arXiv subjects

Mrinal Kumar

Publications and source records attributed to Mrinal Kumar.

At least 37 records · Page 2Linked to original sources

arXiv2024

Time-delayed Dynamic Mode Decomposition for families of periodic trajectories in Cislunar Space

In recent years, the development of the Lunar Gateway and Artemis missions has renewed interest in lunar exploration, including both manned and unmanned missions. This interest necessitates accurate initial orbit determination (IOD) and orbit prediction (OP) in this domain, which faces significant challenges such as severe nonlinearity, sensitivity to initial conditions, large state-space volume, and sparse, faint, and unreliable measurements. This paper explores the capability of data-driven Koopman operator-based approximations for OP in these scenarios. Three stable periodic trajectories from distinct cislunar families are analyzed. The analysis includes theoretical justification for using a linear time-invariant system as the data-driven surrogate. This theoretical framework is supported by experimental validation. Furthermore, the accuracy is assessed by comparing the spectral content captured to period estimates derived from the fast Fourier transform (FFT) and Poincare-like sections.

eess.SY↗

arXiv2024

Accelerating Particle-in-Cell Kinetic Plasma Simulations via Reduced-Order Modeling of Space-Charge Dynamics using Dynamic Mode Decomposition

We present a data-driven reduced-order modeling of the space-charge dynamics for electromagnetic particle-in-cell (EMPIC) plasma simulations based on dynamic mode decomposition (DMD). The dynamics of the charged particles in kinetic plasma simulations such as EMPIC is manifested through the plasma current density defined along the edges of the spatial mesh. We showcase the efficacy of DMD in modeling the time evolution of current density through a low-dimensional feature space. Not only do such DMD-based predictive reduced-order models help accelerate EMPIC simulations, they also have the potential to facilitate investigative analysis and control applications. We demonstrate the proposed DMD-EMPIC scheme for reduced-order modeling of current density, and speed-up in EMPIC simulations involving electron beam under the influence of magnetic field, virtual cathode oscillations, and backward wave oscillator.

physics.plasm-ph↗

arXiv2024

Fast list-decoding of univariate multiplicity and folded Reed-Solomon codes

We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon codes can be made to run in $\tilde{O}(n)$ time. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list decoding. It is known that for every $ε>0$, and rate $r \in (0,1)$, there exist explicit families of these codes that have rate $r$ and can be list decoded from a $(1-r-ε)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above list-decoding tasks in $\tilde{O}(n)$, where $n$ is the block-length of the code. Our algorithms have two main components. The first component builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a $\tilde{O}(n)$ time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design $\tilde{O}(n)$ time algorithms for two natural algebraic problems: given a $(m+2)$-variate polynomial $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i=0}^m Q_i(x)\cdot y_i$ the first algorithm solves order-$m$ linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ while the second solves functional equations of the form $Q\left(x, f(x), f(γx), \dots,f(γ^m x)\right) \equiv 0$, where $m$ is an arbitrary constant and $γ$ is a field element of sufficiently high order. These algorithms can be viewed as generalizations of classical $\tilde{O}(n)$ time algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.

cs.IT↗

arXiv2024

Towards Deterministic Algorithms for Constant-Depth Factors of Constant-Depth Circuits

We design a deterministic subexponential time algorithm that takes as input a multivariate polynomial $f$ computed by a constant-depth circuit over rational numbers, and outputs a list $L$ of circuits (of unbounded depth and possibly with division gates) that contains all irreducible factors of $f$ computable by constant-depth circuits. This list $L$ might also include circuits that are spurious: they either do not correspond to factors of $f$ or are not even well-defined, e.g. the input to a division gate is a sub-circuit that computes the identically zero polynomial. The key technical ingredient of our algorithm is a notion of the pseudo-resultant of $f$ and a factor $g$, which serves as a proxy for the resultant of $g$ and $f/g$, with the advantage that the circuit complexity of the pseudo-resultant is comparable to that of the circuit complexity of $f$ and $g$. This notion, which might be of independent interest, together with the recent results of Limaye, Srinivasan and Tavenas, helps us derandomize one key step of multivariate polynomial factorization algorithms - that of deterministically finding a good starting point for Newton Iteration for the case when the input polynomial as well as the irreducible factor of interest have small constant-depth circuits.

cs.CC↗

arXiv2024

On the Predictive Capability of Dynamic Mode Decomposition for Nonlinear Periodic Systems with Focus on Orbital Mechanics

This paper discusses the predictive capability of Dynamic Mode Decomposition (DMD) in the context of orbital mechanics. The focus is specifically on the Hankel variant of DMD which uses a stacked set of time-delayed observations for system identification and subsequent prediction. A theory on the minimum number of time delays required for accurate reconstruction of periodic trajectories of nonlinear systems is presented and corroborated using experimental analysis. In addition, the window size for training and prediction regions, respectively, is presented. The need for a meticulous approach while using DMD is emphasized by drawing comparisons between its performance on two candidate satellites, the ISS and MOLNIYA-3-50.

eess.SY↗

arXiv2023

Ideal-theoretic Explanation of Capacity-achieving Decoding

In this work, we present an abstract framework for some algebraic error-correcting codes with the aim of capturing codes that are list-decodable to capacity, along with their decoding algorithm. In the polynomial ideal framework, a code is specified by some ideals in a polynomial ring, messages are polynomials and their encoding is the residue modulo the ideals. We present an alternate way of viewing this class of codes in terms of linear operators, and show that this alternate view makes their algorithmic list-decodability amenable to analysis. Our framework leads to a new class of codes that we call affine Folded Reed-Solomon codes (which are themselves a special case of the broader class we explore). These codes are common generalizations of the well-studied Folded Reed-Solomon codes and Multiplicity codes, while also capturing the less-studied Additive Folded Reed-Solomon codes as well as a large family of codes that were not previously known/studied. More significantly our framework also captures the algorithmic list-decodability of the constituent codes. Specifically, we present a unified view of the decoding algorithm for ideal theoretic codes and show that the decodability reduces to the analysis of the distance of some related codes. We show that good bounds on this distance lead to capacity-achieving performance of the underlying code, providing a unifying explanation of known capacity-achieving results. In the specific case of affine Folded Reed-Solomon codes, our framework shows that they are list-decodable up to capacity (for appropriate setting of the parameters), thereby unifying the previous results for Folded Reed-Solomon, Multiplicity and Additive Folded Reed-Solomon codes.

cs.IT↗

arXiv2023

An Improved Line-Point Low-Degree Test

We prove that the most natural low-degree test for polynomials over finite fields is ``robust'' in the high-error regime for linear-sized fields. Specifically we consider the ``local'' agreement of a function $f: \mathbb{F}_q^m \to \mathbb{F}_q$ from the space of degree-$d$ polynomials, i.e., the expected agreement of the function from univariate degree-$d$ polynomials over a randomly chosen line in $\mathbb{F}_q^m$, and prove that if this local agreement is $ε\geq Ω((\frac{d}{q})^τ))$ for some fixed $τ> 0$, then there is a global degree-$d$ polynomial $Q: \mathbb{F}_q^m \to \mathbb{F}_q$ with agreement nearly $ε$ with $f$. This settles a long-standing open question in the area of low-degree testing, yielding an $O(d)$-query robust test in the ``high-error'' regime (i.e., when $ε< \frac{1}{2}$). The previous results in this space either required $ε> \frac{1}{2}$ (Polishchuk \& Spielman, STOC 1994), or $q = Ω(d^4)$ (Arora \& Sudan, Combinatorica 2003), or needed to measure local distance on $2$-dimensional ``planes'' rather than one-dimensional lines leading to $Ω(d^2)$-query complexity (Raz \& Safra, STOC 1997). Our analysis follows the spirit of most previous analyses in first analyzing the low-variable case ($m = O(1)$) and then ``bootstrapping'' to general multivariate settings. Our main technical novelty is a new analysis in the bivariate setting that exploits a previously known connection between multivariate factorization and finding (or testing) low-degree polynomials, in a non ``black-box'' manner. A second contribution is a bootstrapping analysis which manages to lift analyses for $m=2$ directly to analyses for general $m$, where previous works needed to work with $m = 3$ or $m = 4$ -- arguably this bootstrapping is significantly simpler than those in prior works.

cs.CC↗

arXiv2023

Deterministic Algorithms for Low Degree Factors of Constant Depth Circuits

For every constant $d$, we design a subexponential time deterministic algorithm that takes as input a multivariate polynomial $f$ given as a constant depth algebraic circuit over the field of rational numbers, and outputs all irreducible factors of $f$ of degree at most $d$ together with their respective multiplicities. Moreover, if $f$ is a sparse polynomial, then the algorithm runs in quasipolynomial time. Our results are based on a more fine-grained connection between polynomial identity testing (PIT) and polynomial factorization in the context of constant degree factors and rely on a clean connection between divisibility testing of polynomials and PIT due to Forbes and on subexponential time deterministic PIT algorithms for constant depth algebraic circuits from the recent work of Limaye, Srinivasan and Tavenas.

cs.CC↗

arXiv2023

Determinants vs. Algebraic Branching Programs

We show that for every homogeneous polynomial of degree $d$, if it has determinantal complexity at most $s$, then it can be computed by a homogeneous algebraic branching program (ABP) of size at most $O(d^5s)$. Moreover, we show that for $\textit{most}$ homogeneous polynomials, the width of the resulting homogeneous ABP is just $s-1$ and the size is at most $O(ds)$. Thus, for constant degree homogeneous polynomials, their determinantal complexity and ABP complexity are within a constant factor of each other and hence, a super-linear lower bound for ABPs for any constant degree polynomial implies a super-linear lower bound on determinantal complexity; this relates two open problems of great interest in algebraic complexity. As of now, super-linear lower bounds for ABPs are known only for polynomials of growing degree, and for determinantal complexity the best lower bounds are larger than the number of variables only by a constant factor. While determinantal complexity and ABP complexity are classically known to be polynomially equivalent, the standard transformation from the former to the latter incurs a polynomial blow up in size in the process, and thus, it was unclear if a super-linear lower bound for ABPs implies a super-linear lower bound on determinantal complexity. In particular, a size preserving transformation from determinantal complexity to ABPs does not appear to have been known prior to this work, even for constant degree polynomials.

cs.CC↗

arXiv2023

Fast Numerical Multivariate Multipoint Evaluation

We design nearly-linear time numerical algorithms for the problem of multivariate multipoint evaluation over the fields of rational, real and complex numbers. We consider both \emph{exact} and \emph{approximate} versions of the algorithm. The input to the algorithms are (1) coefficients of an $m$-variate polynomial $f$ with degree $d$ in each variable, and (2) points $a_1,..., a_N$ each of whose coordinate has value bounded by one and bit-complexity $s$. * Approximate version: Given additionally an accuracy parameter $t$, the algorithm computes rational numbers $β_1,\ldots, β_N$ such that $|f(a_i) - β_i| \leq \frac{1}{2^t}$ for all $i$, and has a running time of $((Nm + d^m)(s + t))^{1 + o(1)}$ for all $m$ and all sufficiently large $d$. * Exact version (when over rationals): Given additionally a bound $c$ on the bit-complexity of all evaluations, the algorithm computes the rational numbers $f(a_1), ... , f(a_N)$, in time $((Nm + d^m)(s + c))^{1 + o(1)}$ for all $m$ and all sufficiently large $d$. . Prior to this work, a nearly-linear time algorithm for multivariate multipoint evaluation (exact or approximate) over any infinite field appears to be known only for the case of univariate polynomials, and was discovered in a recent work of Moroz (FOCS 2021). In this work, we extend this result from the univariate to the multivariate setting. However, our algorithm is based on ideas that seem to be conceptually different from those of Moroz (FOCS 2021) and crucially relies on a recent algorithm of Bhargava, Ghosh, Guo, Kumar & Umans (FOCS 2022) for multivariate multipoint evaluation over finite fields, and known efficient algorithms for the problems of rational number reconstruction and fast Chinese remaindering in computational number theory.

cs.DM↗

arXiv2022

Monotone Circuit Lower Bounds from Robust Sunflowers

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity, DNF sparsification, randomness extractors, and recent advances on the Erdős-Rado sunflower conjecture. The recent breakthrough of Alweiss, Lovett, Wu and Zhang gives an improved bound on the maximum size of a $w$-set system that excludes a robust sunflower. In this paper, we use this result to obtain an $\exp(n^{1/2-o(1)})$ lower bound on the monotone circuit size of an explicit $n$-variate monotone function, improving the previous best known $\exp(n^{1/3-o(1)})$ due to Andreev and Harnik and Raz. We also show an $\exp(Ω(n))$ lower bound on the monotone arithmetic circuit size of a related polynomial. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an $n^{Ω(k)}$ lower bound on the monotone circuit size of the CLIQUE function for all $k \le n^{1/3-o(1)}$, strengthening the bound of Alon and Boppana.

cs.CC↗

arXiv2022

Fast Multivariate Multipoint Evaluation Over All Finite Fields

Multivariate multipoint evaluation is the problem of evaluating a multivariate polynomial, given as a coefficient vector, simultaneously at multiple evaluation points. In this work, we show that there exists a deterministic algorithm for multivariate multipoint evaluation over any finite field $\mathbb{F}$ that outputs the evaluations of an $m$-variate polynomial of degree less than $d$ in each variable at $N$ points in time \[ (d^m+N)^{1+o(1)}\cdot\poly(m,d,\log|\mathbb{F}|) \] for all $m\in\N$ and all sufficiently large $d\in\mathbb{N}$. A previous work of Kedlaya and Umans (FOCS 2008, SICOMP 2011) achieved the same time complexity when the number of variables $m$ is at most $d^{o(1)}$ and had left the problem of removing this condition as an open problem. A recent work of Bhargava, Ghosh, Kumar and Mohapatra (STOC 2022) answered this question when the underlying field is not \emph{too} large and has characteristic less than $d^{o(1)}$. In this work, we remove this constraint on the number of variables over all finite fields, thereby answering the question of Kedlaya and Umans over all finite fields. Our algorithm relies on a non-trivial combination of ideas from three seemingly different previously known algorithms for multivariate multipoint evaluation, namely the algorithms of Kedlaya and Umans, that of Björklund, Kaski and Williams (IPEC 2017, Algorithmica 2019), and that of Bhargava, Ghosh, Kumar and Mohapatra, together with a result of Bombieri and Vinogradov from analytic number theory about the distribution of primes in an arithmetic progression. We also present a second algorithm for multivariate multipoint evaluation that is completely elementary and in particular, avoids the use of the Bombieri--Vinogradov Theorem. However, it requires a mild assumption that the field size is bounded by an exponential-tower in $d$ of bounded \textit{height}.

cs.DS↗

arXiv2022

Fast, Algebraic Multivariate Multipoint Evaluation in Small Characteristic and Applications

Multipoint evaluation is the computational task of evaluating a polynomial given as a list of coefficients at a given set of inputs. And while \emph{nearly linear time} algorithms have been known for the univariate instance of multipoint evaluation for close to five decades due to a work of Borodin and Moenck \cite{BM74}, fast algorithms for the multivariate version have been much harder to come by. In a significant improvement to the state of art for this problem, Umans \cite{Umans08} and Kedlaya \& Umans \cite{Kedlaya11} gave nearly linear time algorithms for this problem over field of small characteristic and over all finite fields respectively, provided that the number of variables $n$ is at most $d^{o(1)}$ where the degree of the input polynomial in every variable is less than $d$. They also stated the question of designing fast algorithms for the large variable case (i.e. $n \notin d^{o(1)}$) as an open problem. In this work, we show that there is a deterministic algorithm for multivariate multipoint evaluation over a field $\F_{q}$ of characteristic $p$ which evaluates an $n$-variate polynomial of degree less than $d$ in each variable on $N$ inputs in time $$\left((N + d^n)^{1 + o(1)}\text{poly}(\log q, d, p, n)\right)$$ provided that $p$ is at most $d^{o(1)}$, and $q$ is at most $(\exp(\exp(\exp(\cdots (\exp(d)))))$, where the height of this tower of exponentials is fixed. When the number of variables is large (e.g. $n \notin d^{o(1)}$), this is the first {nearly linear} time algorithm for this problem over any (large enough) field.Our algorithm is based on elementary algebraic ideas and this algebraic structure naturally leads to the applications to data structure upper bounds for polynomial evaluation and to an upper bound on the rigidity of Vandermonde matrices.

cs.CC↗

arXiv2021

A Lower Bound on Determinantal Complexity

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P = Det(M)$. We prove that the determinantal complexity of the polynomial $\sum_{i = 1}^n x_i^n$ is at least $1.5n - 3$. For every $n$-variate polynomial of degree $d$, the determinantal complexity is trivially at least $d$, and it is a long standing open problem to prove a lower bound which is super linear in $\max\{n,d\}$. Our result is the first lower bound for any explicit polynomial which is bigger by a constant factor than $\max\{n,d\}$, and improves upon the prior best bound of $n + 1$, proved by Alper, Bogart and Velasco [ABV17] for the same polynomial.

cs.CC↗

arXiv2021

Detection and Prediction of Equilibrium States in Kinetic Plasma Simulations via Mode Tracking using Reduced-Order Dynamic Mode Decomposition

A dynamic mode decomposition (DMD) based reduced-order model (ROM) is developed for tracking, detection, and prediction of kinetic plasma behavior. DMD is applied to the high-fidelity kinetic plasma model based on the electromagnetic particle-in-cell (EMPIC) algorithm to extract the underlying dynamics and key features of the model. In particular, the ability of DMD to reconstruct the spatial pattern of the self electric field from high-fidelity data and the effect of DMD extrapolated self-fields on charged particle dynamics are investigated. An in-line sliding-window DMD method is presented for identifying the transition from transient to equilibrium state based on the loci of DMD eigenvalues in the complex plane. The in-line detection of equilibrium state combined with time extrapolation ability of DMD has the potential to effectively expedite the simulation. Case studies involving electron beams and plasma ball are presented to assess the strengths and limitations of the proposed method.

physics.plasm-ph↗

arXiv2020

If VNP is hard, then so are equations for it

Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient vectors of all polynomials in the class VNP requires algebraic circuits of super-polynomial size. In a recent work of Chatterjee and the authors (FOCS 2020), it was shown that the subclasses of VP and VNP consisting of polynomials with bounded integer coefficients do have equations with small algebraic circuits. Their work left open the possibility that these results could perhaps be extended to all of VP or VNP. The results in this paper show that assuming the hardness of Permanent, at least for VNP, allowing polynomials with large coefficients does indeed incur a significant blow up in the circuit complexity of equations.

cs.CC↗

arXiv2020

Decoding Multivariate Multiplicity Codes on Product Sets

The multiplicity Schwartz-Zippel lemma bounds the total multiplicity of zeroes of a multivariate polynomial on a product set. This lemma motivates the multiplicity codes of Kopparty, Saraf and Yekhanin [J. ACM, 2014], who showed how to use this lemma to construct high-rate locally-decodable codes. However, the algorithmic results about these codes crucially rely on the fact that the polynomials are evaluated on a vector space and not an arbitrary product set. In this work, we show how to decode multivariate multiplicity codes of large multiplicities in polynomial time over finite product sets (over fields of large characteristic and zero characteristic). Previously such decoding algorithms were not known even for a positive fraction of errors. In contrast, our work goes all the way to the distance of the code and in particular exceeds both the unique decoding bound and the Johnson bound. For errors exceeding the Johnson bound, even combinatorial list-decodablity of these codes was not known. Our algorithm is an application of the classical polynomial method directly to the multivariate setting. In particular, we do not rely on a reduction from the multivariate to the univariate case as is typical of many of the existing results on decoding codes based on multivariate polynomials. However, a vanilla application of the polynomial method in the multivariate setting does not yield a polynomial upper bound on the list size. We obtain a polynomial bound on the list size by taking an alternative view of multivariate multiplicity codes. In this view, we glue all the partial derivatives of the same order together using a fresh set $z$ of variables. We then apply the polynomial method by viewing this as a problem over the field $\mathbb{F}(z)$ of rational functions in $z$.

cs.IT↗

arXiv2020

A Polynomial Degree Bound on Equations of Non-rigid Matrices and Small Linear Circuits

We show that there is a defining equation of degree at most $\mathsf{poly}(n)$ for the (Zariski closure of the) set of the non-rigid matrices: that is, we show that for every large enough field $\mathbb{F}$, there is a non-zero $n^2$-variate polynomial $P \in \mathbb{F}[x_{1, 1}, \ldots, x_{n, n}]$ of degree at most $\mathsf{poly}(n)$ such that every matrix $M$ which can be written as a sum of a matrix of rank at most $n/100$ and a matrix of sparsity at most $n^2/100$ satisfies $P(M) = 0$. This confirms a conjecture of Gesmundo, Hauenstein, Ikenmeyer and Landsberg [GHIL16] and improves the best upper bound known for this problem down from $\exp(n^2)$ [KLPS14, GHIL16] to $\mathsf{poly}(n)$. We also show a similar polynomial degree bound for the (Zariski closure of the) set of all matrices $M$ such that the linear transformation represented by $M$ can be computed by an algebraic circuit with at most $n^2/200$ edges (without any restriction on the depth). As far as we are aware, no such bound was known prior to this work when the depth of the circuits is unbounded. Our methods are elementary and short and rely on a polynomial map of Shpilka and Volkovich [SV15] to construct low degree "universal" maps for non-rigid matrices and small linear circuits. Combining this construction with a simple dimension counting argument to show that any such polynomial map has a low degree annihilating polynomial completes the proof. As a corollary, we show that any derandomization of the polynomial identity testing problem will imply new circuit lower bounds. A similar (but incomparable) theorem was proved by Kabanets and Impagliazzo [KI04].

cs.CC↗