The näıve Nyström extension forms a low-rank approximation to a positive-semidefinite matrix by uniformly randomly sampling from its columns. This paper provides the first relativeerror bound on the spectral norm error incurred in this process. This bound follows from a natural connection between the Nyström extension and the column subset selection problem. The main tool is a matrix Chernoff b...

3 Operations and Properties 7 3.1 The Identity Matrix and Diagonal Matrices . . . . . . . . . . . . . . . . . . 8 3.2 The Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.3 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3.4 The Trace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.5 Norms . . . ...

The matrix φ(cd) diag (cl)φ(cd) and hence the expectation ∫ φ(cd) diag (cl)φ(cd)dq(cd) is positive semidefinite for any adversary’s strategy φ(cd). Thus, all eigenvalues λi ≥ 0 are non-negative. The corresponding eigenvectors are additionally eigenvectors of Im + ∫ φ(cd) diag (cl)φ(cd)dq(cd) with eigenvalues 1 + λi > 0. Thus, the inverse matrix exists independently of the choice of φ. This prov...

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in H(Ω), and optimizing functionals arising from some problems in economics. In the continuous setting and assuming smoothness, the convexity constraints may be given locally by asking the Hessian matrix to be positive sem...

In this paper we establish links between, and new results for, three problems that are not usually considered together. The first is a matrix decomposition problem that arises in areas such as statistical modeling and signal processing: given a matrix X formed as the sum of an unknown diagonal matrix and an unknown low rank positive semidefinite matrix, decompose X into these constituents. The ...

We consider the smallest eigenvalue problem for symmetric or Hermitian matrices by properties of semidefinite matrices. The work is based on a floating-point Cholesky decomposition and takes into account all possible computational and rounding errors. A computational test is given to verify that a given symmetric or Hermitian matrix is not positive semidefinite, so it has at least one negative ...

We examine stochastic dynamical systems where the transition matrix, Φ, and the system noise, ΓQΓ T , covariance are nearly block diagonal. When H TR−1H is also nearly block diagonal, where R is the observation noise covariance and H is the observation matrix, our suboptimal filter/smoothers are always positive semidefinite, and have improved numerical properties. Applications for distributed d...

We present a dedicated algorithm for the nonnegative factorization of a correlation matrix from an application in financial engineering. We look for a low-rank approximation. The origin of the problem is discussed in some detail. Next to the description of the algorithm, we prove, by means of a counter example, that an exact nonnegative decomposition of a general positive semidefinite matrix is...

Portfolio risk forecasts are often made by estimating an asset or factor correlation matrix. However, estimation difficulties or exogenous constraints can lead to correlation matrix candidates that are not positive semidefinite (psd). Therefore, practitioners need to reimpose the psd property with the minimum possible correction. Rebonato and Jäckel (2000) raised this question and proposed an a...