arXiv subjects

Eddie Seabrook

Publications and source records attributed to Eddie Seabrook.

4 recordsLinked to original sources

arXiv2025

Slow Feature Analysis on Markov Chains from Goal-Directed Behavior

Slow Feature Analysis is a unsupervised representation learning method that extracts slowly varying features from temporal data and can be used as a basis for subsequent reinforcement learning. Often, the behavior that generates the data on which the representation is learned is assumed to be a uniform random walk. Less research has focused on using samples generated by goal-directed behavior, as commonly the case in a reinforcement learning setting, to learn a representation. In a spatial setting, goal-directed behavior typically leads to significant differences in state occupancy between states that are close to a reward location and far from a reward location. Through the perspective of optimal slow features on ergodic Markov chains, this work investigates the effects of these differences on value-function approximation in an idealized setting. Furthermore, three correction routes, which can potentially alleviate detrimental scaling effects, are evaluated and discussed. In addition, the special case of goal-averse behavior is considered.

cs.LG↗

arXiv2025

What is the relation between Slow Feature Analysis and the Successor Representation?

Slow feature analysis (SFA) is an unsupervised method for extracting representations from time series data. The successor representation (SR) is a method for representing states in a Markov decision process (MDP) based on transition statistics. While SFA and SR stem from distinct areas of machine learning, they share important properties, both in terms of their mathematics and the types of information they are sensitive to. This work studies their connection along these two axes. In particular, both SFA and SR are explored analytically, and in the setting of a one-hot encoded MDP, a formal equivalence is demonstrated in terms of the grid-like representations that occur as solutions/eigenvectors. Moreover, it is shown that the columns of the matrices involved in SFA contain place-like representations, which are formally distinct from place-cell models that have already been defined using SFA.

cs.LG↗

arXiv2022

A Tutorial on the Spectral Theory of Markov Chains

Markov chains are a class of probabilistic models that have achieved widespread application in the quantitative sciences. This is in part due to their versatility, but is compounded by the ease with which they can be probed analytically. This tutorial provides an in-depth introduction to Markov chains, and explores their connection to graphs and random walks. We utilize tools from linear algebra and graph theory to describe the transition matrices of different types of Markov chains, with a particular focus on exploring properties of the eigenvalues and eigenvectors corresponding to these matrices. The results presented are relevant to a number of methods in machine learning and data mining, which we describe at various stages. Rather than being a novel academic study in its own right, this text presents a collection of known results, together with some new concepts. Moreover, the tutorial focuses on offering intuition to readers rather than formal understanding, and only assumes basic exposure to concepts from linear algebra and probability theory. It is therefore accessible to students and researchers from a wide variety of disciplines.

cs.LG↗

arXiv2019

$\mathbb{Z}_2$ vortices in the ground states of classical Kitaev-Heisenberg models

The classical nearest neighbor Kitaev-Heisenberg model on the triangular lattice is known to host $\mathbb{Z}_2$ spin-vortices forming a crystalline superstructure in the ground state. The $\mathbb{Z}_2$ vortices in this system can be understood as distortions of the local $120^\circ$ Néel parent order of the Heisenberg-only Hamiltonian. Here, we explore possibilities of stabilizing further types of $\mathbb{Z}_2$ vortex phases in Kitaev-Heisenberg models including those which rely on more complicated types of non-collinear parent orders such as tetrahedral states. We perform extensive scans through large classes of Kitaev-Heisenberg models on different lattices employing a two-step methodology which first involves a mean-field analysis followed by a stochastic iterative minimization approach. When allowing for longer-range Kitaev couplings we identify several new $\mathbb{Z}_2$ vortex phases such as a state based on the $120^\circ$ Néel order on the triangular lattice which shows a coexistence of different $\mathbb{Z}_2$ vortex types. Furthermore, perturbing the tetrahedral order on the triangular lattice with a suitable combination of first and second neighbor Kitaev interactions we find that a kagome-like superstructure of $\mathbb{Z}_2$ vortices may be stabilized where vortices feature a counter-rotating winding of spins on different sublattices. This last phase may also be extended to honeycomb lattices where it is related to cubic types of parent orders. In total, this analysis shows that $\mathbb{Z}_2$ vortex phases appear in much wider contexts than the $120^\circ$ Néel ordered systems previously studied.

cond-mat.str-el↗