arXiv subjects

Demetrio Stojanoff

Publications and source records attributed to Demetrio Stojanoff.

At least 19 recordsLinked to original sources

arXiv2024

Block subspace expansions for eigenvalues and eigenvectors approximation

Let $A\in\mathbb C^{n\times n}$ and let $\mathcal X\subset \mathbb C^n$ be an $A$-invariant subspace with $\dim \mathcal X=d\geq 1$, corresponding to exterior eigenvalues of $A$. Given an initial subspace $\mathcal V\subset \mathbb C^n$ with $\dim \mathcal V=r\geq d$, we search for expansions of $\mathcal V$ of the form $\mathcal V+A(\mathcal W_0)$, where $\mathcal W_0\subset \mathcal V$ is such that $\dim \mathcal W_0\leq d$ and such that the expanded subspace is closer to $\mathcal X$ than the initial $\mathcal V$. We show that there exist (theoretical) optimal choices of such $\mathcal W_0$, in the sense that $θ_i(\mathcal X,\mathcal V+A(\mathcal W_0))\leq θ_i(\mathcal V+A(\mathcal W))$ for every $\mathcal W\subset \mathcal V$ with $\dim \mathcal W\leq d$, where $θ_i(\mathcal X,\mathcal T)$ denotes the $i$-th principal angle between $\mathcal X$ and $\mathcal T$, for $1\leq i\leq d\leq \dim \mathcal T$. We relate these optimal expansions to block Krylov subspaces generated by $A$ and $\mathcal V$. We also show that the corresponding iterative sequence of subspaces constructed in this way approximate $\mathcal X$ arbitrarily well, when $A$ is Hermitian and $\mathcal X$ is simple. We further introduce computable versions of this construction and compute several numerical examples that show the performance of the computable algorithms and test our convergence analysis.

math.NA↗

arXiv2022

Norm inequalities for the spectral spread of Hermitian operators

In this work we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite dimensional Hilbert space that we call spectral spread. Then, we obtain some submajorization inequalities involving the spectral spread of self-adjoint operators, that are related to Tao's inequalities for anti-diagonal blocks of positive operators, Kittaneh's commutator inequalities for positive operators and also related to the Arithmetic-Geometric mean inequality. In turn, these submajorization relations imply inequalities for unitarily invariant norms (in the compact case).

math.FA↗

arXiv2021

The spectral spread of Hermitian matrices

Let $A$ be a $n\times n$ complex Hermitian matrix and let $λ(A)=(λ_1,\ldots,λ_n)\in \mathbb{R}^n$ denote the eigenvalues of $A$, counting multiplicities and arranged in non-increasing order. Motivated by problems arising in the theory of low rank matrix approximation, we study the spectral spread of $A$, denoted $\text{Spr}^+(A)$, given by $\text{Spr}^+(A) =(λ_1-λ_{n}\, , \, λ_2-λ_{n-1},\ldots, λ_{k}-λ_{n-k+1})\in \mathbb{R}^k$, where $k=[n/2]$ (integer part). The spectral spread is a vector-valued measure of dispersion of the spectrum of $A$, that allows one to obtain several submajorization inequalities. In the present work we obtain inequalities that are related to Tao's inequality for anti-diagonal blocks of positive semidefinite matrices, Zhan's inequalities for the singular values of differences of positive semidefinite matrices, extremal properties of direct rotations between subspaces, generalized commutators and distances between matrices in the unitary orbit of a Hermitian matrix.

math.FA↗

arXiv2021

Absolute variation of Ritz values, principal angles and spectral spread

Let $A$ be a $d\times d$ complex self-adjoint matrix, $\mathcal{X},\mathcal{Y}\subset \mathbb{C}^d$ be $k$-dimensional subspaces and let $X$ be a $d\times k$ complex matrix whose columns form an orthonormal basis of $\mathcal{X}$. We construct a $d\times k$ complex matrix $Y_r$ whose columns form an orthonormal basis of $\mathcal{Y}$ and obtain sharp upper bounds for the singular values $s(X^*AX-Y_r^*\,A\,Y_r)$ in terms of submajorization relations involving the principal angles between $\mathcal{X}$ and $\mathcal{Y}$ and the spectral spread of $A$. We apply these results to obtain sharp upper bounds for the absolute variation of the Ritz values of $A$ associated with the subspaces $\mathcal{X}$ and $\mathcal{Y}$, that partially confirm conjectures by Knyazev and Argentati.

math.FA↗

arXiv2020

Optimal frame designs for multitasking devices with weight restrictions

Let $\mathbf d=(d_j)_{j\in\mathbb I_m}\in\mathbb N^m$ be a finite sequence (of dimensions) and $α=(α_i)_{i\in\mathbb I_n}$ be a sequence of positive numbers (of weights), where $\mathbb I_k=\{1,\ldots,k\}$ for $k\in\mathbb N$. We introduce the $(α\, , \,\mathbf d)$-designs i.e., $m$-tuples $Φ=(\mathcal F_j)_{j\in\mathbb I_m}$ such that $\mathcal F_j=\{f_{ij}\}_{i\in\mathbb I_n}$ is a finite sequence in $\mathbb C^{d_j}$, $j\in\mathbb I_m$, and such that the sequence of non-negative numbers $(\|f_{ij}\|^2)_{j\in\mathbb I_m}$ forms a partition of $α_i$, $i\in\mathbb I_n$. We characterize the existence of $(α\, , \, \mathbf d)$-designs with prescribed properties in terms of majorization relations. We show, by means of a finite-step algorithm, that there exist $(α\, , \, \mathbf d)$-designs $Φ^{\rm op}=(\mathcal F_j^{\rm op})_{j\in\mathbb I_m}$ that are universally optimal; that is, for every convex function $φ:[0,\infty)\rightarrow [0,\infty)$ then $Φ^{\rm op}$ minimizes the joint convex potential induced by $φ$ among $(α\, , \, \mathbf d)$-designs, namely $$ \sum_{j\in\mathbb I_m}\text{P}_φ(\mathcal F_j^{\rm op})\leq \sum_{j\in \mathbb I_m}\text{P}_φ(\mathcal F_j) $$ for every $(α\, , \, \mathbf d)$-design $Φ=(\mathcal F_j)_{j\in\mathbb I_m}$, where $\text{P}_φ(\mathcal F)=tr(φ(S_{\mathcal F}))$; in particular, $Φ^{\rm op}$ minimizes both the joint frame potential and the joint mean square error among $(α\, , \, \mathbf d)$-designs. We show that in this case $\mathcal F_j^{\rm op}$ is a frame for $\mathbb C^{d_j}$, for $j\in\mathbb I_m$. This corresponds to the existence of optimal encoding-decoding schemes for multitasking devices with energy restrictions.

math.FA↗

arXiv2020

Majorization bounds for Ritz values of self-adjoint matrices

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the $\tan Θ$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values.

math.FA↗

arXiv2019

Minimal curves in U(n) and Gl(n)+ with respect to the spectral and the trace norms

Consider the Lie group of n x n complex unitary matrices U(n) endowed with the bi-invariant Finsler metric given by the spectral norm, ||X||_U = ||U*X||_{sp} = ||X||_{sp} for any X tangent to a unitary operator U. Given two points in U(n), in general there exists infinitely many curves of minimal length. The aim of this paper is to provide a complete description of such curves. As a consequence of this description, we conclude that there is a unique curve of minimal length between U and V if and only if the spectrum of U*V is contained in a set of the form \{e^{i θ}, e^{-i θ}\} for some θ\in [0, \infty). Similar studies are done for the Grassmann manifolds. Now consider the cone of n x n positive invertible matrices Gl(n)+ endowed with the bi-invariant Finsler metric given by the trace norm, ||X||_{1, A} = ||A^{-1/2}XA^{-1/2}||_1 for any X tangent to A \in Gl(n)+. In this context, given two points A,B \in Gl(n)+ there exists infinitely many curves of minimal length. In order to provide a complete description of such curves, we provide a characterization of the minimal curves joining two Hermitian matrices X, Y \in H(n). As a consequence of the last description, we provide a way to construct minimal paths in the group of unitary matrices U(n) endowed with the bi-invariant Finsler metric ||X||_{1, U} = ||U*X||_{1} = ||X||_{1} for any X tangent to U \in U(n). We also study the set of intermediate points in all the previous contexts. Between two given unitary matrices U and V we prove that this set is geodesically convex provided ||U - V||_{sp} < 1. In Gl(n)+ this set is geodesically convex for every unitarily invariant norm.

math.FA↗

arXiv2018

Generalized frame operator distance problems

Let $S\in\mathcal{M}_d(\mathbb{C})^+$ be a positive semidefinite $d\times d$ complex matrix and let $\mathbf a=(a_i)_{i\in\mathbb{I}_k}\in \mathbb{R}_{>0}^k$, indexed by $\mathbb{I}_k=\{1,\ldots,k\}$, be a $k$-tuple of positive numbers. Let $\mathbb T_{d}(\mathbf a )$ denote the set of families $\mathcal G=\{g_i\}_{i\in\mathbb{I}_k}\in (\mathbb{C}^d)^k$ such that $\|g_i\|^2=a_i$, for $i\in\mathbb{I}_k$; thus, $\mathbb T_{d}(\mathbf a )$ is the product of spheres in $\mathbb{C}^d$ endowed with the product metric. For a strictly convex unitarily invariant norm $N$ in $\mathcal{M}_d(\mathbb{C})$, we consider the generalized frame operator distance function $Θ_{( N \, , \, S\, , \, \mathbf a)}$ defined on $\mathbb T_{d}(\mathbf a )$, given by $$ Θ_{( N \, , \, S\, , \, \mathbf a)}(\mathcal G) =N(S-S_{\mathcal G }) \quad \text{where} \quad S_{\mathcal G}=\sum_{i\in\mathbb{I}_k} g_i\,g_i^*\in\mathcal{M}_d(\mathbb{C})^+\,. $$ In this paper we determine the geometrical and spectral structure of local minimizers $\mathcal G_0\in\mathbb T_{d}(\mathbf a )$ of $Θ_{( N \, , \, S\, , \, \mathbf a)}$. In particular, we show that local minimizers are global minimizers, and that these families do not depend on the particular choice of $N$.

math.FA↗

arXiv2018

Ergodic theorem in Hadamard spaces in terms of inductive means

Let (G,+) be a compact, abelian, and metrizable topological group. In this group we take $g\in G$ such that the corresponding automorphism t_g is ergodic. The main result of this paper is a new ergodic theorem for functions in L^1(G,M), where M is a Hadamard space. The novelty of our result is that we use inductive means to average the elements of the orbit {t_g^n(h)}_{n\in N}. The advantage of inductive means is that they can be explicitly computed in many important examples. The proof of the ergodic theorem is done firstly for continuous functions, and then it is extended to L^1 functions. The extension is based in a new construction of mollifiers in Hadamard spaces. This construction has the advantage that it only uses the metric structure and the existence of barycenters, and do not require the existence of an underlying vector space. For this reason, it can be used in any Hadamard space, in contrast with those results that need to use the tangent space or some chart to define the mollifier.

math.MG↗

arXiv2018

Local Lidskii's theorems for unitarily invariant norms

Lidskii's additive inequalities (both for eigenvalues and singular values) can be interpreted as an explicit description of global minimizers of functions that are built on unitarily invariant norms, with domains consisting of certain orbits of matrices (under the action of the unitary group). In this paper, we show that Lidskii's inequalities actually describe all global minimizers of such functions and that local minimizers are also global minimizers. We use these results to obtain partial results related to local minimizers of generalized frame operator distances in the context of finite frame theory.

math.FA↗

arXiv2016

Frame completions with prescribed norms: local minimizers and applications

Let $\mathcal F_0=\{f_i\}_{i\in\mathbb{I}_{n_0}}$ be a finite sequence of vectors in $\mathbb C^d$ and let $\mathbf{a}=(a_i)_{i\in\mathbb{I}_k}$ be a finite sequence of positive numbers. We consider the completions of $\cal F_0$ of the form $\cal F=(\cal F_0,\cal G)$ obtained by appending a sequence $\cal G=\{g_i\}_{i\in\mathbb{I}_k}$ of vectors in $\mathbb C^d$ such that $\|g_i\|^2=a_i$ for $i\in\mathbb{I}_k$, and endow the set of completions with the metric $d(\cal F,\tilde {\mathcal F}) =\max\{ \,\|g_i-\tilde g_i\|: \ i\in\mathbb{I}_k\}$ where $\tilde {\cal F}=(\cal F_0,\,\tilde {\cal G})$. In this context we show that local minimizers on the set of completions of a convex potential $\text{P}_φ$, induced by a strictly convex function $φ$, are also global minimizers. In case that $φ(x)=x^2$ then $\text{P}_φ$ is the so-called frame potential introduced by Benedetto and Fickus, and our work generalizes several well known results for this potential. We show that there is an intimate connection between frame completion problems with prescribed norms and frame operator distance (FOD) problems. We use this connection and our results to settle in the affirmative a generalized version of Strawn's conjecture on the FOD.

math.FA↗

arXiv2016

Convex potentials and optimal shift generated oblique duals in shift invariant spaces

We introduce an extension of the convex potentials for finite frames (e.g. the frame potential defined by Benedetto and Fickus) in the framework of Bessel sequences of integer translates of finite sequences in $L^2(\R^k)$. We show that under a natural normalization hypothesis, these convex potentials detect tight frames as their minimizers. We obtain a detailed spectral analysis of the frame operators of shift generated oblique duals of a fixed frame of translates. We use this result to obtain the spectral and geometrical structure of optimal shift generated oblique duals with norm restrictions, that simultaneously minimize every convex potential; we approach this problem by showing that the water-filling construction in probability spaces is optimal with respect to submajorization (within an appropriate set of functions) and by considering a non-commutative version of this construction for measurable fields of positive operators.

math.FA↗

arXiv2016

Frames of translates with prescribed fine structure in shift invariant spaces

For a given finitely generated shift invariant (FSI) subspace $\cW\subset L^2(\R^k)$ we obtain a simple criterion for the existence of shift generated (SG) Bessel sequences $E(\cF)$ induced by finite sequences of vectors $\cF\in \cW^n$ that have a prescribed fine structure i.e., such that the norms of the vectors in $\cF$ and the spectra of $S_{E(\cF)}$ is prescribed in each fiber of $\text{Spec}(\cW)\subset \T^k$. We complement this result by developing an analogue of the so-called sequences of eigensteps from finite frame theory in the context of SG Bessel sequences, that allows for a detailed description of all sequences with prescribed fine structure. Then, given $0<α_1\leq \ldots\leq α_n$ we characterize the finite sequences $\cF\in\cW^n$ such that $\|f_i\|^2=α_i$, for $1\leq i\leq n$, and such that the fine spectral structure of the shift generated Bessel sequences $E(\cF)$ have minimal spread (i.e. we show the existence of optimal SG Bessel sequences with prescribed norms); in this context the spread of the spectra is measured in terms of the convex potential $P^\cW_φ$ induced by $\cW$ and an arbitrary convex function $φ:\R_+\rightarrow \R_+$.

math.FA↗

arXiv2014

Aliasing and oblique dual pair designs for consistent sampling

In this paper we study some aspects of oblique duality between finite sequences of vectors $\cF$ and $\cG$ lying in finite dimensional subspaces $\cW$ and $\cV$, respectively. We compute the possible eigenvalue lists of the frame operators of oblique duals to $\cF$ lying in $\cV$; we then compute the spectral and geometrical structure of minimizers of convex potentials among oblique duals for $\cF$ under some restrictions. We obtain a complete quantitative analysis of the impact that the relative geometry between the subspaces $\cV$ and $\cW$ has in oblique duality. We apply this analysis to compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual of $U\cdot \cF$ minimize every convex potential; we also introduce a notion of aliasing for oblique dual pairs and compute those rigid rotations $U$ for $\cW$ such that the canonical oblique dual pair associated to $U\cdot \cF$ minimize the aliasing. We point out that these two last problems are intrinsic to the theory of oblique duality.

math.FA↗

arXiv2014

Multiplicative Lidskii's inequalities and optimal perturbations of frames

In this paper we study two design problems in frame theory: on the one hand, given a fixed finite frame $\cF$ for $\hil\cong\C^d$ we compute those dual frames $\cG$ of $\cF$ that are optimal perturbations of the canonical dual frame for $\cF$ under certain restrictions on the norms of the elements of $\cG$. On the other hand, for a fixed finite frame $\cF=\{f_j\}_{j\in\In}$ for $\hil$ we compute those invertible operators $V$ such that $V^*V$ is a perturbation of the identity and such that the frame $V\cdot \cF=\{V\,f_j\}_{j\in\In}$ - which is equivalent to $\cF$ - is optimal among such perturbations of $\cF$. In both cases, optimality is measured with respect to submajorization of the eigenvalues of the frame operators. Hence, our optimal designs are minimizers of a family of convex potentials that include the frame potential and the mean squared error. The key tool for these results is a multiplicative analogue of Lidskii's inequality in terms of log-majorization and a characterization of the case of equality.

math.FA↗

arXiv2013

Optimal frame completions with prescribed norms for majorization

Given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we characterize in a complete and explicit way the optimal completions of $\mathcal F_0$ obtained by adding a finite sequence of vectors with prescribed norms, where optimality is measured with respect to majorization (of the eigenvalues of the frame operators of the completed sequence). Indeed, we construct (in terms of a fast algorithm) a vector - that depends on the eigenvalues of the frame operator of the initial sequence $\cF_0$ and the sequence of prescribed norms - that is a minimum for majorization among all eigenvalues of frame operators of completions with prescribed norms. Then, using the eigenspaces of the frame operator of the initial sequence $\cF_0$ we describe the frame operators of all optimal completions for majorization. Hence, the concrete optimal completions with prescribed norms can be obtained using recent algorithmic constructions related with the Schur-Horn theorem. The well known relation between majorization and tracial inequalities with respect to convex functions allow to describe our results in the following equivalent way: given a finite sequence of vectors $\mathcal F_0$ in $\C^d$ we show that the completions with prescribed norms that minimize the convex potential induced by a strictly convex function are structural minimizers, in the sense that they do not depend on the particular choice of the convex potential.

math.FA↗

arXiv2012

Optimal dual frames and frame completions for majorization

In this paper we consider two problems in frame theory. On the one hand, given a set of vectors $\mathcal F$ we describe the spectral and geometrical structure of optimal completions of $\mathcal F$ by a finite family of vectors with prescribed norms, where optimality is measured with respect to majorization. In particular, these optimal completions are the minimizers of a family of convex functionals that include the mean square error and the Bendetto-Fickus' frame potential. On the other hand, given a fixed frame $\mathcal F$ we describe explicitly the spectral and geometrical structure of optimal frames $\mathcal G$ that are in duality with $\mathcal F$ and such that the Frobenius norms of their analysis operators is bounded from below by a fixed constant. In this case, optimality is measured with respect to submajorization of the frames operators. Our approach relies on the description of the spectral and geometrical structure of matrices that minimize submajorization on sets that are naturally associated with the problems above.

math.FA↗

arXiv2011

Robust dual reconstruction systems and fusion frames

We study the duality of reconstruction systems, which are $g$-frames in a finite dimensional setting. These systems allow redundant linear encoding-decoding schemes implemented by the so-called dual reconstruction systems. We are particularly interested in the projective reconstruction systems that are the analogue of fusion frames in this context. Thus, we focus on dual systems of a fixed projective system that are optimal with respect to erasures of the reconstruction system coefficients involved in the decoding process. We consider two different measures of the reconstruction error in a blind reconstruction algorithm. We also study the projective reconstruction system that best approximate an arbitrary reconstruction system, based on some well known results in matrix theory. Finally, we present a family of examples in which the problem of existence of a dual projective system of a reconstruction system of this type is considered.

math.FA↗