RESOLVE: Relational Reasoning with Symbolic and Object-Level Features Using Vector Symbolic Processing

Abhijit Chatterjee; Chandramouli Amarnath; Mohamed Mejri

REVIEW 3 major objections 5 minor 1 cited by

RESOLVE: Relational Reasoning with Symbolic and Object-Level Features Using Vector Symbolic Processing

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read RESOLVE claims that a vector-symbolic layer superposing object and relational codes in high-dimensional bipolar space solves both pure and partial relational reasoning tasks more accurately and more generally than transformer and…

desk verdict A creative VSA twist on relational reasoning whose central attention mechanism is unvalidated and whose efficiency gains don't show up in the data. read the letter →

arxiv 2411.08290 v1 pith:HTDX7STG submitted 2024-11-13 cs.AI cs.LG

Mohamed Mejri , Chandramouli Amarnath , Abhijit Chatterjee This is my paper · ORCID

classification cs.AIcs.LG

keywords vectorsymbolicarchitectureshyperdimensionalcomputingrelationalreasoningbipolarhypervectorsattentionmechanismsequence-to-sequencelearningabstractmathematicalproblemsolving

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

RESOLVE is a proposed neural architecture that aims to fix a known weakness of transformer-based reasoning models: object-level features and relational information are mixed in the same low-dimensional representation and interfere with each other. The paper's central claim is that if both kinds of information are stored as high-dimensional bipolar hypervectors and combined with bundling and binding (elementwise sum and Hadamard product), they can coexist with little interference, so one model handles purely relational tasks (sorting, pairwise ordering) and partially relational tasks (MNIST digit subtraction, math problem-solving) alike. The paper reports higher accuracy and better generalization than the Abstractor, transformer, CorelNet, and PrediNet baselines, and it reports that its bipolar attention-score computation is cheaper in time and memory than standard self-attention. If the claim holds, vector symbolic processing becomes a practical alternative to separating relational from object-level information in sequence-to-sequence reasoning models.

What carries the argument

The load-bearing object is the HD-Attention score $R_{ij} = \cos\bigl(\delta(h_{O_i}), \delta(h_{O_i} \oplus h_{O_j})\bigr)$ computed in a bipolar high-dimensional space, together with bundling $\oplus$ (coordinate-wise summation) and binding $\otimes$ (Hadamard product). The score measures how much the coarse direction of object $i$'s hypervector changes when object $j$ is bundled into it; this directional change is what the paper treats as the relation between the two objects. The binding of the attention-mixed object hypervector with learned symbolic hypervectors is what lets object-level and relational content occupy the same vector with low interference, and the bipolar representation is what makes the score computable with fast sign and addition operations.

What would settle it

Ablate the learned symbols by replacing the learned symbolic hypervectors with random fixed bipolar vectors and retraining only the surrounding encoder and decoder. If accuracy on sorting or ordering stays high, the relational content is not coming from the learned symbols; if accuracy collapses, the symbols are load-bearing. A companion check is to verify on the pairwise-order task that the raw attention scores are order-consistent, i.e., $R_{ij} > R_{ji}$ whenever $o_i \prec o_j$, on held-out pairs.

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Extended reading notes

Core claim

The central discovery claimed is that a vector-symbolic layer can learn abstract relations while preserving object identity, instead of routing relational information through a separate abstractor module. The layer maps each input object to a high-dimensional bipolar vector (entries in $\{-1,+1\}$, dimension $D \sim 10^3$) with a learnable convolution encoder, then computes attention scores $$R_{ij} = \cos\bigl(\delta(h_{O_i}), \delta(h_{O_i} \oplus h_{O_j})\bigr),$$ where $\oplus$ is elementwise bundling and $\delta$ binarizes each coordinate. These scores are softmax-normalized, used to mix the object hypervectors, and the mixed vector is bound (Hadamard product) with learned symbolic hypervectors. The paper's argument is that high dimensionality keeps the superposed object-level and relational codes quasi-orthogonal, so they do not destructively interfere, and that the bipolar form makes attention cheap. On this basis it claims better accuracy and few-sample generalization than the Abstractor and transformer baselines on sorting, pairwise ordering, SET classification, MNIST subtraction, and three math-problem datasets.

Load-bearing premise

The whole method depends on one unproven assumption: that comparing an object with the summed representation of itself and another object, after binarizing, is a trustworthy measure of the relation between the two objects. The paper does not verify this score on its own, only as part of the full model.

Editorial extensions

If this is right

A single RESOLVE module can replace the abstractor in an encoder-decoder, because object and relational codes coexist in one high-dimensional vector rather than being separated into different pathways.
Bipolar attention scores lower the compute and memory cost of self-attention, as shown by the reported roofline bandwidth and peak-performance numbers.
Few-sample generalization improves: the paper reports above 80% accuracy on pairwise ordering with 210 training samples and on low-processed SET classification with 600 samples.
On the math-problem datasets, RESOLVE outperforms both the Abstractor and the transformer on average accuracy and is the strongest model in the 100-sample training regime.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The attention score can be read as a directional-change measure: bundling object $j$ into object $i$ moves $i$'s bipolar hypervector, and the cosine drop is a proxy for how much $j$ alters $i$. This reading suggests the same layer could be tested on other asymmetric relations, such as spatial containment or causal precedence, without redesigning the score.
The quasi-orthogonality argument predicts graceful degradation as sequence length grows, because longer sequences place more bundled objects in superposition; this is testable by running RESOLVE on sorting inputs longer than the six-object training range.
Because the decoder receives object and relational information in one vector, the module may transfer to multimodal reasoning where input objects come from different modalities; the paper mentions multimodal learning only as future work, so this remains an extension, not a claim of the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: this paper is a serious attempt to bring vector symbolic processing into sequence-to-sequence relational reasoning, and the authors deserve credit for shipping code and evaluating on a reasonable spread of tasks. But the load-bearing novelty—the HD-attention score in Eq. 1—is never actually shown to capture relations, and the empirical advantage over the Relational Abstractor mostly evaporates at larger training sizes.

What's new: the combination of learned symbols with high-dimensional bundling and binding inside an encoder-decoder is, as far as I know, not in the literature. Using a bipolar attention score computed over bundled hypervectors is also new, even if the ingredients (Hadamard binding, cosine similarity) are standard. The paper includes code and detailed hyperparameters, which is more than many submissions do.

The experiments are a real attempt to separate pure and partial relational tasks. The low-data regime results are consistently in RESOLVE's favor: it hits above 80% on SET with 600 samples, and the math task gains over the Abstractor are largest at 100 training examples. That's a useful signal.

Now the soft spots, in order of weight.

First, Eq. 1 is the core of the claim but it gets no independent analysis. For iid symmetric inputs, the expected score is 1/2 regardless of relation; the paper never shows that after training the attention matrix actually reflects ordering or subtraction rather than some generic pairwise affinity. This isn't a fatal flaw—the encoder can in principle shape the scores—but the paper's own explanation of why this works is hand-waving. A proper ablation or probe of the attention matrix would settle it.

Second, the scaling story is weaker than advertised. In Table 1, at 10,000 training samples RESOLVE is behind Rel-Abstractor on all three math tasks. The 'better generalizability' claim is really a small-sample advantage. That's worth stating honestly.

Third, the efficiency claim rests on a roofline comparison that shows almost no difference in the reported numbers, and it compares only the attention module, not the full model. With D=1024 and low-dimensional baselines, the complexity argument is not obvious. Actual latency numbers would help.

None of this sinks the paper. It's an architectural proposal with an unverified mechanism, and the empirical evidence is mixed. I'd send it to reviewers, expecting them to push for a real analysis of the attention matrix and a better efficiency evaluation.

Referee Report

3 major / 5 minor

Summary. The paper proposes RESOLVE, a neuro-vector symbolic architecture for relational reasoning. Objects are encoded into high-dimensional bipolar vectors via a learnable HD encoder; a novel HD-attention mechanism computes relation scores using cosine similarity between binarized object vectors and binarized bundles of pairs; the resulting mixed object representation is bound (Hadamard product) with learnable symbol vectors to superpose object-level and relational information. The architecture is evaluated on pairwise ordering, SET classification, MNIST-Math, object sorting, and math problem-solving, comparing against transformers, the Relational Abstractor, CoRelNet, PrediNet, MLP, and LEN. The paper claims higher accuracy, better generalizability, and lower computational overhead than state-of-the-art methods.

Significance. If the claims are substantiated, RESOLVE would be a useful contribution to relational reasoning, particularly for partially relational tasks where object and relational information must coexist. The release of code and detailed hyperparameters in the appendix is a strength. However, the evidence for the central claims is incomplete: the novel HD-attention mechanism is not analyzed or independently validated, and the empirical gains over baselines are mostly confined to the low-data regime with small margins that lack statistical support. The computational overhead comparison also uses mismatched dimensions, weakening the efficiency claim.

major comments (3)

[§5, Eq. (1)] The derivation of the HD-attention score is incorrect. The paper states that because the HD objects are bipolar, their L2 norm is sqrt(D), allowing the cosine similarity to be replaced by ⟨δ(hOi), δ(hOi ⊕ hOj)⟩/D. However, the bundle hOi ⊕ hOj is the real-valued sum of two bipolar vectors; after applying δ, entries where hOi and hOj disagree become zero, so δ(hOi ⊕ hOj) is ternary, not bipolar, and its L2 norm is not sqrt(D) in general. More fundamentally, the paper provides no evidence that this score captures asymmetric relations such as ordering or subtraction: for random i.i.d. inputs the expected score is 1/2, and the paper does not show that training shapes the attention matrix into a meaningful relation. Since contribution 2 and the explanation for the accuracy gains rest on this mechanism, the authors should correct the derivation, provide a theoretical characterization, or analyze the learned attention matrix independently of end-task accuracy.
[§7.4, Table 1] The claim of 'better generalizability' and 'higher accuracy' is only partially supported. At the largest training size (10,000 samples), RESOLVE is behind Rel-Abstractor on all three tasks (Comparison 52.36 vs 52.46, Closest 66.84 vs 69.19, Place Value 98.68 vs 99.43). The reported averages (e.g., 41.86 vs 41.01 overall) are dominated by low-data performance, and no error bars or statistical significance tests are reported for these numbers. The conclusion that RESOLVE 'outperforms the state of the art' should be qualified to the small-data regime, or supported with variance estimates and significance tests.
[§7.5, Table 2] The computational efficiency comparison is not apples-to-apples: HD-Attention operates in D=1024 dimensions while the self-attention baseline uses embedding sizes 32 and 64. The roofline metrics (π and β) therefore reflect different problem sizes, not the claimed advantage. The table also reports no direct latency or energy measurements. To support contribution 3, the comparison should match dimensionality or report end-to-end runtime and memory for the full models.

minor comments (5)

[Throughout] The manuscript contains numerous typos and grammatical errors, e.g., 'an high-dimensional', 'have generated have generated', 'intereference', 'mechansim', and 'descriminative'; a careful proofread is needed.
[§7, Appendix A.1] Section 7 lists LEN as a baseline, but no LEN results are reported in any experiment; Appendix A.1 also refers to 'LARS-VSA' without defining it.
[Figures 8, 10, 12] The figures do not show error bars or confidence intervals even though the appendix reports multiple runs (5 trials for pairwise order, 10 runs for sorting); without these, the accuracy differences are difficult to assess.
[§5, Step 1] The HD-encoder equation hOi[j] = Σ_k Oi[k]·Bi[j-k] is not fully specified: the dimensions of Bi (RN×(D-F+1)) and the relationship between the convolution output length and D should be clarified.
[§5, bundling] The sentence 'It captures the dominant or relevant features of an object pair' is informal and should be replaced with a precise definition of what bundling is intended to encode and how it relates to the attention score.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the accuracy claims are empirical evaluations of a trained architecture, and the few self-citations are provenance/motivation only.

full rationale

The paper's central claims are not derived from its own definitions. The HD-Attention score in Eq. 1 is a proposed architectural form; the paper does not assert that its test accuracies follow analytically from this form, and no equation in Section 5 is defined in terms of the reported outputs. Bundling and binding are standard VSA operations, and the claim that bipolar vectors have L2 norm sqrt(D) is a straightforward identity that does not smuggle in the target result. The experimental results are obtained by fitting the encoder, symbols, and downstream layers to training losses and measuring accuracy on held-out test splits (sorting, SET, MNIST-MATH, math problem-solving); no fitted parameter is renamed as a prediction. The self-citations to Mejri et al. (2024a,b) are used only to attribute the convolution-based encoder implementation and to note that HDC is known to be low-overhead; the latter is also directly measured in Table 2. Neither citation carries the accuracy or generalization claims. No uniqueness theorem from the authors' prior work is invoked, and no ansatz is imported as an external fact. The paper is therefore self-contained against external benchmarks, and no circular step can be exhibited.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard VSA properties (quasi-orthogonality, linear down-projection fidelity) and on the untested assumption that the specific bundling-based attention score in Eq. 1 encodes the required relations. The only free parameter flagged is the hypervector dimension, which is fixed at 1024 without sensitivity analysis.

free parameters (1)

hypervector dimension D = 1024
Set to 1024 in all RESOLVE experiments; no sensitivity analysis is provided, and the architecture's behavior likely depends on this choice.

assumptions (4)

domain assumption High-dimensional bipolar vectors are quasi-orthogonal, allowing superposition of object and relational information with low interference.
Invoked in Sections 3 and 5, based on VSA literature (Kanerva 2009; Menet et al. 2024), not proven for the specific learned HD encoder used here.
ad hoc to paper Bundling (element-wise summation) captures the dominant or relevant features of an object pair.
Stated in Section 5 without proof; the attention score in Eq. 1 relies on this property.
domain assumption A linear projection can map the bound hypervector hS⊗EO to a low-dimensional space with low information loss.
Assumed in Section 6, cited to Yan et al. 2023; not empirically verified for the trained RESOLVE model.
domain assumption The learnable symbols S can represent abstract relations after training.
Standard for learnable parameters, but the paper does not analyze what relations are actually encoded in the symbols.

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Cite this review

Pith. "Pith review of RESOLVE: Relational Reasoning with Symbolic and Object-Level Features Using Vector Symbolic Processing." pith.science (2026). https://pith.science/paper/HTDX7STG

@misc{pith2026241108290,
  author       = {Pith},
  title        = {Pith review of: RESOLVE: Relational Reasoning with Symbolic and Object-Level Features Using Vector Symbolic Processing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTDX7STG}},
  note         = {Machine review of arXiv:2411.08290}
}

read the original abstract

Modern transformer-based encoder-decoder architectures struggle with reasoning tasks due to their inability to effectively extract relational information between input objects (data/tokens). Recent work introduced the Abstractor module, embedded between transformer layers, to address this gap. However, the Abstractor layer while excelling at capturing relational information (pure relational reasoning), faces challenges in tasks that require both object and relational-level reasoning (partial relational reasoning). To address this, we propose RESOLVE, a neuro-vector symbolic architecture that combines object-level features with relational representations in high-dimensional spaces, using fast and efficient operations such as bundling (summation) and binding (Hadamard product) allowing both object-level features and relational representations to coexist within the same structure without interfering with one another. RESOLVE is driven by a novel attention mechanism that operates in a bipolar high dimensional space, allowing fast attention score computation compared to the state-of-the-art. By leveraging this design, the model achieves both low compute latency and memory efficiency. RESOLVE also offers better generalizability while achieving higher accuracy in purely relational reasoning tasks such as sorting as well as partial relational reasoning tasks such as math problem-solving compared to state-of-the-art methods.

Figures

Figures reproduced from arXiv: 2411.08290 by the authors.

**Figure 1.** Example of purely relational task: Pairwise Ordering Analogical reasoning, which involves recognizing abstract relationships between objects, is fundamental to human abstraction and thought. This contrasts with semantic (meaning-based) and procedural (task-based) knowledge acquired from sensory information, which is typically processed through contemporary approaches like deep neural networks (DNNs). However, most… view at source ↗

**Figure 2.** Two examples of partially relational tasks(Figure 2a and 2b) [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Comparison of a relational bottleneck approach applied on the transformer (Figure 3b) separating [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Comparison between SelfAttention Vaswani et al. (2017) 4a, RelationalCrossAttention Altabaa et al. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 5.** Figure 5: HD-Encoder ϕHD and HD-Attention(O1...N) [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: The RESOLVE module inside an encoder-decoder for sequence-to-sequence tasks. It includes abstract (turquoise path) and object level (red path) information that are superposed. The RESOLVE module operates in a high dimensional space (bold arrows). The rest operate in a …

**Figure 7.** Figure 7: RESOLVE pipelines for four tasks In step 2 , the output of the Attentional Encoders, which consists of a set of encoded objects, is mapped to a high-dimensional space using the HD Encoder 2 . These HD Object hypervectors are then mixed using the HD-Attention 3 mechanis…

**Figure 8.** Figure 8: Experiments on single output purely relational tasks and comparison to SOTA. Order relations: modeling asymmetric relations As described in Altabaa et al. (2023), we generated 64 random objects represented by iid Gaussian vectors oi ∼ N (0, I) ∈ R 32, and established a…

**Figure 9.** Figure 9: The SET game SET: modeling multi-dimensional relations with pre-processed objects In the SET (Altabaa et al., 2023) task, players are presented with a sequence of cards. Each card varies along four dimensions: color, number, pattern and shape. A triplet of cards forms …

**Figure 10.** Figure 10: Experiments on single output partially relational tasks and comparison to SOTA. models when trained on small portion of the dataset. RESOLVE outperforms the state of the art with more than 80% accuracy using just 600 training samples. In contrast to the Section 7.1, P…

**Figure 11.** Figure 11: MNIST-Math classification task MNIST-MATH: extracting mathematical rules from a pair of digit images In this case ( [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 12.** Figure 12: Performance of RESOLVE compared to baselines for 6 elements sequence sorting. We generate have generated random objects for the sorting task. First, we create two sets of random attributes: A = a1, a2, a3, a4, where and B = b1, . . . , b12. Each set of attributes has…

**Figure 13.** Figure 13: Examples of input/target sequences from the math problem-solving dataset. [PITH_FULL_IMAGE:figures/full_fig_p010_13.png]

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Reference graph

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Training and Evaluation The models are trained using cross-entropy loss and the Adam optimizer with a learning rate of5 · 10−4. We use a batch size of 128 and train for 500 epochs. To evaluate 14 Submitted to conference paper at ICLR 2025 the learning curves, we vary the training set size, sampling random subsets ranging from 260 to 460 samples in increments of

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Positional symbols are used as the symbol assignment mechanism, which are learned parameters of the model

The relation activation function isσrel = Softmax. Positional symbols are used as the symbol assignment mechanism, which are learned parameters of the model. Transformer Architecture We implemented the standard Transformer architecture as described by (Vaswani et al., 2017). Both the Encoder and Decoder modules share the same hyperparameters, with an incr...

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The outputs are flattened and passed through a feedforward hidden layer with dimension dff = 64, followed by a final layer with a single neuron and sigmoid activation. A dropout rate of 0.4 is used to prevent overfitting. Abstractor Architecture: The Abstractor module uses the following hyperparameters: number of layersL = 1, relation dimensiondr = 4, sym...

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However, inthistask, theinputfeatures used as a sequence of objects are derived from the first convolutional layer of the pre-trainedCNN

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