A multivariate ridge function is a function of the form @math , where @math is univariate and @math . We show that the recovery of an unknown ridge function defined on the hypercube @math with Lipschitz-regular profile @math suffers from the curse of dimensionality when the recovery error is measured in the @math -norm, even if we allow randomized algorithms. If a limited number of components of @math is substantially larger than the others, then the curse of dimensionality is not present and the problem is weakly tractable provided the profile @math is sufficiently regular.
An alternative to the component-wise positivity of the ridge vector that also guarantees polynomial tractability is to assume that for some given @math . This assumption works both for ridge functions defined on the hypercube and for ridge functions defined on the Euclidean ball @math whereas does not lead to a polynomially tractable problem for ridge functions defined on the hypercube, see @cite_25 . Assumption has been studied in @cite_22 @cite_1 @cite_25 , where @cite_1 studies also the effect of noisy measurements. @cite_25 , it has been shown that recovery of ridge functions defined on the Euclidean ball in general suffers from the curse of dimensionality. This finding is based on novel two-sided estimates that reduce the decay behavior of the worst-case recovery error to the decay behavior of entropy numbers of @math -balls. whereas @math implies weak tractability for sufficiently large @math .
@cite_22 @cite_1 @cite_25
2077625299 2138611516 2791521719 1605194072
We present effective algorithms for uniform approximation of multivariate functions satisfying some prescribed inner structure. We extend, in several directions, the analysis of recovery of ridge functions f ( x ) = g ( a , x ) as performed earlier by one of the authors and his coauthors. We consider ridge functions defined on the unit cube - 1 , 1 ] d as well as recovery of ridge functions defined on the unit ball from noisy measurements. We conclude with the study of functions of the type f ( x ) = g ( ? a - x ? l 2 d 2 ) . We study the properties of ridge functions (f(x)=g(a x) ) in high dimensions (d ) from the viewpoint of approximation theory. The function classes considered consist of ridge functions such that the profile (g ) is a member of a univariate Lipschitz class with smoothness ( >0 ) (including infinite smoothness) and the ridge direction (a ) has (p )-norm ( a _p 1 ). First, we investigate entropy numbers in order to quantify the compactness of these ridge function classes in (L_ ). We show that they are essentially as compact as the class of univariate Lipschitz functions. Second, we examine sampling numbers and consider two extreme cases. In the case (p=2 ), sampling ridge functions on the Euclidean unit ball suffers from the curse of dimensionality. Moreover, it is as difficult as sampling general multivariate Lipschitz functions, which is in sharp contrast to the result on entropy numbers. When we additionally assume that all feasible profiles have a first derivative uniformly bounded away from zero at the origin, the complexity of sampling ridge functions reduces drastically to the complexity of sampling univariate Lipschitz functions. In between, the sampling problem’s degree of difficulty varies, depending on the values of ( ) and (p ). Surprisingly, we see almost the entire hierarchy of tractability levels as introduced in the recent monographs by Novak and Woźniakowski. We consider the problem of estimating the discrete clustering structures under Sub-Gaussian Mixture Models. Our main results establish a hidden integrality property of a semidefinite programming (SDP) relaxation for this problem: while the optimal solutions to the SDP are not integer-valued in general, their estimation errors can be upper bounded in terms of the error of an idealized integer program. The error of the integer program, and hence that of the SDP, are further shown to decay exponentially in the signal-to-noise ratio. To the best of our knowledge, this is the first exponentially decaying error bound for convex relaxations of mixture models, and our results reveal the "global-to-local" mechanism that drives the performance of the SDP relaxation. A corollary of our results shows that in certain regimes the SDP solutions are in fact integral and exact, improving on existing exact recovery results for convex relaxations. More generally, our results establish sufficient conditions for the SDP to correctly recover the cluster memberships of @math fraction of the points for any @math . As a special case, we show that under the @math -dimensional Stochastic Ball Model, SDP achieves non-trivial (sometimes exact) recovery when the center separation is as small as @math , which complements previous exact recovery results that require constant separation. Consider the recovery of an unknown signal @math from quantized linear measurements. In the one-bit compressive sensing setting, one typically assumes that @math is sparse, and that the measurements are of the form @math . Since such measurements give no information on the norm of @math , recovery methods typically assume that @math . We show that if one allows more generally for quantized affine measurements of the form @math , and if the vectors @math are random, an appropriate choice of the affine shifts @math allows norm recovery to be easily incorporated into existing methods for one-bit compressive sensing. In addition, we show that for arbitrary fixed @math in the annulus @math , one may estimate the norm @math up to additive error @math from @math such binary measurements through a single evaluation of the inverse Gaussian error function. Finally, all of our recovery guarantees can be made universal over sparse vectors in the sense that with high probability, one set of measurements and thresholds can successfully estimate all sparse vectors @math in a Euclidean ball of known radius.