Lower bounds for invariant statistical models with applications to principal component analysis

Approximation bounds for sparse principal component analysis

We produce approximation bounds on a semidefinite programming relaxation for sparse principal component analysis. These bounds control approximation ratios for tractable statistics in hypothesis testing problems where data points are sampled from Gaussian models with a single sparse leading component. We study approximation bounds for a semidefinite relaxation of the sparse eigenvalue problem, ...

Complexity Theoretic Lower Bounds for Sparse Principal Component Detection

In the context of sparse principal component detection, we bring evidence towards the existence of a statistical price to pay for computational efficiency. We measure the performance of a test by the smallest signal strength that it can detect and we propose a computationally efficient method based on semidefinite programming. We also prove that the statistical performance of this test cannot b...

Nonlinear Principal Component Analysis: Neural Network Models and Applications

1 Competence Centre for Functional Genomics, Institute for Microbiology, Ernst-Moritz-Arndt-University Greifswald, F.-L.-Jahn-Str. 15, 17487 Greifswald, Germany, [email protected] [email protected] 2 Institute for Biochemistry and Biology, University of Potsdam, c/o Max Planck Institute for Molecular Plant Physiology Am Mühlenberg 1, 14424 Potsdam, Germany, Selb...

Perturbation bounds for $g$-inverses with respect to the unitarily invariant norm

Let complex matrices $A$ and $B$ have the same sizes. Using the singular value decomposition, we characterize the $g$-inverse $B^{(1)}$ of $B$ such that the distance between a given $g$-inverse of $A$ and the set of all $g$-inverses of the matrix $B$ reaches minimum under the unitarily invariant norm. With this result, we derive additive and multiplicative perturbation bounds of the nearest per...

Torus Principal Component Analysis with Applications to Rna Structure