{"id":1743,"date":"2024-02-04T22:29:09","date_gmt":"2024-02-04T22:29:09","guid":{"rendered":"https:\/\/www.ethanepperly.com\/?p=1743"},"modified":"2025-03-10T19:01:35","modified_gmt":"2025-03-10T19:01:35","slug":"note-to-self-gaussian-hypercontractivity","status":"publish","type":"post","link":"https:\/\/www.ethanepperly.com\/index.php\/2024\/02\/04\/note-to-self-gaussian-hypercontractivity\/","title":{"rendered":"Note to Self: Gaussian Hypercontractivity"},"content":{"rendered":"\n

The purpose of this note is to describe the Gaussian hypercontractivity inequality. As an application, we’ll obtain a weaker version of the Hanson\u2013Wright inequality<\/a>.<\/p>\n\n\n\n

The Noise Operator<\/h2>\n\n\n\n

We begin our discussion with the following question:<\/p>\n\n\n\n

\n

Let \"f : \real^n \to \real\" be a function. What happens to \"f\", on average, if we perturb its inputs by a small amount of Gaussian noise?<\/p>\n<\/blockquote>\n\n\n\n

Let’s be more specific about our noise model. Let \"x \in \real^n\" be an input to the function \"f\" and fix a parameter \"0 \le \varrho \le 1\" (think of \"\varrho\" as close to 1). We’ll define the noise corruption of \"x\" to be<\/p>\n\n\n\n

(1) <\/span>   <\/span>\"\[\tilde{x}_\varrho = \varrho \cdot x + \sqrt{1-\varrho^2} \cdot g, \quad \text{where } g\sim \operatorname{Normal}(0,I). \]\"<\/p><\/p>\n\n\n\n

Here, \"\operatorname{Normal}(0,I)\" is the standard multivariate Gaussian distribution<\/a>. In our definition of \"\tilde{x}_\varrho\", we both add Gaussian noise \"\sqrt{1-\varrho^2} \cdot g\" and shrink the vector \"x\" by a factor \"\varrho\". In particular, we highlight two extreme cases:<\/p>\n\n\n\n