In this paper, we propose the Compound Mutual Subspace Method (CPMSM) as a theoretical extension of the Mutual Subspace Method, which can efficiently handle multiple sets of patterns by representing them as subspaces. The proposed method is based on the observation that there are two types of subspace perturbations. One type is due to variations within a class and is therefore defined as “withi...

Moment matching theorems for Krylov subspace based model reduction of higherorder linear dynamical systems are presented in the context of higher-order Krylov subspaces. We introduce the definition of a nth-order Krylov subspace Kn k ({Ai} n i=1;u) based on a sequence of n square matrices {Ai}i=1 and vector u. This subspace is a generalization of Krylov subspaces for higher-order systems, incor...

It is well-known that traditional clustering methods considering all dimensions of the feature space usually fail in terms of efficiency and effectivity when applied to high-dimensional data. This poor behavior is based on the fact that clusters may not be found in the high-dimensional feature space, although clusters exist in subspaces of the feature space. To overcome these limitations of tra...

A subspace partition Π of V = V (n, q) is a collection of subspaces of V such that each 1-dimensional subspace of V is in exactly one subspace of Π. The size of Π is the number of its subspaces. Let σ q (n, t) denote the minimum size of a subspace partition of V in which the largest subspace has dimension t, and let ρ q (n, t) denote the maximum size of a subspace partition of V in which the sm...

The quality assessment of results of clustering algorithms is challenging as different cluster methodologies lead to different cluster characteristics and topologies. A further complication is that in high-dimensional data, subspace clustering adds to the complexity by detecting clusters in multiple different lower-dimensional projections. The quality assessment for (subspace) clustering is esp...