Related papers: Improved error bounds for quantization based numer…

Inspired by classical sensitivity results for nonlinear optimization, we derive and discuss new quantitative bounds to characterize the solution map and dual variables of a parametrized nonlinear program. In particular, we derive explicit…

The calculation of the error threshold of quantum error correcting codes typically proceeds as follows. First, syndromes are measured. Then, a decoder infers the error chain and the corresponding correction is applied. The threshold is then…

We establish error estimates of the first-order exponential wave integrator (EWI) for the nonlinear Schr\"odinger equation (NLSE) with a highly singular potential in $\mathbb{R}^d$ with $1\leq d \leq 3$. Our results deal with singular…

We prove lower bounds on the error incurred when approximating any oscillating function using piecewise polynomial spaces. The estimates are explicit in the polynomial degree and have optimal dependence on the meshwidth and frequency when…

This paper examines the asymptotic convergence properties of Lipschitz interpolation methods within the context of bounded stochastic noise. In the first part of the paper, we establish probabilistic consistency guarantees of the classical…

A recent paper by Jordan et al. introduced Decoded Quantum Interferometry (DQI), a novel quantum algorithm that uses the quantum Fourier transform to reduce linear optimization problems -- max-XORSAT and max-LINSAT -- to decoding problems.…

Boundary integral equations are an efficient and accurate tool for the numerical solution of elliptic boundary value problems. The solution is expressed as a layer potential; however, the error in its evaluation grows large near the…

We study the problem of $(\epsilon,\delta)$-differentially private learning of linear predictors with convex losses. We provide results for two subclasses of loss functions. The first case is when the loss is smooth and non-negative but not…

This paper is concerned with the existence and uniqueness of weak solutions to the Cauchy-Dirichlet problem of backward stochastic partial differential equations (BSPDEs) with nonhomogeneous terms of quadratic growth in both the gradient of…

Stochastic parareal (SParareal) is a probabilistic variant of the popular parallel-in-time algorithm known as parareal. Similarly to parareal, it combines fine- and coarse-grained solutions to an ordinary differential equation (ODE) using a…

We suggest a technique for constructing lower (existence) bounds for the fault-tolerant threshold to scalable quantum computation applicable to degenerate quantum codes with sublinear distance scaling. We give explicit analytic expressions…

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. This dissertation makes a threefold contribution to the mathematical theory of quantum error-correcting codes.…

The ODE method has been a workhorse for algorithm design and analysis since the introduction of the stochastic approximation. It is now understood that convergence theory amounts to establishing robustness of Euler approximations for ODEs,…

We study a family of sparse estimators defined as minimizers of some empirical Lipschitz loss function -- which include the hinge loss, the logistic loss and the quantile regression loss -- with a convex, sparse or group-sparse…

Ordinary Differential Equations are a simple but powerful framework for modeling complex systems. Parameter estimation from times series can be done by Nonlinear Least Squares (or other classical approaches), but this can give…

We consider information-theoretic bounds on expected generalization error for statistical learning problems in a networked setting. In this setting, there are $K$ nodes, each with its own independent dataset, and the models from each node…

Quantization is essential for reducing the computational cost and memory usage of deep neural networks, enabling efficient inference on low-precision hardware. Despite the growing adoption of uniform and floating-point quantization schemes,…

State estimation is a classical problem in quantum information. In optimization of estimation scheme, to find a lower bound to the error of the estimator is a very important step. So far, all the proposed tractable lower bounds use…

This paper investigates the stochastic linear-quadratic control problems with affine constraints, in which both equality and inequality constraints are involved. With the help of the Pontryagin maximum principle and Lagrangian duality…

This paper presents preliminary work on computing upper bounds on the estimation error covariance in the framework of the extended Kalman filter. The approach taken is using quadratic constraints to bound the dynamic nonlinearities and use…