Abstract. In the tensor completion problem, one seeks to estimate a low-rank tensor based on a random sample of revealed entries. In terms of the required sample size, earlier work revealed a large gap between estimation with unbounded computational resources (using, for instance, tensor nuclear norm minimization) and polynomial-time algorithms. Among the latter, the best statistical guarantees...

Global Optimization of Polynomial Functions and Applications by Jiawang Nie Doctor of Philosophy in Applied Mathematics University of California, Berkeley Professor James Demmel, Co-Chair Professor Bernd Sturmfels, Co-Chair This thesis discusses the global optimization problem whose objective function and constraints are all described by (multivariate) polynomials. The motivation is to find the...

We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with n-dimensional orthonormal components in the presence of error with constant spectral norm (when viewed as an n2-by-n2 matr...

We consider the asymptotic behavior of regression estimators that minimize the residual sum of squares plus a penalty proportional to ∑ βj γ for some γ > 0. These estimators include the Lasso as a special case when γ = 1. Under appropriate conditions, we show that the limiting distributions can have positive probability mass at 0 when the true value of the parameter is 0. We also consider asymp...

In this work, we propose to use the linear regression partial least square method to predict the output variables of the RA1G boiler. This method consists in finding the regression of an output block regarding an input block. These two blocks represent the outputs and inputs of the process. A criteria of cross validation, based on the calculation of the predicted residual sum of squares, is use...

A wide variety of stability and performance problems for linear and certain classes of nonlinear dynamical systems can be formulated as convex optimization problems involving linear matrix inequalities (LMIs). These formulations can be solved numerically with computationally-efficient interior-point methods. Many of the first LMI-based stability formulations applied to linear systems and the cl...

Despite the rising number of emotional state detection methods motivated by the popularity increase in affective computing techniques in recent years, they are yet faced with subject and domain transferability issues. In this paper, we present an improved methodology for modelling individuals’ emotional states in multimedia interactive environments. Our method relies on a two-layer classificati...

The first and second Zagreb indices of a graph are equal, respectively, to the sum of squares of the vertex degrees, and the sum of the products of the degrees of pairs of adjacent vertices. We now consider analogous graph invariants, based on the second degrees of vertices (number of their second neighbors), called leap Zagreb indices. A number of their basic properties is established.