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pith:LC7PI4DJ

pith:2026:LC7PI4DJXB27BYOG6KMXFP4ILF

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The Expressivity Boundary of Probabilistic Circuits: A Comparison with Large Language Models

Anji Liu, Muhan Zhang, Xuejie Liu, Zhiyu Zhao

Probabilistic circuits match transformer separation rank only on data partitions aligned with their fixed vtree structure and degrade on heterogeneous dependency topologies.

arxiv:2605.12940 v1 · 2026-05-13 · cs.LG · cs.AI

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Claims

C1strongest claim

We prove that structured-decomposable PCs can match Transformer separation rank on vtree-aligned partitions, but show, both theoretically and empirically, that this capacity is limited to partitions aligned with the fixed routing structure, leading to severe degradation when the data exhibits heterogeneous dependency topologies. We further prove that decomposable PCs are strictly more expressive than structured-decomposable ones.

C2weakest assumption

The assumption that language data exhibits heterogeneous dependency topologies that systematically misalign with any fixed vtree structure, and that the separation rank comparison fully captures the practical expressivity gap in autoregressive modeling.

C3one line summary

Probabilistic circuits have an output bottleneck with convex probability combinations and a context bottleneck limited to fixed vtree-aligned partitions, making them less expressive than transformers for language data with heterogeneous dependencies, though decomposable PCs are strictly more capable

References

47 extracted · 47 resolved · 6 Pith anchors

[1] Learning the structure of sum-product net- works via an svd-based algorithm 2015

[2] On the sample complexity of learning sum-product networks

[3] The softmax bottleneck does not limit the probabilities of the most likely tokens 2026

[4] Language mod- els are few-shot learners.Advances in neural information processing systems, 33:1877–1901, 2020 1901

[5] Probabilistic cir- cuits: A unifying framework for tractable probabilistic models.UCLA 2020

Formal links

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Receipt and verification

First computed	2026-05-18T03:09:09.693026Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

58bef47069b875f0e1c6f29972bf88596370f462dea078d26906dc26156bc226

Aliases

arxiv: 2605.12940 · arxiv_version: 2605.12940v1 · doi: 10.48550/arxiv.2605.12940 · pith_short_12: LC7PI4DJXB27 · pith_short_16: LC7PI4DJXB27BYOG · pith_short_8: LC7PI4DJ

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/LC7PI4DJXB27BYOG6KMXFP4ILF \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 58bef47069b875f0e1c6f29972bf88596370f462dea078d26906dc26156bc226

Canonical record JSON

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    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "cs.LG",
    "submitted_at": "2026-05-13T03:22:10Z",
    "title_canon_sha256": "cd2ec51a1e71731ed9259b96b0616a5cf1c338da17bc55b2c0dfc558ffd70e63"
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