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Techniques that rigorously bound the overall rounding error exhibited by a numerical program are of significant interest for communities developing numerical software. However, there are few available tools today that can be used to…
Gaussian process regression generally does not scale to beyond a few thousands data points without applying some sort of kernel approximation method. Most approximations focus on the high eigenvalue part of the spectrum of the kernel…
We establish maximal concentration bounds for the iterates generated by stochastic approximation algorithms with general step sizes, where the noise has a finite-state Markovian component plus a Martingale-difference component. When the…
Quantum information processors promise fast algorithms for problems inaccessible to classical computers. But since qubits are noisy and error-prone, they will depend on fault-tolerant quantum error correction (FTQEC) to compute reliably.…
Local grid refinement aims to optimise the relationship between accuracy of the results and number of grid nodes. In the context of the finite volume method no single local refinement criterion has been globally established as optimum for…
Gaussian Process Regression is a well-known machine learning technique for which several quantum algorithms have been proposed. We show here that in a wide range of scenarios these algorithms show no exponential speedup. We achieve this by…
Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a…
The Knill-Laflamme (KL) conditions distinguish exact quantum error correction codes, and it has played a critical role in the discovery of state-of-the-art codes. However, the family of exact codes is a very restrictive one and does not…
Approximate Bayesian computation (ABC) or likelihood-free inference algorithms are used to find approximations to posterior distributions without making explicit use of the likelihood function, depending instead on simulation of sample data…
We investigate how to train kernel approximation methods that generalize well under a memory budget. Building on recent theoretical work, we define a measure of kernel approximation error which we find to be more predictive of the empirical…
This paper is concerned with the hard thresholding operator which sets all but the $k$ largest absolute elements of a vector to zero. We establish a {\em tight} bound to quantitatively characterize the deviation of the thresholded solution…
Gaussian process regression (GPR) is a popular nonparametric Bayesian method that provides predictive uncertainty estimates and is widely used in safety-critical applications. While prior research has introduced various uncertainty bounds,…
In many instances of fixed-point multiplication, a full precision result is not required. Instead it is sufficient to return a faithfully rounded result. Faithful rounding permits the machine representable number either immediately above or…
In this thesis, we introduce a new quantum Turing machine (QTM) model that supports general quantum operators, together with its pushdown, counter, and finite automaton variants, and examine the computational power of classical and quantum…
We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are…
The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled employing the fixed-stress…
The empirical evidence indicates that stochastic optimization with heavy-tailed gradient noise is more appropriate to characterize the training of machine learning models than that with standard bounded gradient variance noise. Most…
Kernel quadrature is widely used to approximate integrals of smooth functions, with worst-case error typically decaying at the minimax rate $n^{-\alpha/d}$ for smoothness $\alpha$ in dimension $d$. Existing rate-optimal methods often depend…
We introduce an accurate and efficient method for a class of nonlocal potential evaluations with free boundary condition, including the 3D/2D Coulomb, 2D Poisson and 3D dipolar potentials. Our method is based on a Gaussian-sum approximation…
The error coefficient of a linear code is defined as the number of minimum-weight codewords. In an additive white Gaussian noise channel, optimal linear codes with the smallest error coefficients achieve the best possible asymptotic frame…