Journal of Complexity最新文献

We study volumes of sections of convex origin-symmetric bodies in $R^n$ induced by orthonormal systems on probability spaces. The approach is based on volume estimates of John-Löwner ellipsoids and expectations of norms induced by the respective systems. The estimates obtained allow us to establish lower bounds for the radii of sections which gives lower bounds for Gelfand widths (or linear cowidths). As an application we offer a new method of evaluation of Gelfand and Kolmogorov widths of multiplier operators. In particular, we establish sharp orders of widths of standard Sobolev classes $W_p^{\gamma }$, $\gamma >0$ in $L_q$ on two-point homogeneous spaces in the difficult case, i.e. if $1<q\le p\le \infty$.

We investigate non-adaptive methods of deep ReLU neural network approximation in Bochner spaces $L_2(U^{\infty },X,\mu )$ of functions on $U^{\infty }$ taking values in a separable Hilbert space X, where $U^{\infty }$ is either $R^{\infty }$ equipped with the standard Gaussian probability measure, or ${[-1,1]}^{\infty }$ equipped with the Jacobi probability measure. Functions to be approximated are assumed to satisfy a certain weighted ${\ell }_2$-summability of the generalized chaos polynomial expansion coefficients with respect to the measure μ. We prove the convergence rate of this approximation in terms of the size of approximating deep ReLU neural networks. These results then are applied to approximation of the solution to parametric elliptic PDEs with random inputs for the lognormal and affine cases.

The $L_p$-discrepancy is a quantitative measure for the irregularity of distribution of an N-element point set in the d-dimensional unit-cube, which is closely related to the worst-case error of quasi-Monte Carlo algorithms for numerical integration. It's inverse for dimension d and error threshold $\epsilon \in (0,1)$ is the minimal number of points in ${[0,1)}^d$ such that the minimal normalized $L_p$-discrepancy is less or equal ε. It is well known, that the inverse of $L_2$-discrepancy grows exponentially fast with the dimension d, i.e., we have the curse of dimensionality, whereas the inverse of $L_{\infty }$-discrepancy depends exactly linearly on d. The behavior of inverse of $L_p$-discrepancy for general $p\notin \{2,\infty \}$ has been an open problem for many years. In this paper we show that the $L_p$-discrepancy suffers from the curse of dimensionality for all p in $(1,2]$ which are of the form $p=2\ell /(2\ell -1)$ with $\ell \in N$.

This result follows from a more general result that we show for the worst-case error of numerical integration in an anchored Sobolev space with anchor 0 of once differentiable functions in each variable whose first derivative has finite $L_q$-norm, where q is an even positive integer satisfying $1/p+1/q=1$.

A set Q in $Z_{+}^d$ is a lower set if $(k_1,\dots ,k_d)\in Q$ implies $(l_1,\dots ,l_d)\in Q$ whenever $0\le l_i\le k_i$ for all i. We derive new and refine known results regarding the cardinality of the lower sets of size n in $Z_{+}^d$. Next we apply these results for universal discretization of the $L_2$-norm of elements from n-dimensional subspaces of trigonometric polynomials generated by lower sets.

We show that the minimal discrepancy of a point set in the d-dimensional unit cube with respect to the BMO seminorm suffers from the curse of dimensionality.

The dispersion of a point set in the unit square is the area of the largest empty axis-parallel box. In this paper we are interested in the dispersion of lattices in the plane, that is, the supremum of the area of the empty axis-parallel boxes amidst the lattice points. We introduce a framework with which to study this based on the continued fractions expansions of the lattice generators. We give necessary and sufficient conditions under which a lattice has finite dispersion. We obtain an exact formula for the dispersion of the lattices associated to subrings of the ring of integers of quadratic fields. We have tight bounds for the dispersion of a lattice based on the largest continued fraction coefficient of the generators, accurate to within one half. We provide an equivalent formulation of Zaremba's conjecture. Using this framework we are able to give very short proofs of previous results.

To achieve robustness against the outliers or heavy-tailed sampling distribution, we consider an Ivanov regularized empirical risk minimization scheme associated with a modified Huber's loss for nonparametric regression in reproducing kernel Hilbert space. By tuning the scaling and regularization parameters in accordance with the sample size, we develop nonasymptotic concentration results for such an adaptive estimator. Specifically, we establish the best convergence rates for prediction error when the conditional distribution satisfies a weak moment condition.

We present an oracle factorisation algorithm, which in polynomial deterministic time, finds a nontrivial factor of almost all positive integers n based on the knowledge of the number of points on certain elliptic curves in residue rings modulo n.

We study the bit-complexity intrinsic to solving the initial-value and (several types of) boundary-value problems for linear evolutionary systems of partial differential equations (PDEs), based on the Computable Analysis approach. Our algorithms are guaranteed to compute classical solutions to such problems approximately up to error $1/2^n$, so that n corresponds to the number of reliable bits of the output; bit-cost is measured with respect to n. Computational Complexity Theory allows us to prove in a rigorous sense that PDEs with constant coefficients are algorithmically ‘easier’ than general ones. Indeed, solutions to the latter are shown (under natural assumptions) computable using a polynomial number of memory bits, and we prove that the complexity class PSPACE is in general optimal; while the case of constant coefficients can be solved in #P—also essentially optimally so: the Heat Equation ‘requires’ ${\#\text{P}}_1$. Our algorithms raise difference schemes to exponential powers, efficiently: we compute any desired entry of such a power in #P, provided that the underlying exponential-sized matrices are circulant of constant bandwidth. Exponentially powering modular two-band circulant matrices is established even feasible in P; and under additional conditions, also the solution to certain linear PDEs becomes polynomial time computable.

In low-rank matrix recovery, many kinds of measurements fail to meet the standard restricted isometry property (RIP), such as rank-one measurements, that is, ${\left[A\right(X\left)\right]}_i=〈A_i,X〉$ with $\text{rank}\left(A_i\right)=1$, $i=1,...,m$. Historical iterative hard thresholding sequence for low-rank matrix recovery and rank-one measurements was taken as $X^{n+1}=P_s(X^n-{\mu }_n{P}_t(A^{⁎}\text{sign}\left(A\right(X^n)-y)\left)\right)$, which introduced the “tail” and “head” approximations $P_s$ and $P_t$, respectively. In this paper, we remove the term $P_t$ and provide a new iterative hard thresholding algorithm with adaptive step size (abbreviated as AIHT). The linear convergence analysis and stability results on AIHT are established under the ${\ell }_1/{\ell }_2$-RIP. Particularly, we discuss the rank-one Gaussian measurements under the tight upper and lower bounds on $E{\Vert A\left(X\right)\Vert }_1$, and provide better convergence rate and sampling complexity. Besides, several empirical experiments are provided to show that AIHT performs better than the historical rank-one iterative hard thresholding method.