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This paper describes a novel energy-based probabilistic distribution thatrepresents complex-valued data and explains how to apply it to direct featureextraction from complex-valued spectra. The proposed model, the complex-valuedrestricted Boltzmann machine (CRBM), is designed to deal with complex-valuedvisible units as an extension of the well-known restricted Boltzmann machine(RBM). Like the RBM, the CRBM learns the relationships between visible andhidden units without having connections between units in the same layer, whichdramatically improves training efficiency by using Gibbs sampling orcontrastive divergence (CD). Another important characteristic is that the CRBMalso has connections between real and imaginary parts of each of thecomplex-valued visible units that help represent the data distribution in thecomplex domain. In speech signal processing, classification and generationfeatures are often based on amplitude spectra (e.g., MFCC, cepstra, andmel-cepstra) even if they are calculated from complex spectra, and they ignorephase information. In contrast, the proposed feature extractor using the CRBMdirectly encodes the complex spectra (or another complex-valued representationof the complex spectra) into binary-valued latent features (hidden units).Since the visible-hidden connections are undirected, we can also recover(decode) the complex spectra from the latent features directly. Our speechcoding experiments demonstrated that the CRBM outperformed other speech codingmethods, such as methods using the conventional RBM, the mel-log spectrumapproximate (MLSA) decoder, etc.

Complex-Valued Restricted Boltzmann Machine for Direct Speech Parameterization from Complex Spectra