# Adjusting Interpretable Dimensions in Embedding Space
with Human Judgments

Katrin Erk
University of Texas at Austin
<EMAIL_ADDRESS>
&Marianna Apidianaki
University of Pennsylvania
<EMAIL_ADDRESS>

###### Abstract

Embedding spaces contain interpretable dimensions indicating gender, formality
in style, or even object properties. This has been observed multiple times.
Such interpretable dimensions are becoming valuable tools in different areas
of study, from social science to neuroscience. The standard way to compute
these dimensions uses contrasting seed words and computes difference vectors
over them. This is simple but does not always work well. We combine seed-based
vectors with guidance from human ratings of where words fall along a specific
dimension, and evaluate on predicting both object properties like size and
danger, and the stylistic properties of formality and complexity. We obtain
interpretable dimensions with markedly better performance especially in cases
where seed-based dimensions do not work well.

Adjusting Interpretable Dimensions in Embedding Space
with Human Judgments

Katrin Erk University of Texas at Austin<EMAIL_ADDRESS>Marianna
Apidianaki University of Pennsylvania<EMAIL_ADDRESS>

##  1 Introduction

Properties are commonly used in linguistics (Katz and Fodor, 1964; Jackendoff,
1990; Wierzbicka, 1996) as well as in psychology (Murphy, 2002) for
representing word meanings and concepts. Those same properties are
discoverable as interpretable dimensions in word embedding space, and can be
used to explore the patterns and regularities encoded by Large Language Models
(LLMs) (Mikolov et al., 2013b; Bolukbasi et al., 2016). Because LLMs are
trained on texts from many different authors, we can view them as a compact
repository of human utterances. This makes them an interesting resource for
studying linguistic phenomena, analyzing social contexts of words, or as a
stand-in for conceptual knowledge for interpreting brain voxels. Interpretable
dimensions provide an attractive and simple way to access this resource (Grand
et al., 2022; Kozlowski et al., 2019; Garí Soler and Apidianaki, 2020; Lucy et
al., 2022). Compared to probing (Tenney et al., 2019; Conneau et al., 2018),
interpretable dimensions allow for a direct exploration of LLM embedding
space, without external classifiers.

|
---|---
Figure 1: Interpretable dimensions for two object categories and features from
Grand et al. (2022): clothes by wealth (left), animals by size (right). PCA
projection of embeddings, with seed-based (blue) and fit+s (red) dimensions.

The most common way to obtain interpretable dimensions is to specify some seed
pairs of antonyms, and take the average over their vector differences. But it
is unclear what makes good seed pairs, or even how to test whether a
particular property corresponds to a discernible dimension in embedding space.
Antoniak and Mimno (2021) and Lucy et al. (2022) express concerns about the
quality of commonly used hand-curated seed lexicons and propose metrics for
evaluating seeds.

In this paper, we take a different approach to addressing the problem of “bad
seeds” (Antoniak and Mimno, 2021): We propose a method to augment seed-based
interpretable dimensions with additional guidance from human ratings, and we
show that this augmentation is particularly impactful when the original seed-
based dimensions are problematic.111Our code and data are available at
https://github.com/mariannaapi/interpretable-dimensions. Figure 1 shows words
for clothes with dimensions for wealth, and animal names with dimensions for
size, blue for seed-based dimensions and red for our new fitted dimensions.
The fitted dimensions correct overly high wealth estimates for tights and
stockings, and exaggerated size estimates for bee and ant.

While interpretable dimensions are useful both to social science and to
cognitive science, there is an important difference between the fields: In
social science, crowd-sourced datasets cannot be trusted in absolute terms,
because annotators may have social biases of their own; this is the scenario
that Antoniak and Mimno (2021) address. In cognitive science however,
experimental data from human participants is central (though the method used
to solicit data can have an influence on the outcome). We work with data from
cognitive science here, so we can use human ratings to improve seed
dimensions.

Our new method draws inspiration from a completely separate strand of research
on interpretable dimensions, in the context of knowledge graph embeddings
(Derrac and Schockaert, 2015; Jameel and Schockaert, 2016; Bouraoui et al.,
2022). There, interpretable dimensions are learned using labeled training
data. In the current paper, we use a similar learning strategy, and apply it
to a combination of seed-based dimensions and labeled training data. We apply
this technique to predict human ratings on object properties and stylistic
aspects of words, and find that it improves performance particularly in cases
where seed-based dimensions underperform, and that in contrast to seed-based
dimensions it is able to make predictions at the same scale as the original
ratings.

A larger issue is: When can we trust that an interpretable dimension shows us
what the LLM truly “knows” about the property in question, that we are not
misled by noise in our tool? This same worry also arises for probing
classifiers (Hewitt and Liang, 2019; Belinkov, 2022).222There is a related
question of whether information encoded in the model is also relevant for
downstream performance (Ravichander et al., 2021; Lyu et al., 2024). This
question is not so central for linguists or cognitive scientists interested in
knowledge reflected in an LLM. What is central for them is that the dimension
really encodes the property of interest. We take one step towards addressing
this issue: By combining two sources of information, seeds and human
annotation, we hope to reduce noise present in either source.

##  2 Related Work

Interpretable dimensions in word embedding space have first been observed in
NLP (Bolukbasi et al., 2016), and the idea was then taken up in neuroscience
and social science. Grand et al. (2022) discover dimensions for objects’
scalar properties (e.g., danger, size, speed, wealth). Kozlowski et al. (2019)
identify dimensions including affluence, gender, race and morality, and show
that concepts such as sports (e.g., golf, boxing) and music genres (e.g.,
opera, rap, jazz) are ordered along these axes in ways that match cultural
stereotypes. Garg et al. (2018) explore ethnic stereotypes, and Stoltz and
Taylor (2021) go as far as to propose a cultural cartography with word
embeddings. An et al. (2018) use a large number of dimensions to characterize
sentiment, which Kwak et al. (2021) apply to whole documents. Interpretable
dimensions have also been used to represent linguistic notions, such as
complexity and scalar adjective intensity (Garí Soler and Apidianaki, 2020,
2021b; Lyu et al., 2023). In our work, we explore dimensions addressing object
properties in the Grand et al. (2022) datasets, and the abstract notions of
formality and complexity.

In all these studies, dimensions are discovered using the seed-based
methodology, where a few seed pairs of antonyms are specified and the
dimension is computed as the average over vector differences for these pairs.
This method is simpler than alternative representation approaches (e.g., the
multi-task learning framework of Allaway and McKeown (2021)).

Seed pair selection has until now been ad hoc; but some choices, such as the
selected word pairs, their number and order, and the way they are combined,
have a strong impact on the quality of the derived dimension. Antoniak and
Mimno (2021) address the “bad seeds” problem by measuring the coherence of
each seed set pairing after mapping to the bias subspace: When all words in
the vocabulary are projected onto the subspace, the two seed sets must be
drawn as far apart as possible. Lucy et al. (2022) propose to measure the
semantic axis’ self-consistency using a leave-one-out approach, where each
seed is compared to an axis constructed from the remaining seeds. A good seed,
when left out, should be closer to the pole it belongs to.

In our approach, we do not test for seed quality. Instead, we use human
ratings to improve on seed-based dimensions. Our approach is inspired by work
on knowledge graph embeddings (Derrac and Schockaert, 2015; Jameel and
Schockaert, 2016; Bouraoui et al., 2020). Drawing on the conceptual spaces of
Gärdenfors (2014) for intuition, Jameel and Schockaert (2016) learn embeddings
of knowledge graph nodes that include interpretable dimensions for properties.
Like us, they learn interpretable dimensions using labeled training data. Our
objective function is an adaptation of their objective function, but still
different as they also learn the space while we have a fixed space.

For constructing interpretable dimensions, most previous work used static
embeddings (GloVe (Pennington et al., 2014) and word2vec (Mikolov et al.,
2013a)). Recent work extends the methodology to contextualized representations
(Garí Soler and Apidianaki, 2020; Lucy et al., 2022). We experiment with both
kinds of embeddings.

##  3 Methods

###  3.1 Models

#### Seed-based dimensions (seed model).

The seed-based method is the most commonly used for computing interpretable
dimensions (Bolukbasi et al., 2016; Kozlowski et al., 2019; Dev and Phillips,
2019; Garí Soler and Apidianaki, 2021b; Grand et al., 2022). A group of _seed
words_ are chosen which represent opposite ends of the dimension. For the
danger dimension in Grand et al. (2022), for example, the seeds are {safe,
harmless, calm} for the positive side and {dangerous, deadly, threatening} for
the negative side. For each pair of a positive and negative seed word $p,n$
with vectors $\vec{p},\vec{n}$, the difference vector $\vec{p}-\vec{n}$ is
computed; this is a first estimate of the interpretable dimension, but the
vectors can differ substantially across seed pairs. To obtain a more stable
estimate, the vector for the interpretable dimension is then computed as the
average of the difference vectors from individual seed pairs. The rating of
any word $a$ on the property $d$ with interpretable dimension $\vec{d}$ – in
our example from above, danger – is then predicted as the scalar projection
onto the dimension:

$||\text{proj}_{\vec{a}}(\vec{d})||=\frac{\vec{a}\cdot\vec{d}}{||\vec{d}||}$

#### Fitted dimensions (fit model).

Whenever we have gold ratings on some dimension, like human judgments on
degrees of danger of different animals (Grand et al., 2022) or gold ratings
for complexity (Lyu et al., 2023), we can estimate a direction in embedding
space that best matches the gold ratings. We adapted an idea from Jameel and
Schockaert (2016), who learn an embedding space for knowledge graph nodes in
such a way that properties of the nodes correspond to dimensions in space. But
rather than learning a new space, we need to use an existing space spanned by
static or contextualized embeddings, because it is these spaces, and the
patterns in human language use that they encode, that we want to analyze.

We proceed as follows. Let $W=\langle w_{1},\ldots w_{n}\rangle$ be an
annotated dataset of $n$ words with real-valued gold ratings
$\hat{Y}=\langle\hat{y}_{1},\ldots,\hat{y}_{n}\rangle$ for some feature $f$.
Let $\vec{w}_{i}$ be the embedding of word $w_{i}$. For the dimension
$\vec{f}$ to be computed for feature $f$ in that same embedding space, we
stipulate that the scalar projection of $\vec{w_{i}}$ onto $\vec{f}$ be
proportional to the gold rating $\hat{y}_{i}$. For example, say the gold
rating (average human rating) of dolphin on the danger scale (on a scale of
1-5) is 2.1, and the gold rating of tiger is 4.9. Then we want the length of
the projection $\text{proj}_{\vec{\text{dolphin}}}(\vec{\text{{\sc danger}}})$
to be proportional to $c_{\text{{\sc danger}}}\cdot 2.1+b_{\text{{\sc
danger}}}$, and the length of the projection
$\text{proj}_{\vec{\text{tiger}}}(\vec{\text{{\sc danger}}})$ to be
proportional to $c_{\text{{\sc danger}}}\cdot 4.9+b_{\text{{\sc danger}}}$,
for some weight and bias constants $c_{\text{{\sc danger}}},b_{\text{{\sc
danger}}}\in\mathbb{R}$. So in general, we would like to have

$\frac{\vec{w_{i}}\cdot\vec{f}}{||\vec{f}||}=c_{f}\>\hat{y}_{i}+b_{f}$

We turn this into a loss function for computing a _fitted dimension_ $f$,
dropping the denominator $||\vec{f}||$:

$J_{f}=\sum_{w_{i}}\big{(}\vec{w_{i}}\cdot\vec{f}-c_{f}\>\hat{y}_{i}-b_{f}\big{)}^{2}$

#### Fitted dimensions with seed words (fit+sw model).

We also test whether fitted dimensions can be guided by the seed words used to
make seed-based dimensions. The first method follows the intuition of Antoniak
and Mimno (2021) that the scalar projections of seed words should sit “far
out” on an interpretable dimension, further than other words. The fit+sw model
simply extends the collection $W$ of rated words by the seed words. We make
synthetic gold ratings for the seedwords, giving them extreme ratings:
$\max(\hat{Y})+o$ for positive seed words, and $\min(\hat{Y})-o$ for negative
seedwords, for an offset $o$ that is a hyperparameter. We optionally add a
small amount of random jitter (sampled from the interval $[0.001,0.005]$) so
that the seed words don’t all have the same rating.

#### Fitted dimensions with seed dimensions (fit+sd model).

Our second way of extending fitted dimensions with seed word information is
built on the idea that seed-based dimensions and human ratings both provide
useful information for fitting an interpretable dimension, and that they
should be combined. So we use an overall loss function of

$J=\alpha J_{f}+(1-\alpha)J_{d}(D)$

where $J_{f}$ is the loss function from above, and $\alpha$ is a
hyperparameter. $J_{d}(D)$ is a loss that measures distance of the fitted
dimension $\vec{f}$ from a set $D$ of seed-based dimensions, defined as

$J_{d}(D)=\sum_{d\in D}1-cosine(\vec{d},\vec{f})$

#### Fitted dimensions with seeds as words and dimensions (fit+s model).

Our final model uses seeds both as seed words, as in fit+sw, and as seed
dimensions, as in fit+sd.

#### Baselines.

We compare our methods to a baseline which ranks words by frequency (FREQ).
Frequency has been a strong baseline for complexity and formality in previous
work, given that rare words tend to be more complex than frequently used words
(Brooke et al., 2010). We use log-transformed frequency counts in the Google
N-gram corpus (Brants and Franz, 2006). We also use a random baseline, which
assigns to each word a randomly selected score in the range [-3, 3].333This
range was chosen because all gold ratings in our study are z-scores.

###  3.2 Evaluation metrics

In contrast to interpretable dimensions computed from seed words, the fit
models use training data: words with human ratings for the property in
question. When we evaluate these models, we use up some of the data for
training, leaving less for testing. To mitigate the issue, we do cross-
validation, and we focus on evaluation metrics that work well with smaller
test sets. We do not use the correlation metrics used in Garí Soler and
Apidianaki (2020) and Grand et al. (2022), as significance tests become
unreliable with small datasets. Instead, we use a variant of pairwise rank
accuracy, a metric used in Garí Soler and Apidianaki (2020), Cocos et al.
(2018), and Grand et al. (2022).

Pairwise rank accuracy measures the percentage of pairs of words whose
ordering in the gold ratings is the same as in the model predictions. We
define a new variant which we call extended pairwise rank accuracy, r+-acc,
which measures pairwise rank accuracy among words in the test set, and
additionally pairwise rank accuracy between each test word and each training
word. For example, if tiger and butterfly are in the training set for danger,
and cat is in the test, we check whether the score assigned to cat ranks it
after tiger and before butterfly. This metric gives us more evidence on the
quality of predictions than pairwise rank accuracy on its own because it
includes more comparisons, thus making the metric less sparse. Let
$<_{g},<_{m}$ be two complete orderings of the words in $W$, the gold and
model orderings, respectively. For words $w,w^{\prime}$ in $W$, we define an
auxiliary function rm for “rank match”:

$rm_{<_{g},<_{m}}(w,w^{\prime})=\left\\{\begin{array}[]{l@{}l}1&\text{ iff
}(w<_{g}w^{\prime}\wedge w<_{m}w^{\prime})\\\ &~{}\vee(w>_{g}w^{\prime}\wedge
w>_{m}w^{\prime})\\\ 0&\text{ else}\end{array}\right.$

Then standard pairwise rank accuracy on $W=\langle w_{1},\ldots,w_{n}\rangle$
is defined as

$\begin{array}[]{@{}l}\text{r-acc}_{W}(<_{g},<_{m})=\frac{1}{n(n-1)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{1\leq i<j\leq
n}rm_{<_{g},<_{m}}(w_{i},w_{j})\end{array}$

Now assume that $T=\\{k_{1},\ldots,k_{\ell}\\}$, with
$k_{j}\in\\{1,\ldots,n\\}$ for all $j$, is the set of test word indices among
the indices of $W$. Assume both orderings, $<_{g}$ and $<_{m}$, are defined on
all of $W$. Then our new extended pairwise rank accuracy is

$\begin{array}[]{@{}l}\text{r${}^{+}$-acc}_{W}(<_{g},<_{m})=\frac{1}{\ell(\ell-1)+\ell(n-\ell)}\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{1\leq
i<j\leq\ell}rm_{<_{g},<_{m}}(w_{k_{i}},w_{k_{j}})+\\\
~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\sum_{i\in\\{1,\ldots,\ell\\},j\not\in
T}rm_{<_{g},<_{m}}(w_{k_{i}},w_{j})\end{array}$

The first half of this formula measures pairwise rank accuracy among members
of the test set; the second half measures rank accuracy of test words with
respect to training words.

Pairwise rank accuracy and extended pairwise rank accuracy are similar to
correlation metrics in that they measure to what extent gold and model-based
rankings agree. And in fact all three metrics are highly correlated: We tested
correlation between the three metrics on seed-based dimensions for the Grand
et al. (2022) data and obtained highly significant correlations ($p\ll
0.0001,r=0.972$ for pairwise rank accuracy, $r=0.971$ for extended pairwise
rank accuracy).444To measure extended pairwise rank accuracy, the data was
split into training and test folds in 5-fold cross-validation, and r+-acc
scores were averaged over folds. It is not surprising that the values are
almost the same, as in this case extended pairwise rank accuracy is almost the
same as standard pairwise rank accuracy except that the latter omits some
pairwise comparisons, namely between training data points.

As a second evaluation metric, we test how far off from the gold ratings each
individual predicted rating is. We use the mean square error (MSE) of
predicted ratings compared to gold ratings. We can do this because all fit
models learn to predict ratings on the same scale as the gold ratings. In
order to apply the same evaluation to the seed model and the baselines, we
simply use linear regression to map model predictions to ratings on the same
scale as the gold ratings. Linear regression models are fit on the training
portion of each data set, so that test words remain truly unseen.

###  3.3 Data and Vectors

We use the ratings collected by Grand et al. (2022) which describe properties
of objects in nine categories:555The data is available on the Open Science
Framework at https://osf.io/5r2sz/. No license information is given in the
repository. animals, clothing, professions, weather phenomena, sports,
mythological creatures, world cities, states of the United States, and first
names.666Most categories consist of 50 items. Each category is matched with a
subset of these semantic features: age, arousal, cost, danger, gender,
intelligence, location (indoors vs. outdoors), partisanship (liberal vs.
conservative), religiosity, size, speed, temperature, valence, volume, wealth,
weight, and wetness. For style, we use datasets released by Pavlick and
Nenkova (2015) which contain words and phrases with human ratings of formality
and complexity. For each dimension, we sample words777We sample words with
more than three characters to exclude pronouns, articles, numerals, and
multiword phrases. with high annotation confidence (i.e. where annotators
agreed about the word being complex or formal): We calculate the mean standard
deviation for words in our sample, and keep words where deviation between
human scores is lower than that mean. The filtered datasets contain 1,160
words for complexity, and 1,274 words for formality.

We extract seed words from two other datasets released by Pavlick and Nenkova
(2015) which contain pairwise paraphrase judgments of formality and
complexity.888The data is available at
https://cs.brown.edu/people/epavlick/data.html, under “Style Lexicons: Human
and automatic scores of formality and complexity for words, phrases, and
sentences”. No license for the data is given. Annotations reflect which phrase
in a pair (e.g., letter-communication, largely-extensively) is more complex or
formal than the other. We collect five pairs of words for each style dimension
for which inter-rater agreement is high. For complexity, we obtain the seed
pairs work - employment, further - subsequently, strong - powerful, train -
railway, shown - indicated, where the first member of each pair is the
negative seed (the simpler word). For formality, we used winner - recipient,
terrible - disastrous, membership - affiliation, highest - paramount, test -
verify, where again the first member of each pair is the negative seed (the
less formal word).

Following Grand et al. (2022), we averaged over human subject ratings for each
datapoint, then normalized ratings to z-scores separately for each pair of a
category and property.999We would expect the data to show subjective
differences between annotators, and in the future we would like to model the
subjective ratings directly, following Plank et al. (2014). For formality and
complexity, ratings were also converted to z-scores.

As embeddings, we use the same representations as Grand et al. (2022), pre-
trained GLoVE embeddings trained on the Common Crawl (42b tokens), 300
dimensions, uncased (Pennington et al., 2014). We also use contextualized
representations from the BERT (bert-large-uncased) and RoBERTa (roberta-large)
models (Devlin et al., 2019; Liu et al., 2019) with sentences from UkWac
(Baroni et al., 2009).101010Details about sentence selection are given in the
Appendix. For each word instance, we average its contextualized
representations from the top 4 layers of the model.111111 Averaging across
layer subsets is generally better than averaging across all layers or
selecting a single layer (Vulić et al., 2020). If the word is split into
multiple wordpieces during tokenization, we average the representations of its
pieces in order to obtain a single type-level representation for each word, as
is common practice in semantic probing studies (Bommasani et al., 2020; Vulić
et al., 2020; Garí Soler and Apidianaki, 2021a). The final representation for
a word is the average of its representations from the available sentences.
Aggregating representations across multiple contexts is the most common
approach for creating word type-level embeddings from contextualized
representations which serves to null out, to some extent, the impact of
specific contexts (Apidianaki, 2022). It is possible to use more sophisticated
ways for sentence selection, such as language modeling criteria (Garí Soler
and Apidianaki, 2020) and exclusion of contexts where antonyms could occur
(Lucy et al., 2022). However applying such sophisticated context selection
methods is not always better than random selection, which might be due to the
skewed distribution of word senses and the stronger presence of the most
frequent sense of a word in randomly selected sentences (Kilgarriff, 2004;
McCarthy et al., 2004).

##  4 Results and discussion

In this section we evaluate the different interpretable dimension models from
Section 3.1 on the tasks of predicting human ratings of object properties
(Grand et al., 2022), and human ratings of the complexity and formality of
words (Pavlick and Nenkova, 2015).

#### Experimental setup.

To make the most of the limited available data, all models were tested in
5-fold cross-validation. In addition, all models that involve randomness (all
except seed) were re-run three times with different random seeds. For Grand et
al. object features, we first compute mean r+-acc and median MSE for each
category/property pair (averaging over cross-validation runs and random
seeds), then we report averages over those. For formality and complexity we
report overall mean r+-acc and median MSE.121212We use median MSE because
outliers make the mean uninformatively high. Note that because we split the
data into training and test using cross-validation, the numbers reported in
this paper are not comparable with those reported in earlier papers on the
same dataset. We do however compute seed dimensions with the same cross-
validation setup as the fit-based dimensions, so that the numbers that we
report are comparable to each other.

To set hyperparameters, we sample 6 category/property pairs from the Grand et
al. data as development set. Hyperparameters were optimized once per embedding
space; there was no separate hyperparameter optimization for the formality and
complexity data.131313Human ratings on all datasets were on the same scales as
we normalized them all to z-scores. Overall, we find that low values of
$\alpha$ work well, and that it is beneficial to input a single averaged seed
dimension to fit+sd and fit+s, rather than individual seed dimensions. The
choice of offset and jitter does not matter. More details on hyperparameters
and the development set can be found in the Appendix. Results reported below
for Grand et al. data are for all category/property pairs not in the
development set.

|  | seed | fit | fit+sw | fit+sd | fit+s |
---|---|---|---|---|---|---|---
GLoVE | r+-acc | 0.64 | (0.1) | 0.54 | (0.03) | 0.53 | (0.03) | 0.65 | (0.1) | 0.80 | (0.06) |  freq r+-acc: 0.58 rand r+-acc: 0.49
MSE |  $>1000$ | ($>1000$) | 113.2 | (111.7) | 177.1 | (125.4) | 89.6 | (199.6) | 0.7 | (0.36)
BERT | r+-acc | 0.64 | (0.1) | 0.51 | (0.03) | 0.52 | (0.03) | 0.66 | (0.10) | 0.71 | (0.04)
MSE |  $>1000$ | ($>1000$) | 417.4 | (271.7) | 597.4 | (525.6) | 115.0 | (437.4) | 2.0 | (0.6)
RoBERTa | r+-acc | 0.57 | (0.08) | 0.51 | (0.03) | 0.51 | (0.03) | 0.60 | (0.1) | 0.69 | (0.04)
MSE |  $>1000$ | ($>1000$) | 392.5 | (291.3) | 458.0 | (284.3) | 125.2 | (270.5) | 1.9 | (0.6) |
Table 1:  Results on object properties: Extended rank accuracy (abbreviated
r+-acc) and Mean Squared Error (MSE), averaged over category/property pairs.
In brackets: Standard error. Shown for 3 embedding spaces. Bolded: best
performance for each embedding.

|  | Complexity | Formality
---|---|---|---
|  |  6mm fit |  7mm fit+sd |  seed |  1cm fit+sw |  6mm fit+s |  |  |  |  |  |  |
GLoVE |  r+-acc |  0.74 |  0.59 |  0.57 |  0.76 |  0.72 |  freq r+-acc: 0.65 rand r+-acc: 0.50 |  0.73 |  0.53 |  0.37 |  0.68 |  0.69 |  freq r+-acc: 0.63 rand r+-acc: 0.51
MSE |  31.5 |  24.3 |  75.1 |  1.5 |  1.2 |  60.5 |  396.5 |  285.7 |  1.8 |  1.6
BERT |  r+-acc |  0.69 |  0.52 |  0.52 |  0.71 |  0.72 |  |  0.64 |  0.52 |  0.51 |  0.64 |  0.69
MSE |  123.5 |  437.2 |  724.0 |  3.6 |  2.4 |  |  215.6 |  216.3 |  617.1 |  7.8 |  3.2
RoBERTa |  r+-acc |  0.74 |  0.51 |  0.51 |  0.71 |  0.73 |  |  0.67 |  0.53 |  0.53 |  0.66 |  0.71
MSE |  82.7 |  591.9 |  $>1000$ |  3.9 |  2.3 |  |  223.0 |  325.0 |  778.1 |  7.3 |  2.4 |

Table 2: Results on formality and complexity: Extended rank accuracy (r+-acc)
and Mean Squared Error (MSE). Shown for 3 embedding spaces. Bolded: best
performance for each embedding.

#### Overall performance.

Overall results are shown for object properties in Table 1 and for stylistic
features in Table 4. Looking at object properties first, and focusing on
extended rank accuracy, the fit model by itself is not very good, and adding
seeds as words (fit+sw) does not help. fit+sd is better, and outperforms seed
slightly, but the fit+s model, which computes fitted dimensions using seed
information both as words and dimensions, shows the best performance,
outperforming seed strongly. In terms of MSE, even medians are very high for
the seed model, so many seed-based dimensions were not able to predict ratings
on the same scale as the gold ratings. MSE is much lower for both fitted
models that make use of seed dimensions, especially fit+s.141414Though note
that the ratings are z-scores, so the MSE is still larger than half a standard
deviation. Looking at the baselines, the fit and fit+sw models with BERT and
RoBERTa underperform the frequency baseline, and are on par with random
guessing. The frequency baseline is somewhat higher than random, though it is
not entirely clear what kind of signal for object properties can be derived
from word frequency.

On the stylistic data, the relative performance of the fitted models is
similar, but here they mostly do not outperform the seed dimensions. Overall
performance of the seed dimensions is higher here, which raises the question
if fitted models help in particular when seed-based dimensions do not perform
well; we explore this further below. Looking at MSE, we confirm that fitted
models that use seed dimensions have much lower error than the other models.
Comparing embedding spaces, we see consistently the best performance with
GLoVE embeddings. The BERT and RoBERTa fit and fit+sw models in particular are
again at the level of the random baseline. The frequency baseline is
reasonably strong, matching previous findings.

#### Fitted dimensions, by themselves, are underdetermined by human ratings.

The fit model, which computes fitted dimensions from human ratings only, does
not perform well, and we suspect that the size of the embedding space allows
for too many ways to fit a dimension to ratings, causing the model to overfit.
To test this, we first computed fit dimensions, for Grand et al. object
features, from _all_ human ratings, obtaining perfectly fit dimensions in
every single case. We next train dimensions on all human ratings but scramble
the word/rating pairs, making them nonsensical. Again we obtain perfectly fit
dimensions in every single case, which confirms our suspicion. The picture
that emerges is that fit by itself does not have enough information to fit a
good dimension and overfits to the training data. The seed information
provided to fit+sw, fit+sd and fit+s gives the models the additional guidance
needed to make good use of the human ratings, and the combination of seeds and
human ratings on words leads to overall better dimensions – at least in some
cases. We next ask which cases those are.

Figure 2: Increase in r+-acc for fit+s over seed, object properties grouped by
performance of seed.

#### Human ratings help most when seed-based dimensions underperform.

Comparing r+-acc values for seed-based dimensions and fit+s dimensions on the
object property data, we find that fit+s improves over seed in every single
one of the 50 category/property pairs.151515The analysis includes all
conditions not in the development set. For each condition, we again ran 5-fold
cross-validation, each repeated over 3 random seeds, then averaged by
condition. The performance increase is highest when performance of the seed-
based dimensions is lowest, as shown in Figure 2: For the 20% of
category/property pairs with lowest seed performance, average improvement is
27.3 points, while for the 20% of category/property pairs with the highest
seed performance, average improvement is 1.7 points. This could explain the
lack of improvement achieved on stylistic features, as seed already performs
well on this data.

Table 3 further zooms in on the object feature data, showing performance on
some category/property pairs with low, medium, and high performance of the
seed dimensions. We see that fit and fit+sw underperform throughout. fit+s
shows the overall best performance, but the improvement over seed is
particularly high for the first group of conditions, where seed dimensions get
no traction on the data. fit+sd shows good extended rank accuracy on the
conditions with medium to high seed performance, but not on the conditions
that are particularly poorly modeled by seed.

| r+-acc
---|---
Category, Feature | seed | fit | fit+sw | fit+sd | fit+s
sports/speed | 0.46 | 0.55 | 0.56 | 0.52 | 0.78
states/cost | 0.46 | 0.5 | 0.5 | 0.42 | 0.83
cities/arousal | 0.47 | 0.52 | 0.5 | 0.51 | 0.82
animals/intelligence | 0.48 | 0.55 | 0.54 | 0.5 | 0.79
clothing/cost | 0.48 | 0.52 | 0.51 | 0.55 | 0.76
clothing/wealth | 0.62 | 0.52 | 0.53 | 0.6 | 0.82
states/wealth | 0.62 | 0.55 | 0.56 | 0.64 | 0.82
weather/temperature | 0.69 | 0.56 | 0.52 | 0.66 | 0.76
animals/danger | 0.7 | 0.6 | 0.57 | 0.76 | 0.84
clothing/age | 0.71 | 0.52 | 0.55 | 0.71 | 0.8
weather/danger | 0.79 | 0.55 | 0.54 | 0.7 | 0.82
clothing/gender | 0.81 | 0.56 | 0.53 | 0.8 | 0.82
sports/gender | 0.81 | 0.61 | 0.56 | 0.81 | 0.84
professions/gender | 0.85 | 0.56 | 0.56 | 0.87 | 0.86
names/gender | 0.87 | 0.56 | 0.51 | 0.87 | 0.87

Table 3: Detail results for Grand et al. by seed performance: lowest
performance (top box), middling performance (middle), best performance
(bottom). Figure 3: MSE distributions for runs of different models. y-axis:
count of runs.

####  fit+s models are the only ones that predict ratings on the gold scale.

We saw above that median MSE values are extremely high for many models,
especially for seed. We now take a closer look, in particular we want to know
how often we obtain MSE values that are extremely far off from the gold
ratings. We again focus on the object feature data as we there have many
conditions that we can compare. Figure 3 shows, for each model, how many runs
had MSE values in the ranges of $<2,2-10,10-100$, and $>100$. Recall that gold
ratings are z-scores, so they tend to be in a range of -2 to 2. We again only
use the category/property pairs that are not in the development set, but now
count separately each cross-validation run and each random seed. We see that
many runs of seed, fit and fit+sw have very high MSE values. In fit+sd we
first see a considerable percentage of runs with MSE values below 2 (the blue
bar comprises 20% of runs for this model), but strikingly, 97% runs of fit+s
have MSE values below 2, and all have values below 10. So this model is much
more consistent than the other models, and in fact is highly consistent in
fitting dimensions that deliver predictions in the range of the gold data.

#### Zooming in: Examples of predictions.

|
---|---
Figure 4: Clothes rated for wealth (left), animals rated for size (right). Gold ratings: dark purple. seed predictions: red. fit+s prediction: light blue. Datapoints ordered by gold rank. Gold | seed | fit+s
---|---|---
bee | chipmunk | butterfly
ant | hamster | bee
butterfly | monkey | chipmunk
goldfish | butterfly | bird
hamster | goldfish | hamster
chipmunk | dog | ant
bird | bee | crow
crow | bird | goldfish
dog | seal | seal
monkey | crow | dog
seal | ant | monkey
mammoth | whale | whale
whale | mammoth | mammoth
Table 4: Comparing word rankings by humans, seed dimensions, and fit+s dimensions: Animals by size. Italicized: 3 words with highest error in ranking. Gold | seed | fit+s
---|---|---
sweatshirt | shorts | shorts
shorts | sweatshirt | boots
belt | belt | bikini
boots | blouse | tights
hat | boots | skirt
tights | swimsuit | swimsuit
bikini | skirt | stockings
swimsuit | trousers | trousers
skirt | bikini | loafers
blouse | robe | blouse
knickers | cuff | belt
dress | knickers | knickers
collar | hat | dress
trousers | collar | sweatshirt
stockings | vest | robe
robe | tights | collar
vest | loafers | cuff
cuff | stockings | vest
loafers | dress | hat
gown | gown | tuxedo
tuxedo | tuxedo | gown
Table 5: Comparing word rankings by humans, seed dimensions, and fit+s
dimensions: Clothes by wealth. Italicized: 3 words with highest error in
ranking.

We take a closer look at two kinds of object properties: clothes by wealth,
and animals by size. In order to obtain sufficiently many test datapoints to
look at, we divide the data into 2/3 training and 1/3 test (as opposed to the
1/5 we use with 5-fold cross-validation). Figure 1 shows the test data words,
along with seed-based and fit+s dimensions, downprojected into 2 dimensions
using PCA. For the same datapoints, Figure 4 plots gold ratings, seed
predictions, and fit+s predictions. This plot illustrates how the seed
predictions are on a much larger scale than gold ratings, while fit+s is the
only model whose predictions stay on the same scale. (The next to last
datapoint in animals/size is mammoth, which seed largely overestimates – maybe
because mammoth is also an adjective indicating gargantuan size.) Tables 4 and
5 show how the test data words for animals by size, and for clothes by wealth,
are ranked by humans, by the seed dimension, and by the fit+s dimension. The
italicized words are the three words whose model rank is furthest off from
their gold rank. For the animals data, both models mis-rank ant, and overall
seem to struggle more with smaller animals. Among the clothes, both models
overestimate the wealth projected by wearing hats.

##  5 Conclusion

In this paper we have proposed a method for constructing high quality
interpretable dimensions in embedding spaces. We show that by combining seed-
based vectors with guidance from human ratings about properties, it is
possible to induce better dimensions than with the seed-based methodology
alone. We expect the proposed dimensions to be useful in various areas of
study, including linguistics, psychology, and social science.

For the moment, the proposed dimensions address one property at a time. In
future work, we are planning to explore multifaceted properties which would be
better represented through multiple dimensions. Aside from a more elaborate
description of these properties, a space of multiple interpretable dimensions
will offer a rich context of comparison for words that might be similar in
some respect and not in others (e.g., tiger and spider with respect to danger
and size).

##  6 Limitations

In our experiments we use English models and data. The seed-based methodology
has been shown to work well in other languages, so an extension of the
proposed methodology to other languages is possible. A limitation regarding
this extension is the lack of human ratings which are needed for calculating
the fitted dimensions. A possible mitigation would be to translate the
annotated English data into other languages.

The ratings we used in our study were averages over individual human ratings,
possibly obscuring legitimate differences between raters (Plank et al., 2014).
Another limitation of the human ratings used in this study is that they were
out of context, possibly obscuring effects of topic and polysemy.

There are many different ways to use contextualized embeddings. We have
averaged over all token representations generated by BERT and RoBERTa for a
word in a sentence pool, and used the top 4 layers of the models. It is
possible that BERT and RoBERTa would do better, or at least differently, if
other model layers (or layer combinations) were used.

Our approach is not at all compute intensive. All computations were done on a
laptop.

## Acknowledgements

We would like to thank the anonymous reviewers for their valuable feedback.
This research is supported in part by the Office of the Director of National
Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA),
via the HIATUS Program contract #2022-22072200005. The views and conclusions
contained herein are those of the authors and should not be interpreted as
necessarily representing the official policies, either expressed or implied,
of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to
reproduce and distribute reprints for governmental purposes notwithstanding
any copyright annotation therein.

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##  Appendix A Appendix

#### Details on computing.

All experiments were conducted on a MacBook Pro laptop using Python 3.8, with
huggingface version 4.35.2, torch version 2.0.1, sklearn version 1.2.1, numpy
version 1.22.4 and scipy 1.10.0.

#### Hyperparameter estimation.

Development set: 6 conditions sampled at random from the object features
dataset: cities-danger, states-political, animals-wetness, cities-
intelligence, animals-weight, names-age.

As said above, only hyperparameters that made a difference: averaging, always
good, and mixing parameter alpha. Chosen values:

Best parameters:

  * •

$\alpha$ for fit+sd: GLoVE 0.02

  * •

$\alpha$ for fit+s: GLoVE 0.05

#### Embedding spaces, and sentence selection.

The GLoVe embeddings used were trained on Common Crawl (42B tokens, 1.9M
vocab, uncased, 300d vectors), downloaded from
https://nlp.stanford.edu/projects/glove/.

In order to generate embeddings for contextualized instances of words in our
datasets using BERT (bert-large-uncased) and RoBERTa (roberta-large) models
(Devlin et al., 2019; Liu et al., 2019), we used sentences from the UkWac
corpus (Baroni et al., 2009). We collected ten sentences for each word, when
available. We filtered out sentences with more than 100 tokens in order to
avoid including noisy contexts such as webpage menus crawled from the web. If
a word had less than 10 occurrences in UkWac, we used as many sentences as
were available. This was the case for 10 words in the Grand et al. dataset
(nairobi (6), seoul (4), taipei (5), lahore (2), baghdad (9), peyton (9),
tehran (4), johannesburg (4), jaime (5), karachi (7); and for one word
(jazeera (6)) in the formality dataset. For hyphenated words in the Grand et
al. dataset (e.g., new-york, south-carolina, south-dakota), we collected
sentences where they occur without the hyphen.