How Do Artificial Neural Networks Classify Musical Triads? A Case Study in Eluding Bonini's Paradox

How might artificial neural networks (ANNs) inform cognitive science? Often cognitive scientists use ANNs but do not examine their internal structures. In this paper, we use ANNs to explore how cognition might represent musical properties. We train ANNs to classify musical chords, and we interpret network structure to determine what representations ANNs discover and use. We find connection weights between input units and hidden units can be described using Fourier phase spaces, a representation studied in musical set theory. We find the total signal coming through these weighted connection weights is a measure of the similarity between two Fourier structures: the structure of the hidden unit's weights and the structure of the stimulus. This is surprising because neither of these Fourier structures is computed by the hidden unit. We then show how output units use such similarity measures to classify chords. However, we also find different types of units—units that use different activation functions—use this similarity measure very differently. This result, combined with other findings, indicates that while our networks are related to the Fourier analysis of musical sets, they do not perform Fourier analyses of the kind usually described in musical set theory. Our results show Fourier representations of music are not limited to musical set theory. Our results also suggest how cognitive psychologists might explore Fourier representations in musical cognition. Critically, such theoretical and empirical implications require researchers to understand how network structure converts stimuli into responses.     

Construction of CCZ transform for quadratic APN functions

Almost perfect nonlinear (APN) function is an important type of function in cryptography, especially quadratic APN function. Since the notion of CCZ-equivalence developed, the construction of CCZ transform for APN functions to obtain new APN functions became a critical issue in cryptography. Inspired by the result of Budaghyan who used Gold functions, this article gives the construction of CCZ transform for all quadratic vectorial Boolean functions and proves that for quadratic APN functions, the functions transformed have algebraic degree 3, thus EA-inequivalent to all quadratic functions, and have minimum algebraic degree 2, thus EA-inequivalent to all power functions.