Semi-Supervised Local Temporal Poisson Label Propagation on Dynamical Data

Constantin Ruchkin

Keywords: Semi-supervised learning; Poisson Learning; dynamical data; evolving graphs; star–mesh / Kron reduction; classification

Abstract:

We study semi-supervised classification in a dynamic data-stream setting, where objects
and their relations evolve over time while only a small fraction of observations is
labeled. Classical graph-based semi-supervised learning methods, such as label propagation
and Laplacian-based regularization, typically reduce learning to the solution of a global
graph problem. This requires storing the full graph and recomputing the solution whenever
the graph structure changes, which becomes computationally expensive in streaming
environments, especially when noisy, corrupted, or obsolete observations must be removed
promptly from the model. Moreover, classical harmonic formulations degenerate in
extremely low-label regimes.

We propose \emph{Semi-Supervised Local Temporal Poisson Learning} (SLTPL), a local
Poisson-based framework for evolving graphs. The method formulates prediction updates
through a graph Poisson equation with class-dependent sources and sinks induced by labeled
vertices, aggregated through class supernodes. Instead of maintaining the full graph,
SLTPL keeps only a compact active neighborhood, where each newly arriving observation is
connected to a limited set of active neighbors within a temporal window or $k$-NN structure.

The key efficiency mechanism is local graph reduction via the star--mesh transformation
(Kron reduction / Schur complement). We prove that this reduction is exact: under the
zero-sum solvability condition, elimination of zero-forcing unlabeled vertices preserves
Poisson potentials on the active region. We further prove linear convergence of the
iterative Poisson solver on the reduced graph, derive its spectral rate, and bound the
numerical error accumulated over sequential reductions. Computational complexity is
$O(\tau^{2}C)$ per streaming step, where $\tau$ is the active window size and $C$ the
number of classes, compared with $\Omega(n\tau^{2}C)$ for batch recomputation.

We validate SLTPL on two datasets: synthetic Two Moons, ECG arrhythmia classification
(INCART-ECG).
On temporally ordered streams, SLTPL achieves
$88$--$96\%$ accuracy with as few as 2--5 labeled examples per class, consistently
outperforming quantized label propagation and labels-only baselines.
The framework is particularly suitable for sparse-label regimes with local temporal
structure and naturally accommodates concept drift through exponentially decaying
edge weights.

Submission Number: 78