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Open AccessArticle

Extrinsic Bayesian Optimization on Manifolds

by Yihao Fang, Mu Niu, Pokman Cheung and Lizhen Lin

Algorithms 2023, 16(2), 117; https://doi.org/10.3390/a16020117 - 15 Feb 2023

Cited by 1 | Viewed by 3065

We propose an extrinsic Bayesian optimization (eBO) framework for general optimization problems on manifolds. Bayesian optimization algorithms build a surrogate of the objective function by employing Gaussian processes and utilizing the uncertainty in that surrogate by deriving an acquisition function. This acquisition function represents the probability of improvement based on the kernel of the Gaussian process, which guides the search in the optimization process. The critical challenge for designing Bayesian optimization algorithms on manifolds lies in the difficulty of constructing valid covariance kernels for Gaussian processes on general manifolds. Our approach is to employ extrinsic Gaussian processes by first embedding the manifold onto some higher dimensional Euclidean space via equivariant embeddings and then constructing a valid covariance kernel on the image manifold after the embedding. This leads to efficient and scalable algorithms for optimization over complex manifolds. Simulation study and real data analyses are carried out to demonstrate the utilities of our eBO framework by applying the eBO to various optimization problems over manifolds such as the sphere, the Grassmannian, and the manifold of positive definite matrices. Full article

(This article belongs to the Special Issue Gradient Methods for Optimization)

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40 pages, 4062 KB

Open AccessArticle

Fusions of Consciousness

by Donald D. Hoffman, Chetan Prakash and Robert Prentner

Entropy 2023, 25(1), 129; https://doi.org/10.3390/e25010129 - 9 Jan 2023

Cited by 15 | Viewed by 56497

Abstract

What are conscious experiences? Can they combine to form new experiences? What are conscious subjects? Can they combine to form new subjects? Most attempts to answer these questions assume that spacetime, and some of its particles, are fundamental. However, physicists tell us that spacetime cannot be fundamental. Spacetime, they say, is doomed. We heed the physicists, and drop the assumption that spacetime is fundamental. We assume instead that subjects and experiences are entities beyond spacetime, not within spacetime. We make this precise in a mathematical theory of conscious agents, whose dynamics are described by Markov chains. We show how (1) agents combine into more complex agents, (2) agents fuse into simpler agents, and (3) qualia fuse to create new qualia. The possible dynamics of n agents form an

n (n - 1)

-dimensional polytope with

n^{n}

vertices—the Markov polytope

M_{n}

. The total fusions of n agents and qualia form an

(n - 1)

-dimensional simplex—the fusion simplex

F_{n}

. To project the Markovian dynamics of conscious agents onto scattering processes in spacetime, we define a new map from Markov chains to decorated permutations. Such permutations—along with helicities, or masses and spins—invariantly encode all physical information used to compute scattering amplitudes. We propose that spacetime and scattering processes are a data structure that codes for interactions of conscious agents: a particle in spacetime is a projection of the Markovian dynamics of a communicating class of conscious agents. Full article

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