Antonio Falco

Antonio Falco, Wolfgang Hackbusch, Anthony Nouy

The main goal of this paper is to study the topological properties of tensors in tree-based Tucker format. These formats include the Tucker format and the Hierarchical Tucker format. A property of the so-called minimal subspaces is used for obtaining a representation of tensors with either bounded or fixed tree-based rank in the underlying algebraic tensor space. We provide a new characterisation of minimal subspaces which extends the existing characterisations. We also introduce a definition of topological tensor spaces in tree-based format, with the introduction of a norm at each vertex of the tree, and prove the existence of best approximations from sets of tensors with bounded tree-based rank, under some assumptions on the norms weaker than in the existing results.

Antonio Falco, Anthony Nouy

Tensor-based methods are receiving a growing interest in scientific computing for the numerical solution of problems defined in high dimensional tensor product spaces. A family of methods called Proper Generalized Decompositions methods have been recently introduced for the a priori construction of tensor approximations of the solution of such problems. In this paper, we give a mathematical analysis of a family of progressive and updated Proper Generalized Decompositions for a particular class of problems associated with the minimization of a convex functional over a reflexive tensor Banach space.

Antonio Falco, Daniela Falco-Pomares, Hermann G. Matthies

We give a rigorous and self-contained analysis of the Grover--Rudolph quantum state-preparation algorithm, which encodes a probability distribution $\{p_k\}$ as an $n$-qubit amplitude state $\sum_k\sqrt{p_k}\ket{k}$ via a hierarchy of controlled $\RY$ rotations determined by a dyadic refinement of the target. We formalize the dyadic probability tree, derive the trigonometric factorization of conditional masses, and prove by induction that the circuit prepares exactly the desired measurement law. We further prove that perturbing each rotation angle by at most $η$ changes the output distribution by at most $\min(1,nη)$ in total variation, and combine this with a Hoeffding concentration bound to obtain an explicit design rule: $b\ge\log_2(2nπ/\varepsilon)$ bits and $S\ge 2^{n+1}\log(2/δ)/\varepsilon^2$ shots suffice to achieve accuracy $\varepsilon$ with confidence $1-δ$. As a circuit-theoretic complement, we provide an ancilla-free transpilation of each stage into $\{\RY(\cdot),X,\CNOT\}$ via Gray-code ladders and a Walsh--Hadamard angle transform.

Marie Billaud-Friess, Antonio Falco, Anthony Nouy

In this paper, we introduce a new geometric description of the manifolds of matrices of fixed rank. The starting point is a geometric description of the Grassmann manifold $\mathbb{G}_r(\mathbb{R}^k)$ of linear subspaces of dimension $r<k$ in $\mathbb{R}^k$ which avoids the use of equivalence classes. The set $\mathbb{G}_r(\mathbb{R}^k)$ is equipped with an atlas which provides it with the structure of an analytic manifold modelled on $\mathbb{R}^{(k-r)\times r}$. Then we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{k \times r})$ of full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^k)$ and typical fibre $\mathrm{GL}_r$, the general linear group of invertible matrices in $\mathbb{R}^{k\times k}$. Finally, we define an atlas for the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ of non-full rank matrices and prove that the resulting manifold is an analytic principal bundle with base $\mathbb{G}_r(\mathbb{R}^n) \times \mathbb{G}_r(\mathbb{R}^m)$ and typical fibre $\mathrm{GL}_r$. The atlas of $\mathcal{M}_r(\mathbb{R}^{n \times m})$ is indexed on the manifold itself, which allows a natural definition of a neighbourhood for a given matrix, this neighbourhood being proved to possess the structure of a Lie group. Moreover, the set $\mathcal{M}_r(\mathbb{R}^{n \times m})$ equipped with the topology induced by the atlas is proven to be an embedded submanifold of the matrix space $\mathbb{R}^{n \times m}$ equipped with the subspace topology. The proposed geometric description then results in a description of the matrix space $\mathbb{R}^{n \times m}$, seen as the union of manifolds $\mathcal{M}_r(\mathbb{R}^{n \times m})$, as an analytic manifold equipped with a topology for which the matrix rank is a continuous map.

Francisco Chinesta, Antonio Falco, Daniela Falco-Pomares

We study numerical integration on $[0,1]$ by quantum amplitude estimation (QAE), focusing on the cost of constructing the amplitude oracle. Although QAE improves the statistical component of the integration error, this advantage is relevant only when the integrand has low encoding complexity. We introduce a hierarchy of grid function classes $\mathcal{G}_n^{(d)}$, defined by requiring the angle map $Θ_g:\{0,1\}^n\to[0,π]$ to be multilinear of degree at most $d$. Membership is classically checkable in $O(n2^n)$ time by the Walsh--Hadamard transform. For $g\in\mathcal{G}_n^{(d)}$, the encoding operator factorises into $\sum_{k=0}^d\binom{n}{k}$ multi-controlled $R_Y$ gates, interpolating between an affine $O(n)$ regime and the generic exponential regime. Combining this structure with classical discretisation estimates for $g\in C^α[0,1]$, we obtain a depth-versus-accuracy trade-off: gate count $O((\log(1/\varepsilon))^d\varepsilon^{-1})$ suffices to achieve $\varepsilon$-accuracy with constant probability. For $d=1$ this becomes $O(\varepsilon^{-1}\log(1/\varepsilon))$, improving over classical Monte Carlo for every $α\ge1$. We also prove an unconditional separation: $\mathcal{G}_n^{(1)}$ contains functions of Sobolev regularity $s<1/2$ for which the quantum oracle cost is $O(1/\varepsilon)$, whereas classical deterministic or randomised quadrature requires $Ω(\varepsilon^{-1/s})$ evaluations. These results identify explicit integrand classes for which the full cost of QAE-based integration, including state preparation, is asymptotically better than classical methods. Experiments on SpinQ Triangulum and IBM Kingston illustrate the hierarchy at $n=2$: circuits inside $\mathcal{G}_n^{(d)}$ run successfully, while those exceeding the Triangulum coherence budget fail as predicted.

Tarek Frahi, Abel Sancarlos, Matthieu Galle et al.

The present paper aims at analyzing the topological content of the complex trajectories that weeder-autonomous robots follow in operation. We will prove that the topological descriptors of these trajectories are affected by the robot environment as well as by the robot state, with respect to maintenance operations. Topological Data Analysis will be used for extracting the trajectory descriptors, based on homology persistence. Then, appropriate metrics will be applied in order to compare that topological representation of the trajectories, for classifying them or for making efficient pattern recognition.

Antonio Falco, Wolfgang Hackbusch, Anthony Nouy

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the dimensions of the associated minimal subspaces, we introduce, in the underlying algebraic tensor space, the set of tensors in a tree-based format with either bounded or fixed tree-based rank. This class contains the Tucker format and the Hierarchical Tucker format (including the Tensor Train format). In particular, we show that the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank of an algebraic tensor product of normed vector spaces is an analytic Banach manifold. Indeed, the manifold geometry for the set of tensors with fixed tree-based rank is induced by a fibre bundle structure and the manifold geometry for the set of tensors with bounded tree-based rank is given by a finite union of connected components. In order to describe the relationship between these manifolds and the natural ambient space, we introduce the definition of topological tensor spaces in the tree-based format. We prove under natural conditions that any tensor of the topological tensor space under consideration admits best approximations in the manifold of tensors in the tree-based format with bounded tree-based rank. In this framework, we also show that the tangent (Banach) space at a given tensor is a complemented subspace in the natural ambient tensor Banach space and hence the set of tensors in the tree-based format with bounded (respectively, fixed) tree-based rank is an immersed submanifold. This fact allows us to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces.