Structural assumptions behind semi-supervised gains

Here semi-supervised input means labeled examples together with unlabeled examples or their feature marginal $P_X$. The finite harmonic extension uses the supplied graph weights to prefer scores that vary little across large-weight edges; it does not identify the ground-truth conditional label law $P(Y\mid X)$. A cluster assumption posits that points in the same cluster are likely to share a class; low-density separation posits that a classifier boundary lies in a region of small $P_X$-mass; graph smoothness favors scores that vary little across edges with large supplied weights. These are additional modeling assumptions relating $P_X$ and $P(Y\mid X)$, not consequences of observing $P_X$. The pseudo-label procedure is also a training heuristic. This batch states no general population-risk improvement, consistency, minimax, calibration, or graph-limit theorem. The exact harmonic maximum principle concerns the supplied boundary labels, not unknown true labels.

Finite-round confidence-threshold pseudo-labeling: the finite pseudo-label procedure returns a fixed training assignment and makes no risk guarantee.

Finite graph-harmonic label propagation with a residual status: the finite graph solver returns score candidates and residual statuses; only zero-residual exact solutions are the harmonic extension, and neither certifies true labels.

The unlabeled marginal alone does not identify labels: identical feature marginals and observed labels can leave positive-mass labels unidentified.

Zhu, Ghahramani, and Lafferty, Semi-Supervised Learning Using Gaussian Fields and Harmonic Functions, §2

Chapelle, Schölkopf, and Zien, Semi-Supervised Learning, Chapter 1 §1.2.2 and §§1.3.2–1.3.3, and Chapter 11 §11.6