Algorithmic Probability-Guided Machine Learning on Non-Differentiable Spaces

We show how complexity theory can be introduced in machine learning to help bring together apparently disparate areas of current research. We show that this model-driven approach may require less training data and can potentially be more generalizable as it shows greater resilience to random attacks. In an algorithmic space the order of its element is given by its algorithmic probability, which arises naturally from computable processes. We investigate the shape of a discrete algorithmic space when performing regression or classification using a loss function parametrized by algorithmic complexity, demonstrating that the property of differentiation is not required to achieve results similar to those obtained using differentiable programming approaches such as deep learning. In doing so we use examples which enable the two approaches to be compared (small, given the computational power required for estimations of algorithmic complexity). We find and report that 1) machine learning can successfully be performed on a non-smooth surface using algorithmic complexity; 2) that solutions can be found using an algorithmic-probability classifier, establishing a bridge between a fundamentally discrete theory of computability and a fundamentally continuous mathematical theory of optimization methods; 3) a formulation of an algorithmically directed search technique in non-smooth manifolds can be defined and conducted; 4) exploitation techniques and numerical methods for algorithmic search to navigate these discrete non-differentiable spaces can be performed; in application of the (a) identification of generative rules from data observations; (b) solutions to image classification problems more resilient against pixel attacks compared to neural networks; (c) identification of equation parameters from a small data-set in the presence of noise in continuous ODE system problem, (d) classification of Boolean NK networks by (1) network topology, (2) underlying Boolean function, and (3) number of incoming edges.

Space Invaders: Pedestrian Proxemic Utility Functions and Trust Zones for Autonomous Vehicle Interactions

Understanding pedestrian proxemic utility and trust will help autonomous vehicles to plan and control interactions with pedestrians more safely and efficiently. When pedestrians cross the road in front of human-driven vehicles, the two agents use knowledge of each other’s preferences to negotiate and to determine who will yield to the other. Autonomous vehicles will require similar understandings, but previous work has shown a need for them to be provided in the form of continuous proxemic utility functions, which are not available from previous proxemics studies based on Hall’s discrete zones. To fill this gap, a new Bayesian method to infer continuous pedestrian proxemic utility functions is proposed, and related to a new definition of ‘physical trust requirement’ (PTR) for road-crossing scenarios. The method is validated on simulation data then its parameters are inferred empirically from two public datasets. Results show that pedestrian proxemic utility is best described by a hyperbolic function, and that trust by the pedestrian is required in a discrete ‘trust zone’ which emerges naturally from simple physics. The PTR concept is then shown to be capable of generating and explaining the empirically observed zone sizes of Hall’s discrete theory of proxemics.

Analysis of the Discrete Theory of Radiative Transfer in the Coupled “Ocean–Atmosphere” System: Current Status, Problems and Development Prospects

In this paper, we analyze the current state of the discrete theory of radiative transfer. One-dimensional, three-dimensional and stochastic radiative transfer models are considered. It is shown that the discrete theory provides a unique solution to the one-dimensional radiative transfer equation. All approximate solution techniques based on the discrete ordinate formalism can be derived based on the synthetic iterations, the small-angle approximation, and the matrix operator method. The possible directions for the perspective development of radiative transfer are outlined.

Method of cryptologic data transformations

Countering a quantum computer in the process of illegal ultra-high-speed decryption of messages is technically feasible. Information owner must oppose the competitor's computer with tasks, the solution of which requires an infinite number of operations during decryption. For example, the dependence of functions on an infinite number of informative features. The owner encrypts by integrating the functions, the recipient decrypts by solving the integral equations. It is not a discrete but an analog approach that prevails here. The basis for the implementation of this approach was created by Polish scientists. Mathematician Stefan Banach (1892-1945), who created modern functional analysis, and Marian Mazur (1909-1983), the author of " The Qualitative Theory of Information". Their theory was created in contrast with the "Quantitative Information Theory". Cryptologists who have devoted their whole lives to improving the "discrete" theory and found themselves close to power (and finance), try not to recall that Claude Shannon in his basic work "Communication Theory of Secrecy Systems" more than once emphasized the discrete focus of his developments anticipating future research on the specific limitations of his work adapted to the communication theory. Forgetting about the unlimited speeds and amounts of memory of quantum computers the orthodox talk about redundancy and further purely technical issues, including administrative leverages for counteracting against opponents. It is impossible to stop the progress of science. Experiments have shown the reality of creating such post-quantum-level cryptographic systems.

Discrete Two-Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

The theory of the continuous two-dimensional (2D) Fourier transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

Discrete Two Dimensional Fourier Transform in Polar Coordinates Part I: Theory and Operational Rules

The theory of the continuous two-dimensional (2D) Fourier Transform in polar coordinates has been recently developed but no discrete counterpart exists to date. In this paper, we propose and evaluate the theory of the 2D discrete Fourier Transform (DFT) in polar coordinates. This discrete theory is shown to arise from discretization schemes that have been previously employed with the 1D DFT and the discrete Hankel Transform (DHT). The proposed transform possesses orthogonality properties, which leads to invertibility of the transform. In the first part of this two-part paper, the theory of the actual manipulated quantities is shown, including the standard set of shift, modulation, multiplication, and convolution rules. Parseval and modified Parseval relationships are shown, depending on which choice of kernel is used. Similar to its continuous counterpart, the 2D DFT in polar coordinates is shown to consist of a 1D DFT, DHT and 1D inverse DFT.

First-order minisuperspace model for the Faddeev formulation of gravity

Faddeev formulation of general relativity (GR) is considered where the metric is composed of ten vector fields or a ten-dimensional tetrad. Upon partial use of the field equations, this theory results in the usual GR. Earlier we have proposed some minisuperspace model for the Faddeev formulation where the tetrad fields are piecewise constant on the polytopes like 4-simplices or, say, cuboids into which [Formula: see text] can be decomposed. Now we study some representation of this (discrete) theory, an analogue of the Cartan–Weyl connection-type form of the Hilbert–Einstein action in the usual continuum GR.