Polynomial-Time Estimation in Binary Latent-Variable Models

Polynomial-Time Estimation in Binary Latent-Variable Models

Definitions and mathematical statement

Shared notation.

Standard number systems are $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$. Specialized notation and index ranges follow the cited source and the definitions below.

A latent-variable model generates observed variables from unobserved variables and parameters $W$. Identifiability means different parameters produce different observed distributions, modulo the source's allowed symmetries.

Record scope and qualifications.

The catalogue title is retained as metadata. Named certificates, contracts, bounds or frameworks in that title are not asserted as results of the original source extract. The model, sample-access assumptions, parameter encoding, and general identifiability conditions require the source.

Source-record limitation: The binary latent-variable model, parameterization, and general identifiability conditions are absent.

Statement recorded in the supplied source.

Under general identifiability conditions for binary latent-variable models, do statistically consistent procedures exist for estimating the parameter matrix $W$ in time polynomial in the number $d$ of hidden variables? [S2]

Source status.

The supplied catalogue labels this source record open; that metadata was checked against the supplied snapshot on 2026-10-11. The label is preserved for provenance and is not an independent verification of present open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-21.

Short English statement

Estimate the binary latent-model parameter matrix consistently with polynomial dependence on the hidden dimension.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30003870 (OWR-16170-004). Catalogue curation and metadata: CC BY 4.0. The source extract is retained; definitions, English summaries and qualifications are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Matrix Estimation Meets Statistical Network Analysis: Extracting low-dimensional structures in high dimension $2019$, Oberwolfach Reports, DOI 10.4171/owr/2018/29, pp. 1745--1783 https://doi.org/10.4171/owr/2018/29.
• S3 Jaffe et al., ICML 2018 https://proceedings.mlr.press/v80/jaffe18a.html.
• S4 Jaffe et al., preprint https://arxiv.org/abs/1802.09656.
• Source-specific Oberwolfach Reports licences, including CC BY-SA 4.0 where applicable, are retained; see THIRD_PARTY_NOTICES.md.

Preparation and review.

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Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

Identifiability does not supply a uniform sample bound

The primary report restores the model absent from the short extract: $x=W^Th+\sigma\xi$, with $h\in\{0,1\}^d$, $d\le m$ and $W$ of row rank $d$. Its tensor method has exponential dependence on $d$ because it enumerates tensor eigenvectors. We test whether identifiability alone can justify a sample bound polynomial in $d$. Take the noiseless submodel $m=d\ge1$, $\sigma=0$, and independent hidden bits, with $\Pr(h_1=1)=p=2^{-d}$ and all other bits unbiased. Compare $W_0=I_d$ and $W_1=\operatorname{diag}(2,1,\ldots,1)$. Both have full rank and their observed distributions are distinct, hence the two-point parameter family is identifiable.

Verification and remaining gaps

For small $d,N$, every sequence of first-bit observations was enumerated to obtain $1-(1-2^{-d})^N$ exactly, and the bound was checked. This rules out an inference from bare identifiability to a uniform polynomial-in-$d$ sample guarantee. It does not disprove the source's computational target: an algorithm polynomial in the number of supplied samples and in $d$ could still be consistent for every fixed identifiable model as sample size tends to infinity. Quantitative nondegeneracy may also exclude this rare-component family. No such algorithm or computational lower bound is obtained.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2018/29. Relevant passages read: 2018/29 p. 1770, x=W^T h+sigma xi, full row rank, tensor eigenpair enumeration and the polynomial-d question.

https://proceedings.mlr.press/v80/jaffe18a.html. Read primary ICML2018 abstract on moment/tensor estimation, nondegeneracy and consistency; the cached report supplies the explicit model and exponential eigenpair-enumeration dependence.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

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Completion Estimate

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Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, reviewed the written mathematical arguments, and contributed formatting/repository review. This additional model review is not independent human verification. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

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