10,775 research outputs found
Balancing for unstable nonlinear systems
A previously obtained method of balancing for stable nonlinear systems is extended to unstable nonlinear systems. The similarity invariants obtained by the concept of LQG balancing for an unstable linear system can also be obtained by considering a past and future energy function of the system. By considering a past and future energy function for an unstable nonlinear system, the concept of these similarity invariants for linear systems is extended to nonlinear systems. Furthermore the relation of this balancing method with the previously obtained method of balancing the coprime factorization of an unstable nonlinear system is considered. Both methods are introduced with the aim of using it as a tool for model reductio
A Nonconvex Splitting Method for Symmetric Nonnegative Matrix Factorization: Convergence Analysis and Optimality
Symmetric nonnegative matrix factorization (SymNMF) has important
applications in data analytics problems such as document clustering, community
detection and image segmentation. In this paper, we propose a novel nonconvex
variable splitting method for solving SymNMF. The proposed algorithm is
guaranteed to converge to the set of Karush-Kuhn-Tucker (KKT) points of the
nonconvex SymNMF problem. Furthermore, it achieves a global sublinear
convergence rate. We also show that the algorithm can be efficiently
implemented in parallel. Further, sufficient conditions are provided which
guarantee the global and local optimality of the obtained solutions. Extensive
numerical results performed on both synthetic and real data sets suggest that
the proposed algorithm converges quickly to a local minimum solution.Comment: IEEE Transactions on Signal Processing (to appear
Supersymmetry and the formal loop space
For any algebraic super-manifold M we define the super-ind-scheme LM of
formal loops and study the transgression map (Radon transform) on differential
forms in this context. Applying this to the super-manifold M=SX, the spectrum
of the de Rham complex of a manifold X, we obtain, in particular, that the
transgression map for X is a quasi-isomorphism between the [2,3)-truncated de
Rham complex of X and the additive part of the [1,2)-truncated de Rham complex
of LX. The proof uses the super-manifold SSX and the action of the Lie
superalgebra sl(1|2) on this manifold. This quasi-isomorphism result provides a
crucial step in the classification of sheaves of chiral differential operators
in terms of geometry of the formal loop space
Strong Products of Hypergraphs: Unique Prime Factorization Theorems and Algorithms
It is well-known that all finite connected graphs have a unique prime factor
decomposition (PFD) with respect to the strong graph product which can be
computed in polynomial time. Essential for the PFD computation is the
construction of the so-called Cartesian skeleton of the graphs under
investigation.
In this contribution, we show that every connected thin hypergraph H has a
unique prime factorization with respect to the normal and strong (hypergraph)
product. Both products coincide with the usual strong graph product whenever H
is a graph. We introduce the notion of the Cartesian skeleton of hypergraphs as
a natural generalization of the Cartesian skeleton of graphs and prove that it
is uniquely defined for thin hypergraphs. Moreover, we show that the Cartesian
skeleton of hypergraphs can be determined in O(|E|^2) time and that the PFD can
be computed in O(|V|^2|E|) time, for hypergraphs H = (V,E) with bounded degree
and bounded rank
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