Ankit Garg Princeton Univ. Joint work with Leonid Gurvits Rafael Oliveira CCNY Princeton Univ. Avi Wigderson IAS Noncommutative rational identity testing.

Slides:

Advertisements

Similar presentations

Chapter 0 Review of Algebra.

Advertisements

4.1 Introduction to Matrices

Rank Bounds for Design Matrices and Applications Shubhangi Saraf Rutgers University Based on joint works with Albert Ai, Zeev Dvir, Avi Wigderson.

MATRICES. EXAMPLES:

Non-commutative computation with division Avi Wigderson IAS, Princeton Pavel Hrubes U. Washington.

Lecture 6 Matrix Operations and Gaussian Elimination for Solving Linear Systems Shang-Hua Teng.

Arithmetic Hardness vs. Randomness Valentine Kabanets SFU.

Matrices MSU CSE 260.

CE 311 K - Introduction to Computer Methods Daene C. McKinney

MATH – High School Common Core Vs Kansas Standards.

Using Inverse Matrices Solving Systems. You can use the inverse of the coefficient matrix to find the solution. 3x + 2y = 7 4x - 5y = 11 Solve the system.

Numbers and Quantity Extend the Real Numbers to include work with rational exponents and study of the properties of rational and irrational numbers Use.

Jeopardy Algebraic Expressions Q $100 Q $200 Q $300 Q $400 Q $500 Q $100 Q $200 Q $300 Q $400 Q $500 Real Numbers Matrix Usage Math Properties Right Triangles.

Algorithms for a large sparse nonlinear eigenvalue problem Yusaku Yamamoto Dept. of Computational Science & Engineering Nagoya University.

Multilinear NC 1  Multilinear NC 2 Ran Raz Weizmann Institute.

Recent Developments in Algebraic Proof Complexity Recent Developments in Algebraic Proof Complexity Iddo Tzameret Tsinghua Univ. Based on Pavel Hrubeš.

Review of Matrices Or A Fast Introduction.

Finite Mathematics Dr. Saeid Moloudzadeh Solving Polynomial Equations 1 Contents Algebra Review Functions and Linear Models Systems of.

CMPS 1371 Introduction to Computing for Engineers MATRICES.

4.7 Identity and Inverse Matrices and Solving Systems of Equations Objectives: 1.Determine whether two matrices are inverses. 2.Find the inverse of a 2x2.

Inverse and Identity Matrices Can only be used for square matrices. (2x2, 3x3, etc.)

On the computation of the defining polynomial of the algebraic Riccati equation Yamaguchi Univ. Takuya Kitamoto Cybernet Systems, Co. LTD Tetsu Yamaguchi.

Umans Complexity Theory Lectures Lecture 1a: Problems and Languages.

Section 1-4: Solving Inequalities Goal 1.03: Operate with algebraic expressions (polynomial, rational, complex fractions) to solve problems.

Introduction to Linear Algebra Mark Goldman Emily Mackevicius.

Section 5.5 The Real Zeros of a Polynomial Function.

Evaluating Algebraic Expressions 2-7 One-Step Equations with Rational Numbers Additional Example 2A: Solving Equations with Fractions = – 3737 n

Finite Mathematics Dr. Saeid Moloudzadeh Multiplying and Factoring Algebraic Expressions 1 Contents Algebra Review Functions and Linear.

Common Logarithms - Definition Example – Solve Exponential Equations using Logs.

Radical expressions, rational exponents and radical equations ALGEBRA.

Multiply Matrices Chapter 3.6. Matrix Multiplication Matrix multiplication is defined differently than matrix addition The matrices need not be of the.

Holt McDougal Algebra 2 Fundamental Theorem of Algebra How do we use the Fundamental Theorem of Algebra and its corollary to write a polynomial equation.

Happy 80th B’day Dick.

Happy 60 th B’day Noga. Elementary problems encoding computational hardness Avi Wigderson IAS, Princeton or Some problems Noga never solved.

EXPRESSIONS, FORMULAS, AND PROPERTIES 1-1 and 1-2.

Characterizing Propositional Proofs as Non-Commutative Formulas Iddo Tzameret Royal Holloway, University of London Based on Joint work Fu Li (Texas Austin)

Do Now: Perform the indicated operation. 1.). Algebra II Elements 11.1: Matrix Operations HW: HW: p.590 (16-36 even, 37, 44, 46)

Unsupervised Learning II Feature Extraction

If A and B are both m × n matrices then the sum of A and B, denoted A + B, is a matrix obtained by adding corresponding elements of A and B. add these.

MASKS © 2004 Invitation to 3D vision Lecture 6 Introduction to Algebra & Rigid-Body Motion Allen Y. Yang September 18 th, 2006.

Do Now: Graph and state the type of graph and the domain and range if it is a function. Introduction to Analytic Geometry.

1 IAS, Princeton ASCR, Prague. The Problem How to solve it by hand ? Use the polynomial-ring axioms ! associativity, commutativity, distributivity, 0/1-elements.

MTH108 Business Math I Lecture 20.

Lecture 4 Complex numbers, matrix algebra, and partial derivatives

8.1 Determine whether the following statements are correct or not

ECON 213 Elements of Mathematics for Economists

Algebraic Bethe ansatz for the XXZ Heisenberg spin chain

MATH Algebra II Analyzing Equations and Inequalities

Linear independence and matrix rank

Solve Quadratic Equations by the Quadratic Formula

Proving Algebraic Identities

Proving Algebraic Identities

Proving Analytic Inequalities

Proving Analytic Inequalities

Matrix Algebra.

الوحدة السابعة : المصفوفات . تنظيم البيانات فى مصفوفات . الوحدة السابعة : المصفوفات . تنظيم البيانات فى مصفوفات . 1 جمع المصفوفات وطرحها.

Solving Equations with variables on each side

Hiroshi Hirai University of Tokyo

Solving Linear Systems Using Inverse Matrices

Inverse & Identity MATRICES Last Updated: October 12, 2005.

Matrix Algebra.

Analysis Portfolio By: Your Name.

MATH CP Algebra II Analyzing Equations and Inequalities

Panorama of scaling problems and algorithms

MA5242 Wavelets Lecture 1 Numbers and Vector Spaces

3.5 Perform Basic Matrix Operations Algebra II.

Common Core Vs Kansas Standards

PIT Questions in Invariant Theory

Much Faster Algorithms for Matrix Scaling

Presentation transcript:

Ankit Garg Princeton Univ. Joint work with Leonid Gurvits Rafael Oliveira CCNY Princeton Univ. Avi Wigderson IAS Noncommutative rational identity testing (over the rationals)

Outline Introduction to PIT/RIT. Symbolic matrices Algorithm Conclusion/Open problems

Commutative Polynomial Identity Testing (PIT) Arithmetic Circuit Arithmetic Formula

Commutative Polynomial Identity Testing

Non-commutative PIT Arithmetic Circuit Arithmetic Formula

Non-commutative PIT Deterministic polynomial time algorithm for circuits open.

Commutative Rational identity testing (RIT) INV

Commuting RIT

Non-commutative rational identity testing INV

Non-commutative RIT Given two non-commutative rational expressions as formulas/circuits, determine if they represent the same element. What does it mean for two expressions represent the same element? – No easy canonical form. Operational definition [Amitsur `66].

Free Skew Field

Non-commutative rational identity testing

[Cohn-Reutenauer `99]: Reduce to solving a system of (commutative) polynomial equations (for formula representations). Can also be deduced from structural results in [Cohn `71]. Several other algorithms but all exponential time (with or without randomness).

Non-commutative rational identity testing

Outline Introduction to PIT/RIT. Symbolic matrices Algorithm Conclusion/Open problems

Symbolic matrices

Not true in the commutative setting!

Symbolic matrices

SINGULAR

Shrunk Subspaces

Outline Introduction to PIT/RIT. Symbolic matrices Algorithm Conclusion/Open problems

Doubly stochastic operators

Algorithm G

Algorithm already suggested in [Gurvits `04]. Our contribution: prove that it works! “Non-commutative extension” of matrix scaling algorithms [Sinkhorn `64, LSW ‘98].

Analysis - Capacity Main contribution

Fullness dimension

Outline Introduction to PIT/RIT. Symbolic matrices Algorithm Conclusion/Open problems

Conclusion Analytic algorithm for a purely algebraic problem! Polynomial degree bounds not essential to put algebraic geometric problems in P. Not essential for this specific problem [next talk].

Open Problems

Thank You