16,445 research outputs found
Multi-View Clustering via Semi-non-negative Tensor Factorization
Multi-view clustering (MVC) based on non-negative matrix factorization (NMF)
and its variants have received a huge amount of attention in recent years due
to their advantages in clustering interpretability. However, existing NMF-based
multi-view clustering methods perform NMF on each view data respectively and
ignore the impact of between-view. Thus, they can't well exploit the
within-view spatial structure and between-view complementary information. To
resolve this issue, we present semi-non-negative tensor factorization
(Semi-NTF) and develop a novel multi-view clustering based on Semi-NTF with
one-side orthogonal constraint. Our model directly performs Semi-NTF on the
3rd-order tensor which is composed of anchor graphs of views. Thus, our model
directly considers the between-view relationship. Moreover, we use the tensor
Schatten p-norm regularization as a rank approximation of the 3rd-order tensor
which characterizes the cluster structure of multi-view data and exploits the
between-view complementary information. In addition, we provide an optimization
algorithm for the proposed method and prove mathematically that the algorithm
always converges to the stationary KKT point. Extensive experiments on various
benchmark datasets indicate that our proposed method is able to achieve
satisfactory clustering performance
Orthogonal Laurent polynomials in unit circle, extended CMV ordering and 2D Toda type integrable hierarchies
Orthogonal Laurent polynomials in the unit circle and the theory of Toda-like
integrable systems are connected using the Gauss--Borel factorization of a
Cantero-Moral-Velazquez moment matrix, which is constructed in terms of a
complex quasi-definite measure supported in the unit circle. The factorization
of the moment matrix leads to orthogonal Laurent polynomials in the unit circle
and the corresponding second kind functions. Jacobi operators, 5-term recursion
relations and Christoffel-Darboux kernels, projecting to particular spaces of
truncated Laurent polynomials, and corresponding Christoffel-Darboux formulae
are obtained within this point of view in a completely algebraic way.
Cantero-Moral-Velazquez sequence of Laurent monomials is generalized and
recursion relations, Christoffel-Darboux kernels, projecting to general spaces
of truncated Laurent polynomials and corresponding Christoffel-Darboux formulae
are found in this extended context. Continuous deformations of the moment
matrix are introduced and is shown how they induce a time dependant
orthogonality problem related to a Toda-type integrable system, which is
connected with the well known Toeplitz lattice. Using the classical
integrability theory tools the Lax and Zakharov-Shabat equations are obtained.
The dynamical system associated with the coefficients of the orthogonal Laurent
polynomials is explicitly derived and compared with the classical Toeplitz
lattice dynamical system for the Verblunsky coefficients of Szeg\H{o}
polynomials for a positive measure. Discrete flows are introduced and related
to Darboux transformations. Finally, the representation of the orthogonal
Laurent polynomials (and its second kind functions), using the formalism of
Miwa shifts, in terms of -functions is presented and bilinear equations
are derived
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