On the minimal algebraic complexity of the rank-one approximation problem for general inner products

We study the algebraic complexity of Euclidean distance minimization from a generic tensor to a variety of rank-one tensors. The Euclidean Distance (ED) degree of the Segre-Veronese variety counts the number of complex critical points of this optimization problem. We regard this invariant as a function of inner products and conjecture that it achieves its minimal value at Frobenius inner product. We prove our conjecture in the case of matrices. We discuss the above optimization problem for other algebraic varieties, classifying all possible values of the ED degree. Our approach combines tools from Singularity Theory, Morse Theory, and Algebraic Geometry.Comment: 31 pages, 2 table

A Groupwise Multilinear Correspondence Optimization for 3D Faces

The official version of this article is available on the IEEE websiteInternational audienceMultilinear face models are widely used to model the space of human faces with expressions. For databases of 3D human faces of different identities performing multiple expressions, these statistical shape models decouple identity and expression variations. To compute a high-quality multilinear face model, the quality of the registration of the database of 3D face scans used for training is essential. Meanwhile, a multilinear face model can be used as an effective prior to register 3D face scans, which are typically noisy and incomplete. Inspired by the minimum description length approach, we propose the first method to jointly optimize a multilinear model and the registration of the 3D scans used for training. Given an initial registration, our approach fully automatically improves the registration by optimizing an objective function that measures the compactness of the multilinear model, resulting in a sparse model. We choose a continuous representation for each face shape that allows to use a quasi-Newton method in parameter space for optimization. We show that our approach is computationally significantly more efficient and leads to correspondences of higher quality than existing methods based on linear statistical models. This allows us to evaluate our approach on large standard 3D face databases and in the presence of noisy initializations

Advanced Multilinear Data Analysis and Sparse Representation Approaches and Their Applications

Multifactor analysis plays an important role in data analysis since most real-world datasets usually exist with a combination of numerous factors. These factors are usually not independent but interdependent together. Thus, it is a mistake if a method only considers one aspect of the input data while ignoring the others. Although widely used, Multilinear PCA (MPCA), one of the leading multilinear analysis methods, still suffers from three major drawbacks. Firstly, it is very sensitive to outliers and noise and unable to cope with missing values. Secondly, since MPCA deals with huge multidimensional datasets, it is usually computationally expensive. Finally, it loses original local geometry structures due to the averaging process. This thesis sheds new light on the tensor decomposition problem via the ideas of fast low-rank approximation in random projection and tensor completion in compressed sensing. We propose a novel approach called Compressed Submanifold Multifactor Analysis (CSMA) to solve the three problems mentioned above. Our approach is able to deal with the problem of missing values and outliers via our proposed novel sparse Higher-order Singular Value Decomposition approach, named HOSVD-L1 decomposition. The Random Projection method is used to obtain the fast low-rank approximation of a given multifactor dataset. In addition, our method can preserve geometry of the original data. In the second part of this thesis, we present a novel pattern classification approach named Sparse Class-dependent Feature Analysis (SCFA), to connect the advantages of sparse representation in an overcomplete dictionary, with a powerful nonlinear classifier. The classifier is based on the estimation of class-specific optimal filters, by solving an L1-norm optimization problem using the Alternating Direction Method of Multipliers. Our method as well as its Reproducing Kernel Hilbert Space (RKHS) version is tolerant to the presence of noise and other variations in an image. Our proposed methods achieve very high classification accuracies in face recognition on two challenging face databases, i.e. the CMU Pose, Illumination and Expression (PIE) database and the Extended YALE-B that exhibit pose and illumination variations; and the AR database that has occluded images. In addition, they also exhibit robustness on other evaluation modalities, such as object classification on the Caltech101 database. Our method outperforms state-of-the-art methods on all these databases and hence they show their applicability to general computer vision and pattern recognition problems