and Applied Analysis 3 of positive real numbers. By S 2 − X , we denote the space of all sequences defined over X, ‖·, ·‖ . Now we define the following sequence spaces: W ( M,Δ, p, ‖, ·, ‖ ⎧ ⎪⎨ ⎪⎩ x ∈ S 2 −X : ∀ε > 0 { n ∈ : ∞ ∑ k 1 ank [ M (∥ ∥ ∥ ∥ Δxk − L ρ , z ∥ ∥ ∥ ∥ )]pk ≥ ε } ∈ I for some ρ > 0, L ∈ X and each z ∈ X ⎫ ⎪⎬ ⎪⎭ , W 0 ( A,M,Δ, p, ‖, ·, ‖ ⎧ ⎪⎨ ⎪⎩ x ∈ S 2 −X : ∀ε > 0 { n ∈ : ∞ ∑...

Let P be a set of n points in Euclidean space and let 0 < ε < 1. A well-known result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε−2 logn) such that distances change by a factor of 1 + ε at most. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs, but all subsets of at most k p...

Let C be a class of probability distributions over the discrete domain [n] = {1, . . . , n}. We show that if C satisfies a rather general condition – essentially, that each distribution in C can be well-approximated by a variable-width histogram with few bins – then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of k unknow...

We present two new results for the problem of approximating a given real m × n matrix A by a rank-k matrix D, where k < min{m, n}, so as to minimize ||A−D||F . It is known that by sampling O(k/ε) rows of the matrix, one can find a low-rank approximation with additive error ε||A||F . Our first result shows that with adaptive sampling in t rounds and O(k/ε) samples in each round, the additive err...

and Applied Analysis 3 Definition 1.3 see 25 . Let X, μ, ν, ∗, be a intuitionistic fuzzy metric space. Let A be any subset of X. Define φ t inf { μ ( x, y, t ) : x, y ∈ A}, ψ t sup{ν(x, y, t) : x, y ∈ A}, 1.2 i A is said to be q-bounded if limt→∞φ t 1 and limt→∞ψ t 0, ii A is said to be semibounded if limt→∞φ t k and limt→∞ψ t 1 − k, 0 < k < 1 iii A is said to be unbounded if limt→∞φ t 0 and li...

This paper revisits the online PCA problem. Given a stream of n vectors xt ∈ R (columns of X) the algorithm must output yt ∈ R (columns of Y ) before receiving xt+1. The goal of online PCA is to simultaneously minimize the target dimension ` and the error ‖X − (XY +)Y ‖. We describe two simple and deterministic algorithms. The first, receives a parameter ∆ and guarantees that ‖X − (XY +)Y ‖ is ...

In this paper, we show the existence of small coresets for the problems of computing k-median and k-means clustering for points in low dimension. In other words, we show that given a point set P in IR, one can compute a weighted set S ⊆ P , of size O(kε−d log n), such that one can compute the k-median/means clustering on S instead of on P , and get an (1 + ε)-approximation. As a result, we impr...

Given a weighted n-vertex graph G with integer edge-weights taken from a range [−M,M ], we show that the minimum-weight simple path visiting k vertices can be found in time Õ(2poly(k)Mn) = O∗(2kM). If the weights are reals in [1,M ], we provide a (1+ε)-approximation which has a running time of Õ(2poly(k)n(log logM + 1/ε)). For the more general problem of k-tree, in which we wish to find a minim...

We construct pseudorandom generators with improved seed length for several classes of tests. First we consider the class of read-once polynomials over GF(2) in m variables. For error ε we obtain seed length Õ(log(m/ε)) log(1/ε), where Õ hides lower-order terms. This is optimal up to the factor Õ(log(1/ε)). The previous best seed length was polylogarithmic in m and 1/ε. Second we consider produc...