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Classification Annealing Expectation Maximization

Algorithms AIMech:ClassificationAnnealingExpectationMaximization DRAFT

Aliases: Classification Annealing EM, CAEM

A stochastic hard-clustering algorithm that samples component assignments from current mixture posterior probabilities raised to an inverse positive temperature, maximizes the completed-data classification likelihood, and lowers the temperature according to a chosen schedule.

Identity Notes & Concept Scope The Celeux-Govaert CAEM construction changes the random label distribution within classification EM. It differs from deterministic tempered EM, decreasing-step sufficient-statistic SAEM, and Metropolis simulated annealing with an accept/reject transition. Temperature one recovers the SEM assignment law; the low-temperature limit concentrates on maximal posterior components but retains random choice among ties. No normalized identity matches among 442 pinned and 473 maintained records, including the bare acronym probes. Existing supplemented EM and cross-entropy optimization are distinct methods; qualified aliases avoid claiming SEM or CEM as unique names. ICE, MCEM, SAEM, deterministic tempered EM, simulated annealing and k-means provide neighboring but different scopes. Existing stochastic-EM frontier and incidental references are not records of the new identities. The absence assessment is bounded to this snapshot and corpus. Identity-search counts: 227 matched lines in the worktree and 245 in main; all 137 distinct matching-text windows were read, with ignored and hidden files included.

Claims & Grounded Evidence

3 Claims

Mechanism #powered-posterior

At temperature tau_m>0, set s_ik proportional to [p_k f_k(x_i)]^(1/tau_m), equivalently t_ik^(1/tau_m), and normalize across components. Draw a hard label independently for each observation from s_i and maximize the corresponding completed-data likelihood. The power applies to the entire weight-times-density product. The next iteration uses the updated parameters and scheduled temperature; it is not a fractional-responsibility M-step or a Metropolis acceptance test.

supports Source: A classification EM algorithm for clustering and two stochastic versions · Locator: Section 4.2 Eq4.6, p.11 / PDF13; Section 3 M-step, p.5 / PDF7

Scope #endpoint-and-ties

At tau=1, the assignment law equals SEM. Holding positive component scores fixed while tau approaches zero concentrates mass on maximal scores; with multiple tied maxima, their limiting probabilities are equal rather than the smallest-index tie choice specified for deterministic CEM. Literal division by tau=0 is undefined. This fixed-score limit does not establish that a changing-parameter cooling run converges to a global solution.

supports Source: A classification EM algorithm for clustering and two stochastic versions · Locator: Section 4.2 Eq4.6 and SEM/CEM interpolation discussion, p.11 / PDF13; Section 3 tie rule, p.5 / PDF7; curator temperature and tie calculations

Limitation #schedule-and-stopping

The report gives geometric cooling tau_(m+1)=a*tau_m as a practical choice and explicitly says its analogous CAEM convergence result has not yet been proved. The printed range includes a=1, which does not cool. Its experiments use a=0.97 and stop when a partition is unchanged; an unchanged stochastic partition over a finite comparison is not proof of future absorption. Neither a geometric schedule nor the simulated-annealing analogy supplies a global-convergence guarantee.

supports Source: A classification EM algorithm for clustering and two stochastic versions · Locator: Section 4.2 cooling and convergence discussion, p.12 / PDF14; Section 5 experimental settings, pp.12-13 / PDF14-15

⚠️ Curated Limitations & Negative Results

Positive temperature does not repair zero component scores, empty sampled classes or inadmissible completed-data fits. These require the selected model and explicit restart, constraint or deletion rules; such changes alter the procedure.
The source’s informal SAEM blend on p.12 repeats EM in both terms despite referring to a stochastic contribution. The unseen original annealed-SAEM method is not inferred from that printed expression or merged with CAEM.
The 1991 report’s empirical settings and tables do not establish an optimal schedule, a universal ranking, statistical uncertainty calibration or semantic correctness of clusters. The final 1992 journal full text and author code were not obtained.
No author implementation or historical numerical experiment was reproduced. Finite curator calculations check formula boundaries, not empirical performance or a repaired convergence proof.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:UnsupervisedClustering` Unsupervised Clustering	related_match	The defining procedure estimates a hard grouping of unlabeled observations alongside fitted component parameters. The pinned target is a clustering task, so the algorithm is related without an is-a assertion. The word classification in the historical name does not identify the explicitly supervised AIO:Classification class.

Evidence Sources & Primary Literature

1 Source

Source ID	Publication / Document	Type / Access	Reference & Checksum
`cem-source`	A classification EM algorithm for clustering and two stochastic versions Gilles Celeux, Gérard Govaert	author technical report full_text	ID: inria-00075196v1 sha256:30bd7f969d0f…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T04:22:35Z

Updated: 2026-10-12T04:22:35Z

Source File: data/curated/ClassificationAnnealingExpectationMaximization.yaml