Lectures

Following is an outline of lectures given along with references and links to additional reading. For abbreviations, [RP], [SY] and [CKW], check the references page.

\( \def\PAL{\rm PAL} \def\HP{\rm HP} \def\MP{\rm MP} \def\EQ{\rm EQ} \def\USELESS{\rm USELESS} \def\EMPTY{\rm EMPTY} \def\DTIME{\sf DTIME} \def\NTIME{\sf NTIME} \def\DSPACE{\sf DSPACE} \def\NSPACE{\sf NSPACE} \def\P{\sf P} \def\co{\sf co} \def\NP{\sf NP} \def\LOG{\sf LOG} \def\NLOG{\sf NLOG} \def\PSPACE{\sf PSPACE} \def\NPSPACE{\sf NPSPACE} \def\EXP{\sf EXP} \def\F{\mathbb{F}} \)

Theme: Basics - arithmetic models of computation and arithmetic classes

Lec 1 (07 Jan, Wed) : Course information. A brief course overview. Central objects - multivariate polynomials. Computing polynomials diagrammatically, the arithmetic circuit representation. Circuits for computing some simple polynomials. Goals of this course.

^{Reading : Class notes, Chapter 1 of [RP] survey}

Lec 2 (09 Jan, Fri) : Course evaluation policy discussed. The interpolation technique, computing the elementary symmetric polynomial using small sized arithmetic circuits. Determinant polynomial. Algebraic formulation for bipartite perfect matching.

^{Reading : Class notes, Chapter 1 of [RP] survey. Section 7.2 of this notes.}

Lec 3 (12 Jan, Mon) : Two problems expressed via polynomials – primality checking and Hamiltonian cycle. Naive algorithm for primality check is inefficient. Agrawal-Biswas theorem for primality checking. The polynomial identity testing question.

^{Reading : Lemma 3.1 of this paper.}

^{Additional References : A writeup on primality testing by Scott Aaronson.}

Lec 4 (14 Jan, Wed) : Algebraic circuit model. Size and depth. Number of monomials in an $n$ variate degree $d$ polynomial. Naive upper bounds on circuit size. Shannon’s counting lower bound and existence of polynomials requiring large circuit size. The circuit size lower bound question.

^{Reading : Class notes. Section 10.3.2 of this notes.}

Lec 5 (16 Jan, Fri) : Need for explicitness and the need for polynomial families. Permanent and Determinant polynomial families. Valiant’s conjecture and need for a theory. Arithmetic formula model of computation. Size of a formula. The circuit versus formula model.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec () (19 Jan, Mon) : Class cancelled due to Institute day. Compensated on 24 Jan.

Lec 6 (21 Jan, Wed) : Bounds on the formula and circuit size of simple polynomials. Towards an intermediate model - the iterated matrix multiplication polynomial and algebraic branching programs (ABP). Some examples of ABPs computing polynomials.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec 7 (23 Jan, Fri) : Obtaining ABPs from arithmetic formulas and obtaining arithmetic circuits from ABPs with a constant size blowup. Correctness arguments and argument for size bounds.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec 8 (24 Jan, Sat) : (Compensatory class) Characterisation of permanent via cycle covers. Universality of permanent - expressing arithmetic formulas as “projections” of permanent of matrices. Classical complexity classes $\P$ and $\NP$. Polynomial classes VF for formulas and VBP for algebraic branching programs, class VP for circuits. VF $\subseteq$ VBP $\subseteq$ VP. Notion of polynomial being hard for a class.

^{Reading : Class notes. Chapter 2 of [RP] survey.}

Lec () (26 Jan, Mon) : Holiday due to Republic day.

Lec 9 (28 Jan, Wed) : Projection reductions. Expressing any ABP as a projection of iterated matrix multiplication. Iterated matrix multiplication polynomial is VBP-complete. Hardness and completeness. The classical complexity class NP and its analog Valiant’s NP (VNP).

^{Reading : Class notes. Chapter 3 (Section 1) of [RP] survey.}

Lec 10 (30 Jan, Fri) : A depth three arithmetic circuit computing permanent (Ryser’s construction) via inclusion exclusion argument. Permanent is in VNP.

^{Reading : Class notes.}

Lec 11 (02 Feb, Mon) : Matching polynomial is VNP-hard. Towards showing Permanent is VNP-complete. Class VNP stays the same with $g \in $ VP replaced by $g \in $ VF in the definition (statement only). Special case when auxiliary variables occurs once in ABP of $g$.

^{Reading : Class notes, Section 3.3.4 of [RP] survey.}

Lec 12 (04 Feb, Wed) : Showing Permanent is VNP hard. Details of the gadgets - rosette and glue gadget. Correctness and details of the construction.

^{Reading : Class notes, Section 3.3.4 of [RP] survey.}

Lec 13 (06 Feb, Fri) : Expression for determinant - Laplace expansion and its derivation from high school definition of determinant by induction. Sign of permutation and it properties under composition.

^{Reading : Class notes. Also, check this writeup by Aripta.}

Lec 14 (09 Feb, Mon) : Determinant is universal. Expressing determinant using iterated matrix multiplication. Determinants can compute ABPs.

^{Reading : Class notes, Section 3.3.3 of [RP] survey.}

Lec 15 (11 Feb, Wed) : Closed walk (clow) sequences, examples. Visualizing walks in a graph. Visualizing closed walks. Characterization of determinants using closed walk (clow) sequence.

^{Reading : Class notes, Section 3.3.3 of [RP] survey.}

Lec 16 (13 Feb, Fri) : Description of an ABP (Mahajan-Vinay algorithm) computing determinant and correctness argument. Converting a circuit to formula with size blow up. Complete problems for VF - Formula for computing Iterated matrix multiplication of constant sized matrices.

^{Reading : Class notes, Section 2.1.1 and 3.3.3 of [RP] survey.}

Theme: Structural results and Classical lower bounds

Lec 17 (16 Feb, Mon) : IMM$_{3,d}$ is in VF. Tree separator theorem and depth reduction for arithmetic formulas (Brent’s construction).

^{Reading : Class notes, Section 5.3.1 of [RP] survey.}

Lec 18 (18 Feb, Wed) : Width-3 branching programs for computing formulas – Ben-Or and Cleve construction and correctness. Used this construction along with depth reduction to prove that IMM$_{3,d}$ is VF-complete.

^{Reading : Class notes, Lecture notes 1 and 2 by Madhu Sudan.}

Lec 19 (20 Feb, Fri) : Homogenization of arithmetic circuits, Handling divisions. Csanky’s algorithm for computing determinant. A warm-up - computing $m$th Fibonacci number by a $O(\log^2 m)$ depth arithmetic circuit.

^{Reading : Class notes, Section 5.1 and 5.2 of [RP] survey.}

Lec 20 (23 Feb, Mon) : Arithmetic circuits for computing determinant - Newton-Girard identities. Combinatorial proof of the identity. Completed details of Csanky’s algorithm.

^{Reading : Class notes. See here for a combinatorial proof of Newton-Girard identity.}

Lec 21 (25 Feb, Wed) : Recap of techniques so far - homogenization for circuits and ABPs. Valiant’s question of computing linear transforms with linear circuits and log depth. Matrix rigidity - definition and examples.

^{Reading : Class notes.}

Theme: Lower bounds for general and restricted settings

Lec 22 (27 Feb, Fri) : Linear circuits of log depth and linear size can only compute matrices of low rigidity.

^{Reading : Class notes.}

Lec 23 (02 Mar, Mon) : Structural result on depth reduction of DAGs.

^{Reading : Class notes. Slides from Workshop on Matrix Rigidity 2020.}

Lec () (04 Mar, Wed) : Holiday due to Holi.

Lec 24 (06 Mar, Fri) : Existence of matrices of high rigidity (over $\mathbb{Z}p$) by counting arguments.

^{Reading : Class notes. Slides from Workshop on Matrix Rigidity 2020.}

Lec 25 (09 Mar, Mon) : Size lower bounds for circuits due to Baur and Strassen. Bezout’s lemma (statement only) and its use in arguing lower bound. Computing first order partial derivates of a polynomial.

^{Reading : Class notes. Section 6.1 of [RP] survey.}

Lec 26 (11 Mar, Wed) : Showing circuit size lower bounds via transformation to structured circuits. Warm up - lower bound for depth $2$ unbounded fan-in circuits. Making formulas homogeneous.

^{Reading : Class notes. Section 8.1.1 of [RP] survey.}

Lec 27 (13 Mar, Fri) : Depth reduction for formulas (Hyafil). VSBR depth reduction (statement only).

^{Reading : Class notes. Section 5.3.2 of [RP] survey.}

Lec 28 (16 Mar, Mon) : Reduction to depth 4 due to Agarawal-Vinay (statement only). Algebraic independence of polynomials and transcendence degree. Formula size upper bound for determinant polynomial.

^{Reading : Class notes. Section 6.2 and section 26.1.2 of [RP] survey.}

Lec 29 (18 Mar, Wed) : An equivalent way of expressing algebraic independence via linear independence. Using this to show that transcendence degree of $n$ variate polynomials is bounded by $n$. Introduction to Jacobian of a set of polynomials. Intuition behind Kalorokoti’s formula lower bound for determinant.

^{Reading : Class notes. Section 6.2 and section 26.1.2 of [RP] survey.}

Lec () (20 Mar, Fri) : Holiday due to Eid al-Fitr. Class to be compensated.

Lec 30 (23 Mar, Mon) : Testing algebraic independence using Jacobian. Examples.

^{Reading : Class notes. Section 6.2.1 and 6.2.2 of [RP] survey.}

Lec 31 (25 Mar, Wed) : Completed the proof of Jacobian criteria for testing algebraic independence. Definition of the complexity measure in Karorkoti’s argument. Proved largeness of measure for determinant polynomial using the Jacobian criteria.

^{Reading : Class notes. Section 6.2.1 and 6.2.2 of [RP] survey. Chapter 3 of this thesis.}

Lec 32 (27 Mar, Fri) : Measure is small for formulas. Statement and overview. Effect of taking polynomials combinations on transendence degree.

^{Reading : Class notes.}

Lec 33 (28 Mar, Sat) : (Working day, Friday’s timetable) Given a formula $F$ and a subset of variables $Y \subseteq X$, obtaining another formula with leaves being polynomials in $\mathbb{F}[Y\setminus X]$ and variables from $X$. Completed the proof of Kalarkoti’s formula lower bound.

^{Reading : Class notes. Section 6.2.1 and 6.2.2 of [RP] survey.}

Lec 34 (30 Mar, Mon) : Natural lower bound method based on measure. Computing $x_1x_2\ldots x_n$ using an explicit $\Sigma \wedge \Sigma$ circuits of size $2^{O(n)}$. Identifying weakness of such circuits - powers of linear forms have low dimension for the space of partial derivatives.

^{Reading : Class notes. Section 8.1 of [RP] survey. Chapter 10 of [CKW] survey.}

Lec 35 (01 Apr, Wed) : Argued that measure is large for a monomial. Used this in proving size exponential size lower bounds for arbitrary $\Sigma \wedge \Sigma$ computing $x_1x_2\ldots x_n$ which almost matches the upper bound. Small sized depth three arithmetic circuits computing the elementary symmetric polynomials (via interpolation).

^{Reading : Class notes. Section 8.1 of [RP] survey.}

Lec () (03 Apr, Fri) : Holiday due to Good Friday.

Lec 36 (06 Apr, Mon) : Lower bound for homogeneous depth $3$ circuits computing elementary symmetric polynomial (Nisan-Wigderson). Defined the measure. Towards arguing that the measure is large for elementary symmetric polynomials.

^{Reading : Section 9.1 of [RP] survey.}

Lec 37 (08 Apr, Wed) : Structure of the partial derivative space of the elementary symmetric polynomial and its connection to disjointness matrix. Completed the proof assuming that disjointness matrix has full rank.

^{Reading : Section 9.1 of [RP] survey, class notes.}

Theme: Polynomial Identity Testing and connections to circuit lower bounds