Computing derivative-based global sensitivity measures using polynomial chaos expansions

In the field of computer experiments sensitivity analysis aims at quantifying the relative importance of each input parameter (or combinations thereof) of a computational model with respect to the model output uncertainty. Variance decomposition methods leading to the well-known Sobol' indices are recognized as accurate techniques, at a rather high computational cost though. The use of polynomial chaos expansions (PCE) to compute Sobol' indices has allowed to alleviate the computational burden though. However, when dealing with large dimensional input vectors, it is good practice to first use screening methods in order to discard unimportant variables. The {\em derivative-based global sensitivity measures} (DGSM) have been developed recently in this respect. In this paper we show how polynomial chaos expansions may be used to compute analytically DGSMs as a mere post-processing. This requires the analytical derivation of derivatives of the orthonormal polynomials which enter PC expansions. The efficiency of the approach is illustrated on two well-known benchmark problems in sensitivity analysis

Symmetric tensor decomposition

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank

Score Function Features for Discriminative Learning: Matrix and Tensor Framework

Feature learning forms the cornerstone for tackling challenging learning problems in domains such as speech, computer vision and natural language processing. In this paper, we consider a novel class of matrix and tensor-valued features, which can be pre-trained using unlabeled samples. We present efficient algorithms for extracting discriminative information, given these pre-trained features and labeled samples for any related task. Our class of features are based on higher-order score functions, which capture local variations in the probability density function of the input. We establish a theoretical framework to characterize the nature of discriminative information that can be extracted from score-function features, when used in conjunction with labeled samples. We employ efficient spectral decomposition algorithms (on matrices and tensors) for extracting discriminative components. The advantage of employing tensor-valued features is that we can extract richer discriminative information in the form of an overcomplete representations. Thus, we present a novel framework for employing generative models of the input for discriminative learning.Comment: 29 page

The algebra of the box spline

In this paper we want to revisit results of Dahmen and Micchelli on box-splines which we reinterpret and make more precise. We compare these ideas with the work of Brion, Szenes, Vergne and others on polytopes and partition functions.Comment: 69 page

Information Sets of Multiplicity Codes

We here provide a method for systematic encoding of the Multiplicity codes introduced by Kopparty, Saraf and Yekhanin in 2011. The construction is built on an idea of Kop-party. We properly define information sets for these codes and give detailed proofs of the validity of Kopparty's construction, that use generating functions. We also give a complexity estimate of the associated encoding algorithm.Comment: International Symposium on Information Theory, Jun 2015, Hong-Kong, China. IEE