Abstract
This paper presents an energy-based probabilistic model that handles nonnegative data in consideration of both linear and logarithmic scales. In audio applications, magnitude of time-frequency representation, including spectrogram, is regarded as one of the most important features. Such magnitude-based features have been extensively utilized in learning-based audio processing. Since a logarithmic scale is important in terms of auditory perception, the features are usually computed with a logarithmic function. That is, a logarithmic function is applied within the computation of features so that a learning machine does not have to explicitly model the logarithmic scale. We think in a different way and propose a restricted Boltzmann machine (RBM) that simultaneously models linear- and log-magnitude spectra. RBM is a stochastic neural network that can discover data representations without supervision. To manage both linear and logarithmic scales, we define an energy function based on both scales. This energy function results in a conditional distribution (of the observable data, given hidden units) that is written as the gamma distribution, and hence the proposed RBM is termed gamma-Bernoulli RBM. The proposed gamma-Bernoulli RBM was compared to the ordinary Gaussian-Bernoulli RBM by speech representation experiments. Both objective and subjective evaluations illustrated the advantage of the proposed model.
Highlights
L EARNING data representation is a fundamental task, and many methods have been proposed, e.g., variational autoencoders (VAEs) [1]–[3], generative adversarial networks (GANs) [4]–[7], autoregressive (AR) models [8], [9], and normalizing flows [10], [11]
The proposed restricted Boltzmann machine (RBM) represents the conditional distribution of the visible units by the gamma distribution, which naturally limits the domain of data to positive numbers
We introduced a general gamma Boltzmann machine and showed that its conditional distribution is given by the gamma distribution
Summary
L EARNING data representation is a fundamental task, and many methods have been proposed, e.g., variational autoencoders (VAEs) [1]–[3], generative adversarial networks (GANs) [4]–[7], autoregressive (AR) models [8], [9], and normalizing flows [10], [11]. The Gaussian-Bernoulli RBM [17], [18] has been utilized for modeling signals through the magnitude spectra. We propose a variant of RBMs called gammaBernoulli RBM for modeling magnitude spectra in consider-. To manage both linear and logarithmic scales, we define an energy function consisting of the usual quadratic term and an additional log-magnitude term This energy function provides a general gamma Boltzmann machine that simultaneously considers linear- and logmagnitude spectra. The proposed RBM represents the conditional distribution of the visible units (given hidden units) by the gamma distribution, which naturally limits the domain of data to positive numbers. The optimal model among the proposed RBMs was investigated by speech representation experiments Both objective and subjective evaluations illustrated the advantage of the gamma-Bernoulli RBM.
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