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In this paper, the applicability of the HiHTP algorithm for solving the bi-sparse blind deconvolution problem is studied. The bi-sparse blind deconvolution setting here consists of recovering\n \n \n $$\\varvec{h}$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$\\varvec{b}$$<\/jats:tex-math>\n \n \n b<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n from the knowledge of\n \n \n $$\\varvec{h}\\varvec{*}\\varvec{(Qb)}$$<\/jats:tex-math>\n \n \n \n h<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2217<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n Q<\/mml:mi>\n b<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , where\n \n \n $$\\varvec{Q}$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is some linear operator, and both\n \n \n $$\\varvec{b}$$<\/jats:tex-math>\n \n \n b<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$\\varvec{h}$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n are assumed to be sparse. The approach rests upon lifting the problem to a linear one, and then applying HiHTP, through the\n hierarchical sparsity framework<\/jats:italic>\n . Then, for a Gaussian draw of the random matrix\n \n \n $$\\varvec{Q}$$<\/jats:tex-math>\n \n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n , it is theoretically shown that an\n \n \n $$\\varvec{s}$$<\/jats:tex-math>\n \n \n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -sparse\n \n \n $$\\varvec{h} \\varvec{\\in } \\varvec{\\mathbb {K}}^{\\varvec{\\mu }}$$<\/jats:tex-math>\n \n \n \n h<\/mml:mi>\n <\/mml:mrow>\n \n \u2208<\/mml:mo>\n <\/mml:mrow>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n and\n \n \n $$\\varvec{\\sigma }$$<\/jats:tex-math>\n \n \n \u03c3<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -sparse\n \n \n $$\\varvec{b} \\varvec{\\in } \\varvec{\\mathbb {K}}^{\\varvec{n}}$$<\/jats:tex-math>\n \n \n \n b<\/mml:mi>\n <\/mml:mrow>\n \n \u2208<\/mml:mo>\n <\/mml:mrow>\n \n \n K<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n with high probability can be recovered when\n \n \n $$\\varvec{\\mu } \\varvec{\\gtrsim } \\varvec{s}\\, \\varvec{\\log }\\varvec{(s)}^{\\varvec{2}}\\, \\varvec{\\log }\\varvec{(\\mu )}\\, \\varvec{\\log }\\varvec{(\\mu n)} \\varvec{+} \\varvec{s}\\varvec{\\sigma }\\, \\varvec{\\log }\\varvec{(n)}$$<\/jats:tex-math>\n \n \n \n \u03bc<\/mml:mi>\n <\/mml:mrow>\n \n \u2273<\/mml:mo>\n <\/mml:mrow>\n \n s<\/mml:mi>\n <\/mml:mrow>\n \n \n log<\/mml:mo>\n <\/mml:mrow>\n \n \n (<\/mml:mo>\n s<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \n log<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n \u03bc<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n \n log<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n \u03bc<\/mml:mi>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \n +<\/mml:mo>\n <\/mml:mrow>\n \n s<\/mml:mi>\n <\/mml:mrow>\n \n \u03c3<\/mml:mi>\n <\/mml:mrow>\n \n \n log<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n .\n <\/jats:p>","DOI":"10.1007\/s10444-025-10271-7","type":"journal-article","created":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T10:46:46Z","timestamp":1764845206000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Bisparse blind deconvolution through hierarchical sparse recovery"],"prefix":"10.1007","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3370-5528","authenticated-orcid":false,"given":"Axel","family":"Flinth","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ingo","family":"Roth","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gerhard","family":"Wunder","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,12,4]]},"reference":[{"key":"10271_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4948-7","volume-title":"A Mathematical Introduction to Compressive Sensing","author":"S Foucart","year":"2013","unstructured":"Foucart, S., Rauhut, H.: A Mathematical Introduction to Compressive Sensing. 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