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I know one can define a least-squares-fit plane as a point and normal using the centroid of a set of points and the singular vector associated with its least singular value.

However, in doing that, is it possible to compute the residual point-to-plane distance error without simply summing all the point-to-plane distances? That is, as SVD can give the least-squares results, is there any part of the process (a singular value, or some matrix product, perhaps) that quickly produces the value of the minimized error?

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  • $\begingroup$ Are you asking about 3 dimensions? If not, what do you mean by a "plane"? Also, you might find it useful to look up "Principal Component Analysis" $\endgroup$ – Ben Grossmann Jul 31 '17 at 22:50
  • $\begingroup$ Yes, a plane in 3D space. I'm familiar with PCA, but not sure why it's relevant here? SVD is a step in PCA but other than that, I don't know why PCA would be necessary for plane fitting $\endgroup$ – marcman Jul 31 '17 at 23:16
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    $\begingroup$ The reason that PCA works is that it finds the best fit subspaces to the columns of your matrix. The span of the first two "principal components" is exactly the plane that you're looking for. The variance along the remaining component (which is also the square of the third singular value) is the sum-of-squares error. $\endgroup$ – Ben Grossmann Jul 31 '17 at 23:24
  • $\begingroup$ @Omnomnomnom: Ah, I see. I think we're describing the same result. Is it not the case that the plane is also the span of the first two left singular vectors, as those are the first two principal components (assuming ordered singular values)? $\endgroup$ – marcman Aug 1 '17 at 0:19
  • $\begingroup$ Right. If the columns of the matrix are the coordinates, then the left singular vectors are what we want. $\endgroup$ – Ben Grossmann Aug 1 '17 at 1:06
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Let $A$ be a matrix whose columns are the coordinates of the points being fitted, relative to the centroid (that is, every column of $A$ is a point being fitted minus the coordinates of the centroid). From the Eckhart-Young thoeorem, we find that if $A$ has singular value decomposition $A = U \Sigma V^T$ where $$ \Sigma = \pmatrix{\sigma_1\\&\sigma_2 \\ & & \sigma_3}, \qquad \sigma_1 \geq \sigma_2 \geq \sigma_3 $$ Then the coordinates of the projection of the columns on the best fit plane are the columns of the matrix $\tilde A = U \tilde \Sigma V^T$, where $$ \tilde \Sigma = \pmatrix{\sigma_1\\&\sigma_2 \\ & & 0} $$ Now, let $\|A\|$ denote the Frobenius norm, which is to say that $\|A\|^2 = \sum_{i,j}|a_{ij}|^2$. We find that the square of the minimized error is given by $$ \|A- \tilde A\|^2 = \|U(\Sigma - \tilde \Sigma)V^T\|^2 = \|\Sigma - \tilde \Sigma\|^2 = \sigma_3^2 $$

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