Raf Vandebril

A note on the representation and definition of semiseparable matrices

In this paper the definition of semiseparable matrices is investigated. Properties of the frequently used definition and the corresponding representation by generators are deduced. Corresponding to the class of tridiagonal matrices another definition of semiseparable matrices is introduced preserving the nice properties dual to the class of tridiagonal matrices. Several theorems and properties are included showing the viability of this alternative definition. Because of the alternative definition, the standard representation of semiseparable matrices is not satisfying anymore. The concept of a representation is explicitly formulated and a new kind of representation corresponding to the alternative definition is given. It is proved that this representation keeps all the interesting properties of the generator representation. Copyright

A New Truncation Strategy for the Higher-Order Singular Value Decomposition

We present an alternative strategy for truncating the higher-order singular value decomposition (T-HOSVD). An error expression for an approximate Tucker decomposition with orthogonal factor matrices is presented, leading us to propose a novel truncation strategy for the HOSVD, which we refer to as the sequentially truncated higher-order singular value decomposition (ST-HOSVD). This decomposition retains several favorable properties of the T-HOSVD, while reducing the number of operations required to compute the decomposition and practically always improving the approximation error. Three applications are presented, demonstrating the effectiveness of ST-HOSVD. In the first application, ST-HOSVD, T-HOSVD, and higher-order orthogonal iteration (HOOI) are employed to compress a database of images of faces. On average, the ST-HOSVD approximation was only

An implicit QR algorithm for symmetric semiseparable matrices

0.1\%

Implicit double shift QR -algorithm for companion matrices

worse than the optimum computed by HOOI, while cutting the execution time by a factor of

An Orthogonal Similarity Reduction of a Matrix into Semiseparable Form

20

On the Convergence of Rational Ritz Values

. In the second application, classification of handwritten digits, ST-HOSVD achieved a speedup factor of

Fast Computation of the Zeros of a Polynomial via Factorization of the Companion Matrix

50

Chasing Bulges or Rotations? A Metamorphosis of the QR-Algorithm

over T-HOSVD during the training phase, and reduced the classification time and storage costs, while not significantly affecting the classification error. The third application demonstrates the effectiveness of ST-HOSVD in compressing results from a numerical simulation of a partial differential equation. In such problems, ST-HOSVD inevitably can greatly improve the running time. We present an example wherein the

On Generic Nonexistence of the Schmidt--Eckart--Young Decomposition for Complex Tensors

2

Protein fold recognition using geometric kernel data fusion

hour