13 research outputs found
An Efficient Parallel Algorithm for Spectral Sparsification of Laplacian and SDDM Matrix Polynomials
For "large" class of continuous probability density functions
(p.d.f.), we demonstrate that for every there is mixture of
discrete Binomial distributions (MDBD) with
distinct Binomial distributions that -approximates a
discretized p.d.f. for all , where
. Also, we give two efficient parallel
algorithms to find such MDBD.
Moreover, we propose a sequential algorithm that on input MDBD with
for that induces a discretized p.d.f. ,
that is either Laplacian or SDDM matrix and parameter ,
outputs in time a spectral
sparsifier of a matrix-polynomial, where
notation hides factors.
This improves the Cheng et al.'s [CCLPT15] algorithm whose run time is
.
Furthermore, our algorithm is parallelizable and runs in work
and depth . Our main algorithmic contribution is to
propose the first efficient parallel algorithm that on input continuous p.d.f.
, matrix as above, outputs a spectral sparsifier of
matrix-polynomial whose coefficients approximate component-wise the discretized
p.d.f. .
Our results yield the first efficient and parallel algorithm that runs in
nearly linear work and poly-logarithmic depth and analyzes the long term
behaviour of Markov chains in non-trivial settings. In addition, we strengthen
the Spielman and Peng's [PS14] parallel SDD solver
On approximate polynomial identity testing and real root finding
In this thesis we study the following three topics, which share a connection through the (arithmetic) circuit complexity of polynomials. 1. Rank of symbolic matrices. 2. Computation of real roots of real sparse polynomials. 3. Complexity of symmetric polynomials. We start with studying the commutative and non-commutative rank of symbolic matrices with linear forms as their entries. Here we show a deterministic polynomial time approximation scheme (PTAS) for computing the commutative rank. Prior to this work, deterministic polynomial time algorithms were known only for computing a 1/2-approximation of the commutative rank. We give two distinct proofs that our algorithm is a PTAS. We also give a min-max characterization of commutative and non-commutative ranks. Thereafter we direct our attention to computation of roots of uni-variate polynomial equations. It is known that solving a system of polynomial equations reduces to solving a uni-variate polynomial equation. We describe a polynomial time algorithm for (n,k,\tau)-nomials which computes approximations of all the real roots (even though it may also compute approximations of some complex roots). Moreover, we also show that the roots of integer trinomials are well-separated. Finally, we study the complexity of symmetric polynomials. It is known that symmetric Boolean functions are easy to compute. In contrast, we show that the assumption VP \neq VNP implies that there exist hard symmetric polynomials. To prove this result, we use an algebraic analogue of the classical Newton iteration.In dieser Dissertation untersuchen wir die folgenden drei Themen, welche durch die (arithmetische) SchaltkreiskomplexitĂ€t von Polynomen miteinander verbunden sind: 1. der Rang von symbolischen Matrizen, 2. die Berechnung von reellen Nullstellen von dĂŒnnbesetzten (âsparseâ) Polynomen mit reellen Koeffizienten, 3. die KomplexitĂ€t von symmetrischen Polynomen. Wir untersuchen zunĂ€chst den kommutativen und nicht-kommutativen Rang von Matrizen, deren EintrĂ€ge aus Linearformen bestehen. Hier beweisen wir die Existenz eines deterministischem Polynomialzeit-Approximationsschemas (PTAS) fĂŒr die Berechnung des kommutative Ranges. Zuvor waren polynomielle Algorithmen nur fĂŒr die Berechnung einer 1/2-Approximation des kommutativen Ranges bekannt. Wir geben zwei unterschiedliche Beweise fĂŒr den Fakt, dass unser Algorithmus tatsĂ€chlich ein PTAS ist. ZusĂ€tzlich geben wir eine min-max Charakterisierung des kommutativen und nicht-kommutativen Ranges. AnschlieĂend lenken wir unsere Aufmerksamkeit auf die Berechnung von Nullstellen von univariaten polynomiellen Gleichungen. Es ist bekannt, dass das Lösen eines polynomiellem Gleichungssystems auf das Lösen eines univariaten Polynoms zurĂŒckgefĂŒhrt werden kann. Wir geben einen Polynomialzeit-Algorithmus fĂŒr (n, k, \tau)-Nome, welcher AbschĂ€tzungen fĂŒr alle reellen Nullstellen berechnet (in manchen Fallen auch AbschĂ€tzungen von komplexen Nullstellen). ZusĂ€tzlich beweisen wir, dass Nullstellen von ganzzahligen Trinomen stets weit voneinander entfernt sind. SchlieĂlich untersuchen wir die KomplexitĂ€t von symmetrischen Polynomen. Es ist bereits bekannt, dass sich symmetrische Boolesche Funktionen leicht berechnen lassen. Im Gegensatz dazu zeigen wir, dass die Annahme VP \neq VNP bedeutet, dass auch harte symmetrische Polynome existieren. Um dies zu beweisen benutzen wir ein algebraisches Analog zum klassischen Newton-Verfahren
On the Order of Power Series and the Sum of Square Roots Problem
This paper focuses on the study of the order of power series that are linear
combinations of a given finite set of power series. The order of a formal power
series, known as , is defined as the minimum exponent of
that has a non-zero coefficient in . Our first result is that the order
of the Wronskian of these power series is equivalent up to a polynomial factor,
to the maximum order which occurs in the linear combination of these power
series. This implies that the Wronskian approach used in (Kayal and Saha,
TOCT'2012) to upper bound the order of sum of square roots is optimal up to a
polynomial blowup. We also demonstrate similar upper bounds, similar to those
of (Kayal and Saha, TOCT'2012), for the order of power series in a variety of
other scenarios. We also solve a special case of the inequality testing problem
outlined in (Etessami et al., TOCT'2014).
In the second part of the paper, we study the equality variant of the sum of
square roots problem, which is decidable in polynomial time due to (Bl\"omer,
FOCS'1991). We investigate a natural generalization of this problem when the
input integers are given as straight line programs. Under the assumption of the
Generalized Riemann Hypothesis (GRH), we show that this problem can be reduced
to the so-called ``one dimensional'' variant. We identify the key mathematical
challenges for solving this ``one dimensional'' variant
Greedy Strikes Again: A Deterministic PTAS for Commutative Rank of Matrix Spaces
We consider the problem of commutative rank computation of a given matrix space. A matrix space is a (linear) subspace of the (linear) space of n x n matrices over a given field. The problem is fundamental, as it generalizes several computational problems from algebra and combinatorics. For instance, checking if the commutative rank of the space is n, subsumes problems such as testing perfect matching in graphs and identity testing of algebraic branching programs. An efficient deterministic computation of the commutative rank is a major open problem, although there is a simple and efficient randomized algorithm for it. Recently, there has been a series of results on computing the non-commutative rank of matrix spaces in deterministic polynomial time. Since the non-commutative rank of any matrix space is at most twice the commutative rank, one immediately gets a deterministic 1/2-approximation algorithm for the computation of the commutative rank. This leads to a natural question of whether this approximation ratio can be improved. In this paper, we answer this question affirmatively.
We present a deterministic Polynomial-time approximation scheme (PTAS) for computing the commutative rank of a given matrix space B. More specifically, given a matrix space and a rational number e > 0, we give an algorithm, that runs in time O(n^(4 + 3/e)) and computes a matrix A in the given matrix space B such that the rank of A is at least (1-e) times the commutative rank of B. The algorithm is the natural greedy algorithm. It always takes the first set of k matrices that will increase the rank of the matrix constructed so far until it does not find any improvement, where the size of the set k depends on e
Density Independent Algorithms for Sparsifying k-Step Random Walks
We give faster algorithms for producing sparse approximations of the transition matrices of k-step random walks on undirected and weighted graphs. These transition matrices also form graphs, and arise as intermediate objects in a variety of graph algorithms. Our improvements are based on a better understanding of processes that sample such walks, as well as tighter bounds on key weights underlying these sampling processes. On a graph with n vertices and m edges, our algorithm produces a graph with about nlog(n) edges that approximates the k-step random walk graph in about m + k^2 nlog^4(n) time. In order to obtain this runtime bound, we also revisit "density independent" algorithms for sparsifying graphs whose runtime overhead is expressed only in terms of the number of vertices
How many zeros of a random sparse polynomial are real?
We investigate the number of real zeros of a univariate -sparse polynomial
over the reals, when the coefficients of come from independent standard
normal distributions. Recently B\"urgisser, Erg\"ur and Tonelli-Cueto showed
that the expected number of real zeros of in such cases is bounded by
. In this work, we improve the bound to and
also show that this bound is tight by constructing a family of sparse support
whose expected number of real zeros is lower bounded by . Our
main technique is an alternative formulation of the Kac integral by
Edelman-Kostlan which allows us to bound the expected number of zeros of in
terms of the expected number of zeros of polynomials of lower sparsity. Using
our technique, we also recover the bound on the expected number of
real zeros of a dense polynomial of degree with coefficients coming from
independent standard normal distributions
Homogeneous Algebraic Complexity Theory and Algebraic Formulas
We study algebraic complexity classes and their complete polynomials under
\emph{homogeneous linear} projections, not just under the usual affine linear
projections that were originally introduced by Valiant in 1979. These
reductions are weaker yet more natural from a geometric complexity theory (GCT)
standpoint, because the corresponding orbit closure formulations do not require
the padding of polynomials. We give the \emph{first} complete polynomials for
VF, the class of sequences of polynomials that admit small algebraic formulas,
under homogeneous linear projections: The sum of the entries of the
non-commutative elementary symmetric polynomial in 3 by 3 matrices of
homogeneous linear forms.
Even simpler variants of the elementary symmetric polynomial are hard for the
topological closure of a large subclass of VF: the sum of the entries of the
non-commutative elementary symmetric polynomial in 2 by 2 matrices of
homogeneous linear forms, and homogeneous variants of the continuant polynomial
(Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of
circuits with arity-3 product gates.Comment: This is edited part of preprint arXiv:2211.0705
Fixed-parameter debordering of Waring rank
Border complexity measures are defined via limits (or topological closures), so that any function which can approximated arbitrarily closely by low complexity functions itself has low border complexity. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits.
Debordering is at the heart of understanding the difference between Valiant's determinant vs permanent conjecture, and Mulmuley and Sohoni's variation which uses border determinantal complexity. The debordering of matrix multiplication tensors by Bini played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Currently, very few debordering results are known.
In this work, we study the question of debordering the border Waring rank of polynomials. Waring and border Waring rank are very well studied measures in the context of invariant theory, algebraic geometry, and matrix multiplication algorithms. For the first time, we obtain a Waring rank upper bound that is exponential in the border Waring rank and only linear in the degree. All previous known results were exponential in the degree. For polynomials with constant border Waring rank, our results imply an upper bound on the Waring rank linear in degree, which previously was only known for polynomials with border Waring rank at most 5