Teaching a World Model to Play Pokemon

In two dimensions, let a random embedding be \(X=(X_1,X_2)\), where \(X_1\) and \(X_2\) are independent standard Gaussians. Choose the unit direction \(u=(3/5,4/5)\). Its projection is A weighted sum of independent Gaussians is still Gaussian, and its mean is zero. To find its variance, remember that variance averages squared deviations from the mean. Multiplying \(X_1\) by \(3/5\) therefore multiplies its variance by \((3/5)^2\); the same applies to \(X_2\). The mixed term averages to zero because the coordinates are independent and centered at zero, so their variance contributions add: The same argument works in \(D\) dimensions. For any unit direction \(u=(u_1,\ldots,u_D)\), its squared components add to one: \(\sum_j u_j^2=1\). The projection \(u^\top X=\sum_j u_jX_j\) is Gaussian with mean zero and variance \(\sum_j u_j^2=1\), so it follows \(\mathcal N(0,1)\). In compact matrix notation, that variance calculation is \(u^\top I_Du=\|u\|^2=1\).