1. Constrained stress majorization using diagonally scaled gradient projection Tim Dwyer and Kim Marriott Clayton School of Information Technology Monash University Australia

2.  Separation constraints: x 1 +d ≤ x 2, y 1 +d ≤ y 2 can be used with force-directed layout to impose certain spacing requirements Constrained stress majorization layout (x 1,y 1 ) (x 2,y 2 ) (x 3,y 3 ) w1w1 w2w2 h2h2 h3h3 x 1 + ≤ x 2 (w 1 +w 2 ) 2 y 3 + ≤ y 2 (h 2 +h 3 ) 2  In this talk we present:  Diagonal scaling for faster gradient projection  Changes to our active-set solver  Evaluation of the new method  Constrained stress majorization  Stress majorization - reduce overall layout stress  Gradient projection - solve quadratic programs  Active-set solver - projection step

3. “Unix” Graph data From www.graphviz.org

4. Stress majorization stress(X) (x,y)*(x,y)* x* y* x* y*

5.  Instead of solving unconstrained quadratic forms we solve subject to separation constraints  i.e. Quadratic Programming Constrained stress majorization stress(X) x* y* x* y* (x,y)*(x,y)*

6. Gradient projection -g -αg-αg x0x0 x1x1

7. Gradient projection -αg-αg x1x1

8. d x2x2 x1x1 βdβd

9. x*

10.  A badly scaled problem can have poor GP convergence  Condition number of is Convergence

11.  A badly scaled problem can have poor GP convergence  Perfect scaling should give immediate convergence Convergence Newton’s method:

12.  Transform entire problem s.t. Scaled gradient projection

13.  Is itself a quadratic program  Solve with active-set style method  Move each x i to u i  Build blocks of active constraints Projection operation u subj to: x l +d ≤ x r uiui b d a c e

14.  Is itself a quadratic program  Solve with active-set style method  Move each x i to u i  Build blocks of active constraints  Find most violated constraint x l +d ≤ x r Projection operation u subj to: x l +d ≤ x r uiui b d a c e

15.  Is itself a quadratic program  Solve with active-set style method  Move each x i to u i  Build blocks of active constraints:  Find most violated constraint x l +d ≤ x r  Satisfy and add to block B  Move B to average position of constituent vars Projection operation u subj to: x l +d ≤ x r uiui b d a c e

16.  Is itself a quadratic program  Solve with active-set style method  Move each x i to u i  Build blocks of active constraints:  Find most violated constraint x l +d ≤ x r  Add to block B (satisfy constraint)  Move B to average position of constituent vars Projection operation u subj to: x l +d ≤ x r uiui etc… b d a c e

17.  Is itself a quadratic program  Solve with active-set style method  Move each x i to u i  Build blocks of active constraints:  Find most violated constraint x l +d ≤ x r  Add to block B (satisfy constraint)  Move B to average position of constituent vars Projection operation u subj to: x l +d ≤ x r uiui etc… b d a c e

18.  Is itself a quadratic program:  Solve with active-set style method:  Move each x i to u i  Build blocks of active constraints:  Find most violated constraint x l +d ≤ x r  Add to block B (satisfy constraint)  Move B to average position of constituent vars: Projection operation u subj to: x l +d ≤ x r uiui etc… b d a c e

19.  Block structure is preserved between projection operations  Before each projection previous blocks are checked for split points (ensures convergence) Projection operation: incremental b d a c e

20.  Block structure is preserved between projection operations  Before each projection previous blocks are checked for split points (ensures convergence)  In next projection blocks will be moved as one to new weighted average desired positions Projection operation: incremental b d a c e

21. Projection operation b d a c e  Is itself a quadratic program  Scaling by a full n×n matrix turns separation constraints into linear constraints over n variables u subj to: x l +d ≤ x r uiui

22. Scaling for stress majorization  Q is diagonally dominant:  Choose diagonal s.t. ≤

23.  Diagonal scaling:  Separation constraints:  Need new expressions for  Optimal block position  Lagrange multipliers for active constraints Scaled separation constraints

24.  minimize subject to active constraints : where:  minimum at: Optimum block position

25.  Optimum at: where: Optimum block position

26. Test cases unconstrained constrained

27. Test cases unconstrained constrained

28. Results

29. Improved convergence

30.  Diagonal scaling  is cheap to compute  transforms separation constraints into scaled separation constraints  not full linear constraints  so we can still use block tricks  is appropriate for improving condition of graph Laplacian matrices because they are diagonally dominant  particularly improves Laplacian condition if graph has wide variation in degree (often in practical applications) Summary