相关论文: Condition numbers and scale free graphs
In this paper, we consider the explicit expressions of the normwise condition number for the scaled total least squares problem. Some techniques are introduced to simplify the expression of the condition number, and some new results are…
A simple formula is proved to be a tight estimate for the condition number of the full rank linear least squares residual with respect to the matrix of least squares coefficients and scaled 2-norms. The tight estimate reveals that the…
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares…
In this paper, based on the theory of adjoint operators and dual norms, we define condition numbers for a linear solution function of the weighted linear least squares problem. The explicit expressions of the normwise and componentwise…
In this paper, the normwise condition number of a linear function of the equality constrained linear least squares solution called the partial condition number is considered. Its expression and closed formulae are first presented when the…
We derive closed formulas for the condition number of a linear function of the total least squares solution. Given an over determined linear system Ax=b, we show that this condition number can be computed using the singular values and the…
We present an estimation of the condition numbers of the \emph{mass} and \emph{stiffness} matrices arising from shallow ReLU$^k$ neural networks defined on the unit sphere~$\mathbb{S}^d$. In particular, when $\{\theta_j^*\}_{j=1}^n \subset…
In this paper, within a unified framework of the condition number theory we present the explicit expression of the projected condition number of the equality constrained indefinite least squares problem. By setting specific norms and…
This paper concerns singular value decomposition (SVD)-based computable formulas and bounds for the condition number of the Total Least Squares (TLS) problem. For the TLS problem with the coefficient matrix $A$ and the right-hand side $b$,…
This paper is devoted to condition numbers of the total least squares problem with linear equality constraint (TLSE). With novel limit techniques, closed formulae for normwise, mixed and componentwise condition numbers of the TLSE problem…
The condition number of a diagonally scaled matrix, for appropriately chosen scaling matrices, is often less than that of the original. Equilibration scales a matrix so that the scaled matrix's row and column norms are equal. Scaling can be…
Many complex systems--from social and communication networks to biological networks and the Internet--are thought to exhibit scale-free structure. However, prevailing explanations rely on the constant addition of new nodes, an assumption…
The condition number of a linear function of the indefinite least squares solution is called the partial condition number for the indefinite least squares problem. In this paper, based on a new and very general condition number which can be…
A central claim in modern network science is that real-world networks are typically "scale free," meaning that the fraction of nodes with degree $k$ follows a power law, decaying like $k^{-\alpha}$, often with $2 < \alpha < 3$. However,…
We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $\epsilon$) is at least of order $\epsilon$. This allows us to…
In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the linear least squares problem with equality constrains (LSE). We derive the explicit expressions of the mixed and…
The bloom of complex network study, in particular, with respect to scale-free ones, is considerably triggering the research of scale-free graph itself. Therefore, a great number of interesting results have been reported in the past,…
In this paper, we derive the mixed and componentwise condition numbers for a linear function of the solution to the total least squares with linear equality constraint (TLSE) problem. The explicit expressions of the mixed and componentwise…
In this paper, we consider the mixed and componentwise condition numbers for a linear function of the solution to the total least squares (TLS) problem. We derive the explicit expressions of the mixed and componentwise condition numbers…
Many real networks are complex and have power-law vertex degree distribution, short diameter, and high clustering. We analyze the network model based on thresholding of the summed vertex weights, which belongs to the class of networks…