Sum of Squares Planted Clique and PseudoCalibration Joint

Sum of Squares, Planted Clique, and Pseudo-Calibration Joint work with Boaz Barak, Sam Hopkins, Pravesh Kothari, Jonathan Kelner, and Ankur Moitra

Planted Clique Lower Bound •

Talk Outline • Part I: The Sum of Squares Hierarchy • Part II: Planted Clique • Part III: Pseudo-calibration

Part I: The Sum of Squares Hierarchy

The Sum of Squares Hierarchy • Developed independently by Shor, Nesterov, Parrillo, and Lasserre. • Generalization of linear and semidefinite programming • Strictly stronger than the Lovasz-Schrijver hierarchy and the Sherali-Adams hierarchy • Captures the Goemans-Williamson algorithm for MAX-CUT, the Goemens-Linial algorithm for sparsest cut (analyzed by Arora, Rao, Vazirani), and the Arora-Barak-Steurer subexponential time algorithm for Unique Games • Leading candidate for refuting Khot’s Unique Games Conjecture.

Positivstellensatz Proofs •

Pseudo-expectation values •

• Duality

The Moment Matrix •

Lower Bounds Strategy for SOS •

Semidefinite Programs for SOS •

Optimization with SOS • Often have equations with parameter(s) • Want to optimize over green region, SOS optimizes over the blue and green regions. Equations are feasible Infeasible but no proof Positivstellensatz proof of infeasibility • As we increase the degree, the blue region shrinks.

Approximation Algorithms with SOS • Optimal Solution Equations are feasible A Solution Infeasible but no proof Positivstellensatz proof of infeasibility

For more info on SOS • Princeton sum of squares seminar website http: //sos 16. dsteurer. org/lectures/ • Harvard sum of squares seminar website http: //www. boazbarak. org/sos/

Part II: Planted Clique

The Planted Clique Problem j a b i c h d g f e Can you find the 5 -clique?

The Planted Clique Problem j a b i c h d g f e This 5 -clique was planted by adding the red edge.

Part III: Pseudo-Calibration

Pseudo-calibration •

Pseudo-calibrated moments •

Current/Future work •

Acknowledgements • Thanks to the National Science Foundation, Microsoft Research, and the Simons Foundation for supporting this research.

Thank you!