26,088 research outputs found
Bicompletions of distance matrices
In the practice of information extraction, the input data are usually
arranged into pattern matrices, and analyzed by the methods of linear algebra
and statistics, such as principal component analysis. In some applications, the
tacit assumptions of these methods lead to wrong results. The usual reason is
that the matrix composition of linear algebra presents information as flowing
in waves, whereas it sometimes flows in particles, which seek the shortest
paths. This wave-particle duality in computation and information processing has
been originally observed by Abramsky. In this paper we pursue a particle view
of information, formalized in *distance spaces*, which generalize metric
spaces, but are slightly less general than Lawvere's *generalized metric
spaces*. In this framework, the task of extracting the 'principal components'
from a given matrix of data boils down to a bicompletio}, in the sense of
enriched category theory. We describe the bicompletion construction for
distance matrices. The practical goal that motivates this research is to
develop a method to estimate the hardness of attack constructions in security.Comment: 20 pages, 5 figures; appeared in Springer LNCS vol 7860 in 2013; v2
fixes an error in Sec. 2.3, noticed by Toshiki Kataok
On Matrix KP and Super-KP Hierarchies in the Homogeneous Grading
Constrained KP and super-KP hierarchies of integrable equations (generalized
NLS hierarchies) are systematically produced through a Lie algebraic AKS-matrix
framework associated to the homogeneous grading. The role played by different
regular elements to define the corresponding hierarchies is analyzed as well as
the symmetry properties under the Weyl group transformations. The coset
structure of higher order hamiltonian densities is proven.\par For a generic
Lie algebra the hierarchies here considered are integrable and essentially
dependent on continuous free parameters. The bosonic hierarchies studied in
\cite{{FK},{AGZ}} are obtained as special limit restrictions on hermitian
symmetric-spaces.\par In the supersymmetric case the homogeneous grading is
introduced consistently by using alternating sums of bosons and fermions in the
spectral parameter power series.\par The bosonic hierarchies obtained from
and the supersymmetric ones derived from the
affinization of , and are explicitly constructed.
\par An unexpected result is found: only a restricted subclass of the
bosonic hierarchies can be supersymmetrically extended while preserving
integrability.Comment: 36 pages, LaTe
Tensor Decompositions for Signal Processing Applications From Two-way to Multiway Component Analysis
The widespread use of multi-sensor technology and the emergence of big
datasets has highlighted the limitations of standard flat-view matrix models
and the necessity to move towards more versatile data analysis tools. We show
that higher-order tensors (i.e., multiway arrays) enable such a fundamental
paradigm shift towards models that are essentially polynomial and whose
uniqueness, unlike the matrix methods, is guaranteed under verymild and natural
conditions. Benefiting fromthe power ofmultilinear algebra as theirmathematical
backbone, data analysis techniques using tensor decompositions are shown to
have great flexibility in the choice of constraints that match data properties,
and to find more general latent components in the data than matrix-based
methods. A comprehensive introduction to tensor decompositions is provided from
a signal processing perspective, starting from the algebraic foundations, via
basic Canonical Polyadic and Tucker models, through to advanced cause-effect
and multi-view data analysis schemes. We show that tensor decompositions enable
natural generalizations of some commonly used signal processing paradigms, such
as canonical correlation and subspace techniques, signal separation, linear
regression, feature extraction and classification. We also cover computational
aspects, and point out how ideas from compressed sensing and scientific
computing may be used for addressing the otherwise unmanageable storage and
manipulation problems associated with big datasets. The concepts are supported
by illustrative real world case studies illuminating the benefits of the tensor
framework, as efficient and promising tools for modern signal processing, data
analysis and machine learning applications; these benefits also extend to
vector/matrix data through tensorization. Keywords: ICA, NMF, CPD, Tucker
decomposition, HOSVD, tensor networks, Tensor Train
On Weighted Multivariate Sign Functions
Multivariate sign functions are often used for robust estimation and
inference. We propose using data dependent weights in association with such
functions. The proposed weighted sign functions retain desirable robustness
properties, while significantly improving efficiency in estimation and
inference compared to unweighted multivariate sign-based methods. Using
weighted signs, we demonstrate methods of robust location estimation and robust
principal component analysis. We extend the scope of using robust multivariate
methods to include robust sufficient dimension reduction and functional outlier
detection. Several numerical studies and real data applications demonstrate the
efficacy of the proposed methodology.Comment: Keywords: Multivariate sign, Principal component analysis, Data
depth, Sufficient dimension reductio
Symmetric tensor decomposition
We present an algorithm for decomposing a symmetric tensor, of dimension n
and order d as a sum of rank-1 symmetric tensors, extending the algorithm of
Sylvester devised in 1886 for binary forms. We recall the correspondence
between the decomposition of a homogeneous polynomial in n variables of total
degree d as a sum of powers of linear forms (Waring's problem), incidence
properties on secant varieties of the Veronese Variety and the representation
of linear forms as a linear combination of evaluations at distinct points. Then
we reformulate Sylvester's approach from the dual point of view. Exploiting
this duality, we propose necessary and sufficient conditions for the existence
of such a decomposition of a given rank, using the properties of Hankel (and
quasi-Hankel) matrices, derived from multivariate polynomials and normal form
computations. This leads to the resolution of polynomial equations of small
degree in non-generic cases. We propose a new algorithm for symmetric tensor
decomposition, based on this characterization and on linear algebra
computations with these Hankel matrices. The impact of this contribution is
two-fold. First it permits an efficient computation of the decomposition of any
tensor of sub-generic rank, as opposed to widely used iterative algorithms with
unproved global convergence (e.g. Alternate Least Squares or gradient
descents). Second, it gives tools for understanding uniqueness conditions, and
for detecting the rank
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