We study the complexity of high-dimensional approximation in $L_2$-norm when different classes information are available; we compare power function evaluations with arbitrary continuous linear measurements. Here, discuss situation number measurements required to achieve an error $\varepsilon \in (0,1)$ dimension $d\in\mathbb{N}$ depends only poly-logarithmically on $\varepsilon^{-1}$. This corr...

It is well-known that tensor decompositions show separations, is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here we many of these separations disappear the approximate case. Specifically, for every approximation error $\varepsilon$ and norm, define rank minimum element $\varepsilon$-ball with respect to norm. For positive semide...

A quantile summary is a data structure that approximates to ε-relative error the order statistics of a much larger underlying dataset. In this paper we develop a randomized online quantile summary for the cash register data input model and comparison data domain model that uses O( 1 ε log 1 ε ) words of memory. This improves upon the previous best upper bound of O( 1 ε log 1 ε ) by Agarwal et. ...

A simultaneous sparse approximation problem requests a good approximation of several input signals at once using different linear combinations of the same elementary signals. At the same time, the problem balances the error in approximation against the total number of elementary signals that participate. These elementary signals typically model coherent structures in the input signals, and they...

This paper examines asymptotic expansions for estimation errors expressed explicitly as functions of underlying random variables. Taylor series expansions are obtained from which "rst and second moment approximations are derived. While the expansions are essentially equivalent to the traditional Nagar type, the terms are expressed in a form which enables moment approximations to be obtained in ...

In this paper various methods for the estimation of simultaneous equation models with lagged endogenow variables and List order s&fly correlated errors are discussed. The m.&ods dilfer in tbe number of instrumental variables used. The asymptotic and small sample properties of the various metlmds are compared, and the variables which must be included as insuuments to insure consistent estimates ...