Related papers: Efficient Generic Quotients Using Exact Arithmetic
We describe a three precision variant of Newton's method for nonlinear equations. We evaluate the nonlinear residual in double precision, store the Jacobian matrix in single precision, and solve the equation for the Newton step with…
In this paper, we propose a general approach for improving the efficiency of computing distribution functions. The idea is to truncate the domain of summation or integration.
Optimization of quadratic functions and the quotient of those are relevant in subspace and iterative optimization methods. In this paper, the calculation of the generalized operator norm and extremal generalized Rayleigh quotient is…
We present a novel Newton-type method for distributed optimization, which is particularly well suited for stochastic optimization and learning problems. For quadratic objectives, the method enjoys a linear rate of convergence which provably…
Multiplicative inverse is a crucial operation in public key cryptography, and been widely used in cryptography. Public key cryptography has given rise to such a need, in which we need to generate a related public and private pair of…
In this paper we study a new approach in optimization that aims to search a large domain D where a given function takes large, small or specific values via an iterative optimization algorithm based on the gradient. We show that the…
The Cartesian reverse derivative is a categorical generalization of reverse-mode automatic differentiation. We use this operator to generalize several optimization algorithms, including a straightforward generalization of gradient descent…
We show that, under certain circumstances, it is possible to automatically compute Jacobian-inverse-vector and Jacobian-inverse-transpose-vector products about as efficiently as Jacobian-vector and Jacobian-transpose-vector products. The…
We introduce an efficient way, called Newton algorithm, to study arbitrary ideals in C[[x,y]], using a finite succession of Newton polygons. We codify most of the data of the algorithm in a useful combinatorial object, the Newton tree. For…
Derivative-based algorithms are ubiquitous in statistics, machine learning, and applied mathematics. Automatic differentiation offers an algorithmic way to efficiently evaluate these derivatives from computer programs that execute relevant…
Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of…
Using backpropagation to compute gradients of objective functions for optimization has remained a mainstay of machine learning. Backpropagation, or reverse-mode differentiation, is a special case within the general family of automatic…
Most existing work uses dual decomposition and subgradient methods to solve Network Utility Maximization (NUM) problems in a distributed manner, which suffer from slow rate of convergence properties. This work develops an alternative…
We present an algorithm for computing discriminants and prime ideal decomposition in number fields. The algorithm is a refinement of a p-adic factorization method based on Newton polygons of higher order. The running-time and memory…
This paper presents new approaches for finding the determinant and inverse of a matrix. The choice of pivot selection is kept arbitrary and can be made according to the users need. So the ill conditioned matrices can be handled easily. The…
We study two important operations on polynomials defined over complete discrete valuation fields: Euclidean division and factorization. In particular, we design a simple and efficient algorithm for computing slope factorizations, based on…
In recent years many efforts have been devoted to finding bidiagonal factorizations of nonsingular totally positive matrices, since their accurate computation allows to numerically solve several important algebraic problems with great…
We present improved algorithms for fast calculation of the inverse square root for single-precision floating-point numbers. The algorithms are much more accurate than the famous fast inverse square root algorithm and have the same or…
Incremental computation aims to compute more efficiently on changed input by reusing previously computed results. We give a high-level overview of works on incremental computation, and highlight the essence underlying all of them, which we…
Algorithmic differentiation (AD) has become increasingly capable and straightforward to use. However, AD is inefficient when applied directly to solvers, a feature of most engineering analyses. We can leverage implicit differentiation to…