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gowers.wordpress.com | ||
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djalil.chafai.net
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| | | | | This post is mainly devoted to a probabilistic proof of a famous theorem due to Schoenberg on radial positive definite functions. Let us begin with a general notion: we say that \( {K:\mathbb{R}^d\times\mathbb{R}^d\rightarrow\mathbb{R}} \) is a positive definite kernel when \[ \forall n\geq1, \forall x_1,\ldots,x_n\in\mathbb{R}^d, \forall c\in\mathbb{C}^n, \quad\sum_{i=1}^n\sum_{j=1}^nc_iK(x_i,x_j)\bar{c}_j\geq0. \] When \( {K} \) is symmetric, i.e. \( {K(x,y)=K(y,x)} \) for... | |
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terrytao.wordpress.com
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| | | | | [This post is dedicated to Luca Trevisan, who recently passed away due to cancer. Though far from his most significant contribution to the field, I would like to mention that, as with most of my ot... | |
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francisbach.com
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| | | | | [AI summary] This article explores the properties of matrix relative entropy and its convexity, linking it to machine learning and information theory. It discusses the use of positive definite matrices in various contexts, including concentration inequalities and kernel methods. The article also includes a lemma on matrix cumulant generating functions and its proof, as well as references to relevant literature. | |
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theconversation.com
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| | | The newly-discovered penguin species went extinct when the ice age hit, but researchers don't think the cold was to blame for their demise. | ||
