Results 1  10
of
19
Universality in Polytope Phase Transitions and Message Passing Algorithms
, 2012
"... We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are ..."
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Cited by 24 (4 self)
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We consider a class of nonlinear mappings FA,N in R N indexed by symmetric random matrices A ∈ R N×N with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Erwin Bolthausen. Within information theory, they are known as ‘approximate message passing ’ algorithms. We study the highdimensional (large N) behavior of the iterates of F for polynomial functions F, and prove that it is universal, i.e. it depends only on the first two moments of the entries of A, under a subgaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves –for a broad class of random projections – a conjecture by David Donoho and Jared Tanner. 1 Introduction and main results Let A ∈ RN×N be a random Wigner matrix, i.e. a random matrix with i.i.d. entries Aij satisfying E{Aij} = 0 and E{A2 ij} = 1/N. Considerable effort has been devoted to studying the distribution of the eigenvalues of such a matrix [AGZ09, BS05, TV12]. The universality phenomenon is a striking recurring theme in these studies. Roughly speaking, many asymptotic properties of the joint eigenvalues
Tourís, Weierstrass’ Theorem with weights
 J. Approx. Theory
"... Abstract. In this paper we study the set of functions Gvalued which can be approximated by Gvalued continuous functions in the norm L ∞ G (I, w), where I is a compact interval, G is a real and separable Hilbert space and w is certain Gvalued weakly measurable weight. Thus, we obtain a new extensi ..."
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Cited by 8 (6 self)
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Abstract. In this paper we study the set of functions Gvalued which can be approximated by Gvalued continuous functions in the norm L ∞ G (I, w), where I is a compact interval, G is a real and separable Hilbert space and w is certain Gvalued weakly measurable weight. Thus, we obtain a new extension of celebrated Weierstrass approximation theorem. Key words and phrases. Weierstrass ’ theorem, Gvalued weights, Gvalued polynomials, Gvalued continuous functions.
Learning Halfspaces Under LogConcave Densities: Polynomial Approximations and Moment Matching
"... We give the first polynomialtime algorithm for agnostically learning any function of a constant number of halfspaces with respect to any logconcave distribution (for any constant accuracy parameter). This result was not known even for the case of PAC learning the intersection of two halfspaces. We ..."
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Cited by 6 (2 self)
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We give the first polynomialtime algorithm for agnostically learning any function of a constant number of halfspaces with respect to any logconcave distribution (for any constant accuracy parameter). This result was not known even for the case of PAC learning the intersection of two halfspaces. We give two very different proofs of this result. The first develops a theory of polynomial approximation for logconcave measures and constructs a lowdegree ℓ1 polynomial approximator for sufficiently smooth functions. The second uses techniques related to the classical moment problem to obtain sandwiching polynomials. Both approaches deviate significantly from known Fourierbased methods, where essentially all previous work required the underlying distribution to have some product structure. Additionally, we show that in the smoothedanalysis setting, the above results hold with respect to distributions that have subexponential tails, a property satisfied by many natural and wellstudied distributions in machine learning.
Polynomial Approximation and . . .
, 2007
"... About twenty years ago the measure of smoothness ω r ϕ(f,t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time. ..."
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Cited by 1 (0 self)
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About twenty years ago the measure of smoothness ω r ϕ(f,t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time.
CONSTRUCTING NESTED NODAL SETS FOR MULTIVARIATE POLYNOMIAL INTERPOLATION ∗
"... Abstract. We present a robust method for choosing multivariate polynomial interpolation nodes. Our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity, a variant of a weighted Lebesgue function. Nodes are therefore chosen that tend to control oscillations in ..."
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Abstract. We present a robust method for choosing multivariate polynomial interpolation nodes. Our algorithm is an optimization method to greedily minimize a measure of interpolant sensitivity, a variant of a weighted Lebesgue function. Nodes are therefore chosen that tend to control oscillations in the resulting interpolant. This method can produce an arbitrary number of nodes and is not constrained by the dimension of a complete polynomial space. Our method is therefore flexible: nested nodal sets are produced in spaces of arbitrary dimensions, and the number of nodes added at each stage can be arbitrary. The algorithm produces a nodal set given a probability measure on the input space, thus parameterizing interpolants with respect to finite measures. We present examples to show that the method yields nodal sets that behave well with respect to standard interpolation diagnostics: the Lebesgue constant, the Vandermonde determinant, and the Vandermonde condition number. We also show that a nongreedy version of the nodal array has a strong connection with equilibrium measures from weighted pluripotential theory.
A LAGRANGETYPE PROJECTOR ON THE REAL LINE
, 2010
"... We introduce an interpolation process based on some of the zeros of the mth generalized Freud polynomial. Convergence results and error estimates are given. In particular we show that, in some important function spaces, the interpolating polynomial behaves like the best approximation. Moreover the ..."
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Cited by 1 (1 self)
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We introduce an interpolation process based on some of the zeros of the mth generalized Freud polynomial. Convergence results and error estimates are given. In particular we show that, in some important function spaces, the interpolating polynomial behaves like the best approximation. Moreover the stability and the convergence of some quadrature rules are proved.
THE DEGREE OF SHAPE PRESERVING WEIGHTED POLYNOMIAL APPROXIMATION
, 2011
"... We analyze the degree of shape preserving weighted polynomial approximation for exponential weights on the whole real line. In particular, we establish a Jackson type estimate. ..."
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We analyze the degree of shape preserving weighted polynomial approximation for exponential weights on the whole real line. In particular, we establish a Jackson type estimate.
Contents
, 2007
"... About twenty years ago the measure of smoothness ωr ϕ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time. ..."
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About twenty years ago the measure of smoothness ωr ϕ (f, t) was introduced and related to the rate of polynomial approximation. In this article we survey developments about this and related concepts since that time.