Being low-rank in the time-frequency plane

Valentin Emiya (1) , Ronan Hamon (1) , Caroline Chaux (2)

1 QARMA - éQuipe d'AppRentissage de MArseille
2 I2M - Institut de Mathématiques de Marseille

When solving inverse problems and using optimization methods with matrix variables in signal processing and machine learning, it is customary to assume some low-rank prior on the targeted solution. Nonnegative matrix factorization of spectrograms is a case in point in audio signal processing. However, this low-rank prior is not straightforwardly related to complex matrices obtained from a short-time Fourier – or discrete Gabor – transform (STFT), which is generally defined from and studied based on a modulation operator and a translation operator applied to a so-called window. This paper is a first study of the low-rankness property of time-frequency matrices. We characterize the set of signals with a rank-r (complex) STFT matrix in the case of a unit hop size and frequency step with few assumptions on the transform parameters. We discuss the scope of this result and its implications on low-rank approximations of STFT matrices.

Domains: Signal and Image Processing

Valentin Emiya, Ronan Hamon, Caroline Chaux. Being low-rank in the time-frequency plane. ICASSP - IEEE International Conference on Acoustics, Speech and Signal Processing, Apr 2018, Calgary, Canada. ⟨hal-01636111⟩