Rd 中高斯分布的经验近似值

IF 1.5 1区 数学 Q1 MATHEMATICS Advances in Mathematics Pub Date : 2024-11-26 DOI:10.1016/j.aim.2024.110041
Daniel Bartl , Shahar Mendelson
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We show that there is an absolute constant <em>c</em> such that for any <span><math><mi>A</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with probability at least <span><math><mn>1</mn><mo>−</mo><mn>2</mn><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mi>c</mi><mi>Δ</mi><mi>m</mi><mo>)</mo></math></span>, for every <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>,<span><span><span><math><munder><mi>sup</mi><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></munder><mo>⁡</mo><mrow><mo>|</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub><mo>−</mo><mi>P</mi><mo>(</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>)</mo><mo>|</mo></mrow><mo>≤</mo><mi>Δ</mi><mo>+</mo><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo><msqrt><mrow><mi>Δ</mi></mrow></msqrt><mo>.</mo></math></span></span></span> Here <span><math><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the variance of <span><math><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub></math></span> and <span><math><mi>Δ</mi><mo>≥</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is determined by an unexpected complexity parameter of <em>A</em> that captures the set's geometry (Talagrand's <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> functional). 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We show that there is an absolute constant <em>c</em> such that for any <span><math><mi>A</mi><mo>⊂</mo><msup><mrow><mi>S</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, with probability at least <span><math><mn>1</mn><mo>−</mo><mn>2</mn><mi>exp</mi><mo>⁡</mo><mo>(</mo><mo>−</mo><mi>c</mi><mi>Δ</mi><mi>m</mi><mo>)</mo></math></span>, for every <span><math><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>,<span><span><span><math><munder><mi>sup</mi><mrow><mi>x</mi><mo>∈</mo><mi>A</mi></mrow></munder><mo>⁡</mo><mrow><mo>|</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></munderover><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub><mo>−</mo><mi>P</mi><mo>(</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>)</mo><mo>|</mo></mrow><mo>≤</mo><mi>Δ</mi><mo>+</mo><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo><msqrt><mrow><mi>Δ</mi></mrow></msqrt><mo>.</mo></math></span></span></span> Here <span><math><mi>σ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> is the variance of <span><math><msub><mrow><mn>1</mn></mrow><mrow><mo>{</mo><mo>〈</mo><mi>G</mi><mo>,</mo><mi>x</mi><mo>〉</mo><mo>≤</mo><mi>t</mi><mo>}</mo></mrow></msub></math></span> and <span><math><mi>Δ</mi><mo>≥</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> is determined by an unexpected complexity parameter of <em>A</em> that captures the set's geometry (Talagrand's <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> functional). The bound, the probability estimate, and the value of <span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> are all (almost) optimal.</div><div>We use this fact to show that if <span><math><mi>Γ</mi><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup><mo>〈</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mi>x</mi><mo>〉</mo><msub><mrow><mi>e</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is the random matrix that has <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>m</mi></mrow></msub></math></span> as its rows, then the structure of <span><math><mi>Γ</mi><mo>(</mo><mi>A</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>Γ</mi><mi>x</mi><mo>:</mo><mi>x</mi><mo>∈</mo><mi>A</mi><mo>}</mo></math></span> is far more rigid and well-prescribed than was previously expected.</div></div>\",\"PeriodicalId\":50860,\"journal\":{\"name\":\"Advances in Mathematics\",\"volume\":\"460 \",\"pages\":\"Article 110041\"},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2024-11-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0001870824005577\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0001870824005577","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

摘要

假设 G1,...,Gm 是 Rd 中标准高斯随机向量的独立副本。我们证明,对于任何 A⊂Sd-1,对于每 t∈R,supx∈A|1m∑i=1m1{〈Gi,x〉≤t}-P(〈G,x〉≤t)|≤Δ+σ(t)Δ,概率至少为 1-2exp(-cΔm)。这里,σ(t) 是 1{〈G,x〉≤t}的方差,Δ≥Δ0,其中Δ0 由 A 的意外复杂度参数决定,该参数捕捉了集合的几何形状(塔拉格兰德的 γ1 函数)。我们利用这一事实证明,如果Γ=∑i=1m〈Gi,x〉ei 是以 G1,...,Gm 为行的随机矩阵,那么Γ(A)={Γx:x∈A} 的结构远比之前预期的要严格和规范。
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Empirical approximation of the gaussian distribution in Rd
Let G1,,Gm be independent copies of the standard gaussian random vector in Rd. We show that there is an absolute constant c such that for any ASd1, with probability at least 12exp(cΔm), for every tR,supxA|1mi=1m1{Gi,xt}P(G,xt)|Δ+σ(t)Δ. Here σ(t) is the variance of 1{G,xt} and ΔΔ0, where Δ0 is determined by an unexpected complexity parameter of A that captures the set's geometry (Talagrand's γ1 functional). The bound, the probability estimate, and the value of Δ0 are all (almost) optimal.
We use this fact to show that if Γ=i=1mGi,xei is the random matrix that has G1,,Gm as its rows, then the structure of Γ(A)={Γx:xA} is far more rigid and well-prescribed than was previously expected.
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来源期刊
Advances in Mathematics
Advances in Mathematics 数学-数学
CiteScore
2.80
自引率
5.90%
发文量
497
审稿时长
7.5 months
期刊介绍: Emphasizing contributions that represent significant advances in all areas of pure mathematics, Advances in Mathematics provides research mathematicians with an effective medium for communicating important recent developments in their areas of specialization to colleagues and to scientists in related disciplines.
期刊最新文献
Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy Editorial Board A connected sum formula for embedded contact homology Editorial Board Editorial Board
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