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 Aug 4 at 11:52

This question came from our site for professional mathematicians.

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

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.