter
distances to plural vectors than CosClassAvg vectors. What about the Euclidean
length of the predicted plural vectors? Fig. 11 plots the length of predicted
plural vectors against the length of singular vectors, for CosClassAvg (left)
and FRACSS (right). For both methods, length of predicted plural vectors
increases with the length of singular vectors, similar to the trend observed
in Fig. 5 for the length of corpus-extracted plural and singular vectors.
However, there is a striking difference. Most plural vectors predicted by
CosClassAvg are longer than their singular vector (74%). By contrast, as
anticipated above on the basis of an analysis of the $\bm{B}$ matrix, all
plural vectors predicted by FRACSS are shorter than their corresponding
singular vectors.444The signed difference between the length of the target
plural vectors and the length of the predicted plural vectors is lower for
CosClassAvg compared to FRACSS (Wilcoxon signed-rank test $W=69025375.0,p\ll
0.0001,\mathrm{MD}_{\text{CosClassAvg}}=0.09,\mathrm{MD}_{\text{FRACSS}}=0.89,N=11749$).
However, for the corpus-based actual word2vec vectors, 66% of the plural
vectors are longer than the corresponding singular vectors.

Figure 11: Scatter plots depicting the length of predicted plural vectors, on
the vertical axis, versus the length of singular vectors, on the horizontal
axis, by CosClassAvg (left panel) and FRACSS (right panel) with the Locally
WEighted Scatterplot Smoothing (LOWESS) trend lines in red. The dashed black
lines represent the identity line $y=x$.

###  4.2 Discussion

We have seen that FRACSS outperforms CosClassAvg when evaluation is based on
the angle or distance between vectors, but CosClassAvg outperforms FRACSS when
we consider vector lengths. In section 5 we propose another measure for
evaluating the relative merits of the two methods. , we first reflect on some
technical and conceptual problems that come with the FRACSS approach.

One conceptual problem concerns the interpretation of the $\bm{B}$ matrix. Our
t-SNE analysis of shift vectors revealed clustering by semantic class.
However, $\bm{B}$ is calculated by evaluating all singulars and plurals
simultaneously. It is an empirical question whether this is advantageous for
understanding human lexical processing, an issue we pursue in more detail in
section 5. If we assume, for the sake of the argument, that FRACSS is a more
precise version of CosClassAvg, then CosClassAvg provides us with insight into
what FRACSS is actually achieving: semantic-cluster driven local
generalization. In other words, the FRACSS matrix $\bm{B}$ does not represent
a single operation of pluralization that is independent and orthogonal to the
lexical meaning of the singular. To the contrary, $\bm{B}$ captures a wide
range of different local pluralization functions.

Another conceptual problem concerns the directionality of semantic operations.
Baroni . (2014) argue that semantic composition is intrinsically asymmetric
(compare ‘water under the bridge’ and ‘bridge under the water’), and that
therefore implementations of semantic functions using matrix multiplication is
superior to functions using vector addition. Marelli Baroni (2015) likewise
assert that representing affixes as functions over base forms in FRACSS
captures this important asymmetry. However, a FRACSS matrix such as $\bm{B}$
has a pseudo-inverse, and technically it is straightforward to construct a
mapping from plurals to singulars. Such a linear mapping trained in the
inverse direction on the same data correctly predicts singular semantics for
87% of training items and 74% of test items from plural semantics. We conclude
that FRACSS is not intrinsically asymmetric.

There are also some technical issues worth mentioning. First, updating FRACSS
matrices is more costly than updating the CosClassAvg model. Introducing new
word pairs, or even a single pair, to the training dataset necessitates re-
computation of the whole FRACSS network. For CosClassAvg, a new word pair
merely requires re-calibration of the average shift vector for the semantic
class of the pertinent lexeme.

Second, FRACSS requires large numbers of parameters: given vectors of
dimension $n$, it requires a mapping with $n^{2}$ parameters (which can be
conceptualized as the beta weights of a multivariate multiple regression
model). For the present word2vec vectors, we have no less than 300 $\times$
300 = 90,000 parameters. As our current dataset has more than 11,000
datapoints, we have more data than parameters, and FRACSS works just fine.
However, when the number of datapoints is substantially less than the number
of parameters, the FRACSS approach will overfit the data, and not generalize
well. For example, in the study of Lazaridou . (2013), 12 out of 18 affixes
have fewer than 350 words (training samples), whereas the semantic vectors
used had a dimensionality equal to 350. Likewise, 27 out of 34 affixes studied
by Marelli Baroni (2015) are trained on fewer observations than their vectors’
dimensionality.

To avoid the problem of overfitting, one could model the complete set of
derivational affixes of English with one FRACSS mapping. Under the assumption
that the number of derived words is substantially larger than the
dimensionality squared of the embeddings, the model should show good
generalization performance. However, even though data sparsity would no longer
be a problem, the model would not be very informative about the semantics of
the different affixes.

In the light of these considerations, we consider how well the two models for
conceptualizing plurals, FRACSS and CosClassAvg, perform when integrated into
a model of morphology that addresses the mappings between form and meaning,
the Discriminative Lexicon (DL) model proposed by (Baayen ., 2019).

##  5 Conceptualization and the mapping from form to meaning

Up till now, we have considered how the meanings of singulars and plurals are
related to each other, and we have considered two alternative mathematical
formalizations of how to conceptualize a plural given a singular. Both
formalizations provide an account of the semantic productivity of
pluralization. However, for understanding or producing actual words, we need
to consider mappings between form and meaning. In the ensuing paragraphs, we
focus on comprehension, and investigate which of the two formalizations
provides semantic vectors that are better aligned with words’ forms.

An initial question is whether words’ form representations make systematic
contact with the semantic vectors of CosClassAvg and FRACSS. If there exists
systematicity between the form space and these semantic spaces, it should be
possible to find accurate mappings from forms to meanings, not only for
training data, but also for held-out test data.

###  5.1 Comprehension with FRACSS and CosClassAvg

We model comprehension with the discriminative lexicon model of Baayen .
(2019). This model makes use of linear mappings from numeric representations
of words’ forms to numeric representations of words’ meanings. The DL model is
well-suited for our purposes as it has been shown to be successful in modeling
comprehension of morphologically complex words for various languages (Chuang
., 2020; Heitmeier ., 2021; Denistia Baayen, 2022; Heitmeier Baayen, 2020)
and, importantly, because it is flexible in terms of which semantic space is
selected to represent words’ meanings. Keeping form representations and the
representations for singular meanings the same, meaning representations for
plurals can be created according to CosClassAvg, or alternatively, according
to FRACSS.

For our modeling experiments, we extracted all singular and plural tokens from
the vocabulary dataset introduced in section 2. This subset comprises 9541
English singular and plural tokens of 8762 unique orthographic word-form
types. There are more tokens than types because 728 words have two or more
pronunciations in the NewsScape English Corpus. We constructed training data
and test data in such a way that plurals in the test data always had the
corresponding singular in the training data. The training data also included
plural forms that do not have a corresponding singular in the dataset. Of all
plurals with corresponding singulars, 70% were assigned to the training data,
and 30% to the testing data. This resulted in training data comprising 8,507
tokens of 7,886 types, and test data comprising 1034 tokens of 1002 types.
Table 2 provides further information on the composition of the training and
test sets.

Table 2: Number of word-form types and tokens in the datasets used for the DL simulations. Dataset | Word-form Types | Word-form Tokens
---|---|---
Training set |  |
Singular | 5073 | 5511
Plural with seen stem | 2253 | 2412
Plural with unseen stem | 560 | 584
Test set |  |
Plural with seen stem | 1002 | 1034
|  |

Heitmeier . (2021) discuss several methods with which numeric representations
for word forms can constructed. In the present study, we make use of numeric
form vectors that are based on triphones, i.e., context-sensitive phone units
that include information about neighboring segments. For the word cities, the
triphone cues are #s, st, ti, tiz, and iz#, where the # symbol is used to
denote word boundaries. For our dataset, there are 6,375 unique triphones. A
word’s form vector is defined as a vector with length 6,375 that has values
that are either zero or one, depending on whether a triphone is present in a
word (1) or not (0). Words’ form vectors can be brought together in a matrix
$\bm{C}$ with words on rows and triphones on columns (For form vectors derived
from the audio signal, see Shafaei-Bajestan ., 2021). As a result, the matrix
with word form vectors $\bm{C}$ used for deriving mappings from form to
meaning had 8,507 rows and 6,375 columns.

The form vectors for words are based on the phone transcriptions in the
NewsScape English Corpus, which are obtained from the Gentle forced aligner.
Gentle’s ASR backend is kaldi (Povey ., 2011), which is set up to run with a
version of the CMUDict machine-readable pronunciation dictionary
(https://github.com/cmusphinx/cmudict), but with information on stress
removed. For various words, the dictionary offers pronunciation variants, such
as d_B ae_I t_I ah_E and d_B ey_I t_I ah_E for data. Here, CMUDict combines
ARPABET phone representations with additional information on whether a segment
is at the beginning of a word, at an intermediate position, or at the end of a
word (B, I, and E respectively).

We note here that the list of pronunciation variants provided by CMUDict is
far from complete. For instance, for ideology, it provides the transcription
/adildi/ but not the alternative /idildi/. Various reduced forms of function
words as typically found in spoken language are not represented in the
dictionary. For instance, the conjunction and is listed with two variants,
/ænd/ and /nd/, but forms such as /n/ or even /n/ are not included. As a
consequence, the representations we used for words’ forms may not correspond
to the exact way in which these words were actually spoken.

For evaluating the advantages and disadvantages of semantic vectors based on
CosClassAvg and FRACSS, we set up two semantic matrices,
$\bm{S}_{\text{\tiny{CosClassAvg}}}$ and $\bm{S}_{\text{\tiny{FRACSS}}}$ that
were based on word2vec. The vectors for singulars were straightforwardly taken
from word2vec, but the vectors for plurals were calculated either according to
CosClassAvg or according to FRACSS. The two semantic matrices had 8,507 rows
and 300 columns. We then calculated two 6,375$\times$300 mappings,
$\bm{F}_{\text{\tiny{CosClassAvg}}}$ and $\bm{F}_{\text{\tiny{FRACSS}}}$, by
solving the equations

$\displaystyle\bm{S}_{\text{\tiny{CosClassAvg}}}$ $\displaystyle=$
$\displaystyle\bm{C}\bm{F}_{\text{\tiny{CosClassAvg}}}$
$\displaystyle\bm{S}_{\text{\tiny{FRACSS}}}$ $\displaystyle=$
$\displaystyle\bm{C}\bm{F}_{\text{\tiny{FRACSS}}}.$

With these the two mappings, we obtained two sets of predicted semantic
vectors for the training data:

$\displaystyle\hat{\bm{S}}_{\text{\tiny{CosClassAvg}}}$ $\displaystyle=$
$\displaystyle\bm{C}\bm{F}_{\text{\tiny{CosClassAvg}}}$
$\displaystyle\hat{\bm{S}}_{\text{\tiny{FRACSS}}}$ $\displaystyle=$
$\displaystyle\bm{C}\bm{F}_{\text{\tiny{FRACSS}}}.$

Given the form vectors of the held-out plurals, which we collect as the row
vectors of a form matrix $\bm{C}_{\text{\tiny{test}}}$, we also obtain two
matrices with predicted plurals:

$\displaystyle\hat{\bm{S}}_{\text{\tiny{CosClassAvg}},\text{\tiny{test}}}$
$\displaystyle=$
$\displaystyle\bm{C}{\text{\tiny{test}}}\bm{F}_{\text{\tiny{CosClassAvg}}}$
$\displaystyle\hat{\bm{S}}_{\text{\tiny{FRACSS}},\text{\tiny{test}}}$
$\displaystyle=$
$\displaystyle\bm{C}{\text{\tiny{test}}}\bm{F}_{\text{\tiny{FRACSS}}}.$

Prediction accuracy was evaluated by inspecting which gold-standard row vector
is closest to the corresponding predicted semantic vector in terms of
Pearson’s correlation coefficient. If these vectors belong to the same word
(i.e., they have the same row index), prediction is taken to be accurate. In
the same way, we can check whether the gold-standard vector is among the top
$n$ nearest semantic neighbors. Making a prediction for a given test token
always involves choosing among 7,886 $+$ 1 different semantic vectors—the
semantic vectors for the word types in the training set plus the semantic
vector for the current test word. Henceforth, we will refer to the DL model
with FRACSS vectors as DL-FRACSS and the model with CosClassAvg embeddings as
DL-CosClassAvg.

Fig. 12 presents the top 1 to top 5 accuracy of word recognition evaluated on
the training set in dark bars and on the test set in light bars. Recognition
accuracy on the training set by both models is 96% for models’ top 1
predictions and increases to almost 100% as we consider top 2 to top 5
predicted words. With respect to the test data, DL-CosClassAvg outperforms DL-
FRACCS by a wide margin in terms of accuracy (top 1)555The median correlation
between the predicted semantic vectors and the target semantic vector is
larger for the DL-CosClassAvg model compared to DL-FRACSS ($W=214575.0$,
$p<0.0001$, $\mathrm{MD}_{\text{DL-CosClassAvg}}=0.78$,
$\mathrm{MD}_{\text{DL-FRACSS}}=0.77$, $N=1034$). , whereas DL-FRACSS has
slightly better performance when the top 2 or top 3 candidates are considered.

Figure 12: Accuracy of word recognition (%) on the training set ($N=8507$) and
on the test set ($N=1034$) by DL-CosClassAvg on the left panel and by DL-
FRACSS on the right panel.

Recall that our dataset contains words with multiple pronunciations. The
random selection for inclusion in the held-out dataset of seen-stem plural
words may result in either having no instances of the plural word in the
training set (e.g., both pronunciations recorded for reports occur in the test
set), or having one pronunciation in the training data and another
pronunciation in the test set (e.g., results is trained on /zlts/ and tested
on /izlts/). DL-CosClassAvg recognizes at least one instance of a word in the
test set correctly for 63% of words with multiple pronunciations ($N=155$).
DL-FRACSS performs slightly worse at 46%.

Fig. 13 summarizes model accuracy for the training data. DL-FRACSS is slightly
better at recognizing singulars, whereas DL-CosClassAvg performs slightly
better for plurals with unseen stems.

Figure 13: Recognition accuracy on the training set for the singular words,
the plural words with a corresponding singular in the training set (seen-
stem), and the plural words without a singular in the training set (unseen-
stem) by DL-CosClassAvg shown in dark green bars and by DL-FRACSS in light
green bars.

We also examined the kind of errors made by the DL mappings for the words in
the test data. Overall, DL-FRACSS makes 726 errors in the evaluation of the
test set, and DL-CosClassAvg 501 errors. There are 439 word tokens that both
models fail to predict correctly. We distinguished between three types of
errors, tabulated in Fig. 14. First, many seen-stem plural words of the test
set are recognized as their singular word. FRACSS tends to make more errors of
this sort, for which both models frequently get the plural word as their
second-best guess, and they always find the plural word among their first four
guesses. Highly-ranked competitors tend to be synonyms or semantically related
words.

Figure 14: Different types of errors made by the DL-CosClassAvg ($N=501$) and
the DL-FRACSS ($N=726$) model on the test set with 1034 tokens.

Most of the remaining errors are observed for words with similar forms. To
assess this quantitatively, we computed the recall and the overlap indices
between the set of target triphones $t$ and the set of predicted triphones $p$
as follows:

$\displaystyle\text{recall }(t,p)$ $\displaystyle=\frac{\lvert t\cap
p\rvert}{\lvert p\rvert},$ $\displaystyle\text{overlap }(t,p)$
$\displaystyle=\frac{\lvert t\cap p\rvert}{\min(\lvert t\rvert,\lvert
p\rvert)}.$

For example, the word bribes is recognized as tribes by both models. The
predicted and the target word share many form features with a recall and an
overlap index of 0.6. We classified words as ‘similar sounding’ when the
overlap index was greater than 0.3 and the recall index was greater than 0.2.
The remaining words were assigned to the ‘other’ class. The set of words for
which a similar-sounding error was made by DL-FRACSS is a subset of that of
DL-CosClassAvg. The two models are in error for the same 33 words assigned to
the ‘other’ category.

###  5.2 Discussion

Both CosClassAvg and FRACSS generate high-quality plural vectors. FRACSS
plurals are somewhat better aligned with respect to angle, whereas CosClassAvg
plurals are better positioned in terms of length. We used the DL model to
assess whether FRACSS or CosClassAvg vectors are closer aligned with words’
forms. On training data, both models have very similar performance. On held-
out test data, CosClassAvg is more accurate. We take this as evidence that
plural vectors generated by CosClassAvg are better aligned with the
corresponding plural forms.

##  6 General Discussion

Using distributional semantics, visualization with t-SNE, and WordNet, we have
documented for nearly 15,000 pairs of English singulars and their
corresponding plurals that how plural semantics is realized in semantic space
varies with the semantic class of the base word. Instead of there being one
universal shift from singulars to plurals in distributional space, the
direction and length of shift vectors depends on a lexeme’s own semantics. As
a consequence, shift vectors for fruits are substantially different from shift
vectors for instruments.

We proposed the CosClassAvg model to account for the conceptualization of a
plural given the singular. This model proposes that an empirical plural vector
is the sum of four vectors: the vector of the lexeme, the shift vector
appropriate for its semantic class, a lexeme-specific vector representing the
lexeme’s own lexical properties, and an error vector representing measurement
noise. We showed that CosClassAvg provides more precise approximations of
plural vectors than a model based on a general average shift vector (3CosAvg).

We compared the CosClassAvg model with the FRACSS model (Marelli Baroni,
2015). The FRACSS model also takes the semantic vector of the singular as
input, but makes use of matrix multiplication instead of vector addition to
calculate the semantic vector of the plural. The FRACSS model generates plural
vectors that are closer to the target plural vectors. However, the plural
vectors produced by FRACSS are shorter than the target plural vectors.

To better understand the merits of the two models, we also considered how well
the FRACSS plural vectors and the CosClassAvg vectors are aligned with words’
form vectors. We evaluated the quality of the alignment with the
Discriminative Lexicon model (Baayen ., 2019), focusing on its comprehension
network. We created form vectors by first collecting all possible triphones
and then specified, for a given word, in a high-dimensional binary vector,
which triphones are present (1) in that word and which are absent (0). We
created two mappings from form vectors to semantic vectors, one for semantic
vectors that use FRACSS to generate plural vectors, and a second mapping for
semantic vectors that use CosClassAvg to produce plural vectors. For training
data, both types of vectors allowed highly accurate mappings to be set up.
However, for the held-out test data, plural vectors could be predicted with
substantially higher accuracy when plural vectors were created using
CosClassAvg. This suggests that plural vectors created with CosClassAvg are
better aligned with plurals’ forms compared to vectors generated with FRACSS.

CosClassAvg offers two advantages compared to FRACSS. First, FRACSS models run
the risk of overparameterization, especially for small datasets with numbers
of observations that are substantially smaller than the square of the
dimension of the semantic vectors. Second, the FRACSS matrix operation seems
to suggest that pluralization is a unitary operation, represented by one
transformation matrix. However, what this model is actually doing is to
capture, within one highly-parameterized mapping operation, a wide range of
different ways in which plurals are realized, depending on the semantic class
of their lexemes. For comparison, one can set up a single FRACSS model for all
bi-morphemic suffixed derived words of English, with high accuracy on both
training and test data (see the supplementary materials for further details).
However, the different derivational suffixes of English serve different
semantic goals, which emerge immediately from a t-SNE visualization. Thus,
being able to obtain a high-quality mapping between singulars and plurals does
not guarantee that the same semantic operation is governing all transitions
from singulars to their plurals in semantic space.

This conclusion has important consequences for the principle of semantic
compositionality (Pelletier, 2001) as applied to morphology. According to this
principle, the meaning of a plural is determined by the meaning of the
singular and the meaning of the plural suffix, or the meaning that is realized
by the rule that creates plurals from singulars. As we have seen, a general
shift vector that is the same for all lexemes (as formalized by the 3CosAvg
method) has some value, but fails to have the required precision. FRACSS does
not provide a uniform pluralization operation either, as, thanks to its large
numbers of parameters, it can wrap itself around the many individual clusters
of shift vectors that are characteristic of a large number of specific
semantic classes. It is, of course, possible to adjust the CosClassAvg model

$\bm{v}_{\text{pl}}=\bm{v}_{\text{sg}}+\bm{v}_{\textsc{pl}\mid\text{semantic
class}}+\bm{\epsilon}_{\text{lexeme}}+\bm{\epsilon}$

by subtracting the average plural vector $\bar{\bm{v}}$ from all class-
specific vectors:

$\bm{v}_{\text{pl}}=\bm{v}_{\text{sg}}+\bar{\bm{v}}+[\bm{v}_{\textsc{pl}\mid\text{semantic
class}}-\bar{\bm{v}}]+\bm{\epsilon}_{\text{lexeme}}+\bm{\epsilon}.$

This formulation of CosClassAvg isolates what is common to all plurals.
Unfortunately, this common core is a shift vector that is located far outside
the cluster of actual shift vectors (see Fig. 4), and hence it remains unclear
what is gained by incorporating it into the CosClassAvg model. As a
consequence, it is also unclear in what sense English plurals are
‘compositional’ in the sense of, e.g., Pelletier (1994). At the same time, the
present findings dovetail well with the insight from usage-based grammar and
corpus linguistics that individual words, including inflected words, often
have their own highly specific usage profiles (see, e.g., Sinclair, 1991).

Noun pluralization has been characterized as being rather close to derivation:
Booij (1996) characterizes it as inherent inflection, rather than contextual
inflection. It remains an issue for further research to clarify whether the
present conclusions for nominal pluralization generalize to contextual
inflection. Agreement marking on English simple present verbs makes for an
interesting case to pursue in parallel with the present results on nominal
plurals.

We conclude this study of the semantics of English noun pluralization by
placing English in a broader cross-linguistic perspective. Many languages have
rules that are sensitive to semantic subsets of nouns. Some languages split
nouns into a group for which plurality marking is relevant, and a group for
which it is irrelevant. Typically, such splits are made along an animacy
hierarchy, from kinship nouns at the highest rank, to human nouns, to (higher
and lower) animate nouns, to inanimate nouns at the lowest rank (Corbett,
2000).

In Slave, an Athabaskan language in Northwest Territories, Canada, plural
marking occurs optionally only for human nouns and dogs (Rice, 1989). The
World Atlas of Language Structures documents 60 other languages that have an
optional or obligatory plural marking for human nouns and lack a plural for
nouns further down the animacy hierarchy (Haspelmath, 2013).

In Persian, subject-verb agreement in person and number coded on the verb is
obligatory for animate plural nouns but optional for inanimate ones
(Mahootian, 1997, p. 145). Smith-Stark (1974) reports a similar rule in
Georgian. Maori provides a case where number marking is obligatory only for
kinship nouns such as matua ‘parent’ and teina ‘younger sibling’ (Bauer,
1993).

Kiowa, an endangered Tanoan language spoken in Oklahoma, exhibits a strong
relationship between semantically coherent noun classes and number agreement
behavior. Table 3 summarizes the nine classes distinguished by Harbour (2011,
2008), on the basis of which he argues for a morphosemantic theory of number.
Bantu languages are known for their large numbers of semantically motivated
noun classes (see, e.g., Polomé, 1967, for Swahili).

Table 3: Kiowa noun classes based on Harbour (2011, 2008) Class | Semantic characteristics | Example
---|---|---
1 | First person only | ‘I’
2 | Animates | ‘boy’, ‘bird’
| and independently mobile inanimates | ‘leg’, ‘moon’
3 | Default for vegetation | ‘grass’
| and implements | ‘pencil’
4 | Vegetation forming natural collections | ‘tree’
| and implements that act collectively | ‘ember’
5 | Hair types | ‘eyelash’
| and midsize fruit growing in clusters | ‘tomato’
6 | Individuable objects | ‘river’
7 | Non-granular mass nouns | ‘water’
8 | Pluralia tantum nouns, | ‘trousers’
| composite nouns | ‘necklace’
| and granular mass nouns | ‘rice’
9 | Default | ‘shoe’
|  |

English has in few instances grammaticalized the diverse ways in which our
minds perceive and structure the objects and ideas in the world with which we
interact. For English nouns, the distinction between mass and count nouns
comes to mind. Additionally, a major part of present-day English count nouns
that never or occasionally take the suffix -s in their plural form are animal
nouns that are hunted (e.g., duck, woodcock, and elk) or fished (salmon and
crab) (see Quirk ., 1985, for lexemes other than animal names) (see Toupin,
2015, for an extended list of 85 animal nouns).

We kept the stimuli in the present study simple and consistent by focusing on
regular singular and plural forms. Further research is required that
investigates varieties of plurals including irregular plurals (e.g., man
$\sim$ men), zero plurals (fish $\sim$ fish), pluralia tantum (scissors with
no singular variant), singularia tantum (wealth with no plural variant), or
sense-specific plural formations (mouse $\sim$ mice for rodents and mouse
$\sim$ mouses for computer input devices; Acquaviva, 2008).

Many other languages reflect in their grammars a variety of ways in which
nouns are perceived to cluster semantically. Whereas semantic differentiation
in the nominal system is explicitly grammaticalized in these languages,
semantic noun classes also play a role in the grammar of English, albeit
mainly implicitly. By combining distributional semantics, WordNet, and t-SNE
visualization, we have been able to detect that semantic noun clusters also
structure English language use.

## Declarations

### Funding

This research was funded by the European Research Council under the ERC grant
number 742545, Project WIDE, awarded to the last author. The data processing
of the NewsScape corpus was funded by the Competence Network for Scientific
High Performance Computing in Bavaria to the third author.

### Conflict of interest/Competing interests

No potential conflict of interest was reported by the authors.

### Authors’ contributions

E. S-B was responsible for data collection, analysis, interpretation, and
wrote the first draft of the manuscript. M. M-T contributed to the data
collection, analyses, and interpretation. P. U assembled the auditory
resources, and contributed to data collection, interpretation, and writing. R.
H. B contributed to the research planning, data analysis, interpretation, and
writing.

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