arXiv subjects

Antonio Falcó

Publications and source records attributed to Antonio Falcó.

11 recordsLinked to original sources

arXiv2026

Resolution-Consistent Greedy Neural Approximation on Infinite-Dimensional Spaces

We develop constructive approximation and learning guarantees for shallow neural models with infinite-dimensional inputs observed through finitely many coordinates. The analysis is based on a parameter-normalized neural dictionary and its associated weighted variation class. Within this class, the approximation error separates into a distribution-dependent coordinate-truncation term and a greedy finite-width term. For empirical regression, a fully-corrective greedy procedure yields population guarantees whose statistical complexity is uniform in the retained input resolution. The same framework extends to Hilbert-valued responses without an explicit dependence on the output dimension. The dimension-free statements are statistical, not computational: selecting a new neuron still requires solving a nonconvex parameter-search problem. The quasi-Polish construction underlying recent infinite-dimensional universal approximation results provides a motivating example, and synthetic experiments illustrate the predicted resolution, width, and sample-size regimes.

cs.LG↗

arXiv2026

Arnold--Nielsen Geometry for Complexity-Deformed Noncommutative Transport

We deform the Carlen--Maas--Wirth framework for noncommutative dynamical optimal transport by an Arnold--Nielsen type complexity operator. A positive state-independent operator $G$ compatible with the Hilbert bimodule structure of a noncommutative differential calculus $\partial\colon\M\to\Hcal$ can be absorbed into the calculus itself, \[ \partial_G:=G^{1/2}\partial. \] The corresponding complexity-weighted transport problem is exactly the unweighted transport problem generated by $\partial_G$, whenever the deformed quadratic form remains Dirichlet. In finite dimensions we prove existence of minimizers for density-dependent Petz-class metrics and for fixed physical complexity weights, the latter without commutation between $G$ and the state-dependent mobility. On unitary orbits we identify the induced distance with a quotient metric coming from a right-invariant complexity geometry. This yields an exact Bell-state preparation result via Clairaut's relation and an exactly computed restricted-path upper bound for GHZ preparation; the Lindblad detailed-balance case is included only as entropy-gradient-flow background.

quant-ph↗

arXiv2026

Universal optimality of the double-centred matrix under unitarily invariant norms for dissimilarity data

Let $D=(D_{ij})_{i,j=1}^{n}$ be a symmetric dissimilarity matrix and let $D^{(2)}=(D_{ij}^{2})_{i,j=1}^{n}$. We study the affine family of real symmetric matrices \[ A(\mathbf{g})=\tfrac{1}{2}\!\left(\mathbf{1}\mathbf{g}^{\top} +\mathbf{g}\mathbf{1}^{\top}-D^{(2)}\right), \qquad \mathbf{g}\in\mathbb{R}^{n}, \] parametrised by a free diagonal vector $\mathbf{g}\in\mathbb{R}^n$, whose off-diagonal entries satisfy \[ D_{ij}^{2}=a_{ii}+a_{jj}-2a_{ij}, \qquad i\neq j. \] We prove that the double-centred matrix \[ A(\mathbf{g}^F)=-\tfrac{1}{2}J D^{(2)} J, \qquad J=I-\tfrac{1}{n}\mathbf{1}\mathbf{1}^{\top}, \] is the unique minimiser of the Frobenius norm over this family, with minimiser $\mathbf{g}^F$ given explicitly by \[ g^{F}_{k} = \frac{1}{n}\sum_{i=1}^{n}D_{ik}^{2} - \frac{1}{2n^{2}}\sum_{i,j=1}^{n}D_{ij}^{2}, \qquad k=1,\dots,n, \] and that this same representative simultaneously minimises every unitarily invariant norm \[ \min_{\mathbf{g}\in\mathbb{R}^n}\left|\!\left|\!\left|A(\mathbf{g})\right|\!\right|\!\right|, \] including the spectral norm, the nuclear norm, and all Schatten $p$-norms and Ky Fan $k$-norms. Thus the double-centred matrix, central to classical multidimensional scaling, admits a purely variational characterisation that does not depend on the choice of norm and requires no Euclidean realisability assumption on $D$.

math.RA↗

arXiv2026

Dictionary-Restricted First-Order Descent Methods: Bounds and Convergence Rates

This paper develops a general theory for first-order descent methods whose search directions are restricted to a prescribed dictionary in a reflexive Banach space. Instead of assuming that the linear span of the dictionary is dense, as in the classical Proper Generalized Decomposition framework of Falcó and Nouy or in the universality approach of Berná and Falcó, we introduce a geometric condition based on norming sets that guarantees density through a duality argument. This makes it possible to treat dictionaries arising from tensor formats, neural network units, and other nonlinear or parameterized approximation families within a unified setting. On the algorithmic side, we analyze a simple greedy update rule in which each iterate is obtained by minimizing the energy functional along one direction from the dictionary. Under mild differentiability, Lipschitz continuity, and ellipticity assumptions on the objective, we derive explicit quantitative descent bounds and sharp convergence rates. These include algebraic rates that improve those of classical steepest-descent schemes in Banach spaces, as well as arbitrarily high polynomial rates and exponential convergence in a critical regime. The results apply broadly to convex variational problems, high-dimensional approximation, and structured optimization methods that rely on restricted or compressed search directions.

math.OC↗

arXiv2026

Density-Matrix Spectral Embeddings for Categorical Data: Operator Structure and Stability

We introduce a supervised dimensionality reduction methodology for categorical (and discretized mixed-type) data based on a density-matrix construction induced by class-conditional frequencies. Given a labeled dataset encoded in a one-hot survey space, we assemble a frequency matrix whose columns aggregate feature occurrences within each class, and define a normalized Gram-type operator that satisfies the axioms of a density matrix. The resulting representation admits an intrinsic rank bound controlled by the number of classes, enabling low-dimensional spectral embeddings via dominant eigenmodes. Classification is performed in the reduced space through class-conditional kernel density estimation and a maximum-likelihood decision rule. We establish structural invariances, provide complexity estimates, and validate the approach on synthetic benchmarks probing high cardinality, sparsity, noise, and class imbalance.

stat.ML↗

arXiv2026

Vistas of Algebraic Probability: Quantum Computation and Information

Kolmogorov's foundation of probability takes measure spaces, $σ$-algebras, and probability measures as basic objects. It is, however, widely recognized that this classical framework is inadequate for random phenomena involving quantum effects, and more generally for \emph{quantum-like} situations. A broader formulation is provided by an algebraic viewpoint: one starts from an algebra of random variables equipped with a distinguished linear functional -- the \emph{state} -- interpreted as expectation. In this sense, the approach can also be viewed as a modern reading of ideas already implicit in early probability (e.g., the Bernoullis), while its contemporary form has been developed and used extensively in quantum physics. The algebraic framework accommodates both classical and quantum-like behaviours, yet it remains underused in classical probability and uncertainty quantification, where it can nevertheless open new perspectives and clarify structural features. Although the language carries a physics flavor, the subject is purely probabilistic. The key distinction between classical and quantum-like behaviour is \emph{commutativity}: its failure produces the characteristic effects of quantum-like situations. The rise of quantum computing is a prominent setting in which such behaviour may become relevant even for practitioners in computational science. Here we focus on the purely algebraic core of the approach. By restricting attention to finite-dimensional algebras, we avoid many analytical subtleties while retaining the main ideas, their classical limit, and their applicability to quantum-like models and quantum computation.

quant-ph↗

arXiv2026

Geometric Complexity of Quantum Channels via Unitary Dilations

Nielsen's geometric approach to quantum circuit complexity provides a Riemannian framework for quantifying the cost of implementing unitary (closed--system) dynamics. For open dynamics, however, the reduced evolution is described by quantum channels and admits many inequivalent Stinespring realizations, so any meaningful complexity notion must specify which microscopic resources are counted as accessible and which transformations are regarded as gauge. We introduce and analyze a geometric complexity functional for families of quantum channels based on unitary dilations. We distinguish an implementation-dependent complexity, defined relative to explicit dilation data, from an intrinsic channel complexity obtained by minimizing over a physically motivated class of admissible dilations (e.g. bounded environment dimension, energy or norm constraints, and penalty structures). The functional has a subtractive form: it compares the geometric cost of the total unitary realization with a canonical surrogate term that removes purely environmental contributions. We justify this subtraction from concise postulates, including closed-system consistency, environment-only neutrality, and invariance under dilation gauge transformations that leave the channel unchanged. This leads to a companion quantity, noise complexity, quantifying the loss of geometric complexity relative to a prescribed ideal closed evolution. We establish a coherence-based lower bound for unitary geometric complexity, derive structural properties such as linear time scaling under time-homogeneous dilations, and obtain dissipator--controlled bounds in the Markovian (GKSL/Lindblad) regime under a standard dilation construction. Finally, we illustrate the framework on canonical benchmark noise models, including dephasing, amplitude damping, and depolarizing (Pauli) channels.

quant-ph↗

arXiv2025

A mathematical model for a universal digital quantum computer with an application to the Grover-Rudolph algorithm

In this work, we develop a novel mathematical framework for universal digital quantum computation using algebraic probability theory. We rigorously define quantum circuits as finite sequences of elementary quantum gates and establish their role in implementing unitary transformations. A key result demonstrates that every unitary matrix in $\mathrm{U}(N)$ can be expressed as a product of elementary quantum gates, leading to the concept of a universal dictionary for quantum computation. We apply this framework to the construction of quantum circuits that encode probability distributions, focusing on the Grover-Rudolph algorithm. By leveraging controlled quantum gates and rotation matrices, we design a quantum circuit that approximates a given probability density function. Numerical simulations, conducted using Qiskit, confirm the theoretical predictions and validate the effectiveness of our approach. These results provide a rigorous foundation for quantum circuit synthesis within an algebraic probability framework and offer new insights into the encoding of probability distributions in quantum algorithms. Potential applications include quantum machine learning, circuit optimization, and experimental implementations on real quantum hardware.

quant-ph↗

arXiv2023

Universality for non-linear convex variational problems

This article introduces an innovative mathematical framework designed to tackle non-linear convex variational problems in reflexive Banach spaces. Our approach employs a versatile technique that can handle a broad range of variational problems, including standard ones. To carry out the process effectively, we utilize specialized sets known as radial dictionaries, where these dictionaries encompass diverse data types, such as tensors in Tucker format with bounded rank and Neural Networks with fixed architecture and bounded parameters. The core of our method lies in employing a greedy algorithm through dictionary optimization defined by a multivalued map. Significantly, our analysis shows that the convergence rate achieved by our approach is comparable to the Method of Steepest Descend implemented in a reflexive Banach space, where the convergence rate follows the order of $O(m^{-1})$.

math.NA↗

arXiv2023

Geometry of tree-based tensor formats in tensor Banach spaces

In the paper `On the Dirac-Frenkel Variational Principle on Tensor Banach Spaces', we provided a geometrical description of manifolds of tensors in Tucker format with fixed multilinear (or Tucker) rank in tensor Banach spaces, that allowed to extend the Dirac-Frenkel variational principle in the framework of topological tensor spaces. The purpose of this note is to extend these results to more general tensor formats. More precisely, we provide a new geometrical description of manifolds of tensors in tree-based (or hierarchical) format, also known as tree tensor networks, which are intersections of manifolds of tensors in Tucker format associated with different partitions of the set of dimensions. The proposed geometrical description of tensors in tree-based format is compatible with the one of manifolds of tensors in Tucker format.

math.NA↗

arXiv2021

A new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of matrix manifolds

In this paper, we propose a new splitting algorithm for dynamical low-rank approximation motivated by the fibre bundle structure of the set of fixed rank matrices. We first introduce a geometric description of the set of fixed rank matrices which relies on a natural parametrization of matrices. More precisely, it is endowed with the structure of analytic principal bundle, with an explicit description of local charts. For matrix differential equations, we introduce a first order numerical integrator working in local coordinates. The resulting algorithm can be interpreted as a particular splitting of the projection operator onto the tangent space of the low-rank matrix manifold. It is proven to be exact in some particular case. Numerical experiments confirm this result and illustrate the robustness of the proposed algorithm.

math.NA↗