A Polar coordinate system represents syntax in large language models

We thank tm4o for their thorough constructive feedback.

We agree that our discussion of the work by Muller-Eberstein et al. (2022) was not sufficiently detailed, which may have made the novelty of our contributions seem less apparent.

To rectify this, we have expanded our discussion and incorporated four new analyses.

• "Our research shares similarities with Muller-Eberstein et al. (2022), who employed linear probing to classify dependency types in language models (LMs). However, our objectives, method, data and results differ significantly. First, while their study aims to offer 'a lightweight method [to] guide practitioners in selecting an LM encoder for the structured prediction task of dependency parsing,' our goal is to understand how syntactic structures are represented in LMs.

• Second, their method consists of optimizing three separate probes—for structure, labels, and depth—and integrating them with a complex heuristic. In contrast, our method employs contrastive learning to optimize a single probe, trained to identify, in a variety of LMs, a unique subspace that encapsulates all properties of dependency trees.

• Third, the datasets we analyze are not the same. Muller-Eberstein et al. focused on evaluating their probe’s performance on undifferentiated natural sentences. Our study extends beyond this by examining different types of sentences, e.g. synthetic sentences specifically designed to highlight key linguistic phenomena, such as nested structures. In summary, our study builds upon prior research (e.g., Hewitt and Manning, Muller-Eberstein 2022) to offer a more unified, compact, and interpretable representation of dependency trees in contemporary language models."

To substantiate these elements, we now show, in a revised figure (Figure 3 in the attached pdf), that sentences with similar syntactic structures are represented by the same coordinate system.

“In the space of the Polar Probe, the representation of dependency trees appears to be consistent across sentences’ length and substructures. For example, the coordinates of the main phrase are virtually identical whether it is attached to a prepositional phrase and/or a long-nested structure. This invariance supports the notion that dependency trees are represented by a systematic coordinate system that can be recovered with the Polar Probe.”

Relatedly, we now provide new analyses to clarify the representations of subtrees.

“The analysis of synthetic sentences shows that relative and long-nested substructures are represented, for the most part, less precisely than the main phrases. Similarly, the analysis of natural sentences shows that Label Attachment Scores (LAS) decrease with the length and syntactic depth of the sentence (Figure 2A and B in attached pdf). Overall, these results suggest that the Polar representations of dependency trees lose precision as sentences become increasingly complex. This phenomenon corroborates both with their behavior (Lakkretz et al, Linzen et al) and with human behavior (Silverman & Ratner, 1997).

Finally, to evaluate the universality of this representational system, we now evaluate the Polar Probe across a broader range of models and scenarios.

“LAS tend to increase with the size of the LM, irrespective of its family (e.g. Pythia, GPT, BERT). This gain seems to reflect an overall improvement: Indeed, LAS improves with model size for both short and long sentences, as well as both shallow and deep dependency trees. This result suggests that larger and more recent models progressively shape their representations to fit a Polar Coordinate system.”

Questions

Q1: Extended Dependencies: Our current approach is limited to (valid) binary tree structures. However, the notions of (1) extended dependencies, and (2) multiple paths, (3) ambiguous structures (4) semantic graphs could all, in principle, be investigated through a Polar Coordinates, as this proposal can apply to any (e.g. cyclic) directed and labeled graph. We will amend our discussion to highlight these interesting future research directions.

Thanks to the authors for their detailed response, and the additional analyses provided on the paper - and so am bumping the score up from 4 to 5. I still feel like there is room for further analysis -- I really like Reviewer's 9Ubp's comment that 'I think the authors were setting up to explore some questions about the meaningfulness in the "hierarchy" of the tree, especially with the controlled sentence dataset, but then I never saw these really come to fruition.'.

We thank Reviewer tm4o for their response and are grateful that our analyses have been able to address the previous concerns to a satisfactory level.

We acknowledge that the meaningfulness of the Polar Probe's representations on the controlled dataset may not be sufficiently emphasized, as Reviewer 9Ubp noted. To address this issue, we propose adding the following paragraph to the Discussion section:

"One key advantage of the present Polar Probe is its ability to provide linear, compact, and interpretable representation of dependency trees, distinguishing it from other parsing models. Notably, Figure 6 (Figure 3 in the rebuttal PDF) illustrates an unexpected phenomenon: the probe appears to assign the same coordinates to words within identical phrases and syntactic roles, regardless of sentence length, complexity, and semantic variations. This representational invariance offers a promising foundation for further exploration of how recursion and compositionality are effectively represented and processed in neural networks."

We hope this addition clarifies the meaningfulness of hierarchical structures in the Polar Probe and encourages further exploration of its implications for linguistic representation and processing.

We thank reviewer rWeD for their insightful and constructive comments,

Controlled dataset

We agree that reporting performance metrics such as UUAS, LAS, and Balanced Accuracy on the controlled dataset is an important addition. To address this, we will include a new figure in the appendix of the camera-ready version of the manuscript. This figure will display the UUAS, LAS, and Balanced Accuracy for each of the gold edges in the main phrase, with error bars representing standard error. (The figure is available as Figure 4 in the author rebuttal attached pdf) Our result reveals that the degradation in UUAS and LAS is primarily due to the addition of relative clauses and prepositional phrases. Specifically, the accuracy for the ‘nsubj’ relation decreases by approximately 50% after adding these constituents. Unexpectedly, we also observe a degradation of the same magnitude in the ‘obj’ relation, despite the constant distance between words in the ‘obj’ relation across the three sentence levels.

LAS and model size

In response to the reviewer’s recommendation, we have conducted a study on how LAS varies with model size across different families (Pythia, GPT-2, BERT, and SOTA). The updated figure, which will be added to the camera-ready version, demonstrates a clear trend: larger models generally achieve higher LAS scores. (The figure is available as Figure 1 in the author rebuttal attached pdf)

LAS and sentence complexity

We also concur that exploring how performance degrades with sentence complexity is valuable. Our new analysis shows how performance varies with sentence length and depth, indicating that larger models are better at capturing the tails of the syntactic complexity distribution. (The new figures are available as Figure 2A and 2B in the author rebuttal attached pdf)

Finally, we will correct the typos and increase the font size in the legends and axis labels of Figure 3 in the camera-ready version.

We thank reviewer 9UbP for their insightfulreview.

Weaknesses

Probing vs. Parsing:

We agree that the distinction between probing and parsing is insufficiently clear. We will amend the discussion as follows: “The current 'probing' work is related to extensive research on 'parsing'. However, the two fields have distinct objectives, and thus, distinct approaches. Syntactic probes focus on how the vectorial activations of LMs represent the symbolic constructs theorized by linguists (e.g. trees, nodes). For example, the present Polar Probe uses a single linear transformation to provide a compact and interpretable representation of trees through the distance and angle between pairs of word representations. In contrast, parsing studies primarily aim to accurately reconstruct syntactic trees, regardless of the underlying representation. These studies thus prioritize performance over interpretability, which may lead to the adoption of non-linear probes that learn representations that are powerful, but not effectively learnt and utilized by LMs.“

Lack of Baselines:

Previous research, such as Hewitt and Manning (2019), provides a baseline included in our figures. We also include an untrained baseline to show that the Polar Probe extracts syntactic knowledge from self-supervised training, not from tokenization or the probing mechanism. To the best of our knowledge, the only other labeled probing work involving LLMs is by Muller-Eberstein (2022). We will add this baseline to the figures of the camera-ready version. In addition, we now provide a series of new analyses to evaluate our Polar Probe across different LLM families and sizes (attached Figure 1): the results show that the Polar Probe performance improves in more recent and larger models. on variably complex sentences, using the ud-en-ewt test set (attached Figures 2A and 2B): the results show that the Polar Probe gets increasingly imprecise as the length and syntactic depth of the sentences increases. on specific substructures (attached Figure 4): the results show that the Polar Probe is less accurate on nested trees than on the main phrase.

These novel analyses clarify the conditions in which the Polar Probe retrieves dependency trees.

Cosine Similarities:

This is a good point. We will now add the following clarification to the results section.

“Note that dimensionality impacts cosine similarity: Given that random vectors in a dimension D typically have a cosine similarity of approximately 1/sqrt(D), a cosine similarity of 0.5 can be interpreted as relatively high. Overall, these results support the notion that LLMs effectively represent syntactic relationships. “

Pairwise Cosine Similarity Matrix:

We agree that examining the Polar Probe’s pairwise cosine similarity matrix is valuable, especially to linguistics. In particular, we find that the relations that lead to off-diagonal blocks reflect different categories, which in linguistics, can be subtle. For example, both the case-mark and the aux-cop difference are a common source of confusion among linguists. Such is the case that, for clarity, the following examples are provided on the Universal Dependencies website

The following sentence will be added to L196: “We find that higher cosine similarity among off-diagonal blocks in Fig. 2.C reflects subtle linguistic distinctions. For example, distinctions between case-mark relations (case: 'Sue left after the rehearsal' [after - rehearsal] vs. mark: 'Sue left after we did' [after - did]) and aux-cop relations (cop: 'Sue is smart' [is - smart] vs. aux: 'Sue is considered smart' [is - considered]) illustrate these nuanced differences”

Related Work:

Due to space constraints, the Related Work section was omitted in the initial submission. We will add this section in the extra page allowed in the camera-ready version of the paper. A first paragraph can be found at the beginning of our response to reviewer tm4o.

Questions:

Q1: The trees are predicted following Hewitt and Manning’s approach, adapted to labeled trees. Thus, from the probed euclidean distance matrix, we get a tree by computing the Minimum Spanning Tree (MST) with Kruskal’s algorithm. Then, the predicted edges are labeled and directed according to the highest absolute cosine similarity with a set of prototypical vectors for each relation type. We acknowledge that this procedure is designed for interpretation rather than performance. This approach demonstrates that the differences of performance arise from the nature of the representations rather than the heuristics of the prediction method.

Q2: We now revised Figure 5 to clarify its objective (attached Figure 3) and note: “In the space of the Polar Probe, the representation of dependency trees appears to be consistent across sentences’ length and substructures. For example, the coordinates of the main phrase are virtually identical whether it is attached to a prepositional phrase and/or a long-nested structure. This invariance supports the notion that dependency trees are represented by a systematic coordinate system that can be recovered with the Polar Probe.” Furthermore, we now provide (attached Figure 4) a detailed analysis of the results obtained from the controlled dataset. Overall, these results show that subtrees are less well represented than the main phrase.

L33: We agree and will revise it as follows: ”The discrepancy in the system of representations has thus challenged the unification of the theories of human cognitive processes with the underlying computational implementation in the brain.”

Fig1: We agree, and will revise it: “According to the dependency grammar framework …”

L83: We agree and will revise it:: “Following the development of the Structural Probe, squared Euclidean distances between probed word embeddings are not designed to represent both the presence of dependency relations and their types and directions simultaneously. ” We also correct the typos and citations