Alternating Projection Methods
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1 Alternating Projection Methods Failure in the Absence of Convexity Matthew Tam AMSI Vacation Scholar with generous support from CSIRO Supervisor: Prof. Jon Borwein CSIRO Big Day In: 2nd & 3rd February 2012 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
2 The Problem The setting: A Hilbert space, H. The players: r 2 sets S 1,..., S r with corresponding projections P Si. The problem: Given an initial point x 0 we seek a feasible point in r i=1 S i. Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
3 The Problem The setting: A Hilbert space, H. The players: r 2 sets S 1,..., S r with corresponding projections P Si. The problem: Given an initial point x 0 we seek a feasible point in r i=1 S i. von Neumann (1933) Suppose S 1, S 2 are closed subspaces, then x 0 H: (P S2 P S1 ) n x 0 P S1 S 2 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
4 The Problem The setting: A Hilbert space, H. The players: r 2 sets S 1,..., S r with corresponding projections P Si. The problem: Given an initial point x 0 we seek a feasible point in r i=1 S i. von Neumann (1933) Bregman (1965) Suppose S 1, S 2 are closed convex sets, then x 0 H: (P S2 P S1 ) n x 0 w. x where x S 1 S 2 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
5 Why It Works? S 2 x 0 P S1 S 2 (x 0 ) S 1 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
6 Why It Works? S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
7 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
8 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
9 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
10 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
11 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
12 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
13 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
14 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
15 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
16 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
17 Why It Works? P S2 P S1 (x 0 ) S 2 x 0 P S1 S 2 (x 0 ) S 1 P S1 (x 0 ) Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
18 Some Variants Douglas-Rachford (1959) Suppose S 1, S 2 are closed convex sets, then x 0 H: x n+1 := x n + R S2 R S1 (x n ) 2 where R Si (x) := 2P Si (x) x then x n w. x, a fixed point, with P S1 (x) S 1 S 2. Dysktra (1986) Suppose S 1, S 2 are closed convex sets, then x 0 H: x 1 n := x 2 n 1, x i n := P Si (x i 1 n I n 1 i ), In i := xn i (xn i 1 In 1) i with initial values x 2 0 := x 0, I i 0 := 0, then x n P S1 S 2 (x 0 ). Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
19 The Hubble Telescope 1 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
20 The Hubble Telescope Before correction 1 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
21 The Hubble Telescope Before correction After correction 1 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
22 Project Aims Investigate behaviour of three alternating projection variants: von Neumann Dykstra Douglas-Rachford Particularly, cases when the underlying subsets are non-convex. 1 Partially answer the question: When does convergence fail? Develop some tools to help better understand behaviour, which are: Visual Interactive Hands-on (literally!) Even behaviour in R 2 is poorly understood. 2 Douglas Rachford Algorithm in the Absence of Convexity, Borwein, JM & Sims, B, (2011). Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
23 Cinderella in Action a 1 x + a 2 y = b x y 1 2 = 1 Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
24 A Line and 1/2-sphere We consider the case when where the sets are the 1/2-sphere and a line. Projection onto the line is simply the orthogonal projection. The 1/2-sphere is more difficult: 3 x y 1 2 = Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
25 Results Sufficient conditions for failure of von Neumann and Dysktra s methods. Behaviour of Douglas-Rachford in particular cases. Some examples of failure. Despite theoretical justification, these methods appear fairly robust. Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
26 Example: von Neumann Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
27 Example: von Neumann Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
28 Example: von Neumann Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
29 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
30 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
31 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
32 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
33 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
34 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
35 Example: Dykstra Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
36 Example: Douglas-Rachford Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
37 Example: Douglas-Rachford Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
38 Example: Douglas-Rachford Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
39 Where To Next? Explain algorithms, in terms of: Local convergence results. Behaviour of iterations, in the case of divergence. Behaviour for infeasible problems. (Can infeasibility be detected?) Convergence Rate. Acceleration schemes. More examples. Ultimately, a more complete theory for non-convex sets. Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
40 Questions? A big thanks the Big Day In organisers, AMSI and CSIRO; And of course to my supervisor Jon Borwein. Matthew Tam (UoN) Alternating Projection Methods CSIRO Big Day In / 14
The Method of Alternating Projections
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