# Blind Signal Separation for Sparse Signals

Assume we have $N$ measurements $z_1, z_2, \dots, z_N \in \mathbb{R}^{n_z}$ that are generated by

$$z_i = M v_i + e_i$$

where $v_i \in \mathbb{R}^{n_v}$, $n_v < n_z$ and $e_i \in \mathbb{R}^{n_z}$ is an error sampled from normal distribution $\mathcal{N}(0, \sigma^2 I)$. Unknown (tall) matrix $M \in \mathbb R^{n_z \times n_v}$ should have full column rank.

Given only the measurements $z_1, z_2, \dots, z_N$ and no information about $M$, $v_i$ and $e_i$, my task is to find a matrix $B \in \mathbb R^{n_v \times n_z}$ that estimates

$$\hat v_i \approx B \hat z_i$$

for new samples $(\hat v_i, \hat z_i)$ generated by same distributions as before. This is referred to as blind signal separation (BSS). I don't know the distribution of $v_i$ at all, but I have the information that the entries $v_{i,j}$ will always be around $0$ for all indices $j=1,\dots,n_v$ except a small number of indices, up to $k$ many (it's around $\frac{k}{n_v} \leq 0.2$). So, $v_i$ is always "approximately sparse".

1. Is there a good way to use this information to improve certain BSS algorithms to deliver a $B$ that will more probably deliver solutions $\hat v_i$ which tend to have the above sparsity property?

2. Or are there certain algorithms for BSS that are especially well suited for this special case?

3. Is there maybe a clever way to reformulate the BSS problem to contain the sparsity property?

To explain a bit about my application. I want to build software that is filled with piano music and should extract the notes being played. I take several MP3 files and generate samples $z_i$ from wavelet-transforming the sound signal at different time stamps. So a $z_i$ is basically the vector of amplitudes to different frequencies at a certain time. Then the $v_i$ is the vector of the intensities of the different notes being played (obviously unknown resp. not practicable to look up). If you play one note $v_{i,j}=\delta_{ij}$ you get all its harmonic frequencies in $z_i$ which makes it difficult to filter out.

## migrated from mathoverflow.netAug 4 at 11:52

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• It looks similar to Blind Deconvolution problem just without the assumption of $M$ being a Deconvolution operator. – Royi Aug 6 at 9:54