Approximation of Gram-Schmidt Orthogonalization by Data Matrix

Abstract

For a matrix $\mathbf{A}$ with linearly independent columns, this work studies to use its normalization $\bar{\mathbf{A}}$ and $\mathbf{A}$ itself to approximate its orthonormalization $\mathbf{V}$. We theoretically analyze the order of the approximation errors as $\mathbf{A}$ and $\bar{\mathbf{A}}$ approach $\mathbf{V}$, respectively. Our conclusion is able to explain the fact that a high dimensional Gaussian matrix can well approximate the corresponding truncated Haar matrix. For applications, this work can serve as a foundation of a wide variety of problems in signal processing such as compressed subspace clustering.

9 pages