In short, I wonder which kind of spectrum can be modeled using complex number in NMF. And could an imaginary part possibly be a vector?

For detail, inspired by audio processing paper that used complex number in NMF (1). I think it seems possible to use complex number NMF with my biological spectrum:

Let $Y_{k,m} \in \mathbb{C}$ denote the complex spectrogram of an observed mixture signal. The real part of $Y_{k,m}$ at each frame $m$ is a linear sum of the supervised basis spectra $H_{k,1},…,H_{k,L}$ scaled by time-varying intensities $U_{1,m},…,U_{L,m}$. There are some overlapped basis $H_{k,1},…,H_{k,L}$ which can be distinguished by another known basis information $G_{k,1},…,G_{k,L}$. The overlap cannot be removed because this data has group structure and each overlapped basis can belong to different groups.

The imaginary part of $Y_{k,m}$ at each frame $m$ is a linear sum of basis spectra $G_{k,1},…,G_{k,L}$ with $U_{1,m},…,U_{L,m}$ regardless of intensities. The imaginary part of $Y_{k,m}$ can be very incomplete. In $Y_{k,m}$, the existence of basis spectra $H_{k,l}$ is more important than $G_{k,l}$. Thus, I put $G_{k,l}$ information as the imaginary part of $H_{k,l}$. This NMF can be mathematically described as \begin{equation*} Y_{k, m} \simeq\sum_{l}H_{k, l}U_{l, m} \tag{1} \end{equation*}

where $Y_{k,m} = a_{k,m} + b_{k,m}i$ , $H_{k,l} = h_{k,l} + g_{k,l}i$ , and $U_{l,m} = u_{l,m} + 0i$

Let me ask some questions regarding the model:

  1. Is this model logical for the nature of data? If not, what kind of data is compatible with complex NMF?
  2. Could $b_{k,m}$ and $g_{k,l}$ be vectors?
  3. I used to minimize the cost function of NMF with Kullback-Leibler divergence, $D(Y||HU)$, where all matrices are real number. If the matrices are complex, what should I be careful?

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.