Showing papers in "Foundations of Computational Mathematics in 2007"
TL;DR: A complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors defined by the considered kernel.
Abstract: We develop a theoretical analysis of the performance of the regularized least-square algorithm on a reproducing kernel Hilbert space in the supervised learning setting. The presented results hold in the general framework of vector-valued functions; therefore they can be applied to multitask problems. In particular, we observe that the concept of effective dimension plays a central role in the definition of a criterion for the choice of the regularization parameter as a function of the number of samples. Moreover, a complete minimax analysis of the problem is described, showing that the convergence rates obtained by regularized least-squares estimators are indeed optimal over a suitable class of priors defined by the considered kernel. Finally, we give an improved lower rate result describing worst asymptotic behavior on individual probability measures rather than over classes of priors.
728 citations
TL;DR: An adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity and does not rely on a recurrent coarsening of the partitions.
Abstract: In this paper an adaptive finite element method is constructed for solving elliptic equations that has optimal computational complexity. Whenever, for some s > 0, the solution can be approximated within a tolerance e > 0 in energy norm by a continuous piecewise linear function on some partition with O(e-1/s) triangles, and one knows how to approximate the right-hand side in the dual norm with the same rate with piecewise constants, then the adaptive method produces approximations that converge with this rate, taking a number of operations that is of the order of the number of triangles in the output partition. The method is similar in spirit to that from [SINUM, 38 (2000), pp. 466-488] by Morin, Nochetto, and Siebert, and so in particular it does not rely on a recurrent coarsening of the partitions. Although the Poisson equation in two dimensions with piecewise linear approximation is considered, the results generalize in several respects.
467 citations
TL;DR: A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed, and particular attention is paid to the truncated conjugate-gradient technique.
Abstract: A general scheme for trust-region methods on Riemannian manifolds is proposed and analyzed. Among the various approaches available to (approximately) solve the trust-region subproblems, particular attention is paid to the truncated conjugate-gradient technique. The method is illustrated on problems from numerical linear algebra.
452 citations
TL;DR: A Bernstein-type theorem is established, where the smallest separation between data sites plays the role of a Nyquist frequency, and Berstein-type estimates are used to derive inverse estimates for interpolation via SBFs.
Abstract: The purpose of this paper is to get error estimates for spherical basis function (SBF) interpolation and approximation for target functions in Sobolev spaces less smooth than the SBFs, and to show that the rates achieved are, in a sense, best possible. In addition, we establish a Bernstein-type theorem, where the smallest separation between data sites plays the role of a Nyquist frequency. We then use these Berstein-type estimates to derive inverse estimates for interpolation via SBFs.
73 citations
TL;DR: This paper provides a criterion for locating clusters of zeros and a fast algorithm for approximating them, with quadratic convergence, in situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory.
Abstract: Isolated multiple zeros or clusters of zeros of analytic maps with several variables are known to be difficult to locate and approximate. This paper is in the vein of the α-theory, initiated by M. Shub and S. Smale in the beginning of the 1980s. This theory restricts to simple zeros, i.e., where the map has corank zero. In this paper we deal with situations where the analytic map has corank one at the multiple isolated zero, which has embedding dimension one in the frame of deformation theory. These situations are the least degenerate ones and therefore most likely to be of practical significance. More generally, we define clusters of embedding dimension one. We provide a criterion for locating such clusters of zeros and a fast algorithm for approximating them, with quadratic convergence. In the case of a cluster with positive diameter our algorithm stops at a distance of the cluster which is about its diameter.
67 citations
TL;DR: In this article, the authors derived lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise.
Abstract: We study algorithms for approximation of the mild solution of stochastic heat equations on the spatial domain ]0, 1[d. The error of an algorithm is defined in L2-sense. We derive lower bounds for the error of every algorithm that uses a total of N evaluations of one-dimensional components of the driving Wiener process W. For equations with additive noise we derive matching upper bounds and we construct asymptotically optimal algorithms. The error bounds depend on N and d, and on the decay of eigenvalues of the covariance of W in the case of nuclear noise. In the latter case the use of nonuniform time discretizations is crucial.
56 citations
TL;DR: An algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group is provided, which gives rise to algorithms for constructing rational and replacement invariants.
Abstract: We provide an algebraic formulation of the moving frame method for constructing local smooth invariants on a manifold under an action of a Lie group. This formulation gives rise to algorithms for constructing rational and replacement invariants. The latter are algebraic over the field of rational invariants and play a role analogous to Cartan's normalized invariants in the smooth theory. The algebraic algorithms can be used for computing fundamental sets of differential invariants.
54 citations
TL;DR: The paper includes an introduction to Nash-Williams' theory of better-quasi-orderings, which is used to show termination of the algorithm and applies this theory to finiteness results for Hilbert functions.
Abstract: We study Graver test sets for families of linear multistage stochastic integer programs with a varying number of scenarios. We show that these test sets can be decomposed into finitely many "building blocks," independent of the number of scenarios, and we give an effective procedure to compute them. The paper includes an introduction to Nash-Williams' theory of better-quasi-orderings, which is used to show termination of our algorithm. We also apply this theory to finiteness results for Hilbert functions.
44 citations
TL;DR: It is shown that if a polynomial filled Julia set has empty interior, then it is computable and is therefore computable.
36 citations
TL;DR: The existence of relations between volumes and Intersection Theory in the presence of singularities is proved and an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety is proved.
Abstract: We exhibit some new techniques to study volumes of tubes about algebraic varieties in complex projective spaces We prove the existence of relations between volumes and Intersection Theory in the presence of singularities In particular, we can exhibit an average Bezout Equality for equidimensional varieties We also state an upper bound for the volume of a tube about a projective variety As a main outcome, we prove an upper bound estimate for the volume of the intersection of a tube with an equidimensional projective algebraic variety We apply these techniques to exhibit upper bounds for the probability distribution of the generalized condition number of singular complex matrices
27 citations
TL;DR: The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems and immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group.
Abstract: The process of integrating an nth-order scalar ordinary differential equation with symmetry is revisited in terms of Pfaffian systems. This formulation immediately provides a completely algebraic method to determine the initial conditions and the corresponding solutions which are invariant under a one parameter subgroup of a symmetry group. To determine the noninvariant solutions the problem splits into three cases. If the dimension of the symmetry groups is less than the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions can be found by integrating a quotient Pfaffian system on a quotient space, and integrating an equation of fundamental Lie type associated with the symmetry group. If the dimension of the symmetry group is equal to the order of the equation, then there exists an open dense set of initial conditions whose corresponding solutions are obtained either by solving an equation of fundamental Lie type associated with the symmetry group, or the solutions are invariant under a one-parameter subgroup. If the dimension of the symmetry group is greater than the order of the equation, then there exists an open dense set of initial conditions where the solutions can either be determined by solving an equation of fundamental Lie type for a solvable Lie group, or are invariant. In each case the initial conditions, the quotient Pfaffian system, and the equation of Lie type are all determined algebraically. Examples of scalar ordinary differential equations and a Pfaffian system are given.
TL;DR: In this paper, the authors identify a class of n-variate polynomials f for which the integral of any positive integer power fp over the whole space is well approximated by a properly scaled integral over a random subspace of dimension O(log n).
Abstract: We consider the problem of efficient integration of an n-variate polynomial with respect to the Gaussian measure in ℝn and related problems of complex integration and optimization of a polynomial on the unit sphere. We identify a class of n-variate polynomials f for which the integral of any positive integer power fp over the whole space is well approximated by a properly scaled integral over a random subspace of dimension O(log n). Consequently, the maximum of f on the unit sphere is well approximated by a properly scaled maximum on the unit sphere in a random subspace of dimension O(log n). We discuss connections with problems of combinatorial counting and applications to efficient approximation of a hafnian of a positive matrix.
TL;DR: In this article, it was shown that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced to counting the number of common zeros of a finite set of multivariate polynomials.
Abstract: We continue the study of counting complexity begun in [13], [14], [15] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. The reduction is based on a new formula for the coefficients of the Hilbert polynomial of a smooth variety. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #P-lower bound in Bach [1].
TL;DR: A new polynomial-time primal-dual interior-point algorithm for semidefinite programming that is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.
Abstract: Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K, and any positive real number r < 1, we associate, with each direction x e K, a second-order cone Kr(x) containing K. We show that K is the interior of the intersection of the second-order cones Kr(x), as x ranges over all directions in K. Using these second-order cones as approximations to cones of symmetric, positive definite matrices, we develop a new polynomial-time primal-dual interior-point algorithm for semidefinite programming. The algorithm is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.
TL;DR: The difference between the total mean curvatures of two closed surfaces in ${\Bbb R}^3$ in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose is bound.
Abstract: In this paper we bound the difference between the total mean curvatures of two closed surfaces in ℝ3 in terms of their total absolute curvatures and the Frechet distance between the volumes they enclose. The proof relies on a combination of methods from algebraic topology and integral geometry. We also bound the difference between the lengths of two curves using the same methods.
TL;DR: The regression problem is considered and an algorithm approximating the regression function by estimators piecewise constant on the elements of an adaptive partition is described, which gives a risk bound under the assumption that a "weak learning hypothesis" holds.
Abstract: We consider the regression problem and describe an algorithm approximating the regression function by estimators piecewise constant on the elements of an adaptive partition. The partitions are iteratively constructed by suitable random merges and splits, using cuts of arbitrary geometry. We give a risk bound under the assumption that a "weak learning hypothesis" holds, and characterize this hypothesis in terms of a suitable RKHS. Two examples illustrate the general results in two particularly interesting cases.