Open AccessPosted Content
On differentiating eigenvalues and eigenvectors
TLDR
In this paper, the conditions under which unique differentiable functions λ(X) and u(X), respectively, exist in a neighborhood of a square matrix (complex or otherwise) satisfying the equations Xu = λu and Xu = ǫ.Abstract:
Let X0 be a square matrix (complex or otherwise) and u0 a (normalized) eigenvector associated with an eigenvalue λo of X0, so that the triple (X0, u0, λ0) satisfies the equations Xu = λu, We investigate the conditions under which unique differentiable functions λ(X) and u(X) exist in a neighborhood of X0 satisfying λ(X0) = λO, u(X0) = u0, Xu = λu, and We obtain the first and second derivatives of λ(X) and the first derivative of u(X) Two alternative expressions for the first derivative of λ(X) are also presented (This abstract was borrowed from another version of this item)read more
Citations
More filters
Journal ArticleDOI
The rank of demand systems : theory and nonparametric estimation
TL;DR: In this article, a simple nonparametric test of rank using Engel curve data is described and applied to U.S. and U.K. consumer survey data, and the results are used to assess theoretical and empirical aggregation error in representative consumer models, and to explain a representative consumer paradox.
Proceedings ArticleDOI
Universal Spectral Adversarial Attacks for Deformable Shapes
TL;DR: In this article, the existence of universal perturbations for geometric data (shapes) is demonstrated. But the attacks take the form of small perturbation to short eigenvalue sequences; the resulting geometry is then synthesized via shape-from-spectrum recovery.
Posted Content
Universal Spectral Adversarial Attacks for Deformable Shapes
TL;DR: In this paper, the existence of universal attacks for geometric data (shapes) has been demonstrated for images to date, where the attacks take the form of small perturbations to short eigenvalue sequences, which are then synthesized via shape-from-spectrum recovery.
Posted Content
Asymptotic Escape of Spurious Critical Points on the Low-rank Matrix Manifold.
TL;DR: In this paper, the authors show that the Riemannian gradient descent algorithm on the low-rank matrix manifold almost surely escapes some spurious critical points on the boundary of the manifold, which are some rank-deficient matrices that capture only part of the SVD components of the ground truth.