Dmitry Pasechnyuk-Vilensky

Dmitry Pasechnyuk-Vilensky

We study sequential interventions under prerequisite constraints. In this setting, admissible intervention sequences are paths in the ideal lattice of a finite prerequisite poset rather than unconstrained action strings. We give an exact local-to-global theory of order sensitivity on this state space. First, we prove that any two admissible paths with the same endpoints differ by a finite sequence of elementary diamond swaps. Second, for edge-additive path valuations, we show that path-independence is equivalent to vanishing diamond curvature, yielding an endpoint potential with a canonical Möbius parameterization on the ideal lattice. Third, we prove that a local diamond field is induced by an edge-based path model if and only if it satisfies cube consistency, with uniqueness after fixing a reference-tree gauge. Under reduced-state longitudinal assumptions, supported reference paths identify reference-path scores, whereas local order effects require two-sided support of both orders on each diamond. These results yield exact planning consequences, including an order-insensitivity bound and dynamic programming on the truncated ideal lattice.

Dmitry Pasechnyuk-Vilensky, Martin Takáč

We develop a unified theoretical framework for neural architectures whose internal representations evolve as stationary states of dissipative Schrödinger-type dynamics on learned latent graphs. Each layer is defined by a fixed-point Schrödinger-type equation depending on a weighted Laplacian encoding latent geometry and a convex local potential. We prove existence, uniqueness, and smooth dependence of equilibria, and show that the dynamics are equivalent under the Bloch map to norm-preserving Landau--Lifshitz flows. Training over graph weights and topology is formulated as stochastic optimization on a stratified moduli space of graphs equipped with a natural Kähler--Hessian metric, ensuring convergence and differentiability across strata. We derive generalization bounds -- PAC-Bayes, stability, and Rademacher complexity -- in terms of geometric quantities such as edge count, maximal degree, and Gromov--Hausdorff distortion, establishing that sparsity and geometric regularity control capacity. Feed-forward composition of stationary layers is proven equivalent to a single global stationary diffusion on a supra-graph; backpropagation is its adjoint stationary system. Finally, directed and vector-valued extensions are represented as sheaf Laplacians with unitary connections, unifying scalar graph, directed, and sheaf-based architectures. The resulting model class provides a compact, geometrically interpretable, and analytically tractable foundation for learning latent graph geometry via fixed-point Schrödinger-type activations.