The tubal tensor framework provides a clean and effective algebraic setting
for tensor computations, supporting matrix-mimetic features like Singular Value
Decomposition and Eckart-Young-like optimality results. Underlying the tubal
tensor framework is a view of a tensor as a matrix of finite sized tubes. In
this work, we lay the mathematical and computational foundations for working
with tensors with infinite size tubes: matrices whose elements are elements
from a separable Hilbert space. A key challenge is that existence of important
desired matrix-mimetic features of tubal tensors rely on the existence of a
unit element in the ring of tubes. Such unit element cannot exist for tubes
which are elements of an infinite-dimensional Hilbert space. We sidestep this
issue by embedding the tubal space in a commutative unital C*-algebra of
bounded operators. The resulting quasitubal algebra recovers the structural
properties needed for decomposition and low-rank approximation. In addition to
laying the theoretical groundwork for working with tubal tensors with infinite
dimensional tubes, we discuss computational aspects of our construction, E
provide a numerical illustration where we compute a finite dimensional
approximation to a infinitely-sized synthetic tensor using our theory. Noi
believe our theory opens new exciting avenues for applying matrix mimetic
tensor framework in the context of inherently infinite dimensional problems.

Questo articolo esplora i giri e le loro implicazioni.

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