相关论文: Quantum Approximation II. Sobolev Embeddings
The motivation of this paper is the development of an optimisation method for solving optimisation problems appearing in Chebyshev rational and generalised rational approximation problems, where the approximations are constructed as ratios…
Quantum theory predicts probabilities as well as relative phases between different alternatives of the system. A unified description of both probabilities and phases comes through a generalisation of the notion of a density matrix for…
Automating algorithm configuration is growing increasingly necessary as algorithms come with more and more tunable parameters. It is common to tune parameters using machine learning, optimizing performance metrics such as runtime and…
In this paper, we introduce a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only…
We use entropy numbers in combination with the polynomial method to derive a new general lower bound for the n-th minimal error in the quantum setting of information-based complexity. As an application, we improve some lower bounds on…
We study the computational complexity of certain integrable quantum theories in 1+1 dimensions. We formalize a model of quantum computation based on these theories. In this model, distinguishable particles start out with known momenta and…
Random feature approximation is arguably one of the most popular techniques to speed up kernel methods in large scale algorithms and provides a theoretical approach to the analysis of deep neural networks. We analyze generalization…
Quantum statistical models (i.e., families of normalized density matrices) and quantum measurements (i.e., positive operator-valued measures) can be regarded as linear maps: the former, mapping the space of effects to the space of…
A central roadblock to analyzing quantum algorithms on quantum states is the lack of a comparable input model for classical algorithms. Inspired by recent work of the author [E. Tang, STOC'19], we introduce such a model, where we assume we…
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…
In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the nodes (1.5) are investigated. For optimal coefficients explicit form are obtained and norm of the error functional is calculated. In particular, by choosing…
The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point $a\in(0;1)$ and an arbitrary function $y\in\mathring{W}^n_p[0;1]$ set the relations $y^{(k)}(a)=\int_0^1 y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx$. The relation of the…
A rational approximation by a ratio of polynomial functions is a flexible alternative to polynomial approximation. In particular, rational functions exhibit accurate estimations to nonsmooth and non- Lipschitz functions, where polynomial…
The approximate degree of a Boolean function $f(x_{1},x_{2},\ldots,x_{n})$ is the minimum degree of a real polynomial that approximates $f$ pointwise within $1/3$. Upper bounds on approximate degree have a variety of applications in…
This paper studies the problem of computing a linear approximation of quadratic Wasserstein distance $W_2$. In particular, we compute an approximation of the negative homogeneous weighted Sobolev norm whose connection to Wasserstein…
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two…
We establish optimal convergence rates for the continuous piecewise affine finite element approximation of the Sobolev constant in arbitrary dimensions N\geq 2 and for Lebesgue exponents 1<p<N. Our analysis relies on a refined study of the…
One of the basic principles of Approximation Theory is that the quality of approximations increase with the smoothness of the function to be approximated. Functions that are smooth in certain subdomains will have good approximations in…
We characterize all the real numbers a,b,c and 1<= p,q,r<infty such that the weighted Sobolev space W_{a,b}^(q,p)(R^N\{0}) with power weights |x|^a and |x|^b is continuously embedded into L^{r}(R^N;|x|^cdx). Furthermore, we show that this…
For a set $\mathbb{W} \subset L_p(\bT^d)$, $1 < p < \infty$, of multivariate periodic functions on the torus $\bT^d$ and a given function $\varphi \in L_p(\bT^d)$, we study the approximation in the $L_p(\bT^d)$-norm of functions $f \in…