How do you perform a least squares fit for $\Sigma$ in the equation $v = u^T\Sigma u$? The term $v$ is a vector of observed variances of a projected Gaussian distribution. The matrix $u$ is made of columns of 2D vectors representing the vectors along which a 2D normal distribution of covariance matrix $\Sigma$ is projected.

Gaussian Projection

This is inspired from the property of 2D normal distributions that the variance of the distribution projected along a vector $u$ is $\sigma^2 = u^T\Sigma u$ (link). I want to perform tomography, but only care about recovering the 2D covariance matrix.


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