{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/blind-deconvolution-using-convex-programming","title":"Blind Deconvolution using Convex Programming","arxiv_id":"1211.5608","date":"2012-11-21","proceeding":null,"authors":["Ali Ahmed","Benjamin Recht","Justin Romberg"],"abstract":"We consider the problem of recovering two unknown vectors, $\\boldsymbol{w}$ and $\\boldsymbol{x}$, of length $L$ from their circular convolution. We make the structural assumption that the two vectors are members of known subspaces, one with dimension $N$ and the other with dimension $K$. Although the observed convolution is nonlinear in both $\\boldsymbol{w}$ and $\\boldsymbol{x}$, it is linear in the rank-1 matrix formed by their outer product $\\boldsymbol{w}\\boldsymbol{x}^*$. This observation allows us to recast the deconvolution problem as low-rank matrix recovery problem from linear measurements, whose natural convex relaxation is a nuclear norm minimization program. We prove the effectiveness of this relaxation by showing that for \"generic\" signals, the program can deconvolve $\\boldsymbol{w}$ and $\\boldsymbol{x}$ exactly when the maximum of $N$ and $K$ is almost on the order of $L$. That is, we show that if $\\boldsymbol{x}$ is drawn from a random subspace of dimension $N$, and $\\boldsymbol{w}$ is a vector in a subspace of dimension $K$ whose basis vectors are \"spread out\" in the frequency domain, then nuclear norm minimization recovers $\\boldsymbol{w}\\boldsymbol{x}^*$ without error. We discuss this result in the context of blind channel estimation in communications. If we have a message of length $N$ which we code using a random $L\\times N$ coding matrix, and the encoded message travels through an unknown linear time-invariant channel of maximum length $K$, then the receiver can recover both the channel response and the message when $L\\gtrsim N+K$, to within constant and log factors.","url_abs":"https://arxiv.org/abs/1211.5608v3","url_pdf":"https://arxiv.org/pdf/1211.5608v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"blind-deconvolution-using-convex-programming","repo_url":"https://github.com/maiyuxiaoge/blind_deconvolution","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1211.5608","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}