# Using complex number in non-negative matrix factorization (NMF) for signal source separation

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?