相关论文: Effective certification of approximate solutions t…

We present an approach to obtain convergence guarantees of optimization algorithms for deep networks based on elementary arguments and computations. The convergence analysis revolves around the analytical and computational structures of…

The paper presents (human-oriented) specification and (pen-and-paper) verification of the square root function. The function implements Newton method and uses a look-up table for initial approximations. Specification is done in terms of…

This paper deals with the algorithmic aspects of solving feasibility problems of semidefinite programming (SDP), aka linear matrix inequalities (LMI). Since in some SDP instances all feasible solutions have irrational entries, numerical…

We consider the problem of certifying lower bounds for real-valued multivariate transcendental functions. The functions we are dealing with are nonlinear and involve semialgebraic operations as well as some transcendental functions like…

Partial penalized tests provide flexible approaches to testing linear hypotheses in high dimensional generalized linear models. However, because the estimators used in these tests are local minimizers of potentially non-convex…

Successive quadratic approximations (SQA) are numerically efficient for minimizing the sum of a smooth function and a convex function. The iteration complexity of inexact SQA methods has been analyzed recently. In this paper, we present an…

Quantum computers can solve many number theory problems efficiently. Using the efficient quantum algorithm for order finding as an oracle, this paper presents an algorithm that computes the Carmichael function for any integer $N$ with a…

We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…

Farkas' lemma is a fundamental result from linear programming providing linear certificates for infeasibility of systems of linear inequalities. In semidefinite programming, such linear certificates only exist for strongly infeasible linear…

We present a method for constructing global analytical expressions that approximate a function over its entire range. These approximations not only mirror the original function as accurately as desired, but are purposefully created to…

In this paper we consider a family of algorithms for approximate implicitization of rational parametric curves and surfaces. The main approximation tool in all of the approaches is the singular value decomposition, and they are therefore…

Let a Boolean function be available as a black-box (oracle) and one likes to devise an algorithm to test whether it has certain property or it is $\epsilon$-far from having that property. The efficiency of the algorithm is judged by the…

Certifying verification algorithms not only return whether a given property holds or not, but also provide an accompanying independently checkable certificate and a corresponding witness. The certificate can be used to easily validate the…

The concepts of amenable and compatible functions have been introduced in a recent work, in order to state precise mathematical theorems that guarantee that a backward stable algorithm is also forward stable, and that the composition of two…

In the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set C of size M and our task is to identify x. We present a quantum algorithm for the problem that is optimal in its…

We reexamine Smale's alpha theory as a way to certify a numerical solution to an analytic system. For a given point and a system, Smale's alpha theory determines whether Newton's method applied to this point shows the quadratic convergence…

Differentially-algebraic (D-algebraic) functions are solutions of polynomial equations in the function, its derivatives, and the independent variables. We revisit closure properties of these functions by providing constructive proofs. We…

We consider minimization of a smooth nonconvex function with inexact oracle access to gradient and Hessian (without assuming access to the function value) to achieve approximate second-order optimality. A novel feature of our method is that…

In this article, we consider a simple representation for real numbers and propose top-down procedures to approximate various algebraic and transcendental operations with arbitrary precision. Detailed algorithms and proofs are provided to…

Many quantum algorithms make use of oracles which evaluate classical functions on a superposition of inputs. In order to facilitate implementation, testing, and resource estimation of such algorithms, we present quantum circuits for…