Looking for Connections in All the Wrong Places

Project description

NYT Games Connections is a daily puzzle that presents the user with 16 items—generally individual words or short phrases—on 16 "cards" that the user is challenged to group into four categories of four cards according to unspecified (until after solving) commonalities within each category. Most commonly, said commonalities are semantic, often specifically words that appear within some given context. Because the words are often deliberately chosen for their ambiguity, word-disambiguation is a core part of the puzzle.

Meanwhile, Thurnbauer et al. 2023 present a method of unsupervised word-disambiguation by means of density-based clustering (DBSCAN) based on a given target word's context in a corpus.

I have chosen to adapt their approach to word-disambiguation to the task of the Connections solver; that is, exploring their approach as a means to automatically generate a solution to any given Connections puzzle. This is a continuation of an earlier attempt to achieve the same. The first attempt, which simply used a cosine comparison of pre-trained GloVe embeddings, was ineffective. That ineffectiveness is not entirely surprising, because such word embeddings encoded all meanings of a word, which is problematic when a single meaning is the key to a solution.

Because part of the spirit of this project is a re-creation of previous work, it can't be said to be entirely novel; however, the adaptation of that work toward my particular application of course is.

Summary of individual contributions

Approach

While I was able to obtain the code used by Thurnbauer et al. for their published paper, I found that it relied heavily on a pre-existing model, i.e. that which they trained for their own use, and various external files. Indeed, I had no expectation to use their particular model (not only because it used the German wikipedia), this left me to devise the preprocessing and training.

I largely followed what Thurnbauer et al. described in their paper. That is, I also used Wikipedia as a corpus (albeit the English one). Similarly, I followed their lead in using Spacy for preprocessing and CBOW with a window of 5 for the model.

Then, for each word in a given test puzzle, a vector for every context therof was created by averaging the trained embedding for each word in the individual context.

The set of context vectors were then fit to a DBSCAN model—that is, one DBSCAN for the set of one word's context vectors. Roughly following Thurnbauer et al., the parameters used for DBSCAN were an epsilon value of .2 and a minimum count of 10.
The DBSCAN result is then a set of clustered vectors, meant to correspond to a distinct or polysemic sense of a word.

As a first pass of comparing these clusters, I chose to represent the cluster with the centroid of its members, then ran a cosine comparison of each of a word's clusters against all clusters of other words.

Results and Errors

As it stands now, the results are imperfect, but may show some promise. consider the word set below from Connections No. 857 from 2025-10-15:

1. Types of pools: infinity | kiddie | kidney | olympic
2. NATO phonetic alphabet: bravo | delta | golf | lima
3. Ford models: bronco | fiesta | mustang | pinto
4. ___ seal: elephant | great | navy | vacuum

As an exploratory test, below are each word—in no particular order, but to the exclusion of 'olympic' and 'general'—and the top closest (in terms of cosine similarity) cluster centroid vector.

While the result does not conform to the puzzle categories (which is not to be expected, because the types of intuitive similarity are not the same in each group), there are identifiable commonalities within a group's rankings.

These commonalities also indicate some of the possible red herrings of the puzzle. For example, there is a pattern of models of cars being grouped together; while this is an intended categorization, the inclusion of 'golf' (a Volkswagen model) is reasonable. Similarly, 'lima' and 'pinto' are near, being names of beans.

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1fiesta                     mustang
2'bronco'     0.9103        'bronco'     0.9485
3'golf'       0.9020        'bravo'      0.8688
4'bravo'      0.8757        'fiesta'     0.8560 
5
6lima                       bronco
7'pinto'      0.99447       'mustang'    0.9485
8'navy'       0.9251        'fiesta'     0.9103
9'elephant'   0.9120        'golf'       0.8989 
10
11pinto                      elephant
12'lima'       0.9944        'navy'       0.9697 
13'navy'       0.9188        'lima'       0.9120
14'elephant'   0.9109        'pinto'      0.9109 
15
16infinity                   kiddie
17'bravo'      0.9410        'elephant'   0.8395
18'vacuum'     0.9030        'mustang'    0.8335
19'pinto'      0.8841        'bronco'     0.8305 
20
21kidney                     bravo
22'vacuum'     0.8478        'infinity'   0.9410 
23'elephant'   0.7948        'pinto'      0.8906 
24'infinity'   0.7898        'fiesta'     0.8757 
25
26delta                      golf
27'elephant'   0.8671        'fiesta'     0.9020 
28'navy'       0.8549        'bronco'     0.8989 
29'mustang'    0.8531        'mustang'    0.8386 
30
31navy                       vacuum
32'elephant'   0.9697        'infinity'   0.9030
33'lima'       0.9251        'bronco'     0.8938
34'pinto'      0.9188        'kidney'     0.8478

Of course, this is just a start. There are a few ways in which these can be refined. Here, I've only compared cluster centroids, whereas Thurnbauer et al. offer different types of comparisons that yield different similarities: thematic similarity versus similarity in type and function.

Given that the different categorizations from the puzzle use different types of similarity, multiple such means of comparison should be incorporated.

Even then, I will need to devise a way to perform the grouping that best allows for an even classification of four words into four groups. This is a matter of future consideration.

Other future improvements

It is worth acknowledging that there is already a follow-up paper from (many of) the same authors: Reisinger, Goller and Fischer (2024); however, I have not yet delved into this later work.

It is also worth acknowledging that Thurnbauer et al. themselves pointed out that a TF-IDF adjustment on context words may be effective in narrowing words' distinct meanings; I believe that that is a reasonable assessment.

That being said, mine is still very much an incomplete project, so there is much to be done in order to apply these clustered vectors to the puzzle.

One thing that stands out in particular if I were to continue with the DBSCAN approach is that considerations must be made for the computational time required by DBSCAN, which is impractical for common words, appearing to run at O(n^2). For instance, 'navy', with 349,335 occurrences, took a bit over 18 minutes. Meanwhile, extrapolating for 'great', with 1.2 million occurrences, could take as much as 3 hours (to say nothing of memory limitations). For the moment, I've simply excluded from testing any words with excessive counts. It consider it might be reasonable to simply set a cap on the number of occurrences/contexts to use, randomly sampled, which I imagine would still be plenty of data to capture the distinct clusters.

Link to code repo

Reproducibility

See README

References

Thurnbauer, Matthias, Johannes Reisinger, Christoph Goller, and Andreas Fischer. 2023. Towards Resolving Word Ambiguity with Word Embeddings. arXiv. https://arxiv.org/abs/2307.13417

Reisinger, Johaness, Christop Goller, and Andreas Fischer. 2024. 14th International Conference on Advanced Computer Information Technologies. https://ieeexplore.ieee.org/document/10712629