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We propose a general error analysis related to the low-rank approximation of a given real matrix in both the spectral and Frobenius norms. First, we derive deterministic error bounds that hold with some minimal assumptions. Second, we…

Nonlinear kernels can be approximated using finite-dimensional feature maps for efficient risk minimization. Due to the inherent trade-off between the dimension of the (mapped) feature space and the approximation accuracy, the key problem…

State-of-the-art static analysis tools for verifying finite-precision code compute worst-case absolute error bounds on numerical errors. These are, however, often not a good estimate of accuracy as they do not take into account the…

A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In…

Classical simulations of noisy stabilizer circuits are often used to estimate the threshold of a quantum error-correcting code. Physical noise sources are efficiently approximated by random insertions of Pauli operators. For a single qubit,…

Rank-constrained matrix problems appear frequently across science and engineering. The convergence analysis of iterative algorithms developed for these problems often hinges on local error bounds, which correlate the distance to the…

Stochastic rounding (SR) offers an alternative to the deterministic IEEE-754 floating-point rounding modes. In some applications such as PDEs, ODEs and neural networks, SR empirically improves the numerical behavior and convergence to…

Stochastic differential equations are commonly used to describe the evolution of stochastic processes. The state uncertainty of such processes is best represented by the probability density function (PDF), whose evolution is governed by the…

Given the importance of floating-point~(FP) performance in numerous domains, several new variants of FP and its alternatives have been proposed (e.g., Bfloat16, TensorFloat32, and Posits). These representations do not have correctly rounded…

The problem of establishing out-of-sample bounds for the values of an unkonwn ground-truth function is considered. Kernels and their associated Hilbert spaces are the main formalism employed herein along with an observational model where…

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and…

In this work, we build upon the recently introduced Multiscale Robin Coupled Method with Oversampling and Smoothing (MRCM-OS) to develop two highly efficient iterative multiscale methods. The MRCM-OS methodology demonstrated the ability to…

We present the Branch-and-Bound Performance Estimation Programming (BnB-PEP), a unified methodology for constructing optimal first-order methods for convex and nonconvex optimization. BnB-PEP poses the problem of finding the optimal…

Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty…

Simulation-based verification algorithms can provide formal safety guarantees for nonlinear and hybrid systems. The previous algorithms rely on user provided model annotations called discrepancy function, which are crucial for computing…

There are several numerical methods for computing approximate zeros of a given univariate polynomial. In this paper, we develop a simple and novel method for determining sharp upper bounds on errors in approximate zeros of a given…

We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower…

Kernel methods give powerful, flexible, and theoretically grounded approaches to solving many problems in machine learning. The standard approach, however, requires pairwise evaluations of a kernel function, which can lead to scalability…

In the present work, we derive functional upper bounds for the potential error arising from finite-element boundary-element coupling formulations for a nonlinear Poisson-type transmission problem. The proposed a posteriori error estimates…

Verification of programs using floating-point arithmetic is challenging on several accounts. One of the difficulties of reasoning about such programs is due to the peculiarities of floating-point arithmetic: rounding errors, infinities,…