Orthogonal Procrustes problem

The orthogonal Procrustes problem [1] is a matrix approximation problem in linear algebra. In its classical form, one is given two matrices ${\displaystyle A}$ and ${\displaystyle B}$ and asked to find an orthogonal matrix ${\displaystyle \Omega }$ which most closely maps ${\displaystyle A}$ to ${\displaystyle B}$. [2] Specifically,

${\displaystyle R=\arg \min _{\Omega }\|\Omega A-B\|_{F}\quad \mathrm {subject\ to} \quad \Omega ^{T}\Omega =I,}$

where ${\displaystyle \|\cdot \|_{F}}$ denotes the Frobenius norm. This is a special case of Wahba's problem (with identical weights; instead of considering two matrices, in Wahba's problem the columns of the matrices are considered as individual vectors). Another difference is, that wahbas problem tries to find a proper rotation matrix, instead of just an orthogonal one.

The name Procrustes refers to a bandit from Greek mythology who made his victims fit his bed by either stretching their limbs or cutting them off.

Solution

This problem was originally solved by Peter Schönemann in a 1964 thesis, and shortly after appeared in the journal Psychometrika. [3] A proof appeared in 1998. [4]

This problem is equivalent to finding the nearest orthogonal matrix to a given matrix ${\displaystyle M=BA^{T}}$. To find this orthogonal matrix ${\displaystyle R}$, one uses the singular value decomposition (for which the entries of ${\displaystyle \Sigma }$ are non negative)

${\displaystyle M=U\Sigma V^{T}\,\!}$

to write

${\displaystyle R=UV^{T}.\,\!}$

Proof

One proof depends on basic properties of the matrix inner product that induces the Frobenius norm:

${\displaystyle {\begin{aligned}R&=\arg \min _{\Omega }||\Omega A-B\|_{F}^{2}\\&=\arg \min _{\Omega }\langle \Omega A-B,\Omega A-B\rangle \\&=\arg \min _{\Omega }\|\Omega A\|_{F}^{2}+\|B\|_{F}^{2}-2\langle \Omega A,B\rangle \\&=\arg \min _{\Omega }\|A\|_{F}^{2}+\|B\|_{F}^{2}-2\langle \Omega A,B\rangle \\&=\arg \max _{\Omega }\langle \Omega ,BA^{T}\rangle \\&=\arg \max _{\Omega }\langle \Omega ,U\Sigma V^{T}\rangle \\&=\arg \max _{\Omega }\langle U^{T}\Omega V,\Sigma \rangle \\&=\arg \max _{\Omega }\langle S,\Sigma \rangle \quad {\text{where }}S=U^{T}\Omega V\\\end{aligned}}}$

This quantity ${\displaystyle S}$ is an orthogonal matrix (as it is a product of orthogonal matrices) and thus the expression is maximised when ${\displaystyle S}$ equals the identity matrix ${\displaystyle I}$. Thus

${\displaystyle {\begin{aligned}I&=U^{T}RV\\R&=UV^{T}\\\end{aligned}}}$

Generalized/constrained Procrustes problems

There are a number of related problems to the classical orthogonal Procrustes problem. One might generalize it by seeking the closest matrix in which the columns are orthogonal, but not necessarily orthonormal. [5]

Alternately, one might constrain it by only allowing rotation matrices (i.e. orthogonal matrices with determinant 1, also known as special orthogonal matrices). In this case, one can write (using the above decomposition ${\displaystyle M=U\Sigma V^{T}}$)

${\displaystyle R=U\Sigma 'V^{T},\,\!}$

where ${\displaystyle \Sigma '\,\!}$ is a modified ${\displaystyle \Sigma \,\!}$, with the smallest singular value replaced by ${\displaystyle \operatorname {sign} (\det(UV^{T}))}$ (+1 or -1), and the other singular values replaced by 1, so that the determinant of R is guaranteed to be positive. [6] For more information, see the Kabsch algorithm.

See also

• Procrustes analysis
• Procrustes transformation
• Wahba's problem

References

1. Gower, J.C; Dijksterhuis, G.B. (2004), Procrustes Problems, Oxford University Press
2. Hurley, J.R.; Cattell, R.B. (1962), "Producing direct rotation to test a hypothesized factor structure", Behavioral Science, 7 (2): 258–262, doi:10.1002/bs.3830070216
3. Schönemann, P.H. (1966), "A generalized solution of the orthogonal Procrustes problem" (PDF), Psychometrika, 31: 1–10, doi:10.1007/BF02289451, S2CID 121676935.
4. Zhang, Z. (1998), A Flexible New Technique for Camera Calibration (PDF), Microsoft Research Technical Report, 71
5. Everson, R (1997), Orthogonal, but not Orthonormal, Procrustes Problems (PDF)
6. Eggert, DW; Lorusso, A; Fisher, RB (1997), "Estimating 3-D rigid body transformations: a comparison of four major algorithms", Machine Vision and Applications, 9 (5): 272–290, doi:10.1007/s001380050048, S2CID 1611749