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Square root graphical model

Model Families AIMech:SquareRootGraphicalModel DRAFT

Aliases: SQR, Square root Markov random field

A family of finite undirected graphical probability models that retains univariate base measures while coupling specified square roots of sufficient statistics through unary terms and a symmetric quadratic form, with parameters restricted to a finite joint partition function.

Identity Notes & Concept Scope SQR denotes the interaction construction of Inouye et al., not a square-root preprocessing pipeline or a radial-basis neural network. Its Poisson and exponential instantiations retain distinct supports, base measures and validity regimes. For ordinary nonnegative statistics the root is nonnegative; the paper explicitly adopts the signed convention s(x)=x when T(x)=x^2 to recover the Gaussian family. That convention is part of the specification, not the identity sqrt(x^2)=x for arbitrary real x. Indicator statistics remain unchanged by their nonnegative root, so the binary/discrete special case is not automatically a new identity. No normalized identity match was found across all 442 pinned AIO labels and all scoped synonyms and 409 preceding native IDs, names and aliases. Fixed-string full-checkout searches included ignored and hidden files in worktree and main, yielding 15 and 12 matched lines; all nine unique contents were read. They describe prior square-root/temporal-square-root leads and publication links, not existing records of these identities. Search preparation included the three preceding modified-count records before their merge; their bytes are retained unchanged. All 186 semantic label/synonym hits were read. Forty-six fully annotated exact CLASS nodes and outgoing edges were inspected: 45 byte-equivalent to prior inspected nodes and one freshly read. Sixty-four contextual YAMLs comprise 46 generated and 18 native records: 59 byte-verified after prior reading and five fully read during this preparation. The three-character SQR acronym was checked in normalized identities and omitted from broad text searches. Absence is bounded to the pinned snapshot, maintained corpus and inspected semantic neighborhood.

Claims & Grounded Evidence

1 Claim

Mechanism #method

For a finite product domain with specified univariate measure mu, log base measure B and nonnegative scalar sufficient statistic T, set s(x)=sqrt(T(x)). The paper makes one explicit signed-root exception: use s(x)=x for T(x)=x^2 on the real line to recover its Gaussian special case. Let Phi be a finite symmetric matrix, including its diagonal. Define p(x)=Z^(-1)*exp[sum_i theta_i*s(x_i)+sum_(i,j) Phi_ij*s(x_i)*s(x_j)+sum_i B(x_i)] with 0<Z<infinity relative to the product measure. The log partition is A=log Z. The site-i full-conditional kernel is exp[Phi_ii*T(t)+(theta_i+2*sum_(j!=i) Phi_ij*s(x_j))*s(t)+B(t)]. Nonedge matrix entries vanish. With theta=0 and diagonal Phi, the law factors into the original independent univariate exponential family at valid diagonal parameters. Normalizability otherwise depends on the chosen base measure and parameter regime.

supports Source: Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies · Locator: Section 3, PDF p. 3, Equation (3) and root-convention footnote; Section 3.1, pp. 4–5, node conditionals; Section 4, p. 6, discrete and signed-Gaussian special cases; Sections 3.2–3.4, pp. 5–6, normalization and estimation scope
context Source: Square Root Graphical Models — Supplementary Material · Locator: Appendices A–B, supplemental PDF pp. 1–2, base-specific tail arguments and notation differences

⚠️ Curated Limitations & Negative Results

This is a construction with an explicit finite-partition requirement, not a theorem that every exponential-family base allows all finite interactions. The unrestricted Poisson case and the restricted continuous exponential case require separate arguments.
Using the ordinary nonnegative root of x^2 gives |x| and does not recover a general Gaussian with a linear x term. The documented signed convention is essential for that special case.
Several displayed definitions in the source omit log even while calling A a log partition. The record uses Z for the integral or sum and A=log Z, preventing confusion between a normalizer and its logarithm.
The graph is finite and Phi is symmetric; an off-diagonal matrix entry contributes twice in the quadratic form. Separately estimated node regressions need not agree on one symmetric joint parameterization.
The node conditionals usually have two sufficient statistics and differ from the original univariate base family. A base-family name is not a claim about every conditional or marginal distribution, and an edge parameter is not a universal marginal-correlation or causal-effect statement.
The printed radial formulas omit the Jacobian needed to interpret x=z*v as a change of continuous coordinates. In p dimensions the L1-radius transformation contributes z^(p-1); a bare ray kernel is not generally the radius distribution. Discrete directions require a counting-measure treatment. The normalization arguments stated here use direct joint domination instead.
The paper proposes nodewise l1-penalized fitting, numerical conditional partitions and annealed importance sampling. Model existence does not establish finite-data graph recovery, sampler mixing, numerical accuracy or optimizer convergence; no author code, empirical fitting or benchmark was reproduced.
Reported synthetic and airport-delay experiments are contextual. The paper’s synthetic description prints a positive diagonal while calling the matrix negative definite, an inconsistent sign convention that must be resolved before reproducing that generator. Airport associations and fitted likelihoods do not establish causal effects or held-out predictive performance.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:Model` Model	broad_match	This concept denotes a specified family of joint probability distributions and therefore falls within the broader mathematical representation described by Model. Particular estimation algorithms and data applications do not define its identity.
`AIO:ProbabilisticGraphicalModel` Probabilistic Graphical Model	related_match	The construction encodes undirected probabilistic dependencies. The pinned AIO target is task-scoped, so related_match connects the model family to graphical modeling without asserting a model is-a task relation.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`method-source`	Square Root Graphical Models: Multivariate Generalizations of Univariate Exponential Families that Permit Positive Dependencies David I. Inouye, Pradeep Ravikumar, Inderjit S. Dhillon	primary paper full_text	sha256:e864091d4bed…
`supplement`	Square Root Graphical Models — Supplementary Material David I. Inouye, Pradeep Ravikumar, Inderjit S. Dhillon	primary paper full_text	sha256:0e6e7115b6dc…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-11T22:41:59Z

Updated: 2026-10-11T22:42:24Z

Source File: data/curated/SquareRootGraphicalModel.yaml