Low-degree Polynomial Research Articles

Uniform Generation of Spanning Regular Subgraphs of a Dense Graph

Let $H_n$ be a graph on $n$ vertices and let $\overline{H_n}$ denote the complement of $H_n$. Suppose that $\Delta = \Delta(n)$ is the maximum degree of $\overline{H_n}$. We analyse three algorithms for sampling $d$-regular subgraphs ($d$-factors) of $H_n$. This is equivalent to uniformly sampling $d$-regular graphs which avoid a set $E(\overline{H_n})$ of forbidden edges. Here $d=d(n)$ is a positive integer which may depend on $n$.
 Two of these algorithms produce a uniformly random $d$-factor of $H_n$ in expected runtime which is linear in $n$ and low-degree polynomial in $d$ and $\Delta$. The first algorithm applies when $(d+\Delta)d\Delta = o(n)$. This improves on an earlier algorithm by the first author, which required constant $d$ and at most a linear number of edges in $\overline{H_n}$. The second algorithm applies when $H_n$ is regular and $d^2+\Delta^2 = o(n)$, adapting an approach developed by the first author together with Wormald. The third algorithm is a simplification of the second, and produces an approximately uniform $d$-factor of $H_n$ in time $O(dn)$. Here the output distribution differs from uniform by $o(1)$ in total variation distance, provided that $d^2+\Delta^2 = o(n)$.

Computers in mathematical research: the study of three-point root-finding methods

This paper is motivated by results of extensive comparative study of three-point methods for solving nonlinear equations presented in the paper (C. Chun, B. Neta, Comparative study of eighth-order methods for finding simple roots of nonlinear equations, Numer. Algor. 74, 1169–1201, 2017). In the first part, we show that the best ranking method, constructed by Sharma and Arora in 2016, is actually a special case of the family of three-point methods presented in the paper (J. Džunic, M. S. Petkovic, L. D. Petkovic, A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. Math. Comput. 217, 7612–7619, 2011). Since the mentioned Chun-Neta’s comparative study was carried out by testing algebraic polynomials of low degree (not higher than 6), in the second part we have extended comparative study to higher-order polynomials, polynomials with random coefficients, and polynomials with clusters of zeros. The performed experiments confirmed results of Chun and Neta. As in the cited paper of Chun and Neta, we have also applied the quality study based on computer visualization plotting the basins of attraction.

Optimal Sparsification for Some Binary CSPs Using Low-Degree Polynomials

This article analyzes to what extent it is possible to efficiently reduce the number of clauses in NP-hard satisfiability problems without changing the answer. Upper and lower bounds are established using the concept of kernelization. Existing results show that if NP ⊈ coNP/poly, no efficient preprocessing algorithm can reduce n -variable instances of cnf-sat with d literals per clause to equivalent instances with O ( n d −ε ) bits for any ε > 0. For the N ot -A ll -E qual sat problem, a compression to size Õ ( n d −1 ) exists. We put these results in a common framework by analyzing the compressibility of CSPs with a binary domain. We characterize constraint types based on the minimum degree of multivariate polynomials whose roots correspond to the satisfying assignments, obtaining (nearly) matching upper and lower bounds in several settings. Our lower bounds show that not just the number of constraints, but also the encoding size of individual constraints plays an important role. For example, for E xact S atisfiability with unbounded clause length it is possible to efficiently reduce the number of constraints to n +1, yet no polynomial-time algorithm can reduce to an equivalent instance with O ( n 2−ε ) bits for any ε > 0, unless NP ⊆ coNP/poly.

A New Algorithm Based Charge Controller for Global MPPT of Si Solar Cell

This work illustrates the development of a new hybrid MPPT algorithm and the construction of an efficient charge controller based on this algorithm. The existing perturb & observe (P&O) methodology and incremental conductance (IC) method are unable to trace the global maximum power point (GMPP) once there's a frequent modification in solar irradiance. These algorithms stick to local maxima in such scenario so the controller efficiency drops. Here a new hybrid MPPT algorithm was designed to track the GMPP efficiently, which involves the concept of Savitzky Golay digital filter technique. This filter is employed to swish a collection of digital information points that additional will increase the S/N of the signal while not greatly distorting it. The substitute sub-sets of niggardly observations fact are passable about a low-degree polynomial by the come nigh of neat least squares by means of convolution. This algorithm first smoothens the output of photovoltaic using the described technique and removes the local maxima and then further applies the P&O method to find out the GMPP. The circuit was simulated using Proteus and later on, an experimental setup was built using Arduino UNO (ATMEGA328P) to run the algorithm in real-time environment. The controller successfully tracked the GMPP voltages for variable irradiance both in simulation and real-time environment.

On conversions from CNF to ANF

In this paper we discuss conversion methods from the conjunctive normal form (CNF) to the algebraic normal form (ANF) of a Boolean function. Whereas the reverse conversion has been studied before, the CNF to ANF conversion has been achieved predominantly via a standard method which tends to produce many polynomials of high degree. Based on a block-building mechanism, we design a new blockwise algorithm for the CNF to ANF conversion which is geared towards producing fewer and lower degree polynomials. In particular, we look for as many linear polynomials as possible in the converted system and check that our algorithm finds them. As an application, we suggest a new algebraic extension of the resolution calculus which is tailored to the output of the standard CNF to ANF conversion algorithm. Experiments show that the ANF produced by our algorithm outperforms the standard conversion in “real life” examples originating from cryptographic attacks, and that our algebraic resolution algorithm solves some SAT instances which are notoriously hard for SAT solvers.

Derandomized Concentration Bounds for Polynomials, and Hypergraph Maximal Independent Set

A parallel algorithm for maximal independent set (MIS) in hypergraphs has been a long-standing algorithmic challenge, dating back nearly 30 years to a survey of Karp and Ramachandran (1990). The best randomized parallel algorithm for hypergraphs of fixed rank r was developed by Beame and Luby (1990) and Kelsen (1992), running in time roughly (log n ) r ! . We improve the randomized algorithm of Kelsen, reducing the runtime to roughly (log n ) 2 r and simplifying the analysis through the use of more-modern concentration inequalities. We also give a method for derandomizing concentration bounds for low-degree polynomials, which are the key technical tool used to analyze that algorithm. This leads to a deterministic PRAM algorithm also running in (log n ) 2 r +3 time and poly ( m , n ) processors. This is the first deterministic algorithm with sub-polynomial runtime for hypergraphs of rank r > 3. Our analysis can also apply when r is slowly growing; using this in conjunction with a strategy of Bercea et al. (2015) gives a deterministic MIS algorithm running in time exp ( O ( log ( mn ) / log log ( mn )).

On the complexity of testing attainment of the optimal value in nonlinear optimization

We prove that unless $$\hbox {P}=\hbox {NP}$$ , there exists no polynomial time (or even pseudo-polynomial time) algorithm that can test whether the optimal value of a nonlinear optimization problem where the objective and constraints are given by low-degree polynomials is attained. If the degrees of these polynomials are fixed, our results along with previously-known “Frank–Wolfe type” theorems imply that exactly one of two cases can occur: either the optimal value is attained on every instance, or it is strongly NP-hard to distinguish attainment from non-attainment. We also show that testing for some well-known sufficient conditions for attainment of the optimal value, such as coercivity of the objective function and closedness and boundedness of the feasible set, is strongly NP-hard. As a byproduct, our proofs imply that testing the Archimedean property of a quadratic module is strongly NP-hard, a property that is of independent interest to the convergence of the Lasserre hierarchy. Finally, we give semidefinite programming (SDP)-based sufficient conditions for attainment of the optimal value, in particular a new characterization of coercive polynomials that lends itself to an SDP hierarchy.

Low-degree planar polynomials over finite fields of characteristic two

Planar functions are mappings from a finite field Fq to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of q and we consider the case that q is even. We classify polynomials of degree at most q1/4 that induce planar functions on Fq, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over Fq that induce planar functions on infinitely many extensions of Fq. The proof strategy is to study the number of Fq-rational points of an algebraic curve attached to a putative planar function. Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear functions.

Generating Root-Finder Iterative Methods of Second Order: Convergence and Stability

In this paper, a simple family of one-point iterative schemes for approximating the solutions of nonlinear equations, by using the procedure of weight functions, is derived. The convergence analysis is presented, showing the sufficient conditions for the weight function. Many known schemes are members of this family for particular choices of the weight function. The dynamical behavior of one of these choices is presented, analyzing the stability of the fixed points and the critical points of the rational function obtained when the iterative expression is applied on low degree polynomials. Several numerical tests are given to compare different elements of the proposed family on non-polynomial problems.

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a kernel of bitsize {mathcal {O}}(k^{q-1}log {k}) for q-Coloring parameterized by Vertex Cover for any qge 3. This size bound is optimal up to k^{o(1)} factors assuming mathsf {NP} not subseteq mathsf {coNP/poly}, and improves on the previous-best kernel of size {mathcal {O}}(k^q). We generalize this result for deciding q-colorability of a graph G, to deciding the existence of a homomorphism from G to an arbitrary fixed graph H. Furthermore, we can replace the parameter vertex cover by the less restrictive parameter twin-cover. We prove that H-Coloring parameterized by Twin-Cover has a kernel of size {mathcal {O}}(k^{varDelta (H)}log k). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming mathsf {NP} not subseteq mathsf {coNP/poly}, the parameterization by the number of vertices n admits no (generalized) kernel of size {mathcal {O}}(n^{2-varepsilon }) for any varepsilon > 0. Previously, such a lower bound was only known for coloring with q ge 4 colors.

A New Model of Artificial Intelligence

In this article, I'll present a new model of artificial intelligence rooted in information theory that makes use of tractable, low-degree polynomial algorithms that nonetheless allow for the analysis of the same types of extremely high-dimensional datasets typically used in machine learning and deep learning techniques. Specifically, I'll show how these algorithms can be used to identify objects in images, predict complex random paths, predict projectile paths in three-dimensions, and classify three-dimensional objects, in each case making use of inferences drawn from millions of underlying data points, all using low-degree polynomial run time algorithms that can be executed quickly on an ordinary consumer device. In short, the purpose of these algorithms is to commoditize the building blocks of artificial intelligence. All of the code necessary to run these algorithms, and generate the training data, is available on my researchgate homepage, under the project heading, Information Theory. Note that I retain all rights, copyright and otherwise, to all of the algorithms, and other information presented in this paper. In particular, the information contained in this paper may not be used for any commercial purpose whatsoever without my prior written consent.

Harmonicity and invariance on slices of the Boolean cube

In a recent work with Kindler and Wimmer we proved an invariance principle for the slice for low-influence, low-degree harmonic multilinear polynomials (a polynomial in $$x_1,\ldots ,x_n$$ is harmonic if it is annihilated by $$\sum _{i=1}^n \frac{\partial }{\partial x_i}$$ ). Here we provide an alternative proof for general low-degree harmonic multilinear polynomials, with no constraints on the influences. We show that any real-valued harmonic multilinear polynomial on the slice whose degree is $$o(\sqrt{n})$$ has approximately the same distribution under the slice and cube measures. Our proof is based on ideas and results from the representation theory of $$S_n$$ , along with a novel decomposition of random increasing paths in the cube in terms of martingales and reverse martingales. While such decompositions have been used in the past for stationary reversible Markov chains, our decomposition is applied in a non-stationary non-reversible setup. We also provide simple proofs for some known and some new properties of harmonic functions which are crucial for the proof. Finally, we provide independent simple proofs for the known facts that (1) one cannot distinguish between the slice and the cube based on functions of o(n) coordinates and (2) Boolean symmetric functions on the cube cannot be approximated under the uniform measure by functions whose sum of influences is $$o(\sqrt{n})$$ .

Frequency-dependent Template Profiles for High-precision Pulsar Timing

Abstract Pulsar timing experiments require high-fidelity template profiles in order to minimize the biases in pulse time-of-arrival (TOA) measurements and their uncertainties. Efforts to acquire more precise TOAs given the fixed effective area of telescopes, finite receiver noise, and limited integration time have led pulsar astronomers to the solution of implementing ultra-wideband receivers. This solution, however, has run up against the problem that pulse profile shapes evolve with frequency, which raises the question of how to properly measure and analyze TOAs obtained using template-matching methods. This paper proposes a new method for one facet of this problem, that of template profile generation, and demonstrates it on the well-timed millisecond pulsar J1713+0747. Specifically, we decompose pulse profile evolution into a linear combination of basis eigenvectors, the coefficients of which change slowly with frequency such that their evolution is modeled simply by a sum of low-degree piecewise polynomial spline functions. These noise-free, high-fidelity, frequency-dependent templates can be used to make measurements of so-called “wideband TOAs” simultaneously with an estimate of the instantaneous dispersion measure. The use of wideband TOAs is becoming important for pulsar timing array experiments, as the volume of data sets comprised of conventional, subbanded TOAs are quickly becoming unwieldy for the Bayesian analyses needed to uncover latent gravitational wave signals. Although motivated by high-precision timing experiments, our technique is applicable in more general pulsar observations.

MSCryptoNet: Multi-Scheme Privacy-Preserving Deep Learning in Cloud Computing

Privacy in the Internet of Things is a fundamental challenge for the Ubiquitous healthcare systems that depend on the data aggregated and collaborative deep learning among different parties. This paper proposes the MSCryptoNet, a novel framework that enables the scalable execution and the conversion of the state-of-the-art learned neural network to the MSCryptoNet models in the privacy-preservation setting. We also design a method for approximation of the activation function basically used in the convolutional neural network (i.e., Sigmoid and Rectified linear unit) with low degree polynomials, which is vital for computations in the homomorphic encryption schemes. Our model seems to target the following scenarios: 1) the practical way to enforce the evaluation of classifier whose inputs are encrypted with possibly different encryption schemes or even different keys while securing all operations including intermediate results and 2) the minimization of the communication and computational cost of the data providers. The MSCryptoNet is based on the multi-scheme fully homomorphic encryption. We also prove that the MSCryptoNet as a privacy-preserving deep learning scheme over the aggregated encrypted data is secured.

Counting zeros of cosine polynomials: On a problem of Littlewood

We show that if A is a finite set of non-negative integers then the number of zeros of the functionfA(θ)=∑a∈Acos⁡(aθ), in [0,2π], is at least (log⁡log⁡log⁡|A|)1/2−ε. This gives the first unconditional lower bound on a problem of Littlewood and solves a conjecture of Borwein, Erdélyi, Ferguson and Lockhart. We also prove a result that applies to more general cosine polynomials with few distinct rational coefficients. One of the main ingredients in the proof is perhaps of independent interest: we show that if f is an exponential polynomial with few distinct integer coefficients and f “correlates” with a low-degree exponential polynomial P, then f has a very particular structure. This result allows us to prove a structure theorem for trigonometric polynomials with few zeros in [0,2π] and restricted coefficients.

Towards practical privacy-preserving genome-wide association study

BackgroundThe deployment of Genome-wide association studies (GWASs) requires genomic information of a large population to produce reliable results. This raises significant privacy concerns, making people hesitate to contribute their genetic information to such studies.ResultsWe propose two provably secure solutions to address this challenge: (1) a somewhat homomorphic encryption (HE) approach, and (2) a secure multiparty computation (MPC) approach. Unlike previous work, our approach does not rely on adding noise to the input data, nor does it reveal any information about the patients. Our protocols aim to prevent data breaches by calculating the χ2 statistic in a privacy-preserving manner, without revealing any information other than whether the statistic is significant or not. Specifically, our protocols compute the χ2 statistic, but only return a yes/no answer, indicating significance. By not revealing the statistic value itself but only the significance, our approach thwarts attacks exploiting statistic values. We significantly increased the efficiency of our HE protocols by introducing a new masking technique to perform the secure comparison that is necessary for determining significance.ConclusionsWe show that full-scale privacy-preserving GWAS is practical, as long as the statistics can be computed by low degree polynomials. Our implementations demonstrated that both approaches are efficient. The secure multiparty computation technique completes its execution in approximately 2 ms for data contributed by one million subjects.

Savitzky-Golay filtering for Scattered Signal De-noising

Extraction of distinguishing features and decision of classifiers are highly influenced by the low signal-to-noise ratio (SNR) rates, when target identification from scattered electromagnetic waves is considered. In order to increase the correct identification rates, smoothing operations, which should increase the SNR without greatly distorting the signal, are occasionally employed. However, this operation is mostly performed to the complete scattered signal via an over-complete basis. On the other hand, Savitzky-Golay filters can de-noise the signal by fitting successive sub-sets of adjacent scattered data points with a low-degree polynomial through the use of linear least squares. Thus in this study, both computational burden and accuracy of Savitzky-Golay filters are compared to three well-established smoothing techniques in time domain, frequency domain and time-frequency analysis. The analyses are performed with both simulated and measured data from various conductor and dielectric targets having different size, geometry and material type.

Shuffles and Circuits (On Lower Bounds for Modern Parallel Computation)

The goal of this article is to identify fundamental limitations on how efficiently algorithms implemented on platforms such as MapReduce and Hadoop can compute the central problems in motivating application domains, such as graph connectivity problems. We introduce an abstract model of massively parallel computation, where essentially the only restrictions are that the “fan-in” of each machine is limited to s bits, where s is smaller than the input size n , and that computation proceeds in synchronized rounds, with no communication between different machines within a round. Lower bounds on the round complexity of a problem in this model apply to every computing platform that shares the most basic design principles of MapReduce-type systems. We prove that computations in our model that use few rounds can be represented as low-degree polynomials over the reals. This connection allows us to translate a lower bound on the (approximate) polynomial degree of a Boolean function to a lower bound on the round complexity of every (randomized) massively parallel computation of that function. These lower bounds apply even in the “unbounded width” version of our model, where the number of machines can be arbitrarily large. As one example of our general results, computing any nontrivial monotone graph property—such as connectivity—requires a super-constant number of rounds when every machine receives only a subpolynomial (in n ) number of input bits s . Finally, we prove that, in two senses, our lower bounds are the best one could hope for. For the unbounded-width model, we prove a matching upper bound. Restricting to a polynomial number of machines, we show that asymptotically better lower bounds would separate P from NC 1 .

On the Finite Length Scaling of <inline-formula> <tex-math notation="LaTeX">$q$ </tex-math> </inline-formula>-Ary Polar Codes

The polarization process of polar codes over a prime $q$ -ary alphabet is studied. Recently, it has been shown that the blocklength of polar codes with prime alphabet size scales polynomially with respect to the inverse of the gap between code rate and channel capacity. However, except for the binary case, the degree of the polynomial in the bound is extremely large. In this paper, a different approach to computing the degree of this polynomial for any prime alphabet size is shown. This approach yields a lower degree polynomial for various values of the alphabet size that were examined. It is also shown that even lower degree polynomial can be computed with an additional numerical effort.

DMLPG method for numerical simulation of soliton collisions in multi-dimensional coupled damped nonlinear Schrödinger system which arises from Bose–Einstein condensates

In this paper, the direct meshless local Petrov–Galerkin (DMLPG) method is applied to find the numerical solution of coupled damped nonlinear Schrödinger system in one, two and three-dimensional spaces. The propagation properties of single soliton, double and triple solitons of coupled damped nonlinear Schrödinger system are simulated and the interactions between these solitons are studied numerically. The efficient time differencing Runge–Kutta method is utilized for the time discretization. DMLPG shifts the numerical integrations over low-degree polynomials rather than over complicated shape functions and this significantly increases the computational efficiency of DMLPG in comparison with the other meshless local weak form methods especially in two and three dimensions. The main aim of this paper is to show that the DMLPG method can be simply used for solving high-dimensional system of non-linear partial differential equations especially coupled damped nonlinear Schrödinger system. The numerical results confirm the good efficiency of the proposed method for solving our model.