ArXiv Preprint
This paper aims to develop a simple procedure to reduce and control the
condition number of random matrices, and investigate the effect on the
persistent homology (PH) of point clouds of well- and ill-conditioned matrices.
For a square matrix generated randomly using Gaussian/Uniform distribution, the
SVD-Surgery procedure works by: (1) computing its singular value decomposition
(SVD), (2) replacing the diagonal factor by changing a list of the smaller
singular values by a convex linear combination of the entries in the list, and
(3) compute the new matrix by reversing the SVD. Applying SVD-Surgery on a
matrix often results in having different diagonal factor to those of the input
matrix. The spatial distribution of random square matrices are known to be
correlated to the distribution of their condition numbers. The persistent
homology (PH) investigations, therefore, are focused on comparing the effect of
SVD-Surgery on point clouds of large datasets of randomly generated
well-conditioned and ill-conditioned matrices, as well as that of the point
clouds formed by their inverses. This work is motivated by the desire to
stabilise the impact of Deep Learning (DL) training on medical images in terms
of the condition numbers of their sets of convolution filters as a mean of
reducing overfitting and improving robustness against tolerable amounts of
image noise. When applied to convolution filters during training, the
SVD-Surgery acts as a spectral regularisation of the DL model without the need
for learning extra parameters. We shall demonstrate that for several point
clouds of sufficiently large convolution filters our simple strategy preserve
filters norm and reduces the norm of its inverse depending on the chosen linear
combination parameters. Moreover, our approach showed significant improvements
towards the well-conditioning of matrices and stable topological behaviour.
Jehan Ghafuri, Sabah Jassim
2023-02-22
