{"title":"论约翰逊-林登斯特劳斯定理中的稀疏性和次高斯性","authors":"Aurélien GarivierUMPA-ENSL, MC2, Emmanuel PilliatUMPA-ENSL","doi":"arxiv-2409.06275","DOIUrl":null,"url":null,"abstract":"We provide a simple proof of the Johnson-Lindenstrauss lemma for sub-Gaussian\nvariables. We extend the analysis to identify how sparse projections can be,\nand what the cost of sparsity is on the target dimension.The\nJohnson-Lindenstrauss lemma is the theoretical core of the dimensionality\nreduction methods based on random projections. While its original formulation\ninvolves matrices with Gaussian entries, the computational cost of random\nprojections can be drastically reduced by the use of simpler variables,\nespecially if they vanish with a high probability. In this paper, we propose a\nsimple and elementary analysis of random projections under classical\nassumptions that emphasizes the key role of sub-Gaussianity. Furthermore, we\nshow how to extend it to sparse projections, emphasizing the limits induced by\nthe sparsity of the data itself.","PeriodicalId":501379,"journal":{"name":"arXiv - STAT - Statistics Theory","volume":"76 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On Sparsity and Sub-Gaussianity in the Johnson-Lindenstrauss Lemma\",\"authors\":\"Aurélien GarivierUMPA-ENSL, MC2, Emmanuel PilliatUMPA-ENSL\",\"doi\":\"arxiv-2409.06275\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We provide a simple proof of the Johnson-Lindenstrauss lemma for sub-Gaussian\\nvariables. We extend the analysis to identify how sparse projections can be,\\nand what the cost of sparsity is on the target dimension.The\\nJohnson-Lindenstrauss lemma is the theoretical core of the dimensionality\\nreduction methods based on random projections. While its original formulation\\ninvolves matrices with Gaussian entries, the computational cost of random\\nprojections can be drastically reduced by the use of simpler variables,\\nespecially if they vanish with a high probability. In this paper, we propose a\\nsimple and elementary analysis of random projections under classical\\nassumptions that emphasizes the key role of sub-Gaussianity. Furthermore, we\\nshow how to extend it to sparse projections, emphasizing the limits induced by\\nthe sparsity of the data itself.\",\"PeriodicalId\":501379,\"journal\":{\"name\":\"arXiv - STAT - Statistics Theory\",\"volume\":\"76 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - STAT - Statistics Theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.06275\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - STAT - Statistics Theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.06275","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On Sparsity and Sub-Gaussianity in the Johnson-Lindenstrauss Lemma
We provide a simple proof of the Johnson-Lindenstrauss lemma for sub-Gaussian
variables. We extend the analysis to identify how sparse projections can be,
and what the cost of sparsity is on the target dimension.The
Johnson-Lindenstrauss lemma is the theoretical core of the dimensionality
reduction methods based on random projections. While its original formulation
involves matrices with Gaussian entries, the computational cost of random
projections can be drastically reduced by the use of simpler variables,
especially if they vanish with a high probability. In this paper, we propose a
simple and elementary analysis of random projections under classical
assumptions that emphasizes the key role of sub-Gaussianity. Furthermore, we
show how to extend it to sparse projections, emphasizing the limits induced by
the sparsity of the data itself.