The Structure of Relation Decoding Linear Operators in Large Language Models
Authors: Miranda Anna Christ
Adrián Csiszárik
Gergely Becsó
Dániel Varga
Venue: NeurIPS 2025 spotlight ()
Keywords: large language models
relations
tensor networks
interpretability
Primary area: deep_learning
TL;DR: We investigate the structure of relations in large language models, and compress linear relation decoding operators with tensor networks

Abstract: This paper investigates the structure of linear operators introduced in Hernandez et al. [2023] that decode specific relational facts in transformer language models. We extend their single-relation findings to a collection of relations and systematically chart their organization. We show that such collections of relation decoders can be highly compressed by simple order-3 tensor networks without significant loss in decoding accuracy. To explain this surprising redundancy, we develop a cross-evaluation protocol, in which we apply each linear decoder operator to the subjects of every other relation. Our results reveal that these linear maps do not encode distinct relations, but extract recurring, coarse-grained semantic properties (e.g., country of capital city and country of food are both in the country-of-X property). This property-centric structure clarifies both the operators' compressibility and highlights why they generalize only to new relations that are semantically close. Our findings thus interpret linear relational decoding in transformer language models as primarily property-based, rather than relation-specific.

The Structure of Relation Decoding Linear Operators in Large Language Models
Miranda Anna Christ* Adrián Csiszárik Gergely Becsó2.3, Dániel Varga*
Fazekas Mihály High School, Budapest, Hungary
PHUN-REN Alfréd Rényi Insititute of Mathematics, Budapest, Hungary Fötvöö Loránd University, Budapest, Hungary chrm@berkeley edu, {csadrian, begeri, damiel)0renyi. hu
Abstract
This paper investigates the structure of linear operators introduced in Hernandez et al. [2023] that decode specific relational facts in transformer language models. We extend their single-relation findings to a collection relations and systemati- cally chart their organization. We show that such collections of relation decoders can be highly compressed by simple order-3 tensor networks without significant loss in decoding accuracy. To explain this surprising redundancy, cross-evaluation protocol, in which we apply each linear decoder operator to the subjects of every other relation. Our results reveal that these linear maps do not encode distinct relations, but extract recurring, coarse-grained semantic proper- ties (e.g., country of capital city and country of food are both in the country-of-2 property). This property-centre structure clarifies both the operators' compressibil- ity and highlights why they generalize only to new relations that are semantically close. Our findings thus interpret linear relational decoding in transformer language models as primarily property-basee based,, rather than Staaiin--ppecfic///
develop a
Introduction
From ancient philosophy modern cognitive science, knowledge has been framed not merely as isolated entities but as interconnected networks shaped by context and structure. Aristotle's categorical distinctions, Wittgenstein's 's concept of family resemblances [Wittgenstein [953]. and contemporary cognitive theories of prototypes [Rosch] 1973) all emphasize that understanding emerges from recognizing shared properties across diverse contexts. We do not simply store isolated facts, instead, we perceive who does what to whom, what belongs where, and how different ideas interrelate through common attributes. Investigating the underlying structure of how relational knowledge is encoded thus becomes crucial, as it forms the foundation for complex cognitive functions such as generalization, abstraction, and analogical reasoning-ccpabilliice essential for both human cognition and machine learning models.
In this paper, we investigate how transformer language models encode relational knowledge such as Michael Jordan plays basketball by mapping subjects directly to objects through linguistic
predicates. Our investigation builds on the recent work of Hernandez et al. [2023, who show that
for given relation, the transformation from the embedding of a subject to the tokens of the object can be effectively approximated by a single linear operator, called a Linear Relational Embedding (LRE) matrix. Here, we examine relations through the lens of their corresponding lincar decoder
"qqual contributions.
Code and data are available at the project website: htspp://bbit.lyyssruuuurr.od relatioons
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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operators, and perhaps more crucially, rather than analyzing them one-by-one, we consider them collectively. We develop tools uncover their structure and the underlying regularities with the goal of understanding their collective organization.
Beyond interpretability, are also motivated by the possibility that these decoder operators admit more compact, shared representation. Such compression could highlight the core semantic properties shared across different relations, reducing redundancy and perhaps enabling the model to generalize more effectively, analogously to how humans draw on abstract patterns to reason across different contexts. Our experiments with tensor network models explore this hypothesis by seeking compact representations that preserve decoding capacity while exploiting the possible latent structure of relational knowledge.
We summarize our contributions as follows:
We propose novel semantic closeness notion to explore the underlying structure of relations. We reveal that relation decoding functions have a property- -bssed rather than fine-grained relation- specific organization.
We propose order-3 tensor networks compress an entire collection of linear decoder functions into a single, compact model. We demonstrate and analyze the effectiveness of this approach.
We investigate the generalization properties of such tensor network models. We find low level generalization capabilities on generic data, and also provide examples where they excel.
2 Background and Notation
2.1 Relations in Transformer Language Models
One common approach to formalize factual knowledge in language models is through relational triplets (S. R. O), where subject S is linked to object O via relation R [Miller 1995 Berners
Lee et al. 2001 Bollacker et al., 2008 Lenat 1995 Richens 1956 Minsky 1974.. Here, S E S, O O, and R R. with S. O, and R denoting the sets of all possible subjects, objects, and relations, respectively. For instance, the sentence "Paris is the capital city of France" corresponds to the triplet ("Paris", 'capital city of", "France")).
Recent work by Hernandez et al. 2023) demonstrated that for each relation R, the transformation from subject embeddings to object embeddings in transformer-based language models can be effectively modeled by a surprisingly simple decoder: an affine transformation fR : S + O, fR(S)
WRUs bR, where WR E Rddd and bx Rd and d N is the embedding dimension of the transformer, and Us is the embedding of the subject. The effectiveness of such affine approximations suggest surprisingly simple internal organization for relational knowledge within transformer layers.
2.2 Technical Details
Throughout our experiments we study LLMs based on the transformer architecture (Vaswani et al. 2017]. The model maps input tokens into a d-dimensional space via an embedding layer Tembb- This is followed by a sequence of H E N transformer blocks, denoted as La. (h E [H]), and concludes with final transformer head Licad- The entire transformer network is thus the function t == Lemb o L1 o L2 .... o Lk-1 o LA a Lpead- Approximating an affine relation decoder thus equals to modeling the function g L Li+1 o.. Lk-1 Lı (i.e., the function of the transformer from the 1th layer until Lhead) with an affine transformation, where k is the total number of blocks before the Liead component.
One could find many ways to infer an affine approximation the function g. this paper, we infer it with parameterized modeling function, instead of directly approximating it using the Jacobian of g (as shown in Hermandez et al. (2023))
Once we have a linear relation embedding (LRE) matrix WR and a bias bR, we apply the affine function (v)) = W RuR + bg to the subject embedding us. Then we pass the resulting vector to the transformer's 's output head to generate the next token. The performance of the relation decoding function is evaluated using the notion of faithfulness, which essentially measures prediction accuracy.

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Definition 1 (Faithfulness)) Given subject token embedding us R and an inferred linear relation encoder matrix WR R E Raxd and bias bR E R. for a relation R, we define the faithfulness of the affine decoder sn(ns) = Wnws + bn as the top-1 accuracy of the predicted object token. Specifically, we compute to = Inead [n
where Tnead the transformer's output head, and compare the resulting token prediction against the ground truth object token. Faithfulness is then the triplets.
proportion of correct predictions across a set of subject-relation
2.3 Models and Datasets
We used three transformer-based language models in our experiments: GPT-J
[Wang and Komatsuzaki
2021], Llama 3.1 8B (Dubey and et al.) 2024), and GPT-NeoX-2D5 [Black et al. [2022b]. Unless stated otherwise, the results presented in the main text correspond to GPT-J, with additional analyses and details for the other models provided in Appendix [
We use three datasets: 1) The Dataset of Hernandez et al. [2023]: :: consists of 47 distinct, mostly
orthogonal relations (i.e., fruit inside color and adjective antonym). 2) Extended Dataset: our extended version of the dataset of Hernandez et al. [2023] that introduces several new relations, allowing a better understanding of the relational structure. 3) Mathematical Dataset: novel relational dataset containing mathematical operations (i.e., number plus and number times 9) providing more controlled, and in a sense a denser relational structure. For further information, we refer to Appendix D
3 Training Tensor Networks to Represent a Collection of Relations
Matrices representing relations live in Rd where d is the dimension of the embedding space. In the case of GPT-J, = 4096, meaning even a single matrix has more than 16 million parameters. With a collection of 100 relations that amounts approximately 1.6 billion parameters. In this section, explore the following question:
Is it possible to represent a collection of relations in a compact form, and can this representation rely on the striking simplicity of linearity?
From the naïve representation to tensor networks A straightforward way to represent multiple relation matrices is to stack them to form an order-3 tensor. Specifically, a collection of matrices each size Rdxd, can naturally be regarded as a single 3-tensor of dimensions Rdxdxn, primary approach to 'compress' matrices in linear way is via low-rank approximation. A more general, censor-analgoous approach would involve decomposing this order-3 tensor to produce a represen- tation that retains the information content and the functionality of the original tensor, while using significantly fewer parameters. The notion of tensor networks offers a principled way and serves as a remarkably powerful tool for this endeavor: any tensor network having three free legs of dimension can be viewed as a representation of an order-3 tensor. (Regarding tensor networks, we restrict ourselves introducing only the minimal terminology required for our models in Appendix FF For more detailed introduction we refer to Ahle (0024). Furthermore, inner dimension constraints serve as
to rank constraints constrains--siniificann significantly reducing the parameter count. The number of possible internal structures of '3-legged' tensor networks is vast (in fact, theoretically infinite). While this richness of possibilities opens up plethora of opportunities worth for exploration, for the sake of compact exposition, here we explore only two of these.
3.1 Tensor Network Architectures
A tensor network is collection of tensors T T= (T The with a pairing of the legs of the tensors, where each pair connects legs of equal dimensions. Unpaired legs are called 'free legs' and their number is called the order of the network. The 'free legs' of a tensor network are often simply referred to as 'legs'. We can draw these networks as tensor diagrams: dots are tensors, lines are their legs, and joined legs link tensors like edges in a graph. Each leg has label. One labeled picture therefore captures both the layout and the workings of the whole tensor network. In this paper, we implement two tensor network classes. See Fig. lb for the diagrams of these tensor networks.
3

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(a) Average faithfulness a collection of compressed linear operators in the function of paremeter count. Each point corresponds trained tensor network. Filled markers represent models with additional rela- tion embedders. Different marker figures represent tensor networks with different inner structure (sec Fig- Tblon the right). We employ low-rank baseline element-wise low-rank approximation of the decoders.
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(b) Schematic illustrations the tensor networks used in this paper. Left: a basic order-3 tensor network. Right: triangular configuration of tensor network. Both architectures optionally incorporate sepa- rate relation embedding network.
Figure 1: Compressing relation decoding linear operators with order-3 tensor networks.
The network consists of an order-3 tensor T E Xd,, xdo at the core. To connect it with the embeddings of the transformer embeddings, matrices) s,s' E Rdxd,, ,2, E Rddx, o, E Rddx, to project the inputs into the appropriate space. We call dy', d, and do the inner dimensions of the network. The relation-to-matri. mapping implemented by this network can be written out
SimpleOrddrrNetworrk
three order-2 tensors (i.e.,
T-R = T
Σ r,"','',
Dr PepT
P1 p2
p','' S.s 0,0"1
where T. denotes the relation decoder approximation obtained from the tensor network and relation embedding o, for relation R.
The network consists three inner order-3 tensors T7. uu.t T2 T'3 in Rd xd, [Rd xd,, Xd,, Rd, xdyxda xdz respectively, and three 2-tensors (matrices) P1 Rdxd,, p2 E Rdxd,,, priate space. Similarly as above, de,, d, and do are the inner dimensions of the network, serving as a bottleneck. The vector-to-matrix mapping implemented by this network can be written out as:
TriiaggeeenssorNetoorr
E.y.
E Rdxdd projecting the transformer embeddings into the appro-
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The total number of parameters depends on the embedding dimension d of language model and the inner dimensions d,,, dy,, and da of the core tensors. For the Simple- OrderšNelwork, the parameter count is Nsimple (d (d. + d. addd,- + d dx + (dg dy. d The first term corresponds to the three projection matrices, which dominate the total parameter count, while the second term represents the compact order-3 core tensor. For the TriangleTensorNetvonn the total parameters are VTriaagle = (d de + d d, + d do)) + 3 (dg dp do)) reflecting three interconnected order-3 tensors the core.
Parameter counts
From matrices to affine maps The networks above produce matrices linear maps with no bias-brt our task need to produce affine maps. We can embed the bias term directly in the network by enlarging the appropriate bond dimension with inference time, append constant to subject vectors, during training the extra column of weights learns the bias jointly with the rest of the matrix. This simple tweak converts the linear output into an affine map without altering the overall architecture.

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Additional embedder network on the relation leg Optionally, both architectures can be extended with Additional Relation Embedder to loosen the rather strong assumption of linearity on the relation embeddings before entering the tensor network. We experimented with embedder consisting of three dense feed-forward layers with ReLUU [Agarap 2018 activations.
3.2 Training Tensor Networks
tensor network even with a fixed structure can still be obtained and utilized in various ways. Tensor parameters may be determined through factorization optimization algorithms. Once constructed, the resulting tensor network can be used by evaluating contractions on its free legs.
End-to-end training of tensor networks We fix the parameters of the LLM and train only the tensor network. We train our tensor network using the task loss, i.c. the cross-entropy loss for the predicted object token and optimize with SGD. Specifically, 1) we take UR. the embedding of the relation, 2) perform a contraction with the tensor network at the relation leg, 3) read out the matrix representing the LRE, 4) apply it the subject embedding Us, then 5) pass it to the Leaad to get the predicted token distribution, 6) use the gross-entrop loss to match the expected token as a loss function. Formally, our loss function LR. is:
(T
Σ Σ
CE(1o, Lhead ( (Us)),
RER (S,O)ER
where lo denotes the one-hot encoded first token of the object O, CE is the cross-entropy loss and vy is vector representation of subject S. Throughout our paper we always use the ground-truth object values for training.
3.3 Compression Experiments
In these experiments we evaluate the compression capabilities of both SimpleOrder3Networt and TriangleTensorNetworrkk models. We train these models on the dataset Hernandee et al. [2023) until convergence and measure the faithfulness of the resulting decoder functions. We performed a grid search for both models with d, E {2, 4, 6, 8, 30, 100), de', do E {10,50 50, 100, 300]; for the TriangleTensorNeteee
fixed de. dy. d, E {50}. We also trained the tensor networks with and without an extra relation embedder. We discuss hyperparameters in Appendix 4
Baselines As baseline, we train low-rank LRE matrices individually, producing one per relation. Each matrix optimized independently using the same training objective as the tensor networks. This baseline corresponds to low-rank representation of each relation matrix, without sharing parameters across relations or introducing any structured connectivity between them. In addition, as a canonical reference point, we note that linear decoder baseline following the Jacobian-basee procedure of Hernandez et al. [023], which estimates each LRE from a few examples (we used in our setup) achieves a mean faithfulness of 0.41 with approximately 788 million parameters.
Relation decoding matrices are highly compressible Figure La summarizes the results by plotting the mean faithfulness against the parameter count of trained tensor network matrices. We can clearly observe that the relation matrices are highly compressible. Compared to the "vanilla representation" of stacking the 47 relation matrices together (with an overall parameter count of about 788 million) tensor networks even with less than one million parameters significantly outperform the baselines both in terms of parameter count and faithfulness. We can also observe that tensor network models without separate relation embedders consistently outperform those with them, showing that linear structure is sufficient for efficient compression.
Do LREs produced by tensor networks retain their sample-wise generalization capabilities? We also examined the learned relation decoders (LREs) ability to generalize to unseen samples. On the original dataset, the tensor network does outperform or equals the majority baseline in 34 relations out of 47. (See in Appendix I.I) Figure 4a further illustrates the relationship between parameter count and sample-wise test faithfulness. On the extended dataset, the LREs produced by the tensor networks outperform the majority baseline in 49 relations and equal in 8 relations out of the total 79 relations, with an overall mean test faithfulness of 0.42 compared to the majority- serossst-guusss baseline of 0.30. Thus, our models retain sample-wist wise generalization capabilities.
5

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Why Compression Works: Toward a Structural Understanding of Relational Matrices
In the previous section, we demonstrated the compressibility of relations. To gain better understand- ing of this phenomenon, in this section we explore the following question:
What underlying regularities or shared structures allow the affine relational de- coders to be compressed?
4.1 Semantic Similarity of Relations
One natural hypothesis for why compression possible is that the relations are not entirely inde- pendent from each other, and many may share underlying semantic structure. To investigate this, we define a notion semantic similarity between relations. While the embeddings of the relations (calculated from their names prompt templates) could also serve this purpose, we opt for measure more directly connected to the subjects of our investigation: the relation decoders themselves.
Definition 2 (Cross-evaluation protocol) Let {(Ri, Ji)) be a set of k. E N relations R& E R and their corresponding decoder functions ft TRd - TR.. The cross-evaluation protocol proceeds as functtonn follows: for every ordered pair (j. I) [k] x [k), apply the decoder In for relation R3 and record the resulting faithfulness score. Collecting all scores in a ls x array produces the cross-evaluation faithfulness matrix
F = F
F2a faithfullesss RR, Rt, fs).
Cross-evaluation as a measure of semantic similarity The methodology of cross-evaluation allows measure the extent to which specific relation decoder be used map other relations' subjects to their objects. Intuitively, if two relations are semantically related, then applying one decoder to the other relation's 's samples should yield reasonably faithful predictions, producing a relatively high off-diagonal entry in F. Conversely, low or near-zero entries indicate semantic dissimilarity. In this way, the cross-evaluttion matrix may be viewed as empirical similarity kernel over the set of relations-measured through the functional similarity of their decoding functions.
Cross-evaluation results for the dataset of Hernandez et al. [0233 Figure 2a shows the cross- evaluation faithfulness matrix for the 47 relations. We observe that while most of the decoders perform best when evaluated on the relations they were approximated from, there are many off- diagonal elements that are larger than zero zero-sseverl even exceed faithfulness value of 0.7. Also, we can notice places where block structure is apparent. Figure 2b shows a selected subset of the 47 relations to highlight these. Taking closer look at these matrices, there are cases when we can observe an obvious semantic overlap between relations, e.g., characteristic gender, university degree gender, and occupation gender all share the concept of gender while their subjects differ. There are also examples when the connection between relations is not as evident, and the overlap is syntactic rather than semantie: for example, the first letter of word relation proves effective for adjective superlative, likely because many superlatives begin with the same token as their base adjective.
4.2 Uncovering Property-Level Encoding through the Extended Dataset
Motivated by the above observations, obtain more detailed picture, we introduce extended dataset of 79 relations intentionally constructed to contain semantic overlaps between relations. This dataset supplements the original corpus with diverse set of relations designed share coarse-grained properties (e.g., gender, country, antonym) alongside truly orthogonal relations. Details of dataset construction and relation selection criteria are deferred to Appendix D
Cross-evaluation results for the extended dataset Figure 3a shows the cross-evaluation results. Our previous observations extend to this dataset as well, providing further evidence that our initial findings are not merely particularities of specific relations, but reflect more general phenomenon. We can again observe the presence of non-zero off-diagonal elements. Focusing on the relations char- acteristic gender, university degree gender and occupation gender notice similarly competitive

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Linear Approximated Relation
Linear Decoder Approximated on Relation
characteristic gender univ degree gender ecoupation gender fruit inside color fruit outside color adjective antonym adjective comparative adjective superiative
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word last letter
chanacarrittc gander
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occupation genidrr
fruit erdite color
Frait cutsidte calor
adlective anformmr
oelective comparative
adles sdlective superietiv
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word Fext Hetter
world last ketter
Evaluated on Relation
Evaluated on Relation
(a) Cross-cvaluation result using all of the 47 relations.
(b) Subset of relations highlighting the block structure.
Figure 2: Cross-evaluation results for the dataset of Hernandez et al. 2023]. Each cell shows the
faithfulness of matrix obtained using the row relation and evaluated on the column relation.
Linear Decader Approxinated on Rban
Linear Decoder Approximated on Roistion
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Evaluated Relation
Evaluated Relation
(a) Cross-evaluation using the extended dataset.
(b) Subset of relations highlighting the block structure.
Figure 3: Cross-cvaluation results for the extended dataset. The vertical axis indicates the relation used to obtain the matrix, while the horizontal axis indicates the relation it is tested on. Values represent faithfulness scores.
faithfulness in any given permutation during cross-evaluation (around 0.65). The occupation gender decoder even outperforms the characteristic gender decoder when evaluating on the relation charac- teristic gender. Similarly, although the fruit inside color and the fruit outside color operators achieve a faithfulness of only around 0.3, they maintain a similar performance during cross-evaluation.
The resulting matrix also exhibits a block-structure: high intra-block faithfulness among relations sharing a property (e.g., landmark in country, primary language spoken in a country under a country block; adjective antonym, noun anlonym under an antonym block) contrasted to zero inter-block scores where no common property exists. A prominent identity block also emerges, once again reflecting the identity-lik like functionality without further common semantic correspondence (cf. semiconductor chip manufactured by company and mathematical theorem named after mathematician). We observe that inter-block relation decoders work well with a variety of subjects.
Property extractors instead of fine-grained relation decoders Considering all of the above phenomena-most notably, the cross-comattiiility of linear relation decoders, and a diversity of subject types they work on-we hypothesize that linear relation decoders are based on common properties of different subject types. Thus, linear relation decoders function more as property extractors for specific target objects rather than capturing fine-grained relational structure.
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40
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(b) Faithfulness results for the mathemattcss dataset. Blue bars represent relations from the training set, purple bars from the test set split randomly with ratio 75% re- spectively. Green markers denote the majority- base- line, and orange markers represent the faithfulness values individually approximated relation matrices as another baseline.
107
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(a) Sample-wise wise generalization results the dataset of [Hernandez et al. 2023). Purple markers denote test faithfulness and orange markers repre- sent the previously used low-rank baselines.
Figure 4: Test faithfulness results on sample-wise generalization on the dataset of Hernandez et al. 2023 (left), and relation-wise generalization results on the mathematical dataset (right).
With this interpretation, the phenomena above can be explained by the facts that, e.g., the relations characteristic gender, university degree gender and occupation gender are holding common gender property. Likewise, city in country and name birthplace have the country property in common.
In conclusion, our experiments with the extended dataset provide strong empirical support for the hypothesis that linear relation decoders predominantly capture coarse-graine coarse-grained property patterns rather than specific subject-object mappings. This finding underscores the potential for exploiting shared structure across relations, driving more aggressive and semantically principled compression.
4.3 Investigating the Low-Rank Structure of Relation Decoder Operators
How much of compressibility remains if we remove semantically similar relations?
Low-rank structure Posing the question allows to decouple two sources of parameter redun- dancy: 1) overlap stemming from the semantic connection between relations, and 2) any intrinsic low-rank structure that individual decoder matrices might share even when the relations themselves are dissimilar. By isolating the second factor, we can assess whether the high compression ratios reported in earlier sections are merely exploiting semantic redundancy. or whether an additional structure is at play. As the original dataset has near-diagonal faithfulness cross-evaluation matrix, it serves as a good subject of investigation. Through the lens of the above, Figure[Ta]shenn large semantic overlap, substantial compression is possible compared even to the low-rank baseline. This gap suggests that tensor networks capture structural regularities beyond the simple low-rank property of individual (iationss,jointtl exploiting patterns shared across the entire relation set.
that without
Generalization Capabilities of Tensor Networks to Held-Out Relations
Having demonstrated that relations share poare-ggaiined properties enabling drastic compression, in this section we take the final step and ask:
Can tensor networks go beyond mere compression and encode latent properties in a way that enables generalization held-out relations?
Experimental setup In this section, we split the set of relations into training and test subsets, train a tensor network on the training set, and evaluate whether can generate a linear relation decoder matrix for an unseen relation embedding. We assess success using the faithfulness metric. (Note that the generalization thus examined on the level of relations, not on the level of individual subject-object pairs, which would be much casier task.)

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Generalization results for the extended dataset The corresponding results are shown in Ap- Dendix[B.2] We observe that the model produces high-faithfulness decoders for some test elements. but fails to generalize for others. Taking closer look, and building on our cross-evaluation -aasaoo----eddd semantic closeness metric, we find that the model generalizes only to relations that are semantically so close to those in the training set that their decoder matrices are effectively interchangeable. Thus, the model is able identify the matrix that corresponds to the correct coarser-grained relation set (or to put it another way, linear property extractor), but fails to generalize in wider context.
Regarding the vast scale of all possible relations, this outcome not surprising: broader generalization would require denser sample of relations and a linear structure among their decoder matrices assumptions that are quite strong, given the sheer size and diversity of relational semantics. (Note that our tensor network model relies on a simple linear formulation.) We also experimented with additional relation embedders to process the relation input before enters the tensor network (see Figure lb). Our motivation was to give the model an opportunity to reshape the relation space into more lineally organized structure; however, this modification yielded no substantial improvement.
A more realistic setup:
generalization results for the mathematics dataset For more realistic setup, we constructed a relational dataset that more tightly controlled and may have a more linearly arranged relational structure. This dataset also enables the generation of much finer-grained synthetic relation examples.
Our mathematics dataset consists of arithmetic relations modeled as unary operators on integers. (e.g., with relations number plus X and number minus X). (See Appendix for more details.) We ran experiments using three random seeds, presenting one in Figure 4b Across these, our method achieves an average faithfulness of 0.992 with a standard deviation of 00001 on the training set. We observe that the model not only learns the training set perfectly, but also has full generalization capabilities on the test set-the -the tensor network model outputs decoding matrices that achieve an average faithfulness of 0.96 with a standard deviation of #0.031 and maximum of 0.991.
Closing the gap between these generalization results goes beyond the scope of this paper. We identify this as a promising direction for future work, with potential applications related to model performance, compression, and interpretability.
5.1 Ablations the importance of relation, subject, and object embeddings
Randomized relation embeddings To assess the role of relation embeddings, we implemented baseline with randomized relation representations and evaluated it on the mathematical dataset. The results show that removing the semantic information from relation embeddings leads to a drastic performance drop on held-out (subject, object) pairs, while training examples can still be memorized perfectly. These findings confirm that the tensor network relies on meaningful relational representations rather than treating relations simple categorical identifiers. This is further supported by the generalization experiments, as the close to 100% accuracy is attainable only be correctly predicting objects for subjects associated with multiple relations (e.g., 13+6-19, 13-3-10), indicating that it exploits the semantic structure encoded in the relation embeddings.
Randomized subject and object embeddings We also evaluated the effect of randomizing subject and object embeddings. In this setup, each distinet subject and object was mapped to a random vector that remained fixed during training. In our experiments, the tensor network failed to memorize the training set or generalize on held-out pairs (resulting in faithfulness close to O on small tensor networks), indicating that consistent, meaningful entity representations are essential for both memorization and generalization.
6 Related Work
Knowledge representation in deep learning models Knowledge representations have been shown to emerge in neural networks since the original backpropagation results (Rumelhart et al. 1986]. Understanding these representations is widely explored-ex, e.g. implicit entity-models [eva et al.)
2023 dissecting factual recall in attention and MI.P layers [Geva et al. 2021 Li et al. |021 probing
classifiers to predict features from hidden states [Conneau et al. 2018 Hernandez et al. | 2024), and
knowledge-graph embeddings to represent information in low-dimensional spaces [Choudhary et al.

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2021]. Our paper proposes a new. novel method to uncover the underlying structure of knowledge representations with affine relation decoders [Hernandez et al. 2023].
Chanin et al. 2024] observe that the pseudoinverse of the LRE applied to the embedding of the object results in vector (Linear Relational Concept, LRC) that can be used linear probe. For example, the pscudoinverse of LRE "s city is in country o" applied to objects ( ("France", " "Germany", "Italy") results in three vectors that can be used to classify cities of the three countries.
Tensor decompositions and compression of neural networks Tensor decompositions have been widely used to compress and accelerate deep models [Novikov et al. 2015 Anjum al. 2024 Phan ct al. 2020]. Dense weight matrices have been converted to Tensor Train format by Novikov et al.
2015 while et al. [2022] applied Tucker decomposition to reduce Transformer-lat layer parameters
in BERT et al. 2019 A decomposition has been integrated into LoRA adapters (Hu et al.)
2021] by Annum et al. 2024 and Phan et al. [2020) introduced CP decomposition for stable, low-rank
CNN compression. Our setup differs from these methods as we compress approximated relation decoders collectively, rather than decomposing weight tensors individually.
Connections to low-rank adaptation techniques and mixture-of-experts models Applying framework in the context of of low-rank adaptation (LoRA) techniques (Hu et al., [021] is an interesting and natural extension. A single low-rank LoRA matrix can be expressed with an order-2 tensor network with an essentially arbitrary internal structure. An even closer analogy to our work is to consider LoRA matrices collectively- run, or aggregated from several independent LoRA adaptations. In either case, a rensor-network framework can act as a unified, highly compressed representation of all the LoRA matrices. Such setups can also take contextual information into account leg of tensor network. In broader context, this line of reasoning can also be related to mixture-of-experts architectures, where structured compression and modular representations increasingly important. Exploring these connections offers several interesting avenues for future research.
those produced across all layers a single training
7 Limitations
Our study relies linear approximations extracted from relatively small language models (with 6, 8, and 20 billion parameters). How these results generalize to larger, instruction-tuned or mixture- of-experts architectures remains open. Also, the true space of relations among entities in human knowledge is vast, and our dataset represents only a small and biased subset it. Even though large language models encode extensive factual and conceptual information, systematically mapping the relations they implicitly contain would be currently infeasible. Our work should therefore be seen as partial exploration of relational structure rather than a comprehensive map of it.
8 Broader Impact
By compressing hundreds of relation decoders into small tensor-network modules, we reduce param- eter counts by orders of magnitude. This slashes memory and compute needs, broadening access in resource-constrained settings. Exposing coarse-graine grained properties clarifies how facts are organized, making step towards interpretability. Practitioners can more readily understand, debug, and refine factual knowledge in deployed systems. Isolating properties (e.g., gender, nationality, religion) may open a door towards probing or altering broad classes. This raises risks of large-scale fact tampering, or articulation of sensitive attributes without consent, amplifying privacy harms.
Conclusions
We have taken steps toward unveiling the latent structure of relation decoders in transformer language models. We empirically showed that these decoders are compressible via order-3 tensor networks and proposed techniques to achieve high reductions in parameter count. By applying a cross-evas cross-evaluttion protocol, we demonstrated that relation decoders do not act as isolated, relation-specific mappings; instead, they organize into common properties recurring across relations. Finally, we analyzed the generalization properties of models learning to represent a collection of relation decoders, finding limited generalization on general language data but robust performance arithmetic relations.
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Acknowledgements
Supported by the Ministry of Innovation and Technology NRDI Office within the framework of the Artificial Intelligence National Laboratory (RRF-2.3. 1BRF-22311221-22220000044 A. Cs. was partly supported by the project TKP202I-NKTA-6 financed by the National Research, Development and Innovation Fund of the Ministry for Innovation and Technology, Hungary. We thank the anonymous reviewers for their valuable feedback and suggestions.
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A Results with Additional Models
In addition to GPT-J Wang and Komatsuzaki 2021 we cross-cvaluated the relation decoders on the 1.mm--3.1-8B Dubey and et al. 2024] and GPT-NeoX-20B [Black et al. 2022b models. We present the cross-evaluation matrices in Figure S and in Figure respectively. FigureGrespeectvvey The block structure can be clearly observed, demonstrating that our related findings hold across different models.
A.1 Llama 3.1 8B
Linear Decader Approca approximatess erd an Rela tiom
Linrer Decoder Approximates on Relation
Evaluated Relation
Evaluated Relation
(a) Cross-evaluation matrix on the extended dataset.
(b) Subset of relations highlighting the black structure.
Figure 5: Cross-evaluation results for the extended dataset using the Llama-3.1-8B [Dubey and et al. 2024 model. Each cell of the matrix shows the faithfulness score calculated using the decoder obtained from the row relation, and evaluated on the column relation.
A.2 GPT NeoX 20B
Lireer Decoder Agp courooiiment metere on Relation
Linear Deceder Approximeted Relasinn
advero
camputingg concect antrem
nenn antoryt
endancia gander
scor gande
commans the ofcoumc
abnt group
zimaa urguuge councy
otet souncy
unter
Evaluated on Relation
Evaluated on Relation
(a) matrix the extended dataset.
(b) Subset of relations highlighting the block structure.
Figure 6: Cross-evaluation results for the extended dataset using the 20BB [Black et al.) 2022b model. Each cell of the matrix shows the faithfulness score calculated using the decoder obtained from the row relation, and evaluated on the column relation.
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reations
6.3
Tex-rduzior
6.4
63
62
oou aeetiiass
presstiv sist
couit shond
pertoc chatoie you
if
Relations
Figure 7: Sample-wise faithfulness results with tensor networks on the dataset of Hernandez et al. 2023]. All bars represent the test set for a given relation, after splitting all samples with train-test ratio of 75%-25% respectively. Orange markers denote the majority class baseline.
B Additional Generalization Results
B.1 Sample-Wise Generalization on the Dataset of Hernandez et al. 2023
In this section we discuss Figure depicting the sample-wise test faithfulness results the Dataset of [Hernandez et al. [0233] We can observe that although the mean test faithfulness values in Figure seems relatively low, the tensor network does outperform equals the majority baseline in 34 relations out of 47.
On the extended dataset obtain mean test faithfulness of 0.42 compared to the majority- guesss bascline of 0.30. That said, we outperform the majority baseline in 49 relations and equal in relations out of the total 79 relations.
B.2 Relation- Wise Generalization the Exteded Dataset
In presentt the our generalization result on the extended dataset. On the test set the tensor network outperforms the majority baseline in 15 relations, and the Jacobian baseline (reproduced with the method of Hernandez et al. (2023) in 12 relations out of the 20 test relations.
C Models
C.1
GPT-J 6B
GPT-J (published by Wang and Komatsuzaki 22221) comprises 6 billion parameters and adopts a decoder-only transformer architecture. The model consists of 28 transformer layers, each with 16 attention heads. The vocabulary size is 50,257, and a context window of 2,048 tokens is used. The embedding dimension is 4096. In the transformer blocks, the attention layers and the feed- forward neural networks
ran parallel and summed afterwards with the intent of reducing training time. GPT-J was trained on The Pile [Gao et al.) 2020). an 825 GiB English text corpus curated by EleutherAI to support diverse language modeling capabilities. The Pile comprises 22 high-quality subsets, including academic texts (c.g., arXiv, PubMed), web forums (c.g., Stack Exchange), books. legal documents, and code repositories. For further details we refer to Wang and Komatsuzaki
202T
C.2 Llama 3.1 8B
Meta's Llama 3.1 8B model Dubey and et al. 2024 is an autoregressive language model comprising billion parameters. contains transformer layers, each with 32 query-attention heads sharing key/value heads via grouped-query attention (GQA). The embedding dimension is 4096. The gated
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network uses the SwiGLU activation, while all sub-layers employ RMSNorm pre- normalisation and rotary positional embeddings. LLaMA 3.1 8B has a vocabulary size of 128K and was pretrained over 15T tokens that collected from publicly available sources. For further details we refer to Dubey and et al. [024].
C.3 GPT-NeoX 20B
GPT-NeoX Bllck et al. 2022b is a large open source LLM by EleutherAl and is similar to GPT-J. It has 20 billion parameters, 44 layers, hidden dimension size of 6144, and 64 heads. Similarly to GPT-J, attention and worward networks are ram parallel and summed afterwards. For the exact differences please refer to Black al. [2022a] Similarly GPT-J, GPT-Ne was trained on The dataset. For further details we refer to Black et al. [022b]
Pile [ao et al. 2020)
D Datasets
All our datasets describe relations from the world, represented as subject-relaiion-oojjed triplets. For each relation, we record six attributes:
name: the unique identifier of the relation
prompt templates: one more text templates for querying the model, each containing a placeholder for the subject
zero-shot prompt templates: one or more zero-shot templates
relation type: label used to group the relations (e.g.. bias, commonsense, factual, linguistic)
symmetric flag: boolean flag indicating whether swapping subject and object still yields valid statement
samples: datapoints available for that specific relation. Each sample is cobjjctt-bjeer pair, where substituting the subject into the prompt template zero-shol prompt template yields a query that the object naturally completes
D.1 The Dataset of Hernandez et al. 2023
The dataset of Hernandez et al. 2023 consists of 47 relations (listed in Table D. This dataset provides relations that are semantically distinct, showing close identity cross-evaluation matrix (see Section the main text).
Figure 8: Relation-wil wise faithfulness results with tensor networks on the extended dataset. Blue bars represent relations from the training set, purple bars from the test set splitted randomly with a ratio 255% respectively. Green markers denote the majority class baseline, orange markers show the faithfulness values for individually approximated relation matrices as another baseline.
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Table List of the relations in the dataset of Hernandez et 2002
Relation type
Relation name
Number of samples
bias
characteristic gender
30
bias
univ degree gender
38
bias
name birthplace name gender name religion occupation age occupation gender fruit inside color fruit outside color object superclass substance phase of matter task person type task done by tool word sentiment work location city in country company CEO company hq country capital city country currency country language country largest city food from country landmark in country landmark on continent person lead singer of band person father person mother person native language
bias
19
bias
bias
45
bias
19
commonsense
36
commonsense
30
commonsense
76
commonsense
commonsense
32
commonsense
52
commonsense
60
commonsense
38
factual
27
factual
298
factual
674
factual
24
factual
30
factual
24
factual
24
factual
30
factual
836
factual
947
factual
factual
991
factual
994
factual
919
factual
person occupation
821
factual
person plays instrument
513
factual
person sport position
952
factual
plays pro sport
318
factual
person university
factual
pokemon evolution
44
factual
president birth year president election year
19
factual
19
factual
product by company constellation superhero archnemesis
522
factual
362
factual
96
factual
superhero person
100
linguistic
adjective antonym adjective comparative
100
linguistic
68
linguistic
adjective superlative
80
linguistic
verb past tense word first letter word last letter
76
linguistic
241
linguistic
241
D.2 Extended Dataset
We extend the dataset of Hernandzz al. [2023 with 43 newly constructed relations. These relations make the dataset diverse, containing more relations that share common property not almost always semantically distinct ones. We grouped the additional relations by assuming the following six common properties:
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Uneer Awweereaee
(a) Antonym
(b) Color
(c) Country
Ieturddo
Buladen Mddie
(d) Gender
Year
(I) Age
(g) Identity
Figure 9: Cross-evaluation matrices of the new relations grouped by assumed properties.
age: relations that semantically connected to age (e.g., sheep-lamp in yound animal and interns-young in occupation age)
relations, for which the decoder operator only needs to learn an identity mapping. with no other semantic interpretation
identity:
antonym: relations, for which the decoder has to capture the semantic meaning of anytonyms (e.g., adverb antonym and noun antonym)
color: relations related the color property
gender: relations related to the gender property (for experimental purposes woman and man as target objects)
country: relations, which a semantic meaning of country is present, the object being only country names
year: relations, in which the object is always a year
Overlapping structures in the extended dataset Figure 9 displays cross-cvaluation matrices (evaluated with GPT-J) for the additional relations to provide an overview of the semantic relation- ships. The figure highlights the initial assumptions for the construction of the extended dataset (i.e., the presence of relations holding common properties like country, year, or antonym properties). Conversely, the relations expected to form single gender group split into two distinct blocks: one connected to gender role and another that aligns with the property of name. Finally, the relations presumed to have the age properties turned out to be mutually orthogonal.
For the full list of relations and sample sizes Table 2
Table 2: List of the relations in the extended dataset.
Relation type
Relation name
Number of samples
age
grown animal young animal
11
age
16
age
famous person died at age
20
age
life stage progression
15
age
president birth year
19
age
occupation age
45
(continued on next page)
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Relation type
Relation name
Number of samples
antonym
noun antonym
24
antonym
computing concept antonym
30
antonym
adjective antonym
100
antonym
preposition antonym
30
antonym
adverb antonym
30
bias
name religion
31
color
emotion color color mixing with blue fruit inside color warm cool color fruit outside color complementary colors substance phase of matter object superclass work location word sentiment task person type task done by tool food from country ethnic group in a country primary language spoken in country
14
color
6
color
36
color
20
color
30
color
8
commonsense
50
commonsense
76
commonsense
38
commonsense
60
commonsense
32
commonsense
52
country
30
country
40
country
40
country
landmark in country
128
country
electoral commissioner of country city in country
39
country
27
country
name birthplace urban area a country
31
country
40
factual
country language country largest city person occupation person native language superhero person
24
factual
24
factual
145
factual
99
factual
100
factual
person university
91
factual
constellation name person lead singer of band person father product by company person sport position country currency
1 117
factual
21
factual
92
factual
103
factual
146
factual
30
factual
person mother company hq pokemon evolution
104
factual
89
factual
44
factual
company CEO plays pro sport landmark continent person plays instrument superhero archnemesis country capital city characteristic gender author gender
89
factual
117
factual
59
factual
121
factual
96
factual
24
role
30
name
21
name
scientist gender occupation gender character gender athlete gender univ degree gender
24
role
19
name
23
name
25
role
38
name
actor gender
25
(continued next page)
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Relation type
Relation name
year of birth year of death Number of samples
identity
species belonging to a genus semiconductor chip manufactured by company
40
identity
39
identity
independence movement of country
39
identity
mathematical theorem named after mathematician book adapted into film verb past tense a element in the periodic table word first letter word last letter adjective superlative adjective comparative year of coronation
40
identity
40
linguistic
76
linguistic
40
linguistic
125
linguistic
147
linguistic
80
linguistic
68
year
22
year
year war started in
26
year
27
year
28
year
year signed
30
year
year war ended in
25
year
year of coup
26
year
president election year
19
D.3 Mathematical Dataset
The mathematical dataset contains relations that semantically close within mathematical domain. It contains relations on the four basic operations: addition, subtraction, multiplication, and division. We only incorporated the addition and subtraction relations in experiments 50 relations total. We present these 50 relations Table B and refer to the released codebase for the multiplication and division part of the dataset.
D.4 Licensing
We release the extended and the mathematical dataset under the MIT license.
E
Experimantal Details
E.1 Baselines for the Compression Experiments
Below, we discuss how we provide the baselines for the compression experiments. For each relation, obtain the relation decoder using the Jacobian approximation (presented in Hernannez al.) (0233 and described below). For baseline, we average the relation-wise faithfulness of the decoders. The tensor-network models were assessed in the same way: we computed faithfulness for each relation individually and then reported the mean across the entire set.
Jacobian decoder approximations We approximate the relation decoder function o = ( based on the Jacobian W = 0F/0s. Given subject-object pair, we approximate the decoder with the first order Taylor expansion:
F(s) = F(so) + ((s - sp) = Ws b,
where b = F(xo) 8-- To reduce noise, we estimate W and b by averaging a set of subject-object samples rather than relying single pair. For further methodological details, refer the reader to Hernandez et al. [2023].
E.2 Baselines for the Generalization Experiments
To assess the generalization properties of the tensor network-bass based approximation, we compare the faithfulness scores against two baselines for each relation decoder:
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Table List of the addition and subtraction relations of the mathematical dataset.
Relation type
Relation name
Number of samples
addition
number plus 0
201
addition addition
number plus
200
number plus
199
addition
number plus 3
198
addition
number plus
197
addition
number plus 5
196
addition
number plus
195
addition
number plus 7
194
addition
number plus
193
addition
number plus 9
192
addition
number plus
191
addition
number plus 11
190
addition
number plus 12
189
addition
number plus 13
188
addition
number plus 14
187
addition
number plus 15
186
addition
number plus 16
185
addition
number plus 17
184
addition
number plus 18
183
addition
number plus 19
182
addition
number plus 33
168
addition
number plus 50
151
addition
number plus 57
144
addition
number plus 73
128
addition
number plus 100
101
subtraction
number minus
201
subtraction
number minus 2
200
subtraction
number minus?
199
subtraction
number minus 4
198
subtraction
number minus
197
subtraction
number minus 6 number minus
196
subtraction
195
subtraction
number minus
194
subtraction
number minus
193
subtraction
number minus 10
192
subtraction
number minus 11
191
subtraction
number minus 12
190
subtraction
number minus 13
189
subtraction
number minus 14
188
subtraction
number minus 15
187
subtraction
number minus 16
186
subtraction
number minus 17
185
subtraction
number minus 18
184
subtraction
number minus 19
183
subtraction
number minus 20
182
subtraction
number minus 33
168
subtraction
number minus 50
151
subtraction
number minus 57
144
subtraction
number minus 73
128
subtraction
number minus 100
101
1) JJaobban-based decoder approximations, where we measure the faithfulness of decoders approx- imated using Jacobian; and 2) Majority guess, where we select the most frequent object for each relation and compute faithfulness scores if this object were consistently predicted.
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E.3 Hyperparameters
We present all hyperparameters and their respective values in Table We conducted grid search using these values and selected the optimal optimizer, batch size, and learning rate indicated under the "Selected value" column to generate all figures in the paper.
Table 4: List of hyperparameters used in compression experiments.
Hyperparameter
Grid search values
Selected value
General parameters
Optimizer Batch size Learning rate
(SGD, Adam, AdamW) {16, 32}
SGD
16
{0.01, 0.001, 0.0001}
0.001
d,
4096
4096
Compression and sample-wise generalization experiment
d,
{2,4, 6, 8, 30, 100) {10, 50, 100, 300}
d. do
d. = dy' dz'
50
Additional relation embedder
(True, False] 15, 000
Number of iterations
Dataset
Dataset of Hernandez et 2023
Relation-wise generalization experiment
d,r
10
= dof
300
Additional relation embedder
False
Number of iterations Dataset
5000
fextended dataset, mathematical dataset)
Low-rank baselines
Rank
3, 10, 20, 50, 100)
E.4 Hardware
All experiments were internal cluster of either Nvidia A100 40GB Nvidia H100 NVL GPUs. All conducted experiments required cca. 5000 GPU hours.
Tensors and Tensor Networks
Tensors are multidimenssonal arrays, generalizing scalars, vectors, and matrices to higher dimensions.
F1
Basic Concepts
A tensor is defined first and foremost by its order the number of independent indices it carries. Each index often called an axis or, in diagrammatic language, a leg. When we want to emphasise this property, simply say that n-dimensional (order-n) tensor is n-tensor. Every leg receives label.
One can match an index name to each leg: thus 3-tensor T can be denoted Tz where x, V, z are the indices of the legs.
[ xd, xds,
Tensor multiplications are direct generalizations of matrix multiplication. Given matrices A e pdd, xd, and B Xdm, using the tensor notation of An,k and Bk,m. the matrix multiplication
de
(AB) m.
On.
b
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corresponds to the tensor multiplication through the leg k. In general, tensor multiplication of T and U requires a number of legs pairwise the same size be present in both tensors. We can tensor multiply them through any nonempty subset of these pairs. Let (4,, (2,,2..... la be the chosen subset of legs being present in both 7 and U. The other legs of are (4,,42, ........ bn). while the other legs of U are {#1; U2: ... Mm The corresponding tensor multiplication can be calculated via
.. In
I - Ukm.....umm.
Ma
In this paper, only use tensor multiplications through one leg. We call the multiplication of an arbitrary tensor T with a 1-tensor (vector) v a contraction of T' (or contracting T with v). This operation reduces the dimension the number of legs of T by one.
tensor network is a collection of tensors (T*,T,,..) and set of paired legs. We can denote such a network with diagram, which is similar to multigraph, except some edges have only one node as an endpoint, and the other endpoint free. The nodes are the tensors, the edges are legs, and each paired leg connects its two tensors index, but otherwise the names are different, then on the diagram each edge conveniently will have its unique name. One can contract such tensor network by performing tensor multiplications through paired legs in any
graph edge. If one names the paired legs with the same
multiplication corresponds to contracting that edge in the diagram. The result is independent of the order of the contraction, moreover, a single step: the resulting tensor will preserve the free legs of the tensors and each coordinate will be the nested sum of each paired leg.
obtain the same result in
To present this through an example, let take the tensors Ai,k, Bi Cy.alaa Dm,ns Fig We pair the edges denoted with the same index, and the contractions will yield
Yu ΣΣΣΣΣ
which shows how we can treat this tensor network as 2-tensor with respect to tensor multiplication. For further information on tensor networks, we refer to Ahle [2024).
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Reviews:
[{"review_id": "53SMNETETp", "reviewer": "Reviewer_NLdX", "summary": "This paper builds on prior work showcasing that the computation involved in relations like \"The capital of France is___\" (relation=capital, subject=france, object=paris) is well approximated by linear (or affine) maps that generalize across instances of that relation (e.g., the same map works on capital of spain). This work shows that relation decoders can be compressed across different relations that share similar properties, thus viewing these decoders as property extractors rather than single relation decoders. The authors present a novel architecture for training a parameter efficient version of these extractors with tensor networks. The experiments show promising results, but are a little thin in areas. The findings are novel and interesting to the interpretability community.", "questions": "N/A", "limitations": "N/A", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "# Strengths\n* The Hernandez et al., 2023 work is very interesting, and related to a body of interpretability work on related structures (both preceding Hernandez, and currently growing). The property centric view on these relations that allows the compression of what were previously collections of individual relation decoders proposed by this paper would be of great interest to this research community. \n* Shared representational structure within models is an important finding in any domain. This paper provides a nice example with relational structures beyond the single relation level\n\n# Weaknesses\n* Figures 2 and 3 both being devoted to large (slightly illegible) heatmaps is an unfortunate use of space. I think this could be used for more results or visualization that better show the results. Perhaps Figures 2b and 3b become one figure and the 'a' sections go to the appendix?\n* Besides the math dataset, the held out relations don't actually seem to get much benefit from the heldout relations. I count about 8 that look to be substantially far above the majority decision and most underperform the single relation linear decoder. As the authors mention, this result isn't surprising. I think it was worth trying it, but since the result is so strongly expected (this was an extremely hard setup), I feel like it's not the best use of space. The result showing the generalization for math capabilities is very interesting\n\n# Missing citations\n* The authors should provide the proper citations when discussing prototype theory (Rosch, 1971), as well as provide a citation for Wittgenstein when discussing these ideas, although they are so widely known.\n* The authors should cite [Chanin et al., 2024](https://arxiv.org/abs/2311.08968) which extends LREs to concept embeddings. While not redundant with the current work, I think these findings are pretty relevant to this paper. **Specifically, I view the LRCs as property extractors for single relations**, the current paper makes the property extractor idea more salient and introduces compression across relations. Is this a fair account?", "was_revised": true, "final_justification": "See review. I feel positively about this paper", "rebuttal": "Thank you for your valuable suggestions. We will apply the following changes to further improve our paper:\n\n- We will include the missing citations.\n\n- To improve readability in 2b and 3b, we will enlarge fonts and refine labels.\n\n- We will move Figure 4a to the Appendix and only cover its content in the main text. In its place, we will add additional visualizations—e.g., a figure comparing the results of the three evaluated models (GPT‑J, Llama 3.1 8B, and GPT‑NeoX 8B).\n\n- Regarding 2a and 3a, we explored multiple alternatives before settling on the current versions. Our view is that these convey crucial global context: they show that the cross‑evaluation matrices are not diagonal—the surprising observation that motivated our investigation into the semantic structure of relations. Without illustrating this bigger picture, we believe we would lose important context for the reader. With the additional page in the camera‑ready version, we believe that there will be sufficient space to keep this figure in the main text.\n\n- Thank you for drawing our attention to Chanin et al. (2024), as you pointed out, they explore a related but different idea. Their work focuses on (relation, object) pairs with LRCs represented ultimately as vectors, whereas our work focuses on the relations, and the LRE functions themselves. For this reason, we would not interpret LRCs as property extractors in our framework, but there is indeed a meaningful interplay between the notions and approaches presented in the two papers. We will add the reference and a brief paragraph to discuss the relation between the two.\n\nWe are again grateful for your positive feedback. We hope that our responses fully address your suggestions. We are happy to elaborate on any remaining issues throughout the discussion period."}, {"review_id": "ginnEXFgCp", "reviewer": "Reviewer_Jyaq", "summary": "This paper presents an investigation into the decoding of relational knowledge from LLMs. Building based on the Linear Relational Embedding, this work proposes a rank-3 tensor network model to compress an entire collection of linear decoder functions into a single, compact model. The results show that such compression can largely reduce the model sizes.", "questions": "Please see above.", "limitations": "Yes.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths:\n- The paper is quite well written.\n- The designed approach is not too complicated.\n- The motivation is sound.\n- There are some in-depth analysis about why such compression should work.\n- There are some interesting, unobvious findings.\n- Evaluation has included an extended dataset with more relational types and a mathematical dataset.\n\nWeaknesses:\n- Evaluation has mainly focused on GPT-J. Is there any particular reason why this model is chosen? \n- Other models, such as Llama, is shown in the supplementary. How generalizable the results are across different models?\n- In the supplementary, it has been noted that the experiments took a total of 5000 GPU hours. How long would it take for training one instance of the tensor network model?\n- Some GPU memory analysis would be good too, including the effect of the dimension of the embedding space in the original models. \n- Many of the figures are too small to see.\n- Note: all left quotes need to be reformatted. \n- I wonder what the results would be, if instead of using pretrained LLMs, all experiments are conducted on smaller LMs, so that a training from scratch can be conducted using limited datasets. In this way, we can ensure that the LMs have not been exposed to more knowledge present in the test datasets. So we can evaluated whether the compression is not because of it's compressing out knowledge that is not included in the test dataset.", "was_revised": true, "final_justification": "The rebuttal has addressed some of my comments.", "rebuttal": "We thank you for the positive and valuable feedback; below we address each question and detail the updates in the paper we will make based on them.\n\n- We ran most of our experiments on GPT-J simply because it is the model we started to work with.\n\n- Our cross-evaluation experiments with Llama 3.1 8B and GPT-NeoX 8B exhibit strikingly similar behavior to GPT-J, leading us to anticipate similar outcomes across other experiments as well. In the camera-ready version, we will include these figures also in the appendix, and will refer to these additional results more explicitly in the main text of the paper.\n\n- Regarding the legibility of our figures, we will increase the font sizes and refine labels—we can take advantage of the extra page in the camera‑ready version to do so.\n\n- We will also reformat the left quotes.\n\n- We appreciate the suggestion to train smaller LMs from scratch to eliminate any potential pre‑training exposure to the test data. While this is indeed valuable, the scope of our paper is to decode and compactly represent the knowledge already present in pretrained LMs. Exploring this question would address a promising, but rather different research direction: how relational knowledge is acquired under constrained training.\n\n- Wall-clock time: the models train for about 3 hours on a single H100 GPU. Much of the reported 5000 GPU hours were spent on grid searches and frequent evaluations conducted to support a detailed analysis of model behavior.\n\n- GPU memory: for example, in the case of the simple tensor network, the total parameter count is $d \\cdot d_s + d \\cdot d_o + d \\cdot d_r + d_s \\cdot d_r \\cdot d_o$, where $d$ is the embedding dimension of the LLM, and $d_r, d_s, d_o$ are the inner dimensions order-3 tensor in the core. Notably, when substantial compression was achieved, the core tensor parameter count $d_s \\cdot d_r \\cdot d_o$ was significantly lower than that of the projection matrices ($d \\cdot d_s + d \\cdot d_o + d \\cdot d_r$). We will include a brief and exact analysis of the parameter counts and memory usage in the main text, as it would indeed enhance the quality and clarity of the paper.\n\nWe thank you again for the constructive feedback, and we hope we have addressed all your concerns. We remain available to discuss any further questions during the discussion period."}, {"review_id": "gZNo4UZsHh", "reviewer": "Reviewer_Ho1f", "summary": "This paper studies the structure of relation embeddings inside LMs. Past work\nhas found that pre-trained LMs seem to implicitly implement\nHinton-and-Paccanaro-style linear relational embeddings (in the sense that, for\na given relation, you can find a linear map that transforms subject embeddings\ninto object embeddings). Here, we're interested in understanding the structure\nof this embeddings space, in particular whether it's implicitly low dimensional.\nThe paper explores a couple of different low-dimensional tensor network\nparameterizations and finds that all of them significantly improve decoding\naccuracy while reducing the number of parameters needed to represent the full\nset of relations. Additional experiments partly explain this by showing that\nsome relations are highly \"transferable\" (in the sense that applying the\n\"university degree stereotypical gender\" relation to occupations frequently\nyields the \"occupation stereotypical gender\").\n\nI don't have a huge number of comments---overall I thought this was a neat\nempirical paper and likely to be of interest to people working on NN\ninterpretability. That said, there are a couple of technical points that would\nstrengthen the confidence of my recommendation.\n\nFirst, what is being optimized in the equation on line 160, and what is being\nevaluated in Fig 1a? Is faithfulness being measured on a set of training (input,\noutput) pairs, or on a set of held-out pairs? Are the target outputs\n(1) ground-truth values, (2) obtained from model predictions, (3) or obtained by matching\nthe predictions of LREs extracted using the gradient-based method of Hernandez\net al? Even if the outputs are different, are the same inputs being used to\ntrain the tensor networks and the linear decoder baseline?\nIt's quite surprising to me how much faithfulness improves over the\nlinear decoder baseline in Fig 1a, and I'm trying to understand whether this\nresults from a difference in training data / objectives or just from the act of\ncompressing an existing model.\n\nSecond, how much should we interpret these results as telling us something about\n*language model* representations versus linear embeddings in general? The number\nof relations being compressed here is on the order of the number of embeddings\nin the representation space, so it doesn't seem implausible that we would\nachieve high compressibility even starting from random embeddings or embeddings\nfrom a much weaker model (e.g. word2vec). The thing that was surprising about\nthe original Hernandez et al. paper was not that you can embed all these\ndatapoitns, but that an LM's internal computations themselves seem to correspond\nto an LRE that you can read off with just a handful of datapoints. One really\nuseful set of baselines here would be to (1) randomize all the relation\nembeddings, and (2) randomize all the subject / object embeddings, and see how\nmuch this affects your ability to memorize the training set (and with (1), to\ngeneralize to new subject/object pairs)\n\nFinally, a tiny terminological comment---this paper uses \"rank\" to refer to two\ntoally different things: the usual linear-algebraic notion of \"# of linearly\nindependent columns\" (e.g.  \"low-rank embedding\") and \"# of dimensions in a\ntensor\" (e.g. \"rank-3 tensor\"). Consider using \"order\" for the latter to avoid\nconfusion.", "questions": "see above", "limitations": "see above", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "see above", "was_revised": true, "final_justification": "The authors provided a number of new results during the discussion period that clarify the role of test set generalization. My score is not higher mainly because the results seem much stronger on compression than generalization, but I continue to have a positive view of this paper and would be happy to see it accepted.", "rebuttal": "We sincerely appreciate your thoughtful and constructive feedback. We address your questions in detail and provide the requested clarifications below.\n\n**Training details of Figure 1a:**\n\nThroughout our paper we always use the ground-truth values for the target outputs.\n\nIn Figure 1a, we evaluated faithfulness on the set of training (subject, object) pairs. Our intention was to use compressibility as a tool to understand and interpret the decoders’ structural and semantic similarities.\n\n**New results on a (subject, object)-wise train/test split**\n\nTo avoid any confusions however, during the rebuttal period we conducted experiments where we made a train/test split on the (subject, object) pairs. \n\nTo ensure a comprehensive evaluation, we performed experiments on the original dataset as well as the extended and mathematical variants. These results show that on the mathematical dataset the faithfulness value on the test set is close to perfect, 0.97. On the original and extended datasets, we observe weaker generalization: the model exceeds the majority-guess baseline in only about half of the relations and falls short in the remainder, yielding an overall test-faithfulness mean of 0.31. We attribute this drop in performance to having too little training data for our end-to-end training setup (which is naturally expected to be prone to memorization-related issues). This is not the case in the mathematical dataset however: sufficient amount of samples enable this strong generalization (both sample-wise and relation-wise).\n\nIn the camera-ready version of our paper we will elaborate on this level of generalization as well, and include a new figure and analysis with the above content.\n\n**Baselines**\n\nIn Figure 1a, we use the entire set of (subject-object) pairs for a relation. By contrast, the linear decoder baseline follows the Jacobian‑based procedure of Hernandez et al., which estimates each LRE from a couple of examples (in particular, we used 8 examples). We selected this baseline to provide a more canonical reference point. We side with the reviewer that our evaluation could benefit from other baselines as well. We outline several options below that we will include in the camera ready.\n\n**Additional baseline: separately trained LREs**\n\nTwo natural options are:\n- Full-rank LREs: we train the LREs for each given relation separately.\n- Low‑rank LRE variants: we train the LREs for each given relation separately and use a low-rank parametrization.\n\nDuring the rebuttal period we implemented both of these additional baselines to provide a clearer picture. We trained both of these new baselines in the similar end-to-end manner that we used throughout the paper, and also implemented the (subject, object) train/test splitting method discussed above.  \n\n**Tensor networks outperform the “naive approach”**\n\nUnsurprisingly, the results show that indeed separately trained LREs attain high faithfulness—we have an enormous amount of parameters.\n\nNot just more importantly but perhaps even more surprisingly, even with these strong baselines in place, the tensor‑network achieves a better faithfulness‑per‑parameter trade‑off. In particular, the tensor network delivers higher faithfulness than low-rank LREs with even less parameter count.\n\nFor example:\n| Model                       | Parameter count | Mean test faithfulness |\n|----------------------------|-------------------|------------------:|\n|  Low‑rank LRE (rank = 5)   |   2,048,500        | 0.84              |\n|  Low‑rank LRE (rank = 10)  |   4,097,000        | 0.86              |\n|  Low‑rank LRE (rank = 20)  |   8,194,000        | 0.86              |\n|  Tensor network (core size: 47x100x100)  |   1,481,912       | 0.97    |\n\n**Additional baseline: randomize relation embeddings**\n\nDuring the rebuttal period, we have implemented the suggested baseline of randomized relation embeddings and tested it on the mathematical dataset. \nThese results indicate that the absence of the semantic information in relation embeddings results in a drastic performance drop on held-out (subject, object) pairs, whereas training samples can be memorized perfectly.\n\nWe would like to note that the generalization experiments (Figure 4) already demonstrate that the tensor network do make use of the relation embeddings, otherwise, it could not succeed in predicting the object for subjects that correspond to more than one relations in the math dataset (e.g., 13+6=19, 13-3=10). The fact that it exploits even the semantic structure (and do not simply treat relation embeddings as simple discriminative labels) is demonstrated by the positive generalization results on the relation-wise held-out test set.\n\n**Additional baseline: randomize subject/object embeddings**\n\nWe also conducted the suggested experiment with randomized subjects and object embeddings. \n\nSpecifically, we mapped each distinct subject and object to a random vector and used this embedding throughout the training. In our experiments, the tensor network failed to memorize the training set and to generalize on the held-out pairs as well (resulting in a faithfulness of 0).\n\n**Changes to the manuscript**\n\nWe plan to implement the following changes to the manuscript:\n- We will include the sample-wise train/test split results, with a new figure and brief analysis.\n- We will add clarifications regarding the baselines (as discussed above) in a new subsection of Section 3 and in the technical details of Section 2.\n- We will update Figure 1a to reflect the full-rank and low-rank LRE baselines.\n- We will discuss the training with randomized relation and (subject, object) embeddings in a new subsection of Section 3.\n- Thank you for caching the rank/order terminology issue. Will adopt your suggestion in the revised version.\n\nWe thank you again for the constructive feedback, and we hope we have addressed all your concerns. We remain available to discuss any further questions during the discussion period."}, {"review_id": "qX37opzPsW", "reviewer": "Reviewer_svzd", "summary": "The manuscript investigates the collection of Linear Relation Embedding (LRE) matrices within Transformer language models—principally GPT-J-6B.\nThe authors (i) visualize the latent structure of hundreds of relation-specific decoders through a novel semantic proximity metric, and (ii) show that the entire set of matrices can be compressed to a rank-3 tensor network with negligible loss of faithfulness.\nExperiments span three datasets—original (47 relations), expanded (≈100 relations), and a newly constructed mathematical set (hundreds of relations). Despite a reduction of several orders of magnitude in parameter count, the compressed model preserves or even improves faithfulness and acts as a coarse “property extractor.” Generalisation remains narrow: strong transfer occurs mainly for arithmetically well-structured relations, while performance on semantically distant or one-to-many relations is limited.", "questions": "- Have you benchmarked rank-4 or TT-SVD variants, and how do they trade off compression versus fidelity?\n- Could the compressed tensor serve as a base for LoRA-style fine-tuning to further cut memory?", "limitations": "- One-to-many and many-to-many relations remain challenging.\n- Scaling to the full diversity of real-world relations is not yet feasible.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n- Treating relation decoders as a family and clustering them by property similarity is an original perspective that sharpens our understanding of relational structure in LMs.\n- A rank-3 tensor network attains equal or higher faithfulness than individual matrices while reducing parameters by several orders of magnitude.\n\nWeaknesses\n- Limited model diversity – Only a single GPT-J checkpoint is analysed; applicability to other architectures (e.g., Llama-2, Falcon) and languages remains untested.", "was_revised": false, "final_justification": "", "rebuttal": "We are grateful for your valuable and supportive feedback. Below, we provide clarifications and address your questions.\n\n**Model diversity:**\n\nWhile the main text indeed presents experiments using GPT‑J, the supplementary material already includes the cross‑evaluation experiment also with Llama 3.1 8B and GPT‑NeoX 8B, where we observe essentially the same results as for GPT-J.\nFor the camera-ready version:\n- We will extend this analysis and include all of the remaining experiments on Llama 3.1 8B and GPT‑NeoX 8B. \n- We will also explicitly reference these additional results in the main text to ensure readers can easily locate and interpret them.\n- We will also include a figure comparing performance across the different language models.\n\n**Language diversity:**\n\nWhile our method is language-agnostic in principle, evaluating its performance across different languages would require curated cross-lingual relational datasets, which are beyond the scope of this paper. Nonetheless, we recognize that a thorough multilingual analysis is an important and promising direction for future research.\n\n**TT-SVD and order-4 tensor networks:**\n\nOur experimental setup is specific to order-3 tensors (it requires a subject, relation and object embeddings on the free legs). We considered adding additional contextual information via a fourth leg, but this extension is beyond the scope of the present paper.\n\nThank you for suggesting TT-SVD. We hypothesize that the supervised signal provided by the cross-entropy loss in end-to-end training plays a crucial role in guiding what information is retained and compressed within the tensor. Since our training method already proved effective in practice, we did not pursue alternative training or tensor factorization approaches.\n\n**LoRA:**\n\nThank you for bringing up the idea of applying our framework in the context of LoRA. We agree this is a compelling and promising direction, for several reasons:\n\n- A single low-rank LoRA matrix can be expressed with an order-2 tensor network with basically an arbitrary internal structure.\n\n- A setup which is more analogous to our paper is to consider LoRA matrices collectively. We might collect the low‑rank matrices produced by a single LoRA run across an entire network. Alternatively, we could take the matrices obtained from several independent LoRA trainings. In either case, a tensor‑network framework can act as a unified, highly compressed representation of all the LoRA matrices. Such setups can also take contextual information into account on a leg of a tensor network.\n- More broadly, we find this line of thinking highly relevant to mixture-of-experts models, where structured compression and modular representation are increasingly important.\n\nWe appreciate the suggestion, we will mention this in the paper. We believe it opens up several interesting avenues for future work.\n\n\nWe thank you again for the feedback and we hope the clarifications above address all remaining concerns."}]

Author rebuttal:

Meta-review: This is an interesting and well-executed paper that investigates relational knowledge in LLMs and offers several insightful findings. All reviewers spoke positively about the work, and the author rebuttal adequately addressed most of the concerns.

That said, a few limitations remain:

- The studied problem may have a relatively narrow audience within the research community. In some related problems, relational knowledge alone is not a sufficient representation, which suggests that the scope of the proposed interpretation may be limited. Nevertheless, this line of work could potentially open a broader direction in the future.

- The analysis primarily focuses on GPT-J. Extending the study to more modern models would strengthen the generality and impact of the findings.
Decision: Accept (spotlight)