{"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/small-width-low-distortions-quantized-random","title":"Small Width, Low Distortions: Quantized Random Embeddings of Low-complexity Sets","arxiv_id":"1504.06170","date":"2015-04-23","proceeding":null,"authors":["Laurent Jacques"],"abstract":"Under which conditions and with which distortions can we preserve the pairwise-distances of low-complexity vectors, e.g., for structured sets such as the set of sparse vectors or the one of low-rank matrices, when these are mapped in a finite set of vectors? This work addresses this general question through the specific use of a quantized and dithered random linear mapping which combines, in the following order, a sub-Gaussian random projection in $\\mathbb R^M$ of vectors in $\\mathbb R^N$, a random translation, or \"dither\", of the projected vectors and a uniform scalar quantizer of resolution $\\delta>0$ applied componentwise. Thanks to this quantized mapping we are first able to show that, with high probability, an embedding of a bounded set $\\mathcal K \\subset \\mathbb R^N$ in $\\delta \\mathbb Z^M$ can be achieved when distances in the quantized and in the original domains are measured with the $\\ell_1$- and $\\ell_2$-norm, respectively, and provided the number of quantized observations $M$ is large before the square of the \"Gaussian mean width\" of $\\mathcal K$. In this case, we show that the embedding is actually \"quasi-isometric\" and only suffers of both multiplicative and additive distortions whose magnitudes decrease as $M^{-1/5}$ for general sets, and as $M^{-1/2}$ for structured set, when $M$ increases. Second, when one is only interested in characterizing the maximal distance separating two elements of $\\mathcal K$ mapped to the same quantized vector, i.e., the \"consistency width\" of the mapping, we show that for a similar number of measurements and with high probability this width decays as $M^{-1/4}$ for general sets and as $1/M$ for structured ones when $M$ increases. Finally, as an important aspect of our work, we also establish how the non-Gaussianity of the mapping impacts the class of vectors that can be embedded or whose consistency width provably decays when $M$ increases.","url_abs":"http://arxiv.org/abs/1504.06170v3","url_pdf":"http://arxiv.org/pdf/1504.06170v3.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":"small-width-low-distortions-quantized-random","repo_url":"https://github.com/VC86/MLSPbox","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}