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The efficient solution of moderately large-scale linear systems arising from the KKT conditions in optimal control problems (OCPs) is a critical challenge in robotics. With the stagnation of Moore's law, there is growing interest in…
This paper presents new results for the (partial) maximum a posteriori (MAP) problem in Bayesian networks, which is the problem of querying the most probable state configuration of some of the network variables given evidence. First, it is…
Polynomial inequalities lie at the heart of many mathematical disciplines. In this paper, we consider the fundamental computational task of automatically searching for proofs of polynomial inequalities. We adopt the framework of…
Consider a system of $m$ polynomial equations $\{p_i(x) = b_i\}_{i \leq m}$ of degree $D\geq 2$ in $n$-dimensional variable $x \in \mathbb{R}^n$ such that each coefficient of every $p_i$ and $b_i$s are chosen at random and independently…
DAGS scheme is a key encapsulation mechanism (KEM) based on quasi-dyadic alternant codes that was submitted to NIST standardization process for a quantum resistant public key algorithm. Recently an algebraic attack was devised by Barelli…
Recent advances in solving ordinary differential equations (ODEs) with neural networks have been remarkable. Neural networks excel at serving as trial functions and approximating solutions within functional spaces, aided by gradient…
This paper proposes an efficient algorithm for testing copositivity of homogeneous polynomials over the positive semidefinite cone. The algorithm is based on a novel matrix optimization reformulation and requires solving a hierarchy of…
An algorithm and associated strategy for solving polynomial systems within the optimization framework is presented. The algorithm and strategy are named, respectively, the penetrating gradient algorithm and the deepest descent strategy. The…
Many problems of systems control theory boil down to solving polynomial equations, polynomial inequalities or polyomial differential equations. Recent advances in convex optimization and real algebraic geometry can be combined to generate…
Preconditioning has long been a staple technique in optimization, often applied to reduce the condition number of a matrix and speed up the convergence of algorithms. Although there are many popular preconditioning techniques in practice,…
A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead,…
We give the first polynomial time and sample $(\epsilon, \delta)$-differentially private (DP) algorithm to estimate the mean, covariance and higher moments in the presence of a constant fraction of adversarial outliers. Our algorithm…
We introduce Selective Greedy Equivalence Search (SGES), a restricted version of Greedy Equivalence Search (GES). SGES retains the asymptotic correctness of GES but, unlike GES, has polynomial performance guarantees. In particular, we show…
Decision of whether a Boolean equation system has a solution is an NPC problem and finding a solution is NP hard. In this paper, we present a quantum algorithm to decide whether a Boolean equation system FS has a solution and compute one if…
We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function $f: \{0,1\}^n \to \{0,1\}$ is an $s$-sparse GF(2) polynomial versus $\eps$-far from every such polynomial. Our algorithm…
This article considers the problem of solving a system of $n$ real polynomial equations in $n+1$ variables. We propose an algorithm based on Newton's method and subdivision for this problem. Our algorithm is intended only for nondegenerate…
Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…
In this article we provide examples, methods and algorithms to determine conditions on the parameters of certain type of parametric optimization problems, such that among the resulting local minima and maxima there is at least one which…
The (modern) arbitrary derivative (ADER) approach is a popular technique for the numerical solution of differential problems based on iteratively solving an implicit discretization of their weak formulation. In this work, focusing on an ODE…
Mechanistic models with differential equations are a key component of scientific applications of machine learning. Inference in such models is usually computationally demanding, because it involves repeatedly solving the differential…