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?

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