Special Issue "Algorithms for Applied Mathematics "

Guest Editor
Prof. Dr. Doron Levy
Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA
Website: http://www.math.umd.edu/~dlevy
E-Mail:

Submission

All papers should be submitted to algorithms@mdpi.org. To be published continuously until the deadline and papers will be listed together at the special issue website.

Submitted papers should not have been published nor be under consideration for publication elsewhere. All papers are refereed through a peer-review process. A guide for authors is available on the Instructions for Authors page. Algorithms is an international peer-reviewed quarterly journal published by Molecular Diversity Preservation International.

Article Processing Charges (APC) will be waived for well prepared manuscripts of invited papers. For the first three volumes of this new journal the APC are of 300 CHF (or 550 CHF per paper for those papers that require extensive additional formatting and/or English corrections) for papers submitted before 31 December 2010.

• discrete models
• stochastic equations
• ordinary differential equations
  
• time-delayed equations
• partial differential equations
• finite-volume, finite difference, finite elements methods
• particle methods
• spectral methods
• linear and nonlinear optimization
• linear algebra

Type of Paper: Review
Title: Imputing Missing Entries of a Data Matrix: Review and Assessment of Methods
Author: Achiya Dax
Affiliation: Hydrological Service, P.O.B. 36118, Jerusalem 91360, Israel; E-Mail: dax20@water.gov.il
Abstract: The problem of imputing missing entries of a data matrix is easy to state: Some entries of the matrix are unknown and it is needed to assign "appropriate values" to these entries. The need for solving this problem arises in several applications, ranging from traditional fields, such as Statistical analysis of incomplete survey data, Business Reports, Meteorology and Hydrology, to modern fields, such as Machine Learning, Data Mining, Computer Vision, Clustering analysis of DNA microarrays data, Recommender Systems and Collaborative Filtering. The problem is highly interesting and challenging. Many ingenious algorithms have been proposed and there is vast literature on imputing techniques. Yet, most of the papers consider the imputing problem within the context of a specific application. One aim of this paper is to provide a broader view of the problem. Our review considers the problem from a Numerical Linear Algebra and Optimization point of view, concentrating on the main solution methods and their properties. In particular, it is illustrated how different types of data lead to different type of methods.
The first part of our review introduces the problem and provides an overview of existing methods. Starting from simple averaging methods, we outline some basic imputing techniques, such as Iterative Column Regression (ICR), k Nearest Neighbors (KNN) imputing, Restricted SVD imputing, and iterative SVD imputing. Then we move to consider highly sophisticated algorithms, such as minimizing the singular values "tail" (FRAA), Rank minimization, and Nuclear Norm minimization.
The last five methods generate a rank-k matrix that approximates the original data matrix. The second part of the talk presents a direct approach for building a low-rank approximation of an incomplete data matrix. It is based on two effective methods for solving the basic minimum norm problem. Successive rank-one modifications (SROM) and alternating least squares (ALS). Starting from a rank-one approximation the SROM method enables a gradual building of the approximating matrix. This helps to determine the "optimal" rank of the approximating matrix. Numerical experiments illustrate the viability of the proposed approach.

Title: Some structural and algorithmic aspects of bijective components issued from lattice theory
Authors: K. Bertet and Ch. Demko
Affiliation: L3I, Université de La Rochelle, av Michel Crépeau, 17042 La Rochelle, France; E-Mails: karell.bertet@univ-lr.fr; christophe.demko@univ-lr.fr
Abstract: Since the years 2000, the increasing interest carried in Formal Concept Analysis (AFC) in various domains of computer science, such as data-mining and knowledge representation, as well as the fields of ontology or databases, has brought to the surface the structure of concept lattice. A concept lattice is a graph with the lattice property defined from data described by a binary table object x attribute. The nodes of the graph are concepts - a concept is a maximal subset of objects possessing attributes in common. This lattice composed of concepts connected by inclusion, supplies a very
intuitive representation of the data.
In data-mining, new methods of classification select certain concepts of lattice according to a criterion of relevance while other methods propose a use of the structure of the lattice for a classification by navigation similar to that of the decision tree. The structure of the lattice is also used as an ontology scheme built automatically from the data. Not to forget the use of association rules extracted from the concepts which, in data-mining, express correlations between the attributes, while they represent the functional dependencies in databases.
The first part of our review provides an overview of existing
definitions stemming from Formal Concept Analysis, but also from the lattice theory, some structural results and the fundamental bijections which result from it. A direct consequence of these bijections is the existence of a bijective link between a lattice, its binary reduced table or its canonical direct basis of rules which is also denoted as a dependance graph. In a second part, we shall present the main algorithms allowing to generate and to manipulate these objects as well as our lattice library which implements these algorithms for an effective manipulation of these objects in an experimental application context.
Keywords: lattice; concept lattice; Galois lattice; implicational system; closure operator; closure system; canonical
direct basis; dependance graph