We study the problem of designing models for machine learning tasks defined on \emph{sets}. In contrast to traditional approach of operating on fixed dimensional vectors, we consider objective functions defined on sets that are invariant to permutations. Such problems are widespread, ranging from estimation of population statistics \cite{poczos13aistats}, to anomaly detection in piezometer data of embankment dams \cite{Jung15Exploration}, to cosmology \cite{Ntampaka16Dynamical,Ravanbakhsh16ICML1}. Our main theorem characterizes the permutation invariant functions and provides a family of functions to which any permutation invariant objective function must belong. This family of functions has a special structure which enables us to design a deep network architecture that can operate on sets and which can be deployed on a variety of scenarios including both unsupervised and supervised learning tasks. We also derive the necessary and sufficient conditions for permutation equivariance in deep models. We demonstrate the applicability of our method on population statistic estimation, point cloud classification, set expansion, and outlier detection.

Deep Sets

https://hal.archives-ouvertes.fr/hal-01517950/document

Isogeometric analysis using T-splines

https://works.bepress.com/ming-chen_hsu/16/download/

Isogeometric shell analysis: The Reissner-Mindlin shell

We develop finite element data structures for T-splines based on Bezier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bezier elements are defined in terms of a fixed set of polynomial basis functions, the so-called Bernstein basis. The Bezier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T-junctions, referred to as ‘hanging nodes’ in finite element analysis and a fundamental feature of T-splines. A detailed example is presented to illustrate the ideas. Copyright © 2011 John Wiley & Sons, Ltd.

Isogeometric finite element data structures based on Bézier extraction of T-splines

The work is giving estimations of the discrepancy between solutions of the initial and the homogenized problems for a one{dimensional second order elliptic operators with random coeecients satisfying strong or uniform mixing conditions. We obtain several sharp estimates in terms of the corresponding mixing coeecient. Abstract. In the theory of homogenisation it is of particular interest to determine the classes of problems which are stable on taking the homogenisation limits. A notable situation where the limit enlarges the class of original problems is known as memory (nonlocal) eeects. A number of results in that direction has been obtained for linear problems. Tartar (1990) innitiated the study of the eeective equation corresponding to nonlinear equation: @ t u n + a n u 2 n = f: Signiicant progress has been hampered by the complexity of required computations needed in order to obtain the terms in power{series expansion. We propose a method which overcomes that diiculty by introducing graphs representing the domain of integration of the integrals in each term. The graphs are relatively simple, it is easy to calculate with them and they give us a clear image of the form of each term. The method allows us to discuss the form of the eeective equation and the convergence of power{series expansions. The feasibility of our method for other types of nonlinearities will be discussed as well.

Applied Mathematics and Computation

Summary form only given. This paper presents a novel hole-filling algorithm that can fill arbitrary holes in triangular mesh models. First, the advancing front mesh technique is used to cover the hole with newly created triangles. Next, the desirable normals of the new triangles are approximated using our desirable normal computing schemes. Finally, the three coordinates of every new vertex are re-positioned by solving the Poisson equation based on the desirable normals and the boundary vertices of the hole. Many experimental results and error evaluations are given to show the robustness and efficiency of the algorithm.

A Robust Hole-Filling Algorithm for Triangular Mesh

This paper addresses the long-standing problem of the unavoidable gaps that arise when expressing the intersection of two NURBS surfaces using conventional trimmed-NURBS representation The solution converts each trimmed NURBS into an untrimmed T-Spline, and then merges the untrimmed T-Splines into a single, watertight model The solution enables watertight fillets of NURBS models, as well as arbitrary feature curves that do not have to follow iso-parameter curves The resulting T-Spline representation can be exported without error as a collection of NURBS surfaces

/pdf/watertight-trimmed-nurbs-15y8vauib7.pdf

Watertight trimmed NURBS

The progressive and iterative approximation (PIA) method is an efficient and intuitive method for data fitting. However, in the classical PIA method, the number of the control points is equal to that of the data points. It is not feasible when the number of data points is very large. In this paper, we develop a new progressive and iterative approximation for least square fitting (LSPIA). LSPIA constructs a series of fitting curves (surfaces) by adjusting the control points iteratively, and the limit curve (surface) is the least square fitting result to the given data points. In each iteration, the difference vector for each control point is a weighted sum of some difference vectors between the data points and their corresponding points on the fitting curve (surface). Moreover, we present a simple method to compute the practical weight whose corresponding convergence rate is comparable to that of the theoretical best weight. The advantages of LSPIA are two-fold. First, with LSPIA, a very large data set can be fitted efficiently and robustly. Second, in the incremental data fitting procedure with LSPIA, a new round of iterations can be started from the fitting result of the last round of iterations, thus saving great amount of computation. Lots of empirical examples illustrated in this paper show the efficiency and effectiveness of LSPIA.

Progressive and iterative approximation for least squares B-spline curve and surface fitting

In this paper, based on the idea of profit and loss modification, we present the iterative non-uniform B-spline curve and surface to settle a key problem in computer aided geometric design and reverse engineering, that is, constructing the curve (surface) fitting (interpolating) a given ordered point set without solving a linear system. We start with a piece of initial non-uniform B-spline curve (surface) which takes the given point set as its control point set. Then by adjusting its control points gradually with iterative formula, we can get a group of non-uniform B-spline curves (surfaces) with gradually higher precision. In this paper, using modern matrix theory, we strictly prove that the limit curve (surface) of the iteration interpolates the given point set. The non-uniform B-spline curves (surfaces) generated with the iteration have many advantages, such as satisfying the NURBS standard, having explicit expression, gaining locality, and convexity preserving, etc.

/pdf/constructing-iterative-non-uniform-b-spline-curve-and-2l2eduz338.pdf

Constructing iterative non-uniform B-spline curve and surface to fit data points

In this paper, we study the progressive iteration approximation property of a curve (tensor product surface) generated by blending a given data point set and a set of basis functions. The curve (tensor product surface) has the progressive iteration approximation property as long as the basis is totally positive and the corresponding collocation matrix is nonsingular. Thus, the B-spline and NURBS curve (surface) have the progressive iteration approximation property, and Bezier curve (surface) also has the property if the corresponding collocation matrix is nonsingular.

http://www.cad.zju.edu.cn/home/hwlin/pdf_files/Totally-positive-bases-and-progressive-iteration-approximation.pdf

Hongwei Lin

Papers

A Robust Hole-Filling Algorithm for Triangular Mesh

Watertight trimmed NURBS

Progressive and iterative approximation for least squares B-spline curve and surface fitting

Constructing iterative non-uniform B-spline curve and surface to fit data points

Totally positive bases and progressive iteration approximation