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francisbach.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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fa.bianp.net
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| | | | | The Langevin algorithm is a simple and powerful method to sample from a probability distribution. It's a key ingredient of some machine learning methods such as diffusion models and differentially private learning. In this post, I'll derive a simple convergence analysis of this method in the special case when the ... | |
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nhigham.com
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| | | | | A real $latex n\times n$ matrix $LATEX A$ is symmetric positive definite if it is symmetric ($LATEX A$ is equal to its transpose, $LATEX A^T$) and $latex x^T\!Ax > 0 \quad \mbox{for all nonzero vectors}~x. $ By making particular choices of $latex x$ in this definition we can derive the inequalities $latex \begin{alignedat}{2} a_{ii} &>0... | |
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juliasilge.com
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| | | A data science blog | ||
