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In this paper, a new axiomatization for unbounded functional calculi is proposed and the associated theory is elaborated comprising, among others, uniqueness and compatibility results and extension theorems of algebraic and topological…
Unsupervised learning aims at the discovery of hidden structure that drives the observations in the real world. It is essential for success in modern machine learning. Latent variable models are versatile in unsupervised learning and have…
Reversible computing is motivated by both pragmatic and foundational considerations arising from a variety of disciplines. We take a particular path through the development of reversible computation, emphasizing compositional reversible…
Building up on classical linear formulations, we posit that a broad class of problems in signal synthesis and in signal recovery are reducible to the basic task of finding a point in a closed convex subset of a Hilbert space that satisfies…
Considering the question: how non-linear may a non-linear operator be in order to extend the linear regularization theory, we introduce the class of dilinear mappings, which covers linear, bilinear, and quadratic operators between Banach…
We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, that contains all operators of Helffer-Sj\"ostrand type and is closed under the action of smooth proper mappings.…
We here first study the state space realization of a tensor-product of a pair of rational functions. At the expense of "inflating" the dimensions, we recover the classical expressions for realization of a regular product of rational…
We present a general framework for extracting conformal data from critical two-dimensional classical lattice models using finite-size tensor-network flow. The central idea is to identify, from transfer-matrix spectra, a self-consistent…
This paper proposes a novel method for learning highly nonlinear, multivariate functions from examples. Our method takes advantage of the property that continuous functions can be approximated by polynomials, which in turn are representable…
Tensors with unit Frobenius norm are fundamental objects in many fields, including scientific computing and quantum physics, which are able to represent normalized eigenvectors and pure quantum states. While the tensor train decomposition…
In this paper, we show that the semiclassical calculus recently developed on nilpotent Lie groups and nilmanifolds include the functional calculus of suitable subelliptic operators. Moreover, we obtain Weyl laws for these operators. Amongst…
Regarding quaternions as normal matrices, we first characterize the $2\times 2$ matrix-valued functions, defined on subsets of quaternions, whose values are quaternions. Then we investigate the regularity of quaternionic-valued functions,…
Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the…
We present a linear functional calculus with both the safety guarantees expressible with linear types and the rich language of combinators and composition provided by functional programming. Unlike previous combinations of linear typing and…
We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In…
Multimodular functions, primarily used in the literature of queueing theory, discrete-event systems, and operations research, constitute a fundamental function class in discrete convex analysis. The objective of this paper is to clarify the…
The Riesz-Dunford functional calculus over the algebra of octonions, denoted by $\mathbb{O}$, has long been an open problem due to the nonassociativity of octonions. Two core obstacles hinder its development: first, the generalization of…
The development of compositional distributional models of semantics reconciling the empirical aspects of distributional semantics with the compositional aspects of formal semantics is a popular topic in the contemporary literature. This…
For functional data lying on an unknown nonlinear low-dimensional space, we study manifold learning and introduce the notions of manifold mean, manifold modes of functional variation and of functional manifold components. These constitute…
We extend the standard notion of self-concordance to non-convex optimization and develop a family of second-order algorithms with global convergence guarantees. In particular, two function classes -- \textit{weakly self-concordant}…