joint of gaussian random transform of finite vectors

[[concept]]

[!themes] Topics

Evaluation Error: SyntaxError: Unexpected token '>'

at DataviewInlineApi.eval (plugin:dataview:19027:21)

at evalInContext (plugin:dataview:19028:7)

at asyncEvalInContext (plugin:dataview:19038:32)

at DataviewJSRenderer.render (plugin:dataview:19064:19)

at DataviewJSRenderer.onload (plugin:dataview:18606:14)

at DataviewJSRenderer.load (app://obsidian.md/app.js:1:1182416)

at DataviewApi.executeJs (plugin:dataview:19607:18)

at DataviewCompiler.eval (plugin:digitalgarden:10763:23)

at Generator.next (<anonymous>)

at eval (plugin:digitalgarden:90:61)

Proposition

Let $y_{1}, \dots, y_{n} \in R^{m}$ and $G \sim N (0, 1)^{\otimes d \times m}$ . Then

\text{Law}\left(\begin{bmatrix} Gy_{1} \\ \vdots \\ Gy_{n} \end{bmatrix}\right) = {\cal N}\left(0,\, \begin{bmatrix} \lvert \lvert y_{1} \rvert \rvert {#2} I_{d} & \langle y_{1},y_{2} \rangle I_{d} & \dots & \langle y_{1},y_{n} \rangle I_{d} \\ \langle y_{2} , y_{1} \rangle I_{d} & \lvert \lvert y_{2} \rvert \rvert {#2} I_{d} & \dots & \langle y_{2}, y_{n} \rangle I_{d} \\ \vdots & \vdots & \ddots & \vdots \\ \langle y_{n}, y_{1} \rangle I_{d} & \langle y_{n} , y_{2} \rangle I_{d} & \dots & \lvert \lvert y_{n} \rvert \rvert {#2} I_{d} \end{bmatrix}\right)

Or, if we define $Y$ such that the $i$ th column is $y_{i}$ , then the the covariance matrix is given by

Cov ([\begin{matrix} G y_{1} \\ ⋮ \\ G y_{n} \end{matrix}]) = Y^{⊺} Y \otimes I_{d}

Proof

This follows immediately from gaussian random matrix transforms vectors into gaussian random vectors by applying the calculation to each block of the covariance matrix. (the expectation is identical)

References

const modules = await cJS()

const COLUMNS = [  
	{ id: "Name", value: page => page.$link },  
	{ id: "Last Modified", value: page => modules.dateTime.getLastMod(page) },
];  
  
return function View() {  
	const current = dc.useCurrentFile();
// Selecting `#game` pages, for example. 
	let queryString = `@page and linksto(${current.$link})`;
	let pages = dc.useQuery(queryString);
	
	// check types
	pages = pages.filter( (p) => !modules.typeCheck.checkAll(p, current) ).sort()
	
	
	return <dc.Table columns={COLUMNS} rows={pages} paging={20}/>;  
}

const { dateTime } = await cJS()

return function View() {
	const file = dc.useCurrentFile();
	return <p class="dv-modified">Created {dateTime.getCreated(file)}     ֍     Last Modified {dateTime.getLastMod(file)}</p>
}

joint of gaussian random transform of finite vectors

References

See Also

Mentions