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stevensoojin.kim | ||
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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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almostsuremath.com
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| | | | | The martingale property is strong enough to ensure that, under relatively weak conditions, we are guaranteed convergence of the processes as time goes to infinity. In a previous post, I used Doob's upcrossing inequality to show that, with probability one, discrete-time martingales will converge at infinity under the extra condition of $latex {L^1}&fg=000000$-boundedness. Here, I... | |
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fabricebaudoin.blog
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| | | | | In this lecture, we studySobolev inequalities on Dirichlet spaces. The approach we develop is related to Hardy-Littlewood-Sobolev theory The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I give below I learnt it from my colleague RodrigoBañuelos. It bypasses the Marcinkiewicz interpolation theorem,that was originally used... | |
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blog.otoro.net
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| | | [AI summary] This text discusses the development of a system for generating large images from latent vectors, combining Generative Adversarial Networks (GANs) and Variational Autoencoders (VAEs). It explores the use of Conditional Perceptual Neural Networks (CPPNs) to create images with specific characteristics, such as style and orientation, by manipulating latent vectors. The text also covers the ability to perform arithmetic on latent vectors to generate new images and the potential for creating animations by transitioning between different latent states. The author suggests future research directions, including training on more complex datasets and exploring alternative training objectives beyond Maximum Likelihood. | ||
