Decoupling

We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails.Rather than resorting to a concentration-based argument, the method used here relies on a 'small-ball' assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets.The resulting estimates scale correctly with the 'noise level' of the problem, and when applied to the classical, bounded scenario, always improve the known bounds.

show abstract

“…[5]). For example, if p > 2 and h Lp ≤ κ 2 h L 2 then by the PaleyZygmund inequality, for every 0 < u < 1,…”

Section: Some Examplesmentioning

confidence: 99%

Learning without Concentration

Mendelson

2015

J. ACM

221

301

View full text Add to dashboard Cite

show abstract

“…Then, by the lower semicontinuity of the map A → h(A; M), we deduce the supremum-limit bound of Theorem 1. In Section 3 we prove Theorems 2 and 3 by utilizing results on empirical U -processes [18] on the one hand, and compactness properties of the L 1 -metric [25] on the other hand.…”

Section: Contentmentioning

confidence: 99%

The Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous

2012

View full text Add to dashboard Cite

Let M be a bounded domain of R d with a smooth boundary. We relate the Cheeger constant of M and the conductance of a neighborhood graph defined on a random sample from M. By restricting the minimization defining the latter over a particular class of subsets, we obtain consistency (after normalization) as the sample size increases, and show that any minimizing sequence of subsets has a subsequence converging to a Cheeger set of M.

show abstract

“…16i1¡i26n h(X i1 ; X i2 ), n¿2, P 2 h = hd(P×P) and G P (·) is a Gaussian process evaluated at the values Ph, h ∈ H (with covariance function cov(G P (Ph); G P (Pg)), for (Ph)(x) = h(x; u)dP(u)) (see Reference [10,Theorem 5.3.3], specializing to n = 2). In our setting we let S be some bounded region of the plane.…”

Section: −1mentioning

confidence: 99%

“…His proof can be shortened considerably by making use of recent results from the theory of U-processes (see Reference [10]), as shown in the appendix. The ecdf of the set of dependent interpoint distances among n points in the plane is thus a well-deÿned and behaved summary of a conÿguration of points.…”

mentioning

confidence: 98%

The interpoint distance distribution as a descriptor of point patterns, with an application to spatial disease clustering

Bonetti

Pagano

2004

Statistics in Medicine

View full text Add to dashboard Cite

The topic of this paper is the distribution of the distance between two points distributed independently in space. We illustrate the use of this interpoint distance distribution to describe the characteristics of a set of points within some fixed region. The properties of its sample version, and thus the inference about this function, are discussed both in the discrete and in the continuous setting. We illustrate its use in the detection of spatial clustering by application to a well-known leukaemia data set, and report on the results of a simulation experiment designed to study the power characteristics of the methods within that study region and in an artificial homogenous setting.

show abstract

Decoupling

Cited by 225 publications

References 0 publications

Learning without Concentration

Learning without Concentration

The Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous

The interpoint distance distribution as a descriptor of point patterns, with an application to spatial disease clustering

Contact Info

Product

Resources

About