Papers by Douglas S. Gonçalves
ArXiv, 2022
Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. ... more Matrix completion aims to recover an unknown low-rank matrix from a small subset of its entries. In many applications, the rank of the unknown target matrix is known in advance. In this paper, we propose a two-phase algorithm that leverages the rank information to compute both a suitable value for the regularization parameter and a warm-start for an accelerated Soft-Impute algorithm. Properties inherited from proximal gradient algorithms are exploited to propose a parameter tuning to accelerate the method and also to establish a convergence analysis. Numerical experiments with both synthetic and real data show that the proposed algorithm can recover low-rank matrices, with high precision, faster than other well-established matrix completion algorithms.
In this study, we review, compare, analyze and propose new variants of first-order methods for th... more In this study, we review, compare, analyze and propose new variants of first-order methods for the convex hull membership problem (CHMP). Though CHMP can be formulated as a linear or a quadratic programming problem which could then be tackled with standard first-orders methods, here we focus on a geometric algorithm, introduced recently in [17], called Triangle Algorithm (TA). We formally establish TA connections with the well-known conditional gradient (FrankWolfe) methods. Despite that TA has its foundation on a theorem of alternative, known as distance duality, we show that its iteration mechanism is quite similar to a Frank-Wolfe iteration. This new point of view allows us to devise variants of the Triangle and Frank-Wolfe algorithms showing promising results in practice. Furthermore, based on the distance duality theorem, we develop appropriate stopping criteria for first-order methods such as conditional gradient and projected gradient applied to the CHMP. Our numerical experi...
A preparacao, manipulacao e caracterizacao de sistemas quânticos sao tarefas essenciais para a co... more A preparacao, manipulacao e caracterizacao de sistemas quânticos sao tarefas essenciais para a computacao quântica. Em Tomografia de Estados Quânticos, o objetivo e encontrar uma estimativa para a matriz de densidade, associada a um ensemble de estados quânticos identicamente preparados, baseando-se no resultado de medicoes. Este e um importante procedimento em computacao e informacao quântica, sendo aplicado, por exemplo, para verificar a fidelidade de um estado preparado ou em tomografia de processos quânticos. Nesta tese, estudamos metodos matematicos aplicados aos problemas que surgem na reconstrucao de estados quânticos. Na estimacao por Maxima Verossimilhanca, apresentamos dois metodos para a resolucao dos problemas de otimizacao dessa abordagem. O primeiro se baseia em uma reparametrizacao da matriz de densidade e, neste caso, provamos a equivalencia das solucoes locais do problema de otimizacao irrestrita associado. No segundo, relacionado a verossimilhanca multinomial, demonstramos a convergencia global do metodo sob hipoteses mais fracas que as da literatura. Apresentamos tambem duas formulacoes para o caso de tomografia com um conjunto incompleto de medidas: Maxima Entropia e Tomografia Quântica Variacional. Propusemos uma nova formulacao para a segunda, de modo a apresentar propriedades mais parecidas as da Maxima Entropia, mantendo a estrutura de um problema de programacao semidefinida linear. Para outros problemas de otimizacao sobre o espaco de matrizes de densidade alem do problema da Tomografia de Estados Quânticos, apresentamos um metodo de Gradiente Projetado que se mostrou efetivo em testes numericos preliminares. Por fim, discutimos sobre a implementacao de inferencia Bayesiana, atraves de metodos Monte Carlo via cadeias de Markov, no problema de estimacao da matriz de densidade. Abstract
Bioinformatics and Biomedical Engineering, 2019
We propose a coarse-grained representation for the solutions of discretizable instances of the Di... more We propose a coarse-grained representation for the solutions of discretizable instances of the Distance Geometry Problem (DGP). In several real-life applications, the distance information is not provided with high precision, but an approximation is rather given. We focus our attention on protein instances where inter-atomic distances can be either obtained from the chemical structure of the molecule (which are exact), or through experiments of Nuclear Magnetic Resonance (which are generally represented by real-valued intervals). The coarse-grained representation allows us to extend a previously proposed algorithm for the Discretizable DGP (DDGP), the branch-and-prune (BP) algorithm. In the standard BP, atomic positions are fixed to unique positions at every node of the search tree: we rather represent atomic positions by a pair consisting of a feasible region, together with a most-likely position for the atom in this region. While the feasible region is a constant during the search, the associated position can be refined by considering the new distance constraints that appear at further layers of the search tree. To perform the refinement task, we integrate the BP algorithm with a spectral projected gradient algorithm. Some preliminary computational experiments on artificially generated instances show that this new approach is quite promising to tackle real-life DGPs.
Proceedings of the 2020 Federated Conference on Computer Science and Information Systems, 2020
With the most recent releases of MD-JEEP, new relevant features have been included to our softwar... more With the most recent releases of MD-JEEP, new relevant features have been included to our software tool. MD-JEEP solves instances of the class of Discretizable Distance Geometry Problems (DDGPs), which ask to find possible realizations, in a Euclidean space, of a simple weighted undirected graph for which distance constraints between vertices are given, and for which a discretization of the search space can be supplied. Since its version 0.3.0, MD-JEEP is able to deal with instances containing interval data. We focus in this short paper on the most recent release MD-JEEP 0.3.2: among the new implemented features, we will focus our attention on three features: (i) an improved procedure for the generation and update of the boxes used in the coarse-grained representation (necessary to deal with instances containing interval data); (ii) a new procedure for the selection of the so-called discretization vertices (necessary to perform the discretization of the search space); (iii) the implementation of a general parser which allows the user to easily load DDGP instances in a given specified format. The source code of MD-JEEP 0.3.2 is available on GitHub, where the reader can find all additional details about the implementation of such new features, as well as verify the effectiveness of such features by comparing MD-JEEP 0.3.2 with its previous releases.
The existence of an embedding in R satisfying a set of exact distances can be verified by the Cay... more The existence of an embedding in R satisfying a set of exact distances can be verified by the Cayley-Menger conditions [2]. In the MDGP, a set of distances is embeddable if and only if all Cayley-Menger determinants of 3 and 4 points have the correct sign(corresponding to the triangular and tetrangular inequalities) and the ones of 5 and 6 points vanish. Another way to verify the embeddability of a set of distances in a generic K-dimensional space is answering whether a partial distance matrix (Dij = d 2 ij), with missing entries, can be completed to a Euclidean distance matrix D. If so, the Cartesian coordinates can be obtained, in polynomial time, by factoring K†(D) = XX, where K†(D) is a linear transformation of D and X is a K ×N matrix whose columns are the coordinates of the N points [3].
Proceedings of the Tenth International Conference on Motion in Games, 2017
Figure 1: (a) Example pose where the actor's hands come to a close distance. The same pose retarg... more Figure 1: (a) Example pose where the actor's hands come to a close distance. The same pose retargeted on a skeleton with longer forearms by (b) simply transferring joint angles or (c) using our normalized Euclidean distance matrix approach.
Lecture Notes in Computer Science, 2017
We introduce the dynamical distance geometry problem (dynDGP), where vertices of a given simple w... more We introduce the dynamical distance geometry problem (dynDGP), where vertices of a given simple weighted undirected graph are to be embedded at different times t. Solutions to the dynDGP can be seen as motions of a given set of objects. In this work, we focus our attention on a class of instances where motion inter-frame distances are not available, and reduce the problem of embedding every motion frame as a static distance geometry problem. Some preliminary computational experiments are presented.
Quantum Inf. Comput., 2012
Maximum likelihood estimation is one of the most used methods in quantum state tomography, where ... more Maximum likelihood estimation is one of the most used methods in quantum state tomography, where the aim is to reconstruct the density matrix of a physical system from measurement results. One strategy to deal with positivity and unit trace constraints is to parameterize the matrix to be reconstructed in order to ensure that it is physical. In this case, the negative log-likelihood function in terms of the parameters, may have several local minima. In various papers in the field, a source of errors in this process has been associated to the possibility that most of these local minima are not global, so that optimization methods could be trapped in the wrong minimum, leading to a wrong density matrix. Here we show that, for convex negative log-likelihood functions, all local minima of the unconstrained parameterized problem are global, thus any minimizer leads to the maximum likelihood estimation for the density matrix. We also discuss some practical sources of errors.
Distance geometry problem belongs to a class of hard problems in classical computation that can b... more Distance geometry problem belongs to a class of hard problems in classical computation that can be understood in terms of a set of inputs processed according to a given transformation, and for which the number of possible outcomes grows exponentially with the number of inputs. It is conjectured that quantum computing schemes can solve problems belonging to this class in a time that grows only at a polynomial rate with the number of inputs. While quantum computers are still being developed, there are some classical optics computation approaches that can perform very well for specific tasks. Here, we present an optical computing approach for the distance geometry problem in one dimension and show that it is very promising in the classical computing regime.
Journal of Computational and Applied Mathematics, 2021
Algorithmica, 2021
The fundamental inverse problem in distance geometry is the one of finding positions from interpo... more The fundamental inverse problem in distance geometry is the one of finding positions from interpoint distances. The Discretizable Molecular Distance Geometry Problem (DMDGP) is a subclass of the Distance Geometry Problem (DGP) whose search space can be discretized and represented by a binary tree, which can be explored by a Branch-and-Prune (BP) algorithm. It turns out that this combinatorial search space possesses many interesting symmetry properties that were studied in the last decade. In this paper, we present a new algorithm for this subclass of the DGP, which exploits DMDGP symmetries more effectively than its predecessors. Computational results show that the speedup, with respect to the classic BP algorithm, is considerable for sparse DMDGP instances related to protein conformation.
Optimization, 2021
This paper is concerned with the inexact variable metric method for solving convex-constrained op... more This paper is concerned with the inexact variable metric method for solving convex-constrained optimization problems. At each iteration of this method, the search direction is obtained by inexactly minimizing a strictly convex quadratic function over the closed convex feasible set. Here, we propose a new inexactness criterion for the search direction subproblems. Under mild assumptions, we prove that any accumulation point of the sequence generated by the new method is a stationary point of the problem under consideration. In order to illustrate the practical advantages of the new approach, we report some numerical experiments. In particular, we present an application where our concept of the inexact solutions is quite appealing.
Numerical Algorithms, 2020
In this work we consider the problem of minimizing a differentiable functional restricted to the ... more In this work we consider the problem of minimizing a differentiable functional restricted to the set of n × p matrices with orthonormal columns. This problem appears in several fields such as statistics, signal processing, global positioning system, machine learning, physics, chemistry and others. We present an algorithm based on a recent non-monotone variation of the inexact restoration method for nonlinear programming along with its implementation details. We give a simple characterization of the set of tangent directions (with respect to the orthogonality constraints) and we use it for dealing with the minimization (tangent) phase. For the restoration phase we employ the well-known Cayley transform for bringing the computed point back to the feasible set (i.e., the restoration phase is exact). Under standard assumptions we prove that any limit point of the sequence generated by the algorithm is a stationary point. A numerical comparison with a well established algorithm is also presented on three different classes of the problem.
Ciência Florestal, 2020
Visando fornecer informações que sirvam de base para estudos de melhoramento genético de Eucalypt... more Visando fornecer informações que sirvam de base para estudos de melhoramento genético de Eucalyptus foi realizada a análise da diversidade genética usando marcadores ISSR. As espécies estudadas foram Eucalyptus urophylla e Eucalyptus microcorys, ambas com potencial econômico florestal. Os indivíduos estudados pertencem a um teste de espécies e procedências instalado no ano de 1974 e permanecem isentos de tratos silviculturais. Para as análises foram utilizados nove primers ISSR universais. A partir dos resultados avaliou-se a existência de variação intra e interespecífica por meio da porcentagem de polimorfismo, conteúdo de informação polimórfica (PIC) e distância Euclidiana entre indivíduos. A fim de analisar a distância Euclidiana entre os indivíduos foram feitas a análise de coordenadas principais (PCoA) e análise permutacional de dispersão multivariada (PermDisp) seguida pelo teste de Tukey. Observouse elevada porcentagem de polimorfismo (57,14% para Eucalyptus microcorys e 80,9...
Optimization Letters, 2017
Discretizable distance geometry problems (DDGPs) constitute a class of graph realization problems... more Discretizable distance geometry problems (DDGPs) constitute a class of graph realization problems where the vertices can be ordered in such a way that the search space of possible positions becomes discrete, usually represented by a binary tree. Finding such vertex order is an essential step to identify and solve DDGPs. Here we look for discretization orders that minimize an indicator of the size of the search tree. This paper sets the ground for exact solution of the discretization order problem by proposing two integer programming formulations and a constraint generation procedure for an extended formulation. We discuss some theoretical aspects of the problem and numerical experiments on protein-like instances of DDGP are also reported.
Journal of Optimization Theory and Applications, 2019
The Levenberg-Marquardt method (LM) is widely used for solving nonlinear systems of equations, as... more The Levenberg-Marquardt method (LM) is widely used for solving nonlinear systems of equations, as well as nonlinear least-squares problems. In this paper, we consider local convergence issues of the LM method when applied to nonzero-residue nonlinear least-squares problems under an error bound condition, which is weaker than requiring full-rank of the Jacobian in a neighborhood of a stationary point. Differently from the zero-residue case, the choice of the LM parameter is shown to be dictated by (i) the behavior of the rank of the Jacobian, and (ii) a combined measure of nonlinearity and residue size in a neighborhood of the set of (possibly non-isolated) stationary points of the sum of squares function.
Computational Optimization and Applications, 2019
This paper deals with a new variant of the Inexact Restoration Method of Fischer and Friedlander ... more This paper deals with a new variant of the Inexact Restoration Method of Fischer and Friedlander (COAP, 46, pp. 333-346, 2010). We propose an algorithm that replaces the monotone line-search performed in the tangent phase (with regard to the penalty function) by a non-monotone one. Convergence to feasible points satisfying the approximate gradient projection (AGP) condition is proved under mild assumptions. Numerical results on representative test problems validate and show that the proposed approach outperforms the monotone version when a suitable non-monotone parameter is chosen.
Journal of Global Optimization, 2018
Distance Geometry (DG) is based on distances rather than points and angles. The fundamental probl... more Distance Geometry (DG) is based on distances rather than points and angles. The fundamental problem of DG is the Distance Geometry Problem (DGP), which is an inverse problem to "given a set of points in R K , compute some of the pairwise distances". More precisely, given an integer K > 0 and a simple undirected graph G = (V , E) with a non-negative weight function d : E → R + defined on the edges, it asks whether there exists a realization x : V → R K such that ∀{i, j} ∈ E x i − x j = d i j. (1) DIMACS generously supported the Distance Geometry Theory and Applications workshop to which this special issue is dedicated. NSF support (through DIMACS) is also gratefully acknowledged.
CERNE, 2018
HIGHLIGHTS Luminosity levels promoted different results in Khaya senegalensis seedlings developme... more HIGHLIGHTS Luminosity levels promoted different results in Khaya senegalensis seedlings development. Plants in full sun accumulated less biomass than in partial luminosity levels. The nitrogen and magnesium contents changed according to the luminosity. Low interactions of foliar nutrient and luminosity levels in plant development were found.
Uploads
Papers by Douglas S. Gonçalves