Enrique ter Horst

Abel Rodriguez, Henryk Gzyl, German Molina et al.

Mounting empirical evidence suggests that the observed extreme prices within a trading period can provide valuable information about the volatility of the process within that period. In this paper we define a class of stochastic volatility models that uses opening and closing prices along with the minimum and maximum prices within a trading period to infer the dynamics underlying the volatility process of asset prices and compares it with similar models that have been previously presented in the literature. The paper also discusses sequential Monte Carlo algorithms to fit this class of models and illustrates its features using both a simulation study and data form the SP500 index.

Enrique ter Horst, Sridhar Mahadevan, Juan Diego Zambrano

We develop a categorical foundation for belief propagation on factor graphs. We construct the free hypergraph category \(\Syn_Σ\) on a typed signature and prove its universal property, yielding compositional semantics via a unique functor to the matrix category \(\cat{Mat}_R\). Message-passing is formulated using a Grothendieck fibration \(\int\Msg \to \cat{FG}_Σ\) over polarized factor graphs, with schedule-indexed endomorphisms defining BP updates. We characterize exact inference as effective descent: local beliefs form a descent datum when compatibility conditions hold on overlaps. This framework unifies tree exactness, junction tree algorithms, and loopy BP failures under sheaf-theoretic obstructions. We introduce HATCC (Holonomy-Aware Tree Compilation), an algorithm that detects descent obstructions via holonomy computation on the factor nerve, compiles non-trivial holonomy into mode variables, and reduces to tree BP on an augmented graph. Complexity is \(O(n^2 d_{\max} + c \cdot k_{\max} \cdot δ_{\max}^3 + n \cdot δ_{\max}^2)\) for \(n\) factors and \(c\) fundamental cycles. Experimental results demonstrate exact inference with significant speedup over junction trees on grid MRFs and random graphs, along with UNSAT detection on satisfiability instances.

Enrique ter Horst, Juan Diego Zambrano

We introduce Teleodynamic Learning, a new paradigm for machine learning in which learning is not the minimization of a fixed objective, but the emergence and stabilization of functional organization under constraint. Inspired by living systems, this framework treats intelligence as the coupled evolution of three quantities: what a system can represent, how it adapts its parameters, and which changes its internal resources can sustain. We formalize learning as a constrained dynamical process with two interacting timescales: inner dynamics for continuous parameter adaptation and outer dynamics for discrete structural change, linked by an endogenous resource variable that both shapes and is shaped by the trajectory. This perspective reveals three phenomena that standard optimization does not naturally capture: self-stabilization without externally imposed stopping rules, phase-structured learning dynamics that move from under-structuring through teleodynamic growth to over-structuring, and convergence guarantees grounded in information geometry rather than convexity. We instantiate the framework in the Distinction Engine (DE11), a teleodynamic learner grounded in Spencer-Brown's Laws of Form, information geometry, and tropical optimization. On standard benchmarks, DE11 achieves 93.3 percent test accuracy on IRIS, 92.6 percent on WINE, and 94.7 percent on Breast Cancer, while producing interpretable logical rules that arise endogenously from the learning dynamics rather than being imposed by hand. More broadly, Teleodynamic Learning unifies regularization, architecture search, and resource-bounded inference within a single principle: learning as the co-evolution of structure, parameters, and resources under constraint. This opens a thermodynamically grounded route to adaptive, interpretable, and self-organizing AI.