Interpolation theory suggests a generalization of the usual Vandermonde determinant to numbers with multiplicities. We prove a discriminant formula for this generalized Vandermonde determinant and give an application to the Hermite interpolation problem.

In recent years analysis of complexity of learning Gaussian mixture models from sampled data has received significant attention in computational machine learning and theory communities. In this paper we present the first result showing that polynomial time learning of multidimensional Gaussian Mixture distributions is possible when the separation between the component means is arbitrarily small...

The condition number of a Gram matrix defined by a polynomial basis and a set of points is often used to measure the sensitivity of the polynomial approximation. Given a polynomial basis, we consider the problem of finding a set of points and/or weights which minimizes the condition number of the Gram matrix. We use smoothing methods to solve such minimization problems and show the global conve...

For coded distributed computing (CDC), polynomial code is one prevalent encoding method for CDC (called Poly-CDC). It suffers from poor numerical stability due to the Vandermonde matrix serving as coefficient which needs be inverted, and whose condition number increases exponentially with size of or equivalently parallel worker nodes. To improve stability, especially large networks, we propose ...

In this paper, we research on constructing an explicit parametric curve to be taken as the limitation curve of Progressive Iteration Approximation (PIA) which can interpolate some scattered data points by using normalized totally positive (NTP) basis. We prove that when the data points and corresponding parameter list are given, the limitation curve has the unique explicit expression. There is ...

1. Wigner Matrices 2 2. Wigner’s Semicircle Law 4 3. Markov’s Inequality and Convergence of Expectation 6 4. First Proof of Wigner’s Semicircle Law 7 4.1. Convergence of matrix moments in Expectation 7 4.2. Convergence of Matrix Moments in Probability 12 4.3. Weak Convergence 15 5. Removing Moment Assumptions 17 6. Largest Eigenvalue 21 7. The Stiletjes Transform 31 8. The Stieltjes Transform a...

We derive bounds on the extremal singular values and the condition number of N × K, with N > K, Vandermonde matrices with nodes in the unit disk. Such matrices arise in many fields of applied mathematics and engineering, e.g., in interpolation and approximation theory, sampling theory, compressed sensing, differential equations, control theory, and line spectral estimation. The mathematical tec...

Let M be an n × n matrix with i.i.d. CN(0, 1) entries (Complex Normal random variables with mean 0 and variance 1), and define A = M +M ∗ √ 2 , where M∗ denotes the conjugate transpose of M . The resulting random matrix is a member of the Gaussian Unitary Ensemble (GUE), and we would like to classify the distribution of its eigenvalues. In a previous section, the measure of the Gaussian Unitary...