We give algorithms for computing the densest k-dimensional sublattice of an arbitrary lattice, and related problems. This is an important problem in the algorithmic geometry of numbers that includes as special cases Rankin’s problem (which corresponds to the densest sublattice problem with respect to the Euclidean norm, and has applications to the design of lattice reduction algorithms), and th...

Abstract Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell +1}=\bar z_\ell + \textrm {Re}\sum _{k=1}^K\bar b_{\ell k}\,e^{\textrm {i}\omega _{\ell k}\bar }+ c_{\ell ^{\prime}_{\ell k}\cdot x}$. An optimal distribution frequencies $(\omega k},\omega k})$ $e^{\textrm }$ and x}$ is derived. This derivation based on ...

In their fundamental work, Alon, Matias and Szegedy [3] presented celebrated sketching techniques and showed that 4-wise independence is sufficient to obtain good approximations. The question of what random functions are necessary is fundamental for streaming algorithms (see, e.g., Cormode and Muthukrishnan [9].) We present a somewhat surprising fact: on product domain [n], the 4-wise independe...

We consider the following oblivious sketching problem: given ∈ (0, 1/3) and n ≥ d/ 2, design a distribution D over Rk×nd and a function f : R × R → R, so that for any n× d matrix A, Pr S∼D [(1− )‖A‖op ≤ f(S(A), S) ≤ (1 + )‖A‖op] ≥ 2/3, where ‖A‖op = supx:‖x‖2=1 ‖Ax‖2 is the operator norm of A and S(A) denotes S ·A, interpreting A as a vector in R. We show a tight lower bound of k = Ω(d2/ 2) for...

In this contribution, we first introduce the concept of metrical T-norm-based similarity measure for hesitant fuzzy sets (HFSs) {by using the concept of T-norm-based distance measure}. Then,the relationship of the proposed {metrical T-norm-based} similarity {measures} with the {other kind of information measure, called the metrical T-norm-based} entropy measure {is} discussed. The main feature ...

Let K be a symmetric indefinite matrix. Suppose that K 1⁄4 LJL is the generalized Cholesky factorization of K. In this paper we present perturbation analysis for the generalized Cholesky factorization. We obtain the first-order bound on the norm of the perturbation in the generalized Cholesky factor. Also, we give rigorous perturbation bounds. 2002 Elsevier Inc. All rights reserved.

The main result establishes that if a controller C (comprising of a linear feedback of the output and its derivatives) globally stabilizes a (nonlinear) plant P , then global stabilization of P can also be achieved by an output feedback controller C[h] where the output derivatives in C are replaced by an Euler approximation with sufficiently small delay h > 0. This is proved within the conceptu...

We prove sharp mixed norm inequalities for the k-plane transform when acting on radial functions and for potential-like operators supported in k-planes. We also study the Hardy-Littlewood maximal operator on k-planes for radial functions for which we obtain a basic pointwise inequality with interesting consequences. §

In this paper we address the problem of dynamic MRI reconstruction from partially sampled K-space data. Our work is motivated by previous studies in this area that proposed exploiting the spatiotemporal correlation of the dynamic MRI sequence by posing the reconstruction problem as a least squares minimization regularized by sparsity and low-rank penalties. Ideally the sparsity and low-rank pen...

where co means the convex hull and the given functions ai(·) : R n → R, i = 1, 2, . . . , k are supposed to satisfy the following assumptions: A(i): ai(·) are twice continuously differentiable, and A(ii): ai(·) are with linear growth: ‖ai(x)‖ ≤ θ(1 + ‖x‖) for some positive θ, where ‖ · ‖ is the Euclidean norm. For any sequence of k elements (f1, f2, . . . , fk) we use the notation {fi} k i=1. D...