40,116 research outputs found
Learning with Algebraic Invariances, and the Invariant Kernel Trick
When solving data analysis problems it is important to integrate prior
knowledge and/or structural invariances. This paper contributes by a novel
framework for incorporating algebraic invariance structure into kernels. In
particular, we show that algebraic properties such as sign symmetries in data,
phase independence, scaling etc. can be included easily by essentially
performing the kernel trick twice. We demonstrate the usefulness of our theory
in simulations on selected applications such as sign-invariant spectral
clustering and underdetermined ICA
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional problems. The
mathematical developments rely on the theory of gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
Diagonalizing the genome II: toward possible applications
In a previous paper, we showed that the orientable cover of the moduli space
of real genus zero algebraic curves with marked points is a compact aspherical
manifold tiled by associahedra, which resolves the singularities of the space
of phylogenetic trees. In this draft of a sequel, we construct a related
(stacky) resolution of a space of real quadratic forms, and suggest, perhaps
without much justification, that systems of oscillators parametrized by such
objects may may provide useful models in genomics.Comment: 11 pages, 3 figure
Semi-Fredholmness of weighted singular integral operators with shifts and slowly oscillating data
Let be orientation-preserving homeomorphisms of
onto itself, which have only two fixed points at and , and whose
restrictions to are diffeomorphisms, and let
be the corresponding isometric shift operators on the space
given by for
. We prove sufficient conditions for the right and left
Fredholmness on of singular integral operators of the form
, where ,
is a weighted Cauchy singular integral operator,
and
are operators in the Wiener algebras of
functional operators with shifts. We assume that the coefficients for
and the derivatives of the shifts are bounded
continuous functions on which may have slowly oscillating
discontinuities at and .Comment: Accepted for publication in the Proceedings of WOAT 2016 held in
Lisbon in July of 2016, which will be published in the special volume
"Operator Theory, Operator Algebras, and Matrix Theory" of OT series
(Birkh\"auser
Manifold interpolation and model reduction
One approach to parametric and adaptive model reduction is via the
interpolation of orthogonal bases, subspaces or positive definite system
matrices. In all these cases, the sampled inputs stem from matrix sets that
feature a geometric structure and thus form so-called matrix manifolds. This
work will be featured as a chapter in the upcoming Handbook on Model Order
Reduction (P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A.
Schilders, L.M. Silveira, eds, to appear on DE GRUYTER) and reviews the
numerical treatment of the most important matrix manifolds that arise in the
context of model reduction. Moreover, the principal approaches to data
interpolation and Taylor-like extrapolation on matrix manifolds are outlined
and complemented by algorithms in pseudo-code.Comment: 37 pages, 4 figures, featured chapter of upcoming "Handbook on Model
Order Reduction
Note on local structure of Artin stacks
In this note we show that an Artin stack with finite inertia stack is etale
locally isomorphic to the quotient of an affine scheme by an action of a
general linear group
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