Sub-Stiefel Procrustes problem (Problem Procrustesa z macierzami sub-stiefel) Krystyna Ziętak Wrocław April 12, 2016
Outline 1 Joao Cardoso 2 Orthogonal Procrustes problem 3 Unbalanced Stiefel Procrustes problem 4 Sub-Stiefel Procrustes problem 5 Contr-examples 6 Open problems 7 Summary 8 References
Joao Cardoso Orthogonal Procrustes problem Unbalanced Stiefel Procrustes problem Sub-Stiefel Procrustes problem Joao Cardoso, Coimbra, Portugal Manchester, 2013
Procrustes problem Procrustes problems (Frobenius norm): where M: min X M A BX F orthogonal matrices (Green, 1952) symmetric matrices (Higham, 1988) Stiefel matrices (Elden and Park, 1999) sub-stiefel (joint work with Cardoso, 2014)
orthogonal Procrustes problem (square matrices) min X M A BX F = A BQ F where A, B complex square, X M unitary, Q unitary polar factor of C = B A. polar decomposition and SVD C = B A = UΣV = QH (UV )(V ΣV ) Q unitary, H hermitian positive definite C = UΣV SVD
orthogonal rectangular Procrustes problem min X M A BX F = A BQ F where A, B complex, rectangular A, B C m n, X M unitary, Q unitary polar factor of C = B A. Solution is unique, if C nonsingular.
A, B R m n M = {X : X orthogonal, det(x ) = 1} C = B T A = UΣV T, SVD orthogonal rectangular Procrustes problem real rotation variant (see book of Higham) min X M A BX F = A BQ F Q = UDV T, D = diag(1,..., 1, det(uv T )) Solution is unique if det(uv T ) = 1 and σ n 1 (C) 0 or det(uv T ) = 1 and σ n 1 (C) > σ n (C)
symmetric Procrustes problem, Higham 1988 where X M symmetric min X M A BX F solution expressed by means of SVD
Unbalanced Stiefel Procrustes problem min X M A BX F M = {X : X C k n, X X = I } k > n, X orthonormal columns A C m n, B C m k, k > n A and B have different numbers of columns. unbalanced Stiefel Procrustes problem no explicit formulation for solution
unbalanced Stiefel Procrustes problem, real case Necessary conditions for global solution Q (normal equations): B T BQ + QΛ = B T A for some Λ - symmetric matrix of Lagrange multipliers (Elden, Park 1999).
orthogonal Procrustes problem Q is solution iff Q T B T A is positive semi-definite matrix. unbalanced Stiefel Procrustes problem If Q is solution, then Q T B T A is positive semi-definite matrix and there exists symmetric Λ such that B T BQ + QΛ = B T A. Zhang Qui, Du 2007
unbalanced Stiefel Procrustes problem, real case Sufficient conditions (Zhang, Qui, Du 2007): Let orthonormal matrix Q satisfy B T BQ + QΛ = B T A with symmetric Λ. If σ 2 n(b) + λ min (Λ) 0, then Q is solution. If strict inequality, then unique solution.
unbalanced Stiefel Procrustes problem Convergence of some iterative algorithms is not guaranteed (Berg and Knol (1984), Park (1999), Zhang and Du (2006)). Zhang and Du (2006) have proposed successive projection method for B square. Limits of convergent subsequences satisfy certain necessary conditions.
Sub-Stiefel matrix Sub-Stiefel matrix is obtained by taking off the last row and last column of orthogonal matrix of order n + 1 Let S sub denote set of sub-stiefel matrices of order n. sub-stiefel Procrustes problem - A, B real square We find matrix X S sub for which minimum is reached. min X S sub A BX F
Motivation for sub-stiefel matrices: surface unfolding problem in computer vision: reconstructing smooth, flexible and isometrically embedded flat surfaces (Ferreira, Xavier, Costeira (2009), n = 2)
Properties of sub-stiefel matrices. Necessary conditions for solution of sub-stiefel Procrustes problem. When sub-stiefel Procrustes problem has orthogonal solution? Iterative algorithm - in each iteration one solves some orthogonal Procrustes problem.
Sub-Stiefel matrices - characterizations X is sub-stiefel iff σ(x ) = {1,..., 1, s}, 0 s 1 X is sub-stiefel iff where 0 t 1 X = tq 1 + (1 t)q 2, Q 1, Q 2 orthogonal, det(q 1 ) = 1, det(q 2 ) = 1, Q 1 Q2 T symmetric
Let X be solution of sub-stiefel Procrustes problem. Then, for example, we have conditions: A T BX + vv T is symmetric PSD, where vector v such that [X T v] T is Stiefel. If B T (A BX ) is nonsingular, then X is orthogonal. If σ n (B T A) σ 2 1 (B), then there exists orthogonal solution.
B = I Solution: min X S sub A X F X = U A V T A + U A diag(1,..., 1, s )V T A where A = U A Σ A V T A SVD s = min{1, σ min (A)}
Let X be solution of sub-stiefel Procrus. Let min X S sub A BX F = A BX F [ ] X u Y = v T orthogonal for appropriate vectors u, v and number α. α
[ ] X u Y = v T α Then Y is the solution of the orth. Procrus. problem (with extended matrices): [ ] [ ] min A Bu B 0 Y orth v T α 0 T Y 1 = F [ ] [ ] A Bu B 0 F 0 T Y 1 v T α
Iterative algorithm for sub-stiefel Procrustes problem Let X 0 be initial approximation of solution X. The next iterate X 1 is subblock of solution Y 1 of orthogonal Procrustes problem: [ ] [ ] min A Bu0 B 0 Y orth. vo T α 0 0 T Y 1, F where α 0 = σ min (X 0 ) and vectors u 0, v 0 such that [ ] X0 u 0 v0 T orthogonal. α 0 compare iterative algorithms for Stiefel (unbalanced)
Contr-examples Let C = B T A = U C Σ C VC T X 0 = U C VC T, s 0 = 1 Then X 1 = X 0
B nonzero singular, A nonsingular X 0 = U C diag(1,..., 1, s)v T s = σ n(c) (σ 1 (B)) 2 X 1 = X 0
Open problems Necessary and sufficient conditions for solution Conditions for convergence Condition number Acceleration of convergence
γ positive scalar min X γa γbx F = γ min A BX F X Orthogonal polar factor of [ ] B C i = T A B T Bu i (v i ) T s i is solution of extended orthogonal Procrustes problem in each iteration. For scaled problem C i is replaced by [ ] C (γ) γ i = 2 B T A γ 2 B T Bu i (v i ) T s i
C (γ) i = γ 2 ( C i + 1 γ2 γ 2 e n+1 = [0,..., 0, 1] T ) e n+1 (w i ) T C (γ) i is perturbed C i w i = [(v i ) T, s i ] T Orthogonal polar factors can be sensitive for perturbations!
Summary Sub-Stiefel Procrustes problem has been formulated and investigated. Iterative algorithm for Sub-Stiefel Procrustes problem has been proposed. Contr-examples has been presented Open problems were formulated
J.R. Cardoso, K. Ziętak, On a sub-stiefel Procrustes problem arising in computer vision, Numerical Linear Algebra with Applications 22 (2015), 523-547. L. Elden, H. Park, A Procrustes problem on the Stiefel manifold, Numerische Mathematic 82 (1999), 599-619. R. Ferreira, Reconstruction of isometrically embedded flat surfaces from scaled orthographic image data, PhD Thesis, Lisboa, Portugal 2010. B.F. Green, The orthogonal approximation of an oblique structure in factor analysis, Psychometrika 17 (1952), 429-440.
N.J. Higham, Functions of Matrices. Theory and Computation, SIAM, Philadelphia 2008. H. Park, Parallel algorithm for the unbalanced orthogonal Procrustes problem, Parallel Computing 17 (1991), 913 923. Z. Zhang, K. Du, Succesive projection method for solving the unbalanced Procrustes problem, Science in China, Series A, Mathematics 49 (2006), 971 986. Z. Zhang, Y. Qiu, K. Du, Conditions for optimal solutions of unbalanced Procrustes problem on Stiefel manifold, Journal of Computational Mathematics 25 (2007), 661-671.