The sequence of entropy numbers quantifies the degree of compactness of a linear operator acting between quasi-Banach spaces. We determine the asymptotic behavior of entropy numbers in the case of natural embeddings between finite-dimensional Lorentz spaces & ell;np,q in all regimes; our results

We give a new lower bound for the minimal dispersion of a point set in the unit cube and its inverse function in the high dimension regime. This is done by considering only a very small class of test boxes, which allows us to reduce bounding the dispersion to a problem in extremal set theory. Spe

We have provided new results on the structure of optimal transportation networks obtained as minimizers of an energy cost functional posed on a discrete graph. The energy consists of a kinetic (pumping) and a material (metabolic) cost term, constrained by a local mass conservation law. In particu

We consider the trigonometric-like system of piecewise linear functions introduced recently by Daubechies, DeVore, Foucart, Hanin, and Petrova. We provide an alternative proof that this system forms a Riesz basis of L-2([0,1]) based on the Gershgorin theorem. We also generalize this system to hig

We study the integration problem on Hilbert spaces of (multivariate) periodic functions.The standard technique to prove lower bounds for the error of quadrature rules uses bump functions and the pigeon hole principle. Recently, several new lower bounds have been obtained using a different techniq

By the work of P. Levy, the sample paths of the Brownian motion are known to satisfy a certain Holder regularity condition almost surely. This was later improved by Ciesielski, who studied the regularity of these paths in Besov and Besov-Orlicz spaces. We review these results and propose new func

Function values are, in some sense, "almost as good" as general linear information for L2-approximation (optimal recovery, data assimilation) of functions from a reproducing kernel Hilbert space. This was recently proved by new upper bounds on the sampling numbers under the assumption that the si

The dispersion of a point set P subset of[0,1](d) is the volume of the largest box with sides parallel to the coordinate axes, which does not intersect P. It was observed only recently that, for any epsilon > 0, certain randomized constructions provide point sets with dispersion smaller than e an

We prove lower bounds for the worst case error of quadrature formulas that use given sample points X-n = {x(1), ..., x(n)}. We are mainly interested in optimal point sets X-n, but also prove lower bounds that hold with high probability for sets of independently and uniformly distributed points. A

Let 0 < p, q <= infinity and denote by S-p(N) and S-q(N) the corresponding Schatten classes of real N x N matrices. We study the Gelfand numbers of natural identities S-p(N) hooked right arrow S-q(N) between Schatten classes and prove asymptotically sharp bounds up to constants only depending on

We address the structure identification and the uniform approximation of sums of ridge functions f (x) = Sigma(m)(i=1) g(i)(< a(i), x >) on R-d, representing a general form of a shallow feed-forward neural network, from a small number of query samples. Higher order differentiation, as used in our

Entropy numbers and covering numbers of sets and operators are well known geometric notions, which found many applications in various fields of mathematics, statistics, and computer science. Their values for finite-dimensional embeddings id : l(p)(n) -> l(q)(n), 0 < p, q <= infinity, are known (u

We show the following version of the Schur's product theorem. If M = (M-j,M-k)(j,k=1)(n) is an element of R(nxn )is a positive semidefinite matrix with all entries on the diagonal equal to one, then the matrix N = (N-j,N-k)(j,k=1)(n) with the entries N-j,N-k = M-j,k(2) - 1/n is positive semidefin

We study the volume of unit balls B-p,q(n) of finite-dimensional Lorentz sequence spaces l(p,q)(n). We give an iterative formula for vol(B-p,q(n)) for the weak Lebesgue spaces with q = infinity and explicit formulas for q = 1 and q = infinity. We derive asymptotic results for the nth root of vol(

We consider the problem of learning a d-variate function f defined on the cube [-1, 1](d). R-d, where the algorithm is assumed to have black box access to samples of f within this domain. Let S-r.subset of (subset of([d])(r) ) ; r = 1,..., r(0) be sets consisting of unknown r-wise interactions am

In this manuscript we introduce and study an extended version of the minimal dispersion of point sets, which has recently attracted considerable attention. Given a set P-n = (x(1), ..., x(n)} subset of [0, 1](d )and k is an element of {0, 1, ..., n}, we define the k-dispersion to be the volume of

For epsilon is an element of (0, 1/2) and a natural number d >= 2, let N be a natural number with

N >= 2(9) log(2)(d)(log(2)(1/epsilon))/epsilon)(2)

We prove that there is a set of N points in the unit cube [0, 1](d), which intersects all axis-parallel boxes with volume epsilon. That

The classical Jawerth and Franke embeddings

F-p0,q(s0)(R-n) hooked right arrow B-p1,p0(s1)(R-n) and B-p0,p1(s0)(R-n) hooked right arrow F-p1,q1(s1)(R-n)

are versions of Sobolev embedding between the scales of Besov and Triebel-Lizorkin function spaces for s(0) > s(1) and s(0) - n/p(0)

Let 0 < p, q <= infinity and denote by S-P(N), and S-q(N) the corresponding finite-dimensional Schatten classes. We prove optimal bounds, up to constants only depending on p and q, for the entropy numbers of natural embeddings between S-P(N) and S-q(N) This complements the known results in the cl

Support vector machines (SVMs) with the l(1)-penalty became a standard tool in the analysis of high-dimensional classification problems with sparsity constraints in many applications, including bioinformatics and signal processing. We give non-asymptotic results on the performance of l(1)-SVM in

The volume of the unit ball of the Lebesgue sequence space l(p)(m) is very well known since the times of Dirichlet. We calculate the volume of the unit ball of the mixed norm l(q)(n) (l(p)(m)), whose special cases are nowadays popular in machine learning under the name of group Lasso. We give two

Background: High-throughput proteomics techniques, such as mass spectrometry (MS)-based approaches, produce very high-dimensional data-sets. In a clinical setting one is often interested in how mass spectra differ between patients of different classes, for example spectra from healthy patients vs

The availability of big data in materials science offers new routes for analyzing materials properties and functions and achieving scientific understanding. Finding structure in these data that is not directly visible by standard tools and exploitation of the scientific information requires new a

We prove that for any two quasi-Banach spaces X and Y and any alpha > 0 there exists a constant gamma(alpha), > 0 such that

sup(1 <= k <= n) k(alpha)e(k)(T) <= gamma(alpha) sup(1 <= k <= n) k(alpha)c(k)(T)

holds for all linear and bounded operators T : X -> Y. Here e(k)(T) is the k-th

We study the properties of ridge functions in high dimensions from the viewpoint of approximation theory. The function classes considered consist of ridge functions such that the profile is a member of a univariate Lipschitz class with smoothness (including infinite smoothness) and the ridge dire

We present effective algorithms for uniform approximation of multivariate functions satisfying some prescribed inner structure. We extend, in several directions, the analysis of recovery of ridge functions f (x) = g (< a, x >) as performed earlier by one of the authors and his coauthors. We consi

Statistical learning of materials properties or functions so far starts with a largely silent, nonchallenged step: the choice of the set of descriptive parameters (termed descriptor). However, when the scientific connection between the descriptor and the actuating mechanisms is unclear, the causa

Compressed sensing was introduced some ten years ago as an effective way of acquiring signals, which possess a sparse or nearly sparse representation in a suitable basis or dictionary. Due to its solid mathematical backgrounds, it quickly attracted the attention of mathematicians from several dif

We study complex interpolation of weighted Besov and Lizorkin-Triebel spaces. The used weights omega(0), omega(1) are local Muckenhoupt weights in the sense of Rychkov. As a first step we calculate the Calderon products of associated sequence spaces. Finally, as a corollary of these investigation

We introduce Lorentz spaces L-p(.),L-q (R-n) and L-p(),L-q() (R-n) with variable exponents. We prove several basic properties of these spaces including embeddings and the identity L-p(.),(p(.)) (R-n) = L-p(.) (R-n). We also show that these spaces arise through real interpolation between L-p(.) (R

We comment on recent results in the field of information based complexity, which state (in a number of different settings), that the approximation of infinitely differentiable functions is intractable and suffers from the cure of dimensionality. We show that renorming the space of infinitely diff

We address an open problem posed recently by Almeida and Hasto. They defined the spaces l(q)(.)(L-p(.)) of variable integrability and summability and showed that broken vertical bar broken vertical bar . |lq(.)(L-p(.)) broken vertical bar broken vertical bar is a norm if q >= 1 is constant almost

We provide non-smooth atomic decompositions for Besov spaces B-p,q(s)(R-n), s > 0, 0 < p, q <= infinity, defined via differences. The results are used to compute the trace of Besov spaces on the boundary Gamma of bounded Lipschitz domains Omega with smoothness s restricted to 0 < s < 1 and no fur

We study the spaces and of Besov and Triebel-Lizorkin type as introduced recently in Almeida and Hasto (J. Funct. Anal. 258(5):1628-2655, 2010) and Diening et al. (J. Funct. Anal. 256(6):1731-1768, 2009). Both scales cover many classical spaces with fixed exponents as well as function spaces of v

We introduce the concept of average best m-term approximation widths with respect to a probability measure on the unit ball or the unit sphere of . We estimate these quantities for the embedding with 0 < pa parts per thousand currency signqa parts per thousand currency signa for the normalized co

Let us assume that f is a continuous function defined on the unit ball of R-d, of the form f(x) = g(Ax), where A is a k x d matrix and g is a function of k variables for k << d. We are given a budget m is an element of N of possible point evaluations f(x(i)), i = 1, ... , m, of f, which we are al

We deal with decay and boundedness properties of radial functions belonging to Besov and Lizorkin-Triebel spaces. In detail we investigate the surprising interplay of regularity and decay. Our tools are atomic decompositions in combination with trace theorems.

We consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove a homogeneity property for functions with bounded support in the frame of these spaces. As the proof is based on compact embeddings between the studied function spaces, we present also some results on th

We prove a variant of a Johnson-Lindenstrauss lemma for matrices with circulant structure. This approach allows to minimize the randomness used, is easy to implement and provides good running times. The price to be paid is the higher dimension of the target space k = O(epsilon(-2) log(3) n) inste