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

 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

“Unix” Graph data From

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

 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)*

Gradient projection -g -αg-αg x0x0 x1x1

Gradient projection -αg-αg x1x1

d x2x2 x1x1 βdβd

x*

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

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

 Transform entire problem s.t. Scaled gradient projection

 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

 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

 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

 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

 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

 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

 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

 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

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

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

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

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

 Optimum at: where: Optimum block position

Test cases unconstrained constrained

Test cases unconstrained constrained

Results

Improved convergence

 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