A Non-Self-Referential Characterization of the Gram–Schmidt Process via Computational Induction - Chen, Ray-Ming (Mathematics (Basel), 2025)
The Gram–Schmidt process (GSP) plays an important role in algebra. It provides a theoretical and practical approach for generating an orthonormal basis, QR decomposition, unitary matrices, etc. It also facilitates some applications in the fields of communication, machine learning, feature extraction, etc. The typical GSP is self-referential, while the non-self-referential GSP is based on the Gram determinant, which has exponential complexity. The motivation for this article is to find a way that could convert a set of linearly independent vectors u→ij=1n into a set of orthogonal vectors v→j=1n via a non-self-referential GSP (NsrGSP). The approach we use is to derive a method that utilizes the recursive property of the standard GSP to retrieve a NsrGSP. The individual orthogonal vector form we obtain is v→k=∑j=1kβ[k→j]u→j, and the collective orthogonal vectors, in a matrix form, are Vk=Uk(BΔk+). This approach could reduce the exponential computational complexity to a polynomial one. It also has a neat representation. To this end, we also apply our approach on a classification problem based on real data. Our method shows the experimental results are much more persuasive than other familiar methods.