5 research outputs found
Rational invariants of even ternary forms under the orthogonal group
In this article we determine a generating set of rational invariants of
minimal cardinality for the action of the orthogonal group on
the space of ternary forms of even degree . The
construction relies on two key ingredients: On one hand, the Slice Lemma allows
us to reduce the problem to dermining the invariants for the action on a
subspace of the finite subgroup of signed permutations. On the
other hand, our construction relies in a fundamental way on specific bases of
harmonic polynomials. These bases provide maps with prescribed
-equivariance properties. Our explicit construction of these
bases should be relevant well beyond the scope of this paper. The expression of
the -invariants can then be given in a compact form as the
composition of two equivariant maps. Instead of providing (cumbersome) explicit
expressions for the -invariants, we provide efficient algorithms
for their evaluation and rewriting. We also use the constructed
-invariants to determine the -orbit locus and
provide an algorithm for the inverse problem of finding an element in
with prescribed values for its invariants. These are
the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application,
refinement of Definition 3.1. To appear in "Foundations of Computational
Mathematics
Generic separating sets for 3D elasticity tensors
We define what is a generic separating set of invariant functions (a.k.a. a
weak functional basis) for tensors. We produce then two generic separating sets
of polynomial invariants for 3D elasticity tensors, one made of 19 polynomials
and one made of 21 polynomials (but easier to compute) and a generic separating
set of 18 rational invariants. As a byproduct, a new integrity basis for the
fourth-order harmonic tensor is provided
The moving frame method for iterated-integrals: orthogonal invariants
Geometric features, robust to noise, of curves in Euclidean space are of
great interest for various applications such as machine learning and image
analysis. We apply the Fels-Olver's moving frame method (for geometric
features) paired with the log-signature transform (for robust features) to
construct a set of integral invariants under rigid motions for curves in
from the iterated-integral signature. In particular we show that
one can algorithmically construct a set of invariants that characterize the
equivalence class of the truncated iterated-integrals signature under
orthogonal transformations which yields a characterization of a curve in
under rigid motions (and tree-like extensions) and an explicit
method to compare curves up to these transformations.Comment: 37 pages, 4 figure
The moving frame method for iterated-integrals: Orthogonal invariants
We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F. Kiraly and H. Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties