Modeling Perceptual Loudness of Piano Tone: Theory and Applications

The relationship between perceptual loudness and physical attributes of sound is an important subject in both computer music and psychoacoustics. Early studies of "equal-loudness contour" can trace back to the 1920s and the measured loudness with respect to intensity and frequency has been revised many times since then. However, most studies merely focus on synthesized sound, and the induced theories on natural tones with complex timbre have rarely been justified. To this end, we investigate both theory and applications of natural-tone loudness perception in this paper via modeling piano tone. The theory part contains: 1) an accurate measurement of piano-tone equal-loudness contour of pitches, and 2) a machine-learning model capable of inferring loudness purely based on spectral features trained on human subject measurements. As for the application, we apply our theory to piano control transfer, in which we adjust the MIDI velocities on two different player pianos (in different acoustic environments) to achieve the same perceptual effect. Experiments show that both our theoretical loudness modeling and the corresponding performance control transfer algorithm significantly outperform their baselines.

Approximation capabilities of neural networks on unbounded domains

If $p \in (1, \infty)$ and if the activation function belongs to a monotone sigmoid, relu, elu, softplus or leaky relu, we prove that neural networks are universal approximators of $L^{p}(\mathbb{R} \times [0, 1]^n)$. This generalizes corresponding universal approximation theorems on $[0,1]^n.$ Moreover if $p \in (1, \infty)$ and if the activation function belongs to a sigmoid, relu, elu, softplus or leaky relu, we show that neural networks never represents non-zero functions in $L^{p}(\mathbb{R} \times \mathbb{R}^+)$ and $L^{p}(\mathbb{R}^2)$.

The option pricing model based on time values: an application of the universal approximation theory on unbounded domains

Hutchinson, Lo and Poggio raised the question that if learning works can learn the Black-Scholes formula, and they proposed the network mapping the ratio of underlying price to strike $S_t/K$ and the time to maturity $\tau$ directly into the ratio of option price to strike $C_t/K$. In this paper we propose a novel descision function and study the network mapping $S_t/K$ and $\tau$ into the ratio of time value to strike $V_t/K$. Time values' appearance in artificial intelligence fits into traders' natural intelligence. Empirical experiments will be carried out to demonstrate that it significantly improves Hutchinson-Lo-Poggio's original model by faster learning and better generalization performance. In order to take a conceptual viewpoint and to prove that $V_t/K$ but not $C_t/K$ can be approximated by superpositions of logistic functions on its domain of definition, we work on the theory of universal approximation on unbounded domains. We prove some general results which imply that an artificial neural network with a single hidden layer and sigmoid activation represents no function in $L^{p}(\RR^2 \times [0, 1]^{n})$ unless it is constant zero, and that an artificial neural network with a single hidden layer and logistic activation is a universal approximator of $L^{2}(\RR \times [0, 1]^{n})$. Our work partially generalizes Cybenko's fundamental universal approximation theorem on the unit hypercube $[0, 1]^{n}$.