We investigate the online version of Principle Component Analysis (PCA), where in each trial t the learning algorithm chooses a k-dimensional subspace, and upon receiving the next instance vector xt, suffers the “compression loss”, which is the squared Euclidean distance between this instance and its projection into the chosen subspace. When viewed in the right parameterization, this compressio...

We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, O (1/N 2) with respect to initialisation andO (1/N ) with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the e cacy...

In this paper we discuss some basic properties of computable real functions of bounded variation (CBV-functions for short). Especially, it is shown that the image set of semi-computable real numbers under CBV-functions is a proper subset of the class of weakly computable real numbers; Two applications of CBV-functions to semi-computable real numbers produce the whole closure of semi-computable ...

A new method for measuring and designing smooth wavelet basis which dispenses with the need for having a large number of zero moments of the wavelet is given. The method is based on minimizing the “discrete finite variation”, and is a measure of the local “roughness” of a sampled version of the scaling function giving rise to “visually smooth” wavelet basis. Smooth wavelet basis are deemed to b...

The main aim of this paper is to provide an approximation for the function f which possesses continuous derivatives up to the order n−1 (n ≥ 1) and has the n−th derivative of bounded variation, in terms of the chord that connects its end points A = (a, f (a)) and B = (b, f (b)) and some more terms which depend on the values of the k derivatives of the function taken at the end points a and b, w...

In this paper we discuss some basic properties of computable real functions which have bounded variations (CBV-functions for short). Especially, it is shown that the image set of semi-computable real numbers under CBV-functions is a proper subset of weakly computable real number class; Two applications of CBV-functions to semi-computable real numbers produce the whole closure of semi-computable...

We study the evolution of the one dimensional periodic cubic Schrödinger equation (NLS) with bounded variation data. For the linear evolution, it is known that for irrational times the solution is a continuous, nowhere differentiable fractal-like curve. For rational times the solution is a linear combination of finitely many translates of the initial data. Such a dichotomy was fist observed by ...

The absolute loss is the absolute difference between the desired and predicted outcome. This paper demonstrates worst-case upper bounds on the absolute loss for the Perceptron learning algorithm and the Exponentiated Update learning algorithm, which is related to the Weighted Majority algorithm. The bounds characterize the behavior of the algorithms over any sequence of trials, where each trial...

We characterize some major algorithmic randomness notions via differentiability of effective functions. (1) As the main result we show that a real number z ∈ [0, 1] is computably random if and only if each nondecreasing computable function [0, 1] → R is differentiable at z. (2) We prove that a real number z ∈ [0, 1] is weakly 2-random if and only if each almost everywhere differentiable computa...

Some Ostrowski type inequalities are given for the Stieltjes integral where the integrand is absolutely continuous while the integrator is of bounded variation. The case when |f ′| is convex is explored. Applications for the midpoint rule and a generalised trapezoid type rule are also presented.