The widespread deployment of machine learning systems in critical real-world decision-making applications has highlighted the urgent need for counterfactual explainability methods that operate effectively. Global counterfactual explanations, expressed as actions to offer recourse, aim to provide succinct explanations and insights applicable to large population subgroups. High effectiveness, measured by the fraction of the population that is provided recourse, ensures that the actions benefit as many individuals as possible. Keeping the cost of actions low ensures the proposed recourse actions remain practical and actionable. Limiting the number of actions that provide global counterfactuals is essential to maximize interpretability. The primary challenge, therefore, is to balance these trade-offs—maximizing effectiveness, minimizing cost, while maintaining a small number of actions. We introduce GLANCE, a versatile and adaptive algorithm that employs a novel agglomerative approach, jointly considering both the feature space and the space of counterfactual actions, thereby accounting for the distribution of points in a way that aligns with the model's structure. This design enables the careful balancing of the trade-offs among the three key objectives, with the size objective functioning as a tunable parameter to keep the actions few and easy to interpret. Our extensive experimental evaluation demonstrates that GLANCE consistently shows greater robustness and performance compared to existing methods across various datasets and models.

Impressive progress in generative models and implicit representations gave rise to methods that can generate 3D shapes of high quality. However, being able to locally con-trol and edit shapes is another essential property that can unlock several content creation applications. Local control can be achieved with part-aware models, but existing meth-ods require 3D supervision and cannot produce textures. In this work, we devise PartNeRF, a novel part-aware gener-ative model for editable 3D shape synthesis that does not require any explicit 3D supervision. Our model generates objects as a set of locally defined NeRFs, augmented with an affine transformation. This enables several editing op-erations such as applying transformations on parts, mixing parts from different objects etc. To ensure distinct, manip-ulable parts we enforce a hard assignment of rays to parts that makes sure that the color of each ray is only determined by a single NeRF. As a result, altering one part does not af-fect the appearance of the others. Evaluations on various ShapeNet categories demonstrate the ability of our model to generate editable 3D objects of improved fidelity, compared to previous part-based generative approaches that require 3D supervision or models relying on NeRFs.

In this work, we present Fairness Aware Counterfactuals for Subgroups (FACTS), a framework for auditing subgroup fairness through counterfactual explanations. We start with revisiting (and generalizing) existing notions and introducing new, more refined notions of subgroup fairness. We aim to (a) formulate different aspects of the difficulty of individuals in certain subgroups to achieve recourse, i.e. receive the desired outcome, either at the micro level, considering members of the subgroup individually, or at the macro level, considering the subgroup as a whole, and (b) introduce notions of subgroup fairness that are robust, if not totally oblivious, to the cost of achieving recourse. We accompany these notions with an efficient, model-agnostic, highly parameterizable, and explainable framework for evaluating subgroup fairness. We demonstrate the advantages, the wide applicability, and the efficiency of our approach through a thorough experimental evaluation on different benchmark datasets.

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined by the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest for the public in general and for policy makers in particular. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be estimated by computing volumes of convex bodies. We design and implement practical algorithms in the exact and approximate setting, and experimentally juxtapose them in order to study the trade-off of exactness and accuracy for speed. We also experimentally find an efficient parameter-tuning to achieve a sufficiently good estimation of the probability density of each copula. Our C++ software, based on Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.

Determining the number of solutions of a multi-homogeneous polynomial system is a fundamental problem in algebraic geometry. The multi-homogeneous Bézout (m-Bézout) number bounds from above the number of non-singular solutions of a multi-homogeneous system, but its computation is a #P>-hard problem.

The Canny-Emiris formula [3] gives the sparse resultant as a ratio between the determinant of a Sylvester-type matrix and a minor of it, by a subdivision algorithm. The most complete proof of the formula was given by D'Andrea et al. in [9] under general conditions on the underlying mixed subdivision. Before the proof, Canny and Pedersen had proposed [5] a greedy algorithm which provides smaller matrices, in general. The goal of this paper is to give an explicit class of mixed subdivisions for the greedy approach such that the formula holds, and the dimensions of the matrices are reduced compared to the subdivision algorithm. We measure this reduction for the case when the Newton polytopes are zonotopes generated by n line segments (where n is the rank of the underlying lattice), and for the case of multihomogeneous systems. This article comes with a JULIA implementation of the treated cases.

Randomized dimensionality reduction has been recognized as one of the fundamental techniques in handling high-dimensional data. Starting with the celebrated Johnson-Lindenstrauss Lemma, such reductions have been studied in depth for the Euclidean (l_2) metric, but much less for the Manhattan (l_1) metric. Our primary motivation is the approximate nearest neighbor problem in l_1. We exploit its reduction to the decision-with-witness version, called approximate near neighbor, which incurs a roughly logarithmic overhead. In 2007, Indyk and Naor, in the context of approximate nearest neighbors, introduced the notion of nearest neighbor-preserving embeddings. These are randomized embeddings between two metric spaces with guaranteed bounded distortion only for the distances between a query point and a point set. Such embeddings are known to exist for both l_2 and l_1 metrics, as well as for doubling subsets of l_2. The case that remained open were doubling subsets of l_1. In this paper, we propose a dimension reduction by means of a near neighbor-preserving embedding for doubling subsets of l_1. Our approach is to represent the pointset with a carefully chosen covering set, then randomly project the latter. We study two types of covering sets: c-approximate r-nets and randomly shifted grids, and we discuss the tradeoff between them in terms of preprocessing time and target dimension. We employ Cauchy variables: certain concentration bounds derived should be of independent interest.

We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies. We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.

The problem of Approximate Nearest Neighbor (ANN) search is fundamental in computer science and has benefited from significant progress in the past couple of decades. However, most work has been devoted to pointsets whereas complex shapes have not been sufficiently treated. Here, we focus on distance functions between discretized curves in Euclidean space: they appear in a wide range of applications, from road segments to time-series in general dimension. For $\ell_p$-products of Euclidean metrics, for any $p$, we design simple and efficient data structures for ANN, based on randomized projections, which are of independent interest. They serve to solve proximity problems under a notion of distance between discretized curves, which generalizes both discrete Fréchet and Dynamic Time Warping distances. These are the most popular and practical approaches to comparing such curves. We offer the first data structures and query algorithms for ANN with arbitrarily good approximation factor, at the expense of increasing space usage and preprocessing time over existing methods. Query time complexity is comparable or significantly improved by our algorithms, our algorithm is especially efficient when the length of the curves is bounded.

The number of embeddings of minimally rigid graphs in RD is (by definition) finite, modulo rigid transformations, for every generic choice of edge lengths. Even though various approaches have been proposed to compute it, the gap between upper and lower bounds is still enormous. Specific values and its asymptotic behavior are major and fascinating open problems in rigidity theory. Our work considers the maximal number of real embeddings of minimally rigid graphs in R3. We modify a commonly used parametric semi-algebraic formulation that exploits the Cayley-Menger determinant to minimize the a priori number of complex embeddings, where the parameters correspond to edge lengths. To cope with the huge dimension of the parameter space and find specializations of the parameters that maximize the number of real embeddings, we introduce a method based on coupler curves that makes the sampling feasible for spatial minimally rigid graphs. Our methodology results in the first full classification of the number of real embeddings of graphs with 7 vertices in R3, which was the smallest open case. Building on this and certain 8-vertex graphs, we improve the previously known general lower bound on the maximum number of real embeddings in R3.

Approximate nearest neighbor search (ϵ-ANN) in high dimensions has been mainly addressed by Locality Sensitive Hashing (LSH), which has complexity with polynomial dependence in dimension, sublinear query time, but subquadratic space requirement. We introduce a new “low-quality” embedding for metric spaces requiring that, for some query, there exists an approximate nearest neighbor among the pre-images of its k > 1 approximate nearest neighbors in the target space. In Euclidean spaces, we employ random projections to a dimension inversely proportional to k . Our approach extends to the decision problem with witness of checking whether there exists an approximate near neighbor; this also implies a solution for ϵ-ANN. After dimension reduction, we store points in a uniform grid of side length ϵ /√ d ′ , where d ′ is the reduced dimension. Given a query, we explore cells intersecting the unit ball around the query. This data structure requires linear space and query time in O ( d n ρ ), ρ ≈ 1-ϵ 2 i>/log(1ϵ), where n denotes input cardinality and d space dimension. Bounds are improved for doubling subsets via r -nets. We present our implementation for ϵ-ANN in C++ and experiments for d ≤ 960, n ≤ 10 6 , using synthetic and real datasets, which confirm the theoretical analysis and, typically, yield better practical performance. We compare to FALCONN, the state-of-the-art implementation of multi-probe LSH: our prototype software is essentially comparable in terms of preprocessing, query time, and storage usage.

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear halfspaces. We implement and evaluate randomized polynomial-time algorithms for accurately approximating the polytope’s volume in high dimensions (e.g., few hundreds) based onhit-and-run random walks. To carry out this efficiently, we experimentally correlate the effect of parameters, such as random walk length and number of sample points, with accuracy and runtime. Our method is based on Monte Carlo algorithms with guaranteed speed and provably high probability of success for arbitrarily high precision. We exploit the problem’s features in implementing a practical rounding procedure of polytopes, in computing only partial “generations” of random points, and in designing fast polytope boundary oracles. Our publicly available software is significantly faster than exact computation and more accurate than existing approximation methods. For illustration, volume approximations of Birkhoff polytopesB11,…,B15are computed, in dimensions up to 196, whereas exact methods have only computed volumes of up toB10.

We examine implicit representations of parametric or point cloud models, based on interpolation matrices, which are not sensitive to base points. We show how interpolation matrices can be used for ray shooting of a parametric ray with a surface patch, including the case of high-multiplicity intersections. Most matrix operations are executed during pre-processing since they solely depend on the surface. For a given ray, the bottleneck is equation solving. Our Maple code handles bicubic patches in < 1 sec, though numerical issues might arise. Our second contribution is to extend the method to parametric space curves and, generally, to codimension > 1, by computing the equations of (hyper)surfaces intersecting precisely at the given object. By means of Chow forms, we propose a new, practical, randomized algorithm that always produces correct output but possibly with a non-minimal number of surfaces. For space curves, we typically obtain 3 surfaces whose polynomials are of near-optimal degree; in this case, computation reduces to a Sylvester resultant. Our Maple prototype is not faster but yields fewer equations and seems more robust than Maple's implicitize.

The construction of r-nets offers a powerful tool in computational and metric geometry. We focus on high- dimensional spaces and present a new randomized algorithm which efficiently computes approximate r-nets with respect to Euclidean distance. For any fixed ∊ > 0, the approximation factor is 1 + ∊ and the complexity is polynomial in the dimension and subquadratic in the number of points. The algorithm succeeds with high probability. Specifically, we improve upon the best previously known (LSH- based) construction of Eppstein et al. [EHS15] in terms of complexity, by reducing the dependence on ∊, provided that ∊ is sufficiently small. Our method does not require LSH but, instead, follows Valiant's [Val15] approach in designing a sequence of reductions of our problem to other problems in different spaces, under Euclidean distance or inner product, for which r-nets are computed efficiently and the error can be controlled. Our result immediately implies efficient solutions to a number of geometric problems in high dimension, such as finding the (1 + ∊)-approximate kth nearest neighbor distance in time subquadratic in the size of the input.

It has by now become a standard approach to use the theory of sparse (or toric) elimination, based on the Newton polytope of a polynomial, in order to reveal and exploit the structure of algebraic systems. This talk surveys compact formulae, including older and recent results, in sparse elimination. We start with root bounds and juxtapose two recent formulae: a generating function of the m-Bezout bound and a closed-form expression for the mixed volume by means of a matrix permanent. For the sparse resultant, a bevy of results have established determinantal or rational formulae for a large class of systems, starting with Macaulay. The discriminant is closely related to the resultant but admits no compact formula except for very simple cases. We offer a new determinantal formula for the discriminant of a sparse multilinear system arising in computing Nash equilibria. We introduce an alternative notion of compact formula, namely the Newton polytope of the unknown polynomial. It is possible to compute it efficiently for sparse resultants, discriminants, as well as the implicit equation of a parameterized variety. This leads us to consider implicit matrix representations of geometric objects.

We bound the Boolean complexity of computing isolating hyperboxes for all complex roots of systems of bilinear polynomials. The resultant of such systems admits a family of determinantal Sylvester-type formulas, which we make explicit by means of homological complexes. The computation of the determinant of the resultant matrix is a bottleneck for the overall complexity. We exploit the quasi-Toeplitz structure to reduce the problem to efficient matrix-vector products, corresponding to multivariate polynomial multiplication. For zero-dimensional systems, we arrive at a primitive element and a rational univariate representation of the roots. The overall bit complexity of our probabilistic algorithm is OB(n4 D4 + n2D4 τ), where n is the number of variables, D equals the bilinear Bezout bound, and τ is the maximum coefficient bitsize. Finally, a careful infinitesimal symbolic perturbation of the system allows us to treat degenerate and positive dimensional systems, thus making our algorithms and complexity analysis applicable to the general case.

The approximate nearest neighbor problem (epsilon-ANN) in Euclidean settings is a fundamental question, which has been addressed by two main approaches: Data-dependent space partitioning techniques perform well when the dimension is relatively low, but are affected by the curse of dimensionality. On the other hand, locality sensitive hashing has polynomial dependence in the dimension, sublinear query time with an exponent inversely proportional to (1+epsilon)^2, and subquadratic space requirement. We generalize the Johnson-Lindenstrauss Lemma to define "low-quality" mappings to a Euclidean space of significantly lower dimension, such that they satisfy a requirement weaker than approximately preserving all distances or even preserving the nearest neighbor. This mapping guarantees, with high probability, that an approximate nearest neighbor lies among the k approximate nearest neighbors in the projected space. These can be efficiently retrieved while using only linear storage by a data structure, such as BBD-trees. Our overall algorithm, given n points in dimension d, achieves space usage in O(dn), preprocessing time in O(dn log n), and query time in O(d n^{rho} log n), where rho is proportional to 1 - 1/loglog n, for fixed epsilon in (0, 1). The dimension reduction is larger if one assumes that point sets possess some structure, namely bounded expansion rate. We implement our method and present experimental results in up to 500 dimensions and 10^6 points, which show that the practical performance is better than predicted by the theoretical analysis. In addition, we compare our approach with E2LSH.

Large scale duplicate detection, clustering and mining of documents or images has been conventionally treated with seed detection via hashing, followed by seed growing heuristics using fast search. Principled clustering methods, especially kernelized and spectral ones, have higher complexity and are difficult to scale above millions. Under the assumption of documents or images embedded in Euclidean space, we revisit recent advances in approximate k-means variants, and borrow their best ingredients to introduce a new one, inverted-quantized k-means (IQ-means). Key underlying concepts are quantization of data points and multi-index based inverted search from centroids to cells. Its quantization is a form of hashing and analogous to seed detection, while its updates are analogous to seed growing, yet principled in the sense of distortion minimization. We further design a dynamic variant that is able to determine the number of clusters k in a single run at nearly zero additional cost. Combined with powerful deep learned representations, we achieve clustering of a 100 million image collection on a single machine in less than one hour.

Based on the computation of a polytope Q, called the predicted polytope, containing the Newton polytope P of the implicit equation, implicitization of a parametric hypersurface is reduced to computing the nullspace of a numeric matrix. Polytope Q may contain P as a Minkowski summand, thus jeopardizing the efficiency of sparse implicitization. Our contribution is twofold. On one hand we tackle the aforementioned issue in the case of 2D curves and 3D surfaces by Minkowski decomposing Q, thus detecting the Minkowski summand relevant to implicitization: we design and implement in Sage a new, public domain, practical, potentially generalizable and worst-case optimal algorithm for Minkowski decomposition in 3D based on integer linear programming. On the other hand, we formulate basic geometric predicates, namely membership and sidedness for given query points, as rank computations on the interpolation matrix, thus avoiding to expand the implicit polynomial. This approach is implemented in Maple.

We experimentally study the fundamental problem of computing the volume of a convex polytope given as an intersection of linear inequalities. We implement and evaluate practical randomized algorithms for accurately approximating the polytope's volume in high dimensions (e.g. one hundred). To carry out this efficiently we experimentally correlate the effect of parameters, such as random walk length and number of sample points, on accuracy and runtime. Moreover, we exploit the problem's geometry by implementing an iterative rounding procedure, computing partial generations of random points and designing fast polytope boundary oracles. Our publicly available code is significantly faster than exact computation and more accurate than existing approximation methods. We provide volume approximations for the Birkhoff polytopes B11, …, B15, whereas exact methods have only computed that of B10.

Semi-mixed algebraic systems are those where the equations can be partitioned into subsets with common Newton polytopes. We are interested in counting roots of semi-mixed multihomogeneous systems, where both variables and equations can be partitioned into blocks, and each block of equations has a given degree in each block of variables. The motivating example is counting the number of totally mixed Nash equilibria in games of several players. Firstly, this paper relates and unifies the BKK and multivariate Bézout bounds for semi-mixed systems, through mixed volumes and matrix permanents. Permanent expressions for BKK bounds hold for all multihomogeneous systems, without any requirement of semi-mixed structure, as well as systems whose Newton polytopes are products of polytopes in complementary subspaces. Secondly, by means of a novel asymptotic analysis, the complexity of a combinatorial geometric algorithm for semi-mixed volumes (i.e., mixed volumes of semi-mixed systems) is explored and juxtaposed to the complexities of computing permanents, or using generating functions (via MacMahon's Master theorem), or orthogonal polynomials.

The Newton polytope of the resultant, or resultant polytope, characterizes the resultant polynomial more precisely than total degree. The combinatorics of resultant polytopes are known in the Sylvester case [Gelfand et al.90] and up to dimension 3 [Sturmfels 94]. We extend this work by studying the combinatorial characterization of 4-dimensional resultant polytopes, which show a greater diversity and involve computational and combinatorial challenges. In particular, our experiments, based on software respol for computing resultant polytopes, establish lower bounds on the maximal number of faces. By studying mixed subdivisions, we obtain tight upper bounds on the maximal number of facets and ridges, thus arriving at the following maximal f-vector: (22,66,66,22), i.e. vector of face cardinalities. Certain general features emerge, such as the symmetry of the maximal f-vector, which are intriguing but still under investigation. We establish a result of independent interest, namely that the f-vector is maximized when the input supports are sufficiently generic, namely full dimensional and without parallel edges. Lastly, we offer a classification result of all possible 4-dimensional resultant polytopes.

We develop an incremental algorithm to compute the Newton polytope of the resultant, aka resultant polytope, or its projection along a given direction. The resultant is fundamental in algebraic elimination and in implicitization of parametric hypersurfaces. Our algorithm exactly computes vertex- and halfspace-representations of the desired polytope using an oracle producing resultant vertices in a given direction. It is output-sensitive as it uses one oracle call per vertex. We overcome the bottleneck of determinantal predicates by hashing, thus accelerating execution from 18 to 100 times. We implement our algorithm using the experimental CGAL package triangulation. A variant of the algorithm computes successively tighter inner and outer approximations: when these polytopes have, respectively, 90% and 105% of the true volume, runtime is reduced up to 25 times. Our method computes instances of 5-, 6- or 7-dimensional polytopes with 35K, 23K or 500 vertices, resp., within 2hr. Compared to tropical geometry software, ours is faster up to dimension 5 or 6, and competitive in higher dimensions.

Constructive methods for matrices of multihomogeneous (or multigraded) resultants for unmixed systems have been studied by Weyman, Zelevinsky, Sturmfels, Dickenstein and Emiris. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bézout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for all corresponding degree vectors, and specify domains that will surely contain such vectors; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, for a general scaled system, thus including unmixed systems. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bézout-type matrices, known as partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

Our probabilistic analysis sheds light to the following questions: Why do random polynomials seem to have few, and well separated real roots, on the average? Why do exact algorithms for real root isolation may perform comparatively well or even better than numerical ones?

In this paper we derive aggregate separation bounds, named after Davenport-Mahler-Mignotte (DMM), on the isolated roots of polynomial systems, specifically on the minimum distance between any two such roots. The bounds exploit the structure of the system and the height of the sparse (or toric) resultant by means of mixed volume, as well as recent advances on aggregate root bounds for univariate polynomials, and are applicable to arbitrary positive dimensional systems. We improve upon Canny's gap theorem [7] by a factor of O(dn-1), where d bounds the degree of the polynomials, and n is the number of variables. One application is to the bitsize of the eigenvalues and eigenvectors of an integer matrix, which also yields a new proof that the problem is polynomial. We also compare against recent lower bounds on the absolute value of the root coordinates by Brownawell and Yap [5], obtained under the hypothesis there is a 0-dimensional projection. Our bounds are in general comparable, but exploit sparseness; they are also tighter when bounding the value of a positive polynomial over the simplex. For this problem, we also improve upon the bounds in [2, 16]. Our analysis provides a precise asymptotic upper bound on the number of steps that subdivision-based algorithms perform in order to isolate all real roots of a polynomial system. This leads to the first complexity bound of Milne's algorithm [22] in 2D.

Constructive methods for matrices of multihomogeneous resultants for unmixed systems have been studied in [7, 13, 15]. We generalize these constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. Bézout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for the corresponding degree vectors, as well as precise domains where these concentrate; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, in the general case. The encountered matrices are classified; these include a new type of Sylvester-type matrix as well as Bézout-type matrices, which we call partial Bezoutians. Our public-domain Maple implementation includes efficient storage of complexes in memory, and construction of resultant matrices.

We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Delaunay graph is constructed incrementally. Our first contribution is to propose robust end efficient algorithms for all required predicates, thus generalizing our earlier algorithms for ellipses, and we analyze their algebraic complexity, under the exact computation paradigm. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 X 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of ellipses, which is the first exact implementation for the problem. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when few degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k > 15.

This paper investigates several robotic mechanisms for ankle function evaluation, measurement and physiotherapy. For the choice, design and operation of the mechanism the kinematics of the foot is described. This is based on a kinematics model of foot adopted from biomechanics literature, under the hypothesis that foot kinematics is similar to that of a 2R serial robot. A 3D scanner and an inertial sensor were used in order to fully specify the design framework by studying a larger sample of healthy subjects. Our experimental analysis confirms and enhances the 2R foot model, and leads us to the choice of the specific mechanism. We compute the required workspace and thus address the issues required for a complete and efficient design. We compare mechanisms based on serial and parallel robots, and choose a parallel tripod with an extra rotation axis for its simplicity, accuracy and generality. The robot must be capable to perform several multi-axis motions and sustain a significant range of forces and torques. The kinematic analysis of the robot confirms that it can follow all the range of foot movements.

Based on precomputed Sturm–Habicht sequences, discriminants and invariants, we classify, isolate with rational points, and compare the real roots of polynomials of degree up to 4. In particular, we express all isolating points as rational functions of the input polynomial coefficients. Although the roots are algebraic numbers and can be expressed by radicals, such representation involves some roots of complex numbers. This is inefficient, and hard to handle in applications in geometric computing and quantifier elimination. We also define rational isolating points between the roots of the quintic. We combine these results with a simple version of Rational Univariate Representation to isolate all common real roots of a bivariate system of rational polynomials of total degree ≤2 and to compute the multiplicity of these roots. We present our software within library synaps and perform experiments and comparisons with several public-domain implementations. Our package is 2–10 times faster than numerical methods and exact subdivision-based methods, including software with intrinsic filtering.

We consider exact real solving of well-constrained, bivariate systems of relatively prime polynomials. The main problem is to compute all common real roots in isolating interval representation, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of ÕB(N14) for the purely projection-based method, and ÕB(N12) for two subresultants-based methods: these ignore polylogarithmic factors, and N bounds the degree and the bitsize of the polynomials. The previous record bound was ÕB(N14).

We study the Voronoi diagram, under the Euclidean metric, of a set of ellipses, given in parametric representation. We use an efficient incremental algorithm and focus on the required predicates. The paper concentrates on InCircle, which is the hardest predicate: it decides the position of a query ellipse relative to the Voronoi circle of three given ellipses. We describe an exact, real-time, and complete implementation for InCircle, combining a certified numeric algorithm with algebraic computation. The numeric part leads to a real-time implementation for non-degenerate inputs. It relies on a geometric preprocessing that guarantees a unique solution in a box of parametric space, where a customized subdivision-based method approximates the Voronoi circle tracing the bisectors. Our subdivision method achieves quadratic convergence by exploiting the geometric characteristics of the problem. To achieve robustness, we develop interval-arithmetic techniques, based on the C++ package Alias. We switch to an algebraic approach for handling the degeneracies fast. Based on a different algebraic system to model InCircle, we apply real solving and resultant theory. The latter relies on certain symbolic routines which are efficiently implemented in Maple. Our approach readily generalizes to arbitrary conics. The paper concludes with experiments showing that most instances run in less than 0.1 sec, on a 2.6GHz Pentium-4, whereas degenerate cases may take up to 13 sec.

This paper examines the computation of the Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete algorithms, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of 2 given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of 2 given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of 3 given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by 4 given ellipses, and a query ellipse. The paper is restricted to non-intersecting ellipses, but the extension to arbitrary ones is straightforward. The ellipses are input in parametric representation or constructively. For κ1 and κ2 we derive optimal algebraic conditions, solve them exactly and provide efficient implementations in C++. For κ3 we compute a tight bound on the number of complex tritangent circles and use the parametric form of the ellipses in order to design an exact subdivision-based algorithm, which is implemented on Maple. This approach essentially answers κ4 as well. We conclude with current work on optimizing κ3 and implementing it in C++.

This paper is concerned with symbolic computations in Volterra system identification using higher order cumulants. An efficient method that implements the Leonov-Shiryaev theorem is introduced. The proposed method relies on the exploitation of recursive relations between cumulants. The method is applied on the problem of blind identification of Volterra-Hammerstein systems excited by stationary higher order white noise. It solves previously intractable instances

Our work goes towards answering the growing need for the robust and efficient manipulation of curved objects in numerous applications. The kernel of the CGAL library provides several functionalities which are, however, mostly restricted to linear objects. We focus here on the arrangement of conic arcs in the plane. Our first contribution is the design, implementation and testing of a kernel for computing arrangements of circular arcs.A preliminary C++ implementation exists also for arbitrary conic curves. We discuss the representation and predicates of the geometric objects. Our implementation is targeted for inclusion in the CGAL library. Our second contribution concerns exact and efficient algebraic algorithms for the case of conics. They treat all inputs, including degeneracies, and they are implemented as part of the library SYNAPS 2.1.Our tools include Sturm sequences, resultants, Descartes' rule, andisolating points. Thirdly, our experiments on circular arcs show that our methods compare favorably to existing alternatives using CORE 1.6x and LEDA 4.5.

Multihomogeneous structure in algebraic systems is the first step away from the classical theory of homogeneous equations towards fully exploiting arbitrary supports. We propose constructive methods for resultant matrices in the entire spectrum of resultant formulae, ranging from pure Sylvester to pure Bezout types, including hybrid matrices. Our approach makes heavy use of the combinatorics of multihomogeneous systems, inspired by and generalizing certain joint results by Zelevinsky, and Sturmfels or Weyman [15, 18]. One contribution is to provide conditions and algorithmic tools so as to classify and construct the smallest possible determinantal formulae for multihomogeneous resultants. We also examine the smallest Sylvester-type matrices, generically of full rank, which yield a multiple of the resultant. The last contribution is to characterize the systems that admit a purely Bezout-type matrix and show a bijection of such matrices with the permutations of the variable groups. Interestingly, it is the same class of systems admitting an optimal Sylvester-type formula. We conclude with an example showing all kinds of matrices that may be encountered, and illustrations of our MAPLE implementation.

A new algorithm for calibrating Gough platforms is proposed. It requires internal sensor measurements and only the position information is provided by external sensors. It removes the need to measure orientation, which is intricate and error-prone, by algebraic elimination. This approach, relying on resultant and dialytic elimination, produces an equivalent, yet simpler, set of equations. A numerical simulation is given to compare the existing techniques with our method using partial information, which proves to be significantly more robust, without compromising accuracy. It reduces initial error in pose determination by 99% and 80-98%, in two sets of experiments with realistic conditions. We compare different choices for the measured configurations and show the relevance of configurations at the workspace's boundary. This increases reliability by avoiding to use any random measured configurations.

Our main contribution is an explicit construction of square resultant matrices, which are submatrices of those introduced by Cattani, Dickenstein and Sturmfels [4]. The determinant is a nontrivial multiple of the sparse (or toric) resultant. The matrix is hybrid in that it contains a submatrix of Sylvester type and an additional row expressing the toric Jacobian. If we restrict attention to such matrices, the algorithm yields the smallest possible matrix in general. This is achieved by strongly exploiting the combinatorics of sparse elimination. The algorithm uses a new piecewise-linear lifting, defined for bivariate systems of 3 polynomials with Newton polygons being scaled copies of a single polygon. The major motivation comes from systems encountered in CAD. Our MAPLE implementation, applied to certain examples, illustrates our construction and compares with alternative matrices.

The relatively recent theory of sparse variable elimination exploits the structure in polynomial equations in order to define tighter bounds on the number of common roots and faster methods for numerically approximating them.The model of sparsity is of combinatorial nature, which leads naturally to certain problems in convex geometry in generaldimensional euclidean spaces.This work addresses one such problem, namely the computation of certain integer points in the interior of integer convex polytopes.These polytopes are Minkowski sums, but avoiding their explicit construction is precisely one of the main features of the algorithm.Output-sensitive bounds are derived for our algorithm, in terms of the sparsity parameters of the problem.A public domain implementation is described and its performance illustrated on certain families of inputs.

Multivariate resultants generalize the Sylvester resultant of two polynomials and characterize the solvability of a polynomial system. They also reduce the computation of all common roots to a problem in linear algebra. We propose a determinantal formula for the sparse resultant of an arbitrary system of n + 1 polynomials in n variables. This resultant generalizes the classical one and has significantly lower degree for polynomials that are sparse in the sense that their mixed volume is lower than their Bézout number. Our algorithm uses a mixed polyhedral subdivision of the Minkowski sum of the Newton polytopes in order to construct a Newton matrix. Its determinant is a nonzero multiple of the sparse resultant and the latter equals the GCD of at most n + 1 such determinants. This construction implies a restricted version of an effective sparse Nullstellensatz. For an arbitrary specialization of the coefficients, there are two methods that use one extra variable and yield the sparse resultant. This is the first algorithm to handle the general case with complexity polynomial in the resultant degree and simply exponential in n . We conjecture its extension to producing an exact rational expression for the sparse resultant.

The problem of computing zeros of a system of polynomial equations has been well studied in the computational literature.A n umber of algorithms have been proposed and many computer algebra and public domain packages provide the capability of computing the roots of polynomial equations.Most of these implementations are based on Gr obner bases which can be slow for even small problems.In this paper, we present a new system, MARS, to compute the roots of a zero dimensional polynomial system.It is based on computing the resultant of a system of polynomial equations followed by eigendecomposition of a generalized companion matrix.MARS includes a robust library of Maple functions for constructing resultant matrices, an ecient library of Matlab routines for numerically solving the eigenproblem, and C code generation routines and a C library for incorporating the numerical solver into applications.We illustrate the usage of MARS on various examples and utilize dierent resultant formulations.

International audience

Resultants characterize the existence of roots of systems of multivariate nonlinear polynomial equations, while their matrices reduce the computation of all common zeros to a problem in linear algebra. Sparse elimination theory has introduced the sparse resultant, which takes into account the sparse structure of the polynomials. The construction of sparse resultant, or Newton, matrices is a critical step in the computation of the resultant and the solution of the system. We exploit the matrix structure and decrease the time complexity ofconstructing such matrices to roughly quadratic in the matrix dimension, whereas the previous methods had cubic complexity. The space complexity is also decreased by one order of magnitude. These results imply similar improvements in the complexity of computing the resultant itself and of solving zero-dimensional systems. We apply some novel techniques for determining the rank of rectangular matrices by an exact or numerical computation. Finally, we improve the existing complexity for polynomial multiplication under our model of sparseness, o ering bounds linear in the number of variables and the number of nonzero terms.

Sparse elimination exploits the structure of a multivariate polynomial by considering its Newton polytope instead of its total degree. We concentrate on polynomial systems that generate zero-dimensional ideals. A monomial basis for the coordinate ring is defined from a mixed subdivision of the Minkowski sum of the Newton polytopes. We offer a new simple proof relying on the construction of a sparse resultant matrix, which leads to the computation of a multiplication map and all common zeros. The size of the monomial basis equals the mixed volume and its computation is equivalent to computing the mixed volume, so the latter is a measure of intrinsic complexity. On the other hand, our algorithms have worst-case complexity proportional to the volume of the Minkowski sum. In order to derive bounds in terms of the sparsity parameters, we establish new bounds on the Minkowski sum volume as a function of mixed volume. To this end, we prove a lower bound on mixed volume in terms of Euclidean volume which is of independent interest.

We wish to increase the power of an arbitrary algorithm designed for nondegenerate input by allowing it to execute on all inputs. We concentrate on infinitesimal symbolic perturbations that do not affect the output for inputs in general position. Otherwise, if the problem mapping is continuous, the input and output space topology are at least as coarse as the real euclidean one, and the output space is connected, then our perturbations make the algorithm produce an output arbitrarily close or identical to the correct one. For a special class of algorithms, which includes several important algorithms in computational geometry, we describe a deterministic method that requires no symbolic computation. Ignoring polylogarithmic factors, this method increases the worst-case bit complexity only by a multiplicative factor which is linear in the dimension of the geometric space. For general algorithms, a randomized scheme with an arbitrarily high probability of success is proposed; the bit complexity is then bounded by a small-degree polynomial in the original worst-case complexity. In addition to being simpler than previous ones, these are the first efficient perturbation methods.

This paper addresses the problem of efficient construction of monomial bases for the coordinate rings of zero-dimensional varieties.

We propose an efficient method for computing the resultant, of a sparse polynomial system of n + 1 equations in n unknowns.Our approach carries over from [(UE93] and constructs a matrix whose determinant is a nonzero multiple of the resultant, and from which the latter is easily extracted.For certain classes of syskms, it attains optimality by expressing the resultant, as a sillgle determinant.An illll>lenlelltatioll of the algorithm is described and empirical results presented and conlpared with those from [CE93] and [SZ].In addition, the important subproblem of computiug Mixed [Tolumes is examined and an efficient algorithm is inlplemeuted.Dixon ~esultant, of a perturbed system leads to an algorit,hln that terminates successfully in about 30 minutes, while all major Clrobner bases methods seem to run out of memory after running for a few days, even }vhen wor]iing on a homomorphic image of the problem over a finite field [MC92aj.In general, Grobner bases algorit)hlns can also exploit, sparseness; yet, when a sparse resultant, is known, the solution is much faster, as seen iu these applications.There exist some classes of problems, such as the kinematics of mechanisms and the generalized kinematics problem with ods aw expected 183 constraints, for which sparse methto be very efficient.So, ideally, we would like to have a sparse resultant for every problem, which calls for a general algorithm to construct, sparse resultants.The first efficient, algorithm was proposed in [CE93], while here we take a different, tack at

We wish to increase the power of an arbitrary geometric algorithm designed for non-degenerate input, by allowing it to execute over arbitrary inputs. This paper describes a deterministic direct perturbation of the input which applies to algorithms whose branching decisions depend on determinants in the input parameters. We concentrate on four predicates that cover most important algorithms in computational geometry. The method alters the input parameters by infinitesimal amounts to guarantee that the perturbed input lies in general position. It is an attractive candidate for practical use, and is considerably simpler as well as more efficient than existing approaches. Specifically, it is the first method with a complexity overhead that is polynomial in the input size and, in most cases, a small constant. Under our real computation model, the asymptotic complexity remains unaffected; the bit complexity is at worst increased by a factor proportional to d^2+alpha, where d is the dimension of the geometric space of the input objects and alpha an arbitrarily small positive constant. A variation of the perturbation, applied to two predicates, achieves optimal bit size for the perturbation quantities and is even more efficient. Lastly, we illustrate the applicability of our approach

Algorithms modeled as algebraic branching programs, with inputs from an infinite ordered field, are studied. Direct perturbations on the input, so that an algorithm designed under the assumption of nondegeneracy can be applied to all inputs, are described. A deterministic method for algorithms with determinant tests and a randomized one for arbitrary test expressions are defined. They both incur extra complexity factors that are constant in several cases. Moreover, polynomial and exponential time algorithms always remain in the same complexity class while being enhanced with the power to execute on arbitrary inputs. Both methods are distinguished by their conceptual elegance and are significantly faster than previous ones.>