A Polar coordinate system represents syntax in large language models
Authors: Pablo J. Diego Simon
Stéphane d'Ascoli
Emmanuel Chemla
Yair Lakretz
Jean-Remi King
Venue: NeurIPS 2024 poster (2024)
Keywords: Natural Language Processing
Large Language Models
Interpretability
Syntax
Linguistics
Cognitive Science
Primary area: natural_language_processing
TL;DR: We show that the presence and type of syntactic relations in a sentence can be inferred respectively from distances and orientations in the activation space of language models

Abstract: Originally formalized with symbolic representations, syntactic trees may also be effectively represented in the activations of large language models (LLMs). Indeed, a ''Structural Probe'' can find a subspace of neural activations, where syntactically-related words are relatively close to one-another. However, this syntactic code remains incomplete: the distance between the Structural Probe word embeddings can represent the \emph{existence} but not the type and direction of syntactic relations. Here, we hypothesize that syntactic relations are, in fact, coded by the relative direction between nearby embeddings. To test this hypothesis, we introduce a ''Polar Probe'' trained to read syntactic relations from both the distance and the direction between word embeddings. Our approach reveals three main findings. First, our Polar Probe successfully recovers the type and direction of syntactic relations, and substantially outperforms the Structural Probe by nearly two folds. Second, we confirm that this polar coordinate system exists in a low-dimensional subspace of the intermediate layers of many LLMs and becomes increasingly precise in the latest frontier models. Third, we demonstrate with a new benchmark that similar syntactic relations are coded similarly across the nested levels of syntactic trees. Overall, this work shows that LLMs spontaneously learn a geometry of neural activations that explicitly represents the main symbolic structures of linguistic theory.

A polar coordinate system represents syntax in large language models
Pablo Diego-Simón
Stéphane D'Ascoli Meta AL, Paris, France stephane www.aaallggaall.ooooooooooooo
ENS, PSL University, Paris, France pablo-diego. simon@psl.e.
Emmanuel Chemla
Yair Lakretz
ENS, PSL University, Paris, France emmanuel. chemla@ens www.aaeeenenssee.e.ee psl.eu
ENS, PSL University, Paris, France yair www.eatttgaaal..coomm
Jean-Rémi King Meta AI, Paris, France j Janremi @meta com
Abstract
Originally formalized with symbolic representations, syntactic trees may also be effectively represented in the activations of large language models (LLMs). Indeed, a "Structural Probe" can find subspace of neural activations, where syntactically- related words are relatively close one-another. However, this syntactic code remains incomplete: the distance between the Structural Probe word embeddings can represent the existence but the rype and direction of syntactic relations. Here, we hypothesize that syntactic relations are, in fact, coded by the relative direction between nearby embeddings. To test this hypothesis, we introduce a "Polar Probe" trained to read syntactic relations from both the distance and the direction between word embeddings. Our approach reveals three main findings. First, our Polar Probe successfully recovers the type and direction of syntactic relations, and substantially outperforms the Structural Probe by nearly two folds. Second, we confirm that this polar coordinate system exists in low- -ww-dmensiioal subspace of the intermediate layers many LLMs and becomes increasingly precise the latest frontier models. Third, demonstrate with new benchmark that similar syntactic relations are coded similarly across the nested levels of syntactic trees. Overall, this work shows that LLMs spontaneously learn geometry of neural activations that explicitly represents the main symbolic structures of linguistic theory.
Introduction
Human languages have long been proposed to systematically follow tree-like structures (Chomsky 1957 Tesnière 1953). In sentence, words that are far apart can be syntactically linked. For example cats is the subject of "chase" in the sentence ''The cats in cities chase the mice". In dependency grammar, the edges of such trees are directed and labelled to indicate the type of syntactic relation between words ("cats" the subject of "chase". Fig. IB).
Despite their conceptual soundness and alignment with human behavior (Robins) 2013). syntactic trees have long been the crux of core challenge in cognitive science (Smolensky 1987): trees symbolic representations, which can superficially appear incompatible with the vectorial representations of neural networks. This opposition between symbols and vectors has been major challenge to the
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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unification of linguistic theories on the one hand, and neuroscience and connectionist Al on the other hand.
Recently, Hewitt and Manning (Hewitt and Manning 2019) proposed an important concept for this issue, by suggesting that the existence of syntactic link between two words may be represented by the distance between their corresponding embeddings. Specifically, their "Structural Probe" consists in finding subspace of contextualized word embeddings such that the squared euclidean distance between words represents their distance in the dependency tree. They showed that the Structural Probe most powerful in the intermediate layers of language models: these layers contain subspace where yyntacicalll--eelted words are closer together.
This Structural Probe, however, can only reveal one aspect of dependency trees: namely, the existence of syntactic relations, between word pairs. However, whether and how the direction and the type of syntactic relations represented in language models remains unknown.
Here, we hypothesize that syntactic relations are represented by a polar coordinate system, where the existence and type of syntactic relations are represented by distances and direction, respectively (Fig. D. To test this hypothesis, we introduce a "Polar Probe": a linear transformation trained such that pairs of words linked by the same dependency type are collinear, while remaining orthogonal to different dependency types.
A Unlabeled undirected graph B
Dependency trees
rimod
case
The
cot
chases
the
mouse
The
cat
chases
the
mouse
The
cots
dites
chase
mice
Structural Probe
D
Polar Probe
Prise
spacee
mioe
chuse
cata
The
Figure 1: Dependency trees hypothesized in linguisties and in neural networks. A. According to the dependency grammar framework, the sentences can he deseribed as lincar sequences of words connected by an acyelic graph. B. More precisely, such acyclic graph is both labeled and directed, where each edge has a direction. representing the hierarchy of the syntactic relation, and a label, representing the type of syntactic relation. C. The Structural Probe (Hewitt and Manning 2019) finds linear transform (gray plane) of the language model's activations (here simplified their dependency tree. In the Structural Probe subspace, however, it not possible to distinguish whether "The cat chases the mouse" or "The mouse chases the cat." D. Our Polar Probe finds a linear transformation where the angle between syntactically-relate word additionally represents the type and direction of these syntactic relations, and the distance codes its presence. The colored arrows indicate orthogonal directions in the Polar-Prohe subspace.
3D space), such that the distance between word embeddings is predicted by
2
Methods
The goal of this work is to find a linear readout of the activations of pretrained language models which explicitly represents both the presence and the types of syntactic relations between words.
2.1 Problem statement
Dependency grammar represents the syntax of a sentence symbolic tree. Accordingly, let:
wi be the 1th word in a sentence,
* d W4, Wi + + indicate the syntactic distance between two words,

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C be the set of syntactic types,
L wi, wy + C indicate the type of syntactic relation between directly-coneetedd words.
wy; wy { {w,, wy} } indicate the head word (and thus direction) of the syntactic relation between directly-connected words.
Language models based on neural networks represent sentences as sequences of word vectors (a.k.a word embeddings] | they incorporate information about the sentence they belong within, becoming contextualized word embeddings.
As these embeddings propagate through the layers of the network,
Let h E R be the 1th contextualized word embedding of w, obtained at the output of layer & of a neural network. For simplicity, we will drop the layer index l in what follows, keeping in mind that different layers yield different representations.
Retrieving syntactic trees from a neural network requires to obtain three read-out functions d R R + Z+, u
R R R and R$ Idt + C such that the following conditions are met:
d(b,, hj) so d(w;, wy),
T(, by) vo t(wx, wy))
@(h; h2) ss u(wi, wyy,
While d, and could be any complex functions, the goal the present work to identify a simple, interpretable "code" of how syntactic trees may be represented in vectorial systems. Following the classic definition of representation as a linearly readable information, we focus on linear operators
(DiCarlo and Cox 2007
Kriegeskorte and Bandettini
2007 King and Dehaene 2014).
Structural Probe
In (Hewitt and Manning 2019). the authors propose to solve d as distance in subspace of the contextualized word embeddings. Their "Structural Probe" is linear transform, Bs g + g which projects word embeddings such that their relative distances correspond to their distances in the syntactic tree. Formally, if s; E R the (directed) "edge embedding" between words w, and wy:
S.j = h, hy R
Then, the predicted distance d E R+ between two words can computed directly as function of this and no other information coming from the individual word embedding
d(, h,) = |s
Given a set of sentences, the authors extract the set Uss pairs words belonging to the same sentence and optimize Bs to minimize the absolute difference between the distances within the syntactic tree and the distances between the probed word embeddings:
(s
Id(w,, wy) - d(w,, wy))
(3)
Lggl
Following the development of the Structural Probe, squared Euclidean distances between probed word embeddings are not designed represent both the presence of dependency relations and their types and directions simultaneously. (Hewitt and Manning 2019) thus only propose a representational system to solve d, but not é and à. Whether and how the directed and labeled syntactic tree is encoded in neural networks, thus remains unknown.
"Sequences built from "tokens", which sometimes correspond subwords. When this is the can simply average the subword embeddings to obtain word embeddings (Hewitt and Manning 2019).
"See (Chen et al. 2021) for an explanation of how hyperbolic spaces are best measured through square Euclidean distances.

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2.3 Angular Probe
Here, we hypothesize that neural networks use the orientation the relations formed by connected word pairs to represent the type and direction of their syntactic dependency.
To test this hypothesis,
first introduce "Angular Probe" consisting of linear transform B R + RR By abuse of notation, denote f(s..) = t(w;, wy) the syntactic type of the corresponding edge. For the Structural Probe, the function d to be recovered is defined on all pairs of words; here the function & be recovered is only defined on pairs of syntactically linked words, hence we only consider word pairs (w;, wy) which are indeed syntactically linked.
We use contrastive learning to align relations of the same type and ensure that different types pointing to different directions. This approach designed such that linear readout could explicitly categorize dependeney types.
Specifically, the objective for the Angular Probe is to ensure that given two edge embeddings s and s' of syntactic types C = t(s) and = +(ss), the linear transforms BA and A are colinear c c', and orthogonal if + c'.
Formally, the Angular Probe is the linear transform which minimizes the contrastive loss:
1 Σ (4(BAs, (((s) ((s))))
CA(BA)
22 is the set of edge embeddings of syntactically connected words,
xx,,y= +
x-
is the cosine similarity,
[xilly
if X' is true is the indicator function.
. 1 1:
otherwise
We can then construct prototypical vector for each dependency type, by averaging all the probed edge embeddings belonging to that type:
Ve
Σ BAS,
2 = {s E Ral((s) = cc.
(5)
SEQ
The Angular Probe gives us the following function to retrieve the syntactic type of any given edge:
i(b; hy) = argmax A(BASij, V.)l
(6)
We also get the function predict the head word (and direction) of the syntactic relation given predicted type f:
h; LB Vi(hhh)) 2
à a(h, h,) =
h if i(B Vibh,))) <
2.4 Polar Probe
The Angular Probe and the Structural Probe have independent objectives applied to two different datasets: Cs relies on Uss, which includes all word pairs from any sentence, whereas C relies on QA which contains only pairs of words that are syntactically linked.
Consequently, we define the Polar Probe single linear transformation Bp RR* &R" which results from the joint optimization of both the the Angular and Structural objectives.
This Polar Probe defines the final functions to identify syntactic distances and relation types:

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d(b,, by) pS ||
(8)
t(, hy) = argmax (Bps: Ve)l
(9)
h if (Bppss V Vih..b,) 0
a(h, by) hy if (BpSsy, Vi(h,hy)) <0
(10)
Therefore, the Polar Probe minimizes the following loss function, with the weighing the Angular objective.
argmin (s( + (B)
(11)
Bp
2.5 Data
Natural dataset. We consider natural sentences extracted from the English Web Treehank dataset (Silveira et al. 2014 This corpus contains 254,820 words from 16,622 sentences, sourced from diverse array of web media genres, including weblogs, newsgroups, emails reviews. All the sentences in the dataset are manually annotated according to the Universal Dependencies frame- work (Nivre et 2017), where each word is node, each syntactic link labelled directed edge, and the syntactic tree is acyclic.
Sentences containing email or web addresses excluded from the dataset. Such filter removes noisy sentences not interesting from a syntactic point of view. We follow the default splitting provided by the English Web TreeBank resulting in total of 1127 sentences for training, 1851 for validation, and 1869 for testing.
Controlled dataset. To precisely evaluate our approach on well-controlled sentences, we designed dataset, extending previous work (Lakretz al. 2021b), comprising 100 sentences built with long-nested structure (e.g., "The book that the boy besides the car reads fascinates my teacher"). In these sentences, a subordinate clause branches off the main phrase. There also constituent 'besides the car' that forms branch inside the subordinate clause, adding further complexity to the syntactic structure.
This controlled dataset is designed to clarify how the Polar Probe reconstructs the syntactic tree in complex conditions, but conditions well theorized in linguistics. In particular, we can create several variations of these long-nested nested sentences.
Short: "The book fascinates my teacher"
Relative clause: "The book that the boy reads fascinates my teacher"
Long-nested: "The book that the boy besides the car reads fascinates my teacher"
Thanks to this dataset we can study whether the Polar Probe maps embeddings of different syntactically-identic sentences to consistent latent locations and orientations. addition, with the different "sentence levels", ,word position and syntactic role can be disentangled to compare the Polar Probe's activations accross levels, as shown in Fig:
2.6 Training
train the Polar Probe on the neural activations of Mistral-7B-vO.11 and 2lama-2-7b-hf (Touvron) et al. 2023 Jiang et al. 2023). in response to sentences of the "Natural Dataset" described above. Both models are nuto-Regresssve Language Models, they aim to identify future words from input sentences. Furthermore, these models read all the words simultaneously, building representations which depend on the whole sequence tokens.
We trained the Polar Probes with gradient descent, using the Adam optimizer Kingma and Ba 2014) with a learning rate of 0.005, and a batch size of 200 sentences. The duration of the training is 30 epochs, we perform model selection using the validation set. Hyperparameter À is set to 10.0, ensuring an optimal balance between the Angular and Structural objective.
http:://unniverssaalpeeeeeeiiii org/ treebanks/ www.eeeeeeeeee.ittteeeeeee
5

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A
B Dependency Type evaluation
PCA of probed relations
obj
The
chasse
the
rocse
det
nsubj
chance
charce
1.0
Balacced Accurasy
AUC
C
No Probe
Structural Probe
Polar Probe
red.j
nauty
Cosine Similarity
c350
cose
0050
ob
oN
cti
cane
cet wovinn
punzt
caso
pursi
caso
Out
advmrdd and
Figure The Polar Probe reliably identifies dependency types.
visualization edges linearly read
by the Polar Probe. The color of cach edge corresponds to one of three diffcrent dependency types ('nsubj'. . "obj". 'det'): y the linear readouts point in systematic dircctions. B. AUC and Balanced Accuracy metrics obtained for dependency type classification. C. Pairwise cosine similarity (0=orthogonal, !=collinear) matrices obtained without prohe (left) the Structural Probe (middle) and the Polar Probe (right).
2.7 Evaluation
We evaluate each probe either its ability faithfully represent (i) the unlabelled and undirected dependency tree ("structure"), (ii) the type and direction of dependencies and (iii) both of these elements.
Dependency structure. Following (Hewitt and Manning, 2019), we evaluate whether the probes accurately predicts the existence of each syntactic relation by using the Undirected Unlabeled Attachment Score (UUAS). UUAS quantifies the proportion of dependency relations (directly con- nected words) in the dependency tree that are correctly identified by the probe, irrespective of their dependency types and direction.
To evaluate the accuracy of the predicted dependency types, we use three distinct metrics: Area Under the Curve (AUC), Dependency Type Accuracy and Dependency Type Balanced Accuracy.
Dependency type.
For AUC, we compute the cosine similarity between each edge s; and all other edges sj. that are cither from the same dependency type, or not. This procedure ends with a distance vector æ E 8 of edge pairs and binary vector of y 1m We can finally input these two vectors into scikit-learn's poc_auc_scors (Pedregosa et al. 201T).
For Accuracy and Balanced Accuracy, we classify dependency types by comparing relations to prototypical relations. For this, from the training set, we pool 10,000 relations and define a prototype Sy for each dependency type k. by computing the centroid of all relations belonging to the same type Then, we predict dependency types from the cosine similarity between each relation s and each prototype V., using scikit-learn's use seikit-learn accuracy_ score or balanced, accuracy_ score to limit the effect of imbalance between dependency types.
wewgghbbrrCaassiiifiii (Pedregosa et al. 2011). Finally, we

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Combined dependency type and structure. Finally, to provide metric which evaluates both dependency structures and dependency types, we compute the Labeled Attachment Score (LAS). LAS is defined as the proportion of correctly predicted labeled and directed edges in sentence.
Baselines. We compare the Polar Probe to a variety of baselines: (i) Structural Probe, (ii) "Polar Probe Random LLM": Polar Probe trained top of random language model and (iii) "No Probe": the raw activations of the Language Model without any transformation.
3 Results
Reliable coding of dependency types. We first analyze the Polar Probe on the 16th layer of Llama- 2-7b-hf, Mistral-7B--0..1 and BERT-large Touvron et al. 2023 Jiang et al., 2023 Devlin et al. 2019) on the English Web Treebank (EWT) sentences (Silveira et al. (014)) annotated with dependency trees. We evaluate, similar syntactic relations point towards similar orientations in the probe's ' representational space (Fig.
independent test set with 10000 relations, whether pairs of words linked by
Fig. A shows a Principal Component Analysis (PCA) projection of the dependency relation embeddings from the test set, once linearly read by the Polar Probe. For readability, we restrict ourselves to three the most common types of dependencies in the dataset. As expected, the three types of dependency consistently point in different directions.
We then compute the cosine similarity between all pairs of edge embeddings probed with the Polar Probe (Fig. C (right)), indeed showing that relations of the same types are collinear, while relations of different types are orthogonal. This is much clearer for the Polar Probe than for baselines (Fig. C
Comparison with baselines. In Fig. 2 B, we summarize with AUC and Balanced Accuracy the extent to which the orientations of these edge embeddings reliably represent the syntactic types.
On average across dependency types, the Polar Probe reaches a AUC score of 95%, well above the Structural Probe (AUC=74%), the No Probe (AUC=80%) and the Polar Probe trained on a Randomly Initialized LLM (AUC=50%). The same relative results across probes are conserved for the Balanced Accuracy score. Importantly, these results confirm that the Polar Probe outperforms the Structural Probe in predicting dependency types from the probe embeddings. The latter is therefore something not emergent in the Structural Probe.
Unexpectedly, "No Probe" predicts syntactic types well above chance, and significantly better than the Structural. This hints to the fact that syntactic types are already represented in the raw activations. A likely explanation for this is that words belonging to the same part-of-speech (such as verbs, nouns) are clustered in the embedding space, thus partially guiding the inference of syntactic dependencies.
Moreover, training a Polar Probe on a random initialization of a language model does not accomplish the contrastive objective better than chance. Resulting in near chance-level Balanced Accuracy and AUC scores. This confirms that the linear probe needs a rich underlying representation space to work, and cannot learn to cluster the different syntactic types on its own.
Layer-wise analysis. To evaluate how the Angular and Structural performance interact in the Polar Probe, and whether the mechanism generalizes to both Mistral-7B-v0.11 1, Llama-2-7b-l and BERT-large, we evaluate the layers of the three models on Labeled Attachment Score (LAS) (Fig. 3 (See Supplementary for BERT-large and Mistral-7B-vO.11
Interestingly, BERT-,arge, Mistral-7B-vO. 1 and Llama-2-7b-hf all peak at layer 16, which is the same layer reported in (Hewitt and Manning 2019). At layer 16, the models achieve LAS on the test set of 70.2, 60.6 and 62.9 respectively. As recently reported in (Fisape et al., 2022) for the Structural Probe. these results suggest that the Polar Probe also works best with Masked Language Models. The Polar Probe, despite its conceptual simplicity matches performance with a more intricate and modular labeled probe (Müller-Eberstein et al. 2022).
Structural evaluation. The Polar Probe is optimized with both a Structural and an Angular objective. This means that the optimization of such probe might affect the original performance of the Structural Probe. To verify that the gap in performance, compute the UUAS between the
7

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predicted tree and the annotated tree for both the Structural and Polar probe (Fig. B). The results confirm that the Polar Probe preserves (but does not improve) the syntactic distances between the linear readout of the probed word embeddings.
A
B
C
Dependency Existence Score
Polar Structural
Depondency Type Score
80
Existence and Type Score
70
80-
40-
50-
40
20
20
60-
10
10
0
8
16
24
32
24
0
16
24
32
Layer
Layer
Layer
Figure 3: The Polar Probe outperforms the Structural Probe at identifying labeled and directed dependen- Structural probe,
cies. A. For dependency existence, Polar Probe matches the UUAS performance peaking at layer 16. For dependency type, the Polar Probe outperforms Label Accuracy the Structural (LAS) Probe by around 80% accross the different layers of Llama-2-7b-hf, C. For both dependency existence and type, the Polar Probe outperforms in LAS the Structural Probe by around 90% accross the different layers of Llama-2-7b-hf.
Dimensionality analysis. How many dimensions necessary successfully represent the full syntactic tree with the proposed polar coordinate system? To address this question, we varied k", namely the dimensionality of the space of the Polar Probe (Fig. 4. Analogously to the rank analysis of Structural Probe (Hewitt and Manning (211)), we observe a peak around k." 128. Contrary theoretical predictions (Smolensky) 1987). these results suggest that the space representing the complete syntactic tree needs not be unreasonably large. For dependency types, this phenomenon could be relatively intuitive, the unit circle (i.e. only 2 dimensions) can easily separate many different dependency types, such that a weakly non-linear readout would isolate these categories.
A
B
C
Dependency Existence Score
Palar
Dependency Type Score
Existence and Type Score
70
80
70
10
100
1000
10
100
1000
10
100
1000
Polar Probe Dimension
Polar Probe Dimension
Polar Probe Dimension
Figure 4: The optimal dimensionality the Polar Probe is an order magnitude small than model's layer size. Polar Probe performance as a function of dimensionality. measured by A. UUAS, B. Dependency Type Accuracy and C. LAS for Llama-2-7b-Be as function of k", the dimensionality of the probe's space. The optimal dimensionality for the Polar Probe 128, achieving the highest LAS.
Controlled sentences. Natural sentences are highly variable in structure and content. To verify more precisely the behavior of the Polar Probe, we evaluate on the "Controlled Dataset" and its different sentence levels: Short, Relative Clause and Long-Nested.
First, we observe that in the space of the Polar Probe, the representation of dependency trees appears to be consistent across sentences' length and substructures. For example, as shown in Fig. [3] the PCA visualization Polar Probe word embeddings belonging to the main phrase virtually identical whether it is attached to a relative clause or to 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.

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We also observe that dependency types are robustly identified, whether they are part of the main phrase or not.
Overall, these results visually confirm that the Polar Prohe reliably represents the dependency structure and dependency types of complex syntactic structures. The latter agrees with the extensive Structural and Angular evaluations performed.
A
Main Phrase
Relative Clause
Prepositional Phrase
The
The
Phll
that
RC
book
book
kdt
PP
reads
beachee
ants leacher
Tosg nates
Je ther
Cark
the
B
det nsubj obj nmod
True
lecomaate
Busk
maae
dosk
masy
The
Predicted
Figure S: Visualization of the dependency tree uncovered by the Polar Probe on a set of sentences with increasingly complex hierarchical structures. A. We display a PCA visualization of the distributions of word embeddings (once linearly read out by the Polar Probe), for the different syntactic levels in the "Controlled Dataset". Each individual distribution corresponds to specific role of the word the sentence. The centroids are linked with colored lines, displaying the true syntactic tree of the corresponding sentence. B. Most frequent syntactic tree prediction by the Polar Probe for the different syntactic levels. The relations between words are color coded according to the type of syntactic dependency. The incorrectly predicted relations represented with dashed arrows. That is, either dependency relation existence (no arrow), dependency type (with arrow) was erroneously identified.
4 Discussion
Summary. We show that within the activation space of language models, there exists a subspace, where syntactic trees are fully represented by polar coordinate system. There, the presence and type of a syntactic relation between two words is represented by their distance and relative direction, respectively. Importantly, the Polar Probe preserves the structural properties of the Structural Probe (Hewitt and Manning 2019), but better represents the type of syntactic relations.
Limitations. The present work presents four main limitations. First, we only investigate the English language. Yet, human languages use different grammatical rules, and may, consequently, be structured according to different types of trees. Interestingly, as language models become increasingly able to process a wide spectrum of languages (Costa-jussia exciting possibility to explore universal (or divergent) grammatical representations in artificial neural networks, following EMiller-Eberstein et al. 2022 Chi al. 2020).
al. 2022), the present framework opens the
Second, syntactic structures are not necessarily restricted to the description of relations between words. In particular, morphology predicts that words themselves may be represented as trees of morphemes. Consequently, whether and how the present framework generalizes to the different scales of linguistic structures remains be further investigated.
Third, like the Structural Probe, the Polar Probe is based on a supervised task: we optimize linear transformation that maximally retrieves known syntactic structure from the neural activations.

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Developing an unsupervised probe would be important to help discovers unsuspected syntactic structures. In addition, we here focused on dependency structures. Yet, other formalisms, based on phrase structures (Chomsky T957 Joshi and Schabes, 1997 Cinque and Rizzi 2009) Chomsky 2014) could offer alternative trees, and could be equally probed through the present framework. This approach could thus offer the possibility of experimentally testing which of these linguistic theories best account for the representations of human languages in neural networks.
Finally, we assume that syntactic trees can be best read out using Euclidean probes. However, alternative assumptions, such as hyperbolic representations, have been a fruitful tool to interpret deep learning models' representations in both text and image modalities (Dhingra et al. 2018)|Ncca and Kiela, 2017: Desai et al.) (203). We speculate that this direction could provide a valuable avenue for extending the current work.
5 Related work
Syntax in artificial neural networks. Overall, this study complements previous research on syntax in artificial neural networks. Originally, (Smolensky) 1987) demonstrated that vectorial systems could, in principle, represent symbolic structures with tensor products but did not provide an empirical demonstration that neural networks did, in fact, demonstrate this property. More recently, language models were tested their capacity to process syntactic structures by evaluating their behavior on grammatical and ungrammatical sentences (Lakretz et al. 20200 202 [ewiit and Manning
2019 Hale et al. 2022. Evanson et al. 2023 Linzen et al.) (016). Finally, several groups explored
how this capacity was instantiated in the neural activations (Huang et al.) 2017 Palangi et al. 2017]
Soulos et al., 2019 Lakretz et al.
2019), culminating in (Hewitt and Manning 2019) 's Structural
Probe. Since the discovery of the Structural Probe different adaptations have been developed, notably
including hyperbolic (Chen et al.
2021), orthoghonal (Limisiewiez and Mareček] 2021), ) nonlinear
(Eisape et al.] 2022] White et al.
2021) )) variants. The present work completes this long effort by
showing how interpretable syntactic code based both distances and orientations spontaneously
emerges in language models.
Syntax in biological neural networks. This link between linguistics and artificial neural networks holds significant potential for neuroscience. In particular, until the latest rise of large language models, many experimental neuroimaging studies aimed identify the neural bases of syntax in the human brain (Hale et al.) (022). For example, (Pallier et al. 2011) showed with functional Magnetic Resonance Imaging (fMRI that several regions of the superior temporal lobe and prefrontal cortex responded proportionally to constitent size. Critically, language models are now becoming standard bases predict and explain the brain responses natural language processing: The activations of these artificial neural networks have indeed been shown to linearly map onto fMRI, intracranial and MEG recordings of the brain in responses to the same words and sentences (Jain and Huth 2018 Caucheteux and King 2022, Reddy and Wehbe [021] [20 Caucheteux et al.) [022]] Pasquiou et al. 2022 2023). However, this mapping remains difficult to interpret, and the neural code for syntax in the brain remains major unknown. The present work thus provides testable hypothesis to understand how swwtacticcccccccomc trees may be explicitly represented in the brain.
Broader impact. Combined with the works outlined above, our results open the exciting possibility that the polar coordinate system may, in fact, explain how syntax is encoded in the human brain. Criti- cally, this framework may generalize beyond syntactic tree structures, and apply to any compositional problem, including compositional semanties, object-feature binding in vision, and representation of knowledge graphs. Above all, while many have long opposed symbolic and connectionist formalisms, this work contributes to show how these two systems of representations may be largely compatible with one another, as long predicted (Smolensky) 1990. (Smolensky et al.) (022). This reconciliation thus holds great promises to understand the brain mechanisms of language and composition.
Acknowledgments
This project was provided with computer and storage resources by GENCI at IDRIS thanks to the grant 2023-AD011014766 on the supercomputer Jean Zay's the V100 and A100 partition (PDS).
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This project has received funding from the European Union's 's Horizon 2020 research and innovation program under the Marie Sklodowska-Cun Curie grant agreement No 945304 (PDS).

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White, J. C., Pimentel, T., Saphra, N., and Cotterell, R. (2021). A non-linear structural probe. In Toutanova, K., Rumshisky, A., Zettlemoyer, L., Hakkani-Tur, D., Beltagy, I., Bethard, S., Cotterell, R., Chakraborty, T. and Zhou, Y., editors, Proceedings of the 2021 Conference of the North American Chapter of the Association for Computational Linguisties: Human Language Technologies, pages 132-138, Online. Association for Computational Linguistics.
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Medel Famil GFT
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Parameter Count (Million parameters)
Figure 6: Polar Probe performance the EN-EWT dataset for Language Models with different families and sizes
Strurtural (aabet
60
Ramkand model size
50-
Strudurall and Label
50
40-
Ranked
20-
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Sentence length
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Figure 7: Comparative analysis of Polar Probe performance the N-EWT dataset a function of sentence length (left) and sentence depth (right). The scores shown across various model sizes (ranked by model size), with darker lines indicating larger models.
LAS
(RAI)
Accuraccy (%6
Short
Relative
Long-nested
Figure 8: Polar Probe performance across different sentence structures and dependency types controlled dataset. The three categories (Short, Relative clause, Long-nested) show the performance breakdown by Unlabeled Attachment Score (UUAS), Labeled Attachment Score (LAS), and specific dependency relations in the main phrase. Error bars represent the standard error across relations.
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22


Reviews:
[{"review_id": "RelYgyPWeD", "reviewer": "Reviewer_9UbP", "summary": "Previous work introduced linear probes to explore how syntactic relationships are encoded in LLM embeddings.  This work aims to take it a step further and examine how types of syntactic relationships are encoded in the LLMs.  They introduce a polar probe that when optimized can predict the type of syntactic relations via the angle between them.  In a multi-faceted evaluation, the model outperforms baselines (which are essentially ablations of the model) in terms of stronger cosine similarity between the same relations, and in terms of tree accuracy (UAS/LAS).", "questions": "Suggestions / Questions:\n\nQ1 - How the trees are predicted is not clear / whether these are a series of independent predictions or whether they are processed sequentially or decoded jointly?\n\nQ2 Do the same syntactic relations that occur at different levels of the sentence have distinct embeddings?  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.  Especially the talk of short/relative/long-nested partitions -- where are these discussed?  Fig. 5 is mentioned (L241) but Fig 5 is fluff.\n\nL33: \"and its neuronal substrate.\"  What?  It's just a model.\n\nL36: \"to settle\"?  Though the sentence is bizarre regardless\n\nL24-L253, improper citation formats almost everywhere\n\n\"According to linguistics, sentences can be described as linear sequences of words connected by a dependency tree\", there isn't a \"linguistics\" -- there are many competing syntactic frameworks and heated debate as to the pros/cons of each framework, but at the end of the day, these are merely formalisms\n\nL83: Squared euclidean distances (between two word embeddings) cannot trivially represent the presence\n84 of dependency relations and their types and directions simultaneously.\n\nWhy not?  If these are represented in separate subspaces, why is it not possible to represent these three concepts in a vector space?", "limitations": "Yes, the limitations are described.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- An interesting paper with a clear contribution, building on existing probing work while asking a couple new research questions\n\n- The results appear convincing\n\n- The potential to explore syntax through the lens of LLMs, especially when LLMs can be easily trained on unlabeled text, or especially when LLMs are increasingly multilingual, points to some exciting future directions.\n\n- The evaluation also includes some linguistically interesting example cases.  Essentially exactly what I would have asked for (in addition to the larger corpora studies)", "weaknesses": "- I find the distinction between probing and parsing to be not entirely clear.  At the point where the evaluation is in terms of UAS/LAS, could this not be compared directly to parsers on the same data (especially since building on top of LLM embeddings would be the most performant solution)?  And where would the discrepancies be, and what would that mean?  Do LLMs not encode those relationships?\n\n- In general the paper seems to suffer from a lack of convincing baselines.  The baselines presented -- the structural or angular probe, are steps along the path to the polar probe.\n\n- Cosine similarity between identical syntactic categories is surprisingly low (to me).  The ranking of categories in terms of the strength of that correlation is also surprising, ith things like 'case' being quite strong.  In general there are many \"odd\" patterns that I don't have an intuitive explanation for why they occur, and aren't discussed in detail in this work.\n\n- There is no dedicated related work.  I do think the parsing literature, and especially the parsing-on-top-of-LLMs literature is relevant.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer 9UbP for their insightfulreview.\n\n## Weaknesses\n\n### Probing vs. Parsing:\n\nWe agree that the distinction between probing and parsing is insufficiently clear. We will amend the discussion as follows:\n“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.“\n\n### Lack of Baselines: \n\nPrevious 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.\nIn addition, we now provide a series of new analyses to evaluate our Polar Probe\nacross 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.\non specific substructures (attached Figure 4): the results show that the Polar Probe is less accurate on nested trees than on the main phrase.\n\nThese novel analyses clarify the conditions in which the Polar Probe retrieves dependency trees.\n\n### Cosine Similarities: \n\nThis is a good point. We will now add the following clarification to the results section.\n\n“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. “\n\n### Pairwise Cosine Similarity Matrix:\nWe 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\n\nThe following sentence will be added to L196:\n“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”\n\n### Related Work: \nDue 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.\n\n### Questions:\n\nQ1:\nThe 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. \nWe 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.\n\nQ2:\nWe now revised Figure 5 to clarify its objective (attached Figure 3) and note:\n“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.”\nFurthermore, 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.\n\nL33:\nWe 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.”\n\nFig1:\nWe agree, and will revise it: “According to the dependency grammar framework …”\n\nL83:\nWe agree and will revise it::\n“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. ”\nWe also correct the typos and citations", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "JtI5pA6U8Y", "reviewer": "Reviewer_tm4o", "summary": "Whereas prior work (Hewitt and Manning 2018) probed syntactic distance and depth, this work proposed to push that forward by also probing headedness and dependency type.  Specifically, this doesn't separately probe those three, but aims for a single vector space where euclidean distance defines syntactic distance but the difference vector maps to a UD dependency label (optimized with a contrastive learning objective).", "questions": "- Have the authors looked at extended dependencies?  The notion of differences as dependency type seems more specific (and to imply more interesting consequences) if there are multiple \"paths\" through a tree.", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "It is a pretty well-written paper, and the framing of the angular probe idea seems well explained and to have some elegance to it (in aiming for a single underlying representation); parts of the implementation seem well-considered to get that single representation.", "weaknesses": "- If viewed merely as a combination of probing structure and labeling, it is very similar to a work like Muller-Eberstein et al. 2022.   The advantage of this paper -  having more of a shared representation -- is appealing, but I wish the consequences of that shared space were better explored.\n- Analysis was somewhat lacking: for a probing paper, there were relatively work showing what this tells us about the syntactic behavior of models.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank tm4o for their thorough constructive feedback.\n\nWe 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. \n\nTo rectify this, we have expanded our discussion and incorporated four new analyses.\n\n* \"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.\n\n* 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.\n\n* 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.\nIn 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.\"\n\n\nTo 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. \n\n“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.”\n\nRelatedly, we now provide new analyses to clarify the representations of subtrees. \n\n“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).\n\nFinally, to evaluate the universality of this representational system, we now evaluate the Polar Probe across a broader range of models and scenarios.\n\n“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.”\n\n### Questions\n\nQ1:\nExtended Dependencies:\nOur 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.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LlRzQBPXjp", "reviewer": "Reviewer_rWeD", "summary": "This paper proposes polar probes, a kind of structural probe that learns a distance and rotation function that can more accurately classify syntactic structure from language model representations than previous approaches. In particular, the question of whether direction can represent the type of syntactic relationship is answered. The authors find that their polar probe outperforms previous methods quite significantly and show", "questions": "N/a", "limitations": "Yes", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "This paper is very well presented and a pleasure to read. The empirical findings are strong and clearly support the hypothesis that the direction, as well as the angle of the representations of an LM projected on a plane represent the syntactic relationships encoded by the model. The authors show that this interpretation is able to much more strongly reconstruct ground truth syntactic parses from hidden state representations than structured probes. The controlled dataset provides a clean comparison in a challenging setting and is a useful resource for future work. A major finding of this work is that it vastly raises the upper bar for how well we should consider syntax to be encoded by language models.", "weaknesses": "Weaknesses like the focus on dependency parses and drawbacks of current tokenizers are addressed in limitations, but are still weaknesses nonetheless.\n\nPlease include  UUAS, LAS and Balanced Accuracy for the evaluation on the controlled dataset separately for comparison.\n\nAs thorough as this paper is, I think it could go deeper on the model analysis. It's nice that the layer-wise analysis is consistent with previous work, but this would be mostly expected. For example, could the authors show that models of different sizes capture more/less syntactic complexity? Is there a critical point where syntax becomes well represented and gains are diminishing after more scaling? Do larger models capture more of the \"tail\" of rare syntactic constructions? This could be carried out on the GPT2 or Pythia family of models.\n\nNits:\n\n- please make the text in the legend/axis labels for figure 3 bigger\n\n- Typo L36: \"proposed settle\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank reviewer rWeD for their insightful and constructive comments,\n\n\n### Controlled dataset\n\nWe 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)\nOur 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.\n\n\n### LAS and model size\n\nIn 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)\n\n\n### LAS and sentence complexity\n\nWe 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)\n\nFinally, we will correct the typos and increase the font size in the legends and axis labels of Figure 3 in the camera-ready version.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
We would like to thank the reviewers for the detailed and relevant comments,

## The strengths pointed by the reviewers are:

* The paper is well-written (tm4o, rWeD)
* Clear contribution, the work opens several avenues of research (9UbP, rWeD)
* Convincing results (9UbP, rWeD)
* Paper provides with a controlled dataset with linguistically interesting sentences (9UbP, rWeD)
* Elegant implementation (tm4o)

## The weaknesses pointed by the reviewers are:

* Findings on the controlled dataset are unclear. (tm4o, rWeD )
* Show the universality of the Polar Probe (rWeD)
* Unclear difference with Muller-Eberstein, 2022 (tm4o)
* Lack of baselines (9UbP)



## To address the weaknesses pointed out by the reviewers, we have made four major  modifications:

### Understand better the role of the controlled dataset, (Fig. 5 is not clear)

First, we have now replaced Fig. 5 with a new figure (Fig. 3 in the attached pdf) to clarify the role of the controlled dataset. Our updated results show that sentences are consistently represented in the same coordinates of the probe’s space, irrespective of their length and substructure. This result illustrate the validity of the polar coordinate system proposed in the present study.

Second, we provide new analyses to evaluate the precision (UUAS and LAS) of different substructures of the syntactic tree (Fig. 4 in the attached pdf). The results show that deep structures (preposition phrases, nested phrases) lead to lower performance than the main phrase – A phenomenon which echos with the behavior of both LLMs’ and humans (Silverman,& Ratner, 1997). In addition, the subject-verb agreement performance gets deteriorated as the linear distance in the sentence between both words increases.

Overall, these new elements clarify how syntactic structure can be linearly decoded from LLMs.

### Show the universality of the Polar Probe

We now added new analyses to demonstrate the universality of the Polar Probe representation system across different model families and sizes (Fig. 1 in the attached pdf). Our results show that the Polar Probe's performance is robust across different models and sizes, and that larger models consistently achieve higher LAS performance.

In addition, we also show how the LAS deteriorates with respect to the length and the depth of the sentence. We have added a new figure (Fig. 2 A and B in the attached pdf) showing the better performance of larger models in the tail of the syntactic complexity distribution.
Unclear difference with (Muller-Eberstein, 2022)

We now clarify the differences between our work and (Muller-Eberstein et al. 2022) in terms of goal, method, data, and consequences. Specifically, we have highlighted our goal to understand whether and how syntactic structures are organized in LMs, whereas Muller-Eberstein et al.'s goal is to propose a lightweight method to “guide practitioners in their choice of LM encoder for the structured prediction task of dependency parsing”. In particular, we now clarify in the discussion that our approach optimizes a single and interpretable probe to find a unique subspace that jointly represents the existence, directionality and labels of dependency trees.


### Lack of baselines

To better contextualize the results, we will add the Muller-Eberstein baseline in Figure 3 of the camera-ready version. Additionally, we added extra references with the performance analysis on the controlled dataset and these scores will be a useful resource for the community to better interpret LAS performance. Lastly, we also added an analysis of the performance of the Polar Probe across different model sizes and families (Fig. 1 in the attached pdf). 

Overall, these new analyses strengthen our original conclusion. 

Once again, we would like to thank our reviewers for their help in improving the study.

Meta-review: Overall, there is a consensus among the reviewers that the paper should be accepted, with a clear contribution and results that support the main idea. There is a need for better baselines and distinguishing the work from previous work in the literature. The authors clarified the difference between their work and previous work (M&E 2023), mentioning that they aim to show that syntax can be represented using polar coordinates, while previous work tried to build encoders for dependency parsing.
Decision: Accept (poster)