A Potpourri of Nonlinear Algebra Chris Hillar 2 Det 2 , 2 , 2 ( A ) - PowerPoint PPT Presentation

1 a 1 a 2 a 3 + c 1 c 2 c 3 = 2 , a 1 a 3 b 2 + c 1 c 3 d 2 = 0 , a 2 a 3 b 1 + c 2 c 3 d 1 = 0 , X d i = a 3 b 1 b 2 + c 3 d 1 d 2 = − 4 , a 1 a 2 b 3 + c 1 c 2 d 3 = 0 , a 1 b 2 b 3 + c 1 d 2 d 3 = − 4 , θ i + θ j a 2 b 1 b 3 + c 2 d 3 d 1 = 4 , b 1 b 2 b 3 + d 1 d 2 d 3 = 0 j 6 = i A Potpourri of Nonlinear Algebra Chris Hillar �◆� 2 Det 2 , 2 , 2 ( A ) = 1  ✓ �  �◆ ✓ �  a 000 a 010 a 100 a 110 a 000 a 010 a 100 a 110 det + − det − 4 a 001 a 011 a 101 a 111 a 001 a 011 a 101 a 111  �  � a 000 a 010 a 100 a 110 − 4 det det . a 001 a 011 a 101 a 111 a 1 c 1 − b 1 d 1 − u 2 , b 1 c 1 + a 1 d 1 , c 1 u − a 2 1 + b 2 1 , d 1 u − 2 a 1 b 1 , a 1 u − c 2 1 + d 2 1 , b 1 u − 2 d 1 c 1 , a 2 c 2 − b 2 d 2 − u 2 , b 2 c 2 + a 2 d 2 , c 2 u − a 2 2 + b 2 2 , d 2 u − 2 a 2 b 2 , a 2 u − c 2 2 + d 2 2 , b 2 u − 2 d 2 c 2 , a 3 c 3 − b 3 d 3 − u 2 , b 3 c 3 + a 3 d 3 , c 3 u − a 2 3 + b 2 3 , d 3 u − 2 a 3 b 3 , a 3 u − c 2 3 + d 2 3 , b 3 u − 2 d 3 c 3 , a 4 c 4 − b 4 d 4 − u 2 , b 4 c 4 + a 4 d 4 , c 4 u − a 2 4 + b 2 4 , d 4 u − 2 a 4 b 4 , a 4 u − c 2 4 + d 2 4 , b 4 u − 2 d 4 c 4 , a 2 1 − b 2 1 + a 1 a 3 − b 1 b 3 + a 2 3 − b 2 3 , a 2 1 − b 2 1 + a 1 a 4 − b 1 b 4 + a 2 4 − b 2 4 , a 2 1 − b 2 1 + a 1 a 2 − b 1 b 2 + a 2 2 − b 2 2 , a 2 2 − b 2 2 + a 2 a 3 − b 2 b 3 + a 2 3 − b 2 3 , a 2 3 − b 2 3 + a 3 a 4 − b 3 b 4 + a 2 4 − b 2 4 , 2 a 1 b 1 + a 1 b 2 + a 2 b 1 + 2 a 2 b 2 , 2 a 2 b 2 + a 2 b 3 + a 3 b 2 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 3 + a 2 b 1 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 4 + a 4 b 1 + 2 a 4 b 4 , 2 a 3 b 3 + a 3 b 4 + a 4 b 3 + 2 a 4 b 4 , w 2 1 + w 2 2 + · · · + w 2 17 + w 2 18 . a 000 x 0 y 0 + a 010 x 0 y 1 + a 100 x 1 y 0 + a 110 x 1 y 1 = 0 , a 001 x 0 y 0 + a 011 x 0 y 1 + a 101 x 1 y 0 + a 111 x 1 y 1 = 0 , a 000 x 0 z 0 + a 001 x 0 z 1 + a 100 x 1 z 0 + a 101 x 1 z 1 = 0 , a 010 x 0 z 0 + a 011 x 0 z 1 + a 110 x 1 z 0 + a 111 x 1 z 1 = 0 , a 000 y 0 z 0 + a 001 y 0 z 1 + a 010 y 1 z 0 + a 011 y 1 z 1 = 0 , a 100 y 0 z 0 + a 101 y 0 z 1 + a 110 y 1 z 0 + a 111 y 1 z 1 = 0 , ICERM | Computational Nonlinear Algebra | June 2014

Outline Computational complexity of nonlinear algebra “Real-life” examples: Tensor problems - graph theory, optimization, Groebner bases Neuroscience: The Retina Equations - bipartite graphs, probability, matrix analysis

Computational Nonlinear Algebra Problem: Solve on a finite computer in finite time a finite set of polynomial (quadratic) equations computability ring reference [Hilbert’s 10th Problem] Undecidable [Davis, Putnam, Robinson, Z (“Uncomputable”) Matijasevi č ’61/’70] ????? [Poonen ’03] Q Decidable [Tarski–Seidenberg] R (“Computable”) [Hironaka ’64, Buchberger ’70] C

Some “Random” Polynomial Systems: a) a 1 a 2 a 3 + c 1 c 2 c 3 = 2 , a 1 a 3 b 2 + c 1 c 3 d 2 = 0 , a 2 a 3 b 1 + c 2 c 3 d 1 = 0 , a 3 b 1 b 2 + c 3 d 1 d 2 = − 4 , a 1 a 2 b 3 + c 1 c 2 d 3 = 0 , a 1 b 2 b 3 + c 1 d 2 d 3 = − 4 , a 2 b 1 b 3 + c 2 d 3 d 1 = 4 , b 1 b 2 b 3 + d 1 d 2 d 3 = 0 b) a 1 c 1 − b 1 d 1 − u 2 , b 1 c 1 + a 1 d 1 , c 1 u − a 2 1 + b 2 1 , d 1 u − 2 a 1 b 1 , a 1 u − c 2 1 + d 2 1 , b 1 u − 2 d 1 c 1 , a 2 c 2 − b 2 d 2 − u 2 , b 2 c 2 + a 2 d 2 , c 2 u − a 2 2 + b 2 2 , d 2 u − 2 a 2 b 2 , a 2 u − c 2 2 + d 2 2 , b 2 u − 2 d 2 c 2 , a 3 c 3 − b 3 d 3 − u 2 , b 3 c 3 + a 3 d 3 , c 3 u − a 2 3 + b 2 3 , d 3 u − 2 a 3 b 3 , a 3 u − c 2 3 + d 2 3 , b 3 u − 2 d 3 c 3 , a 4 c 4 − b 4 d 4 − u 2 , b 4 c 4 + a 4 d 4 , c 4 u − a 2 4 + b 2 4 , d 4 u − 2 a 4 b 4 , a 4 u − c 2 4 + d 2 4 , b 4 u − 2 d 4 c 4 , a 2 1 − b 2 1 + a 1 a 3 − b 1 b 3 + a 2 3 − b 2 3 , a 2 1 − b 2 1 + a 1 a 4 − b 1 b 4 + a 2 4 − b 2 4 , a 2 1 − b 2 1 + a 1 a 2 − b 1 b 2 + a 2 2 − b 2 2 , a 2 2 − b 2 2 + a 2 a 3 − b 2 b 3 + a 2 3 − b 2 3 , a 2 3 − b 2 3 + a 3 a 4 − b 3 b 4 + a 2 4 − b 2 4 , 2 a 1 b 1 + a 1 b 2 + a 2 b 1 + 2 a 2 b 2 , 2 a 2 b 2 + a 2 b 3 + a 3 b 2 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 3 + a 2 b 1 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 4 + a 4 b 1 + 2 a 4 b 4 , 2 a 3 b 3 + a 3 b 4 + a 4 b 3 + 2 a 4 b 4 , w 2 1 + w 2 2 + · · · + w 2 17 + w 2 18 . c) a 000 x 0 y 0 + a 010 x 0 y 1 + a 100 x 1 y 0 + a 110 x 1 y 1 = 0 , a 001 x 0 y 0 + a 011 x 0 y 1 + a 101 x 1 y 0 + a 111 x 1 y 1 = 0 , a 000 x 0 z 0 + a 001 x 0 z 1 + a 100 x 1 z 0 + a 101 x 1 z 1 = 0 , a 010 x 0 z 0 + a 011 x 0 z 1 + a 110 x 1 z 0 + a 111 x 1 z 1 = 0 , a 000 y 0 z 0 + a 001 y 0 z 1 + a 010 y 1 z 0 + a 011 y 1 z 1 = 0 , a 100 y 0 z 0 + a 101 y 0 z 1 + a 110 y 1 z 0 + a 111 y 1 z 1 = 0 ,

A Briefer on Computational Complexity I. Model of Computation Alan Turing - What are inputs / outputs? - What is a computation? II. Model of Complexity Stephen Cook - Cost of computation? Dick Karp III. Model of Reducibility - What are equivalent problems? Leonid Levin

I. Model of Computation: Turing Machine [Turing ’37][Turing 1936] Inputs: finite list of rational numbers Outputs: YES/NO or rational vectors II. Model of Complexity: Time complexity Turing Machine (Mike Davey) Number of Tape-Level moves III. Model of Reducibility: Classes: P (polynomial-time), NP ,, NP-Hard NP-complete, NP-hard, ... Tensor Problems P2 input I input I’ P1 NP-Complete polynomial-sized transformation NP P2 P1 P Matrix Problems = YES/NO YES/NO the world of all computational problems

NP-complete decision problems [Cook-Karp-Levin 1971/2] Graph coloring: Given graph G , is there a proper 3-coloring? 1 2 1 2 4 3 4 3 YES NO is an NP-complete (can verify quickly) problem 1 Million $$$ prize (Clay Math)

Connection to nonlinear algebra Theorem [Bayer ‘82]: Whether or not a graph is 3- colorable can be encoded as whether a system of cubic equations over has a nonzero solution C Reformulation [H., Lim ’13]: Whether or not a graph G on v vertices with edges E is 3-colorable can be encoded as whether the following homogeneous quadratics has a nonzero solution in C ( x i y i − u 2 , y i u − x 2 x i u − y 2 i = 1 , . . . , v, i , i , C G = j : { i,j } ∈ E ( x 2 i + x i x j + x 2 P j ) , i = 1 , . . . , v. Quadratic System x i = a i + i b i over the reals R y i = c i + i d i

primitive cube root of 1 Example: The following ω graph is 3-colorable: ω 2 1 1 2 x i = 1 x i = ω 4 3 x i = ω 2 1 ω The system has a (nonzero) solution over the reals: 35 homogeneous quadratics in 35 indeterminates: A ∈ Q 35 × 35 × 35 b) a 1 c 1 − b 1 d 1 − u 2 , b 1 c 1 + a 1 d 1 , c 1 u − a 2 1 + b 2 1 , d 1 u − 2 a 1 b 1 , a 1 u − c 2 1 + d 2 1 , b 1 u − 2 d 1 c 1 , a 2 c 2 − b 2 d 2 − u 2 , b 2 c 2 + a 2 d 2 , c 2 u − a 2 2 + b 2 2 , d 2 u − 2 a 2 b 2 , a 2 u − c 2 2 + d 2 2 , b 2 u − 2 d 2 c 2 , a 3 c 3 − b 3 d 3 − u 2 , b 3 c 3 + a 3 d 3 , c 3 u − a 2 3 + b 2 3 , d 3 u − 2 a 3 b 3 , a 3 u − c 2 3 + d 2 3 , b 3 u − 2 d 3 c 3 , a 4 c 4 − b 4 d 4 − u 2 , b 4 c 4 + a 4 d 4 , c 4 u − a 2 4 + b 2 4 , d 4 u − 2 a 4 b 4 , a 4 u − c 2 4 + d 2 4 , b 4 u − 2 d 4 c 4 , a 2 1 − b 2 1 + a 1 a 3 − b 1 b 3 + a 2 3 − b 2 3 , a 2 1 − b 2 1 + a 1 a 4 − b 1 b 4 + a 2 4 − b 2 4 , a 2 1 − b 2 1 + a 1 a 2 − b 1 b 2 + a 2 2 − b 2 2 , a 2 2 − b 2 2 + a 2 a 3 − b 2 b 3 + a 2 3 − b 2 3 , a 2 3 − b 2 3 + a 3 a 4 − b 3 b 4 + a 2 4 − b 2 4 , 2 a 1 b 1 + a 1 b 2 + a 2 b 1 + 2 a 2 b 2 , 2 a 2 b 2 + a 2 b 3 + a 3 b 2 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 3 + a 2 b 1 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 4 + a 4 b 1 + 2 a 4 b 4 , 2 a 3 b 3 + a 3 b 4 + a 4 b 3 + 2 a 4 b 4 , w 2 1 + w 2 2 + · · · + w 2 17 + w 2 18 .

1 2 Example: The following graph is not 3-colorable: 4 3 The system does not have (nonzero) solution over : R a 2 2 − b 2 2 + a 2 a 4 − b 2 b 4 + a 2 4 − b 2 4 , 2 a 2 b 2 + a 2 b 4 + a 4 b 2 + 2 a 4 b 4 b) a 1 c 1 − b 1 d 1 − u 2 , b 1 c 1 + a 1 d 1 , c 1 u − a 2 1 + b 2 1 , d 1 u − 2 a 1 b 1 , a 1 u − c 2 1 + d 2 1 , b 1 u − 2 d 1 c 1 , a 2 c 2 − b 2 d 2 − u 2 , b 2 c 2 + a 2 d 2 , c 2 u − a 2 2 + b 2 2 , d 2 u − 2 a 2 b 2 , a 2 u − c 2 2 + d 2 2 , b 2 u − 2 d 2 c 2 , a 3 c 3 − b 3 d 3 − u 2 , b 3 c 3 + a 3 d 3 , c 3 u − a 2 3 + b 2 3 , d 3 u − 2 a 3 b 3 , a 3 u − c 2 3 + d 2 3 , b 3 u − 2 d 3 c 3 , a 4 c 4 − b 4 d 4 − u 2 , b 4 c 4 + a 4 d 4 , c 4 u − a 2 4 + b 2 4 , d 4 u − 2 a 4 b 4 , a 4 u − c 2 4 + d 2 4 , b 4 u − 2 d 4 c 4 , a 2 1 − b 2 1 + a 1 a 3 − b 1 b 3 + a 2 3 − b 2 3 , a 2 1 − b 2 1 + a 1 a 4 − b 1 b 4 + a 2 4 − b 2 4 , a 2 1 − b 2 1 + a 1 a 2 − b 1 b 2 + a 2 2 − b 2 2 , a 2 2 − b 2 2 + a 2 a 3 − b 2 b 3 + a 2 3 − b 2 3 , a 2 3 − b 2 3 + a 3 a 4 − b 3 b 4 + a 2 4 − b 2 4 , 2 a 1 b 1 + a 1 b 2 + a 2 b 1 + 2 a 2 b 2 , 2 a 2 b 2 + a 2 b 3 + a 3 b 2 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 3 + a 2 b 1 + 2 a 3 b 3 , 2 a 1 b 1 + a 1 b 4 + a 4 b 1 + 2 a 4 b 4 , 2 a 3 b 3 + a 3 b 4 + a 4 b 3 + 2 a 4 b 4 , w 2 1 + w 2 2 + · · · + w 2 17 + w 2 18 .

Example: The graph G below is uniquely 3-colorable [Example of Akbari, Mirrokni, Sadjad ‘01 disproving a conjecture of Xu ’90] The coloring ideal is trivial (< 2 sec computation) I G [H., Windfeldt ’08]