arXiv subjects

Zhiqiang Xu

Publications and source records attributed to Zhiqiang Xu.

122 records · Page 7Linked to original sources

arXiv2011

A remark about orthogonal matching pursuit algorithm

In this note, we investigate the theoretical properties of Orthogonal Matching Pursuit (OMP), a class of decoder to recover sparse signal in compressed sensing. In particular, we show that the OMP decoder can give $(p,q)$ instance optimality for a large class of encoders with $1\leq p\leq q \leq 2$ and $(p,q)\neq (2,2)$. We also show that, if the encoding matrix is drawn from an appropriate distribution, then the OMP decoder is $(2,2)$ instance optimal in probability.

cs.IT↗

arXiv2011

Deterministic Sampling of Sparse Trigonometric Polynomials

One can recover sparse multivariate trigonometric polynomials from few randomly taken samples with high probability (as shown by Kunis and Rauhut). We give a deterministic sampling of multivariate trigonometric polynomials inspired by Weil's exponential sum. Our sampling can produce a deterministic matrix satisfying the statistical restricted isometry property, and also nearly optimal Grassmannian frames. We show that one can exactly reconstruct every $M$-sparse multivariate trigonometric polynomial with fixed degree and of length $D$ from the determinant sampling $X$, using the orthogonal matching pursuit, and $# X$ is a prime number greater than $(M\log D)^2$. This result is almost optimal within the $(\log D)^2 $ factor. The simulations show that the deterministic sampling can offer reconstruction performance similar to the random sampling.

math.NA↗

arXiv2010

Multivariate Splines and Polytopes

In this paper, we use multivariate splines to investigate the volume of polytopes. We first present an explicit formula for the multivariate truncated power, which can be considered as a dual version of the famous Brion's formula for the volume of polytopes. We also prove that the integration of polynomials over polytopes can be dealt with by the multivariate truncated power. Moreover, we show that the volume of the cube slicing can be considered as the maximum value of the box spline. Based on this connection, we give a simple proof for Good's conjecture, which has been settled by probability methods.

math.NA↗

arXiv2010

Application of multivariate splines to discrete mathematics

Using methods developed in multivariate splines, we present an explicit formula for discrete truncated powers, which are defined as the number of non-negative integer solutions of linear Diophantine equations. We further use the formula to study some classical problems in discrete mathematics as follows. First, we extend the partition function of integers in number theory. Second, we exploit the relation between the relative volume of convex polytopes and multivariate truncated powers and give a simple proof for the volume formula for the Pitman-Stanley polytope. Third, an explicit formula for the Ehrhart quasi-polynomial is presented.

math.CO↗

arXiv2010

Hierarchical zonotopal spaces

Zonotopal algebra interweaves algebraic, geometric and combinatorial properties of a given linear map X. Of basic significance in this theory is the fact that the algebraic structures are derived from the geometry (via a non-linear procedure known as "the least map"), and that the statistics of the algebraic structures (e.g., the Hilbert series of various polynomial ideals) are combinatorial, i.e., computable using a simple discrete algorithm known as "the valuation function". On the other hand, the theory is somewhat rigid since it deals, for the given X, with exactly two pairs each of which is made of a nested sequence of three ideals: an external ideal (the smallest), a central ideal (the middle), and an internal ideal (the largest). In this paper we show that the fundamental principles of zonotopal algebra as described in the previous paragraph extend far beyond the setup of external, central and internal ideals by building a whole hierarchy of new combinatorially defined zonotopal spaces.

math.AC↗

arXiv2009

Subspace modelling for structured noise suppression

The problem of structured noise suppression is addressed by i)modelling the subspaces hosting the components of the signal conveying the information and ii)applying a non-extensive nonlinear technique for effecting the right separation. Although the approach is applicable to all situations satisfying the hypothesis of the proposed framework, this work is motivated by a particular scenario, namely, the cancellation of low frequency noise in broadband seismic signals.

math-ph↗

arXiv2009

Adaptive non-uniform B-spline dictionaries on a compact interval

Non-uniform B-spline dictionaries on a compact interval are discussed. For each given partition, dictionaries of B-spline functions for the corresponding spline space are constructed. It is asserted that, by dividing the given partition into subpartitions and joining together the bases for the concomitant subspaces, slightly redundant dictionaries of B-splines functions are obtained. Such dictionaries are proved to span the spline space associated to the given partition. The proposed construction is shown to be potentially useful for the purpose of sparse signal representation. With that goal in mind, spline spaces specially adapted to produce a sparse representation of a given signal are considered.

math.FA↗

arXiv2009

Marginal Likelihood Integrals for Mixtures of Independence Models

Inference in Bayesian statistics involves the evaluation of marginal likelihood integrals. We present algebraic algorithms for computing such integrals exactly for discrete data of small sample size. Our methods apply to both uniform priors and Dirichlet priors. The underlying statistical models are mixtures of independent distributions, or, in geometric language, secant varieties of Segre-Veronese varieties.

stat.CO↗

arXiv2009

A remark about Mahler's conjecture and the maximum value of box splines

In this paper, we recast a special case of Mahler'c conjecture by the maximum value of box splines. This is the case of polytopes with at most $2n+2$ facets. An asymptotic formula for univariate box splines is given. Based on the formula, Mahler's conjecture is proved in this case provided $n$ is big enough.

math.MG↗

arXiv2008

A spline interpretation of Eulerian numbers

In this paper, we explore the interrelationship between Eulerian numbers and B splines. Specifically, using B splines, we give the explicit formulas of the refined Eulerian numbers, and descents polynomials. Moreover, we prove that the coefficients of descent polynomials $D_d^n(t)$ are log-concave. This paper also provides a new approach to study Eulerian numbers and descent polynomials.

math.NA↗

arXiv2008

Refinement Equations and Spline Functions

In this paper, we exploit the relation between the regularity of refinable functions with non-integer dilations and the distribution of powers of a fixed number modulo 1, and show the nonexistence of a non-trivial {\bf C}^{\infty} solution of the refinement equation with non-integer dilations. Using this, we extend the results on the refinable splines with non-integer dilations and construct a counterexample to some conjecture concerning the refinable splines with non-integer dilations. Finally, we study the box splines satisfying the refinement equation with non-integer dilation and translations. Our study involves techniques from number theory and harmonic analysis.

math.NA↗

arXiv2008

Discrete schemes for Gaussian curvature and their convergence

In this paper, several discrete schemes for Gaussian curvature are surveyed. The convergence property of a modified discrete scheme for the Gaussian curvature is considered. Furthermore, a new discrete scheme for Gaussian curvature is resented. We prove that the new scheme converges at the regular vertex with valence not less than 5. By constructing a counterexample, we also show that it is impossible for building a discrete scheme for Gaussian curvature which converges over the regular vertex with valence 4. Finally, asymptotic errors of several discrete scheme for Gaussian curvature are compared.

cs.CV↗

arXiv2008

Sagbi Bases of Cox-Nagata Rings

We degenerate Cox-Nagata rings to toric algebras by means of sagbi bases induced by configurations over the rational function field. For del Pezzo surfaces, this degeneration implies the Batyrev-Popov conjecture that these rings are presented by ideals of quadrics. For the blow-up of projective n-space at n+3 points, sagbi bases of Cox-Nagata rings establish a link between the Verlinde formula and phylogenetic algebraic geometry, and we use this to answer questions due to D'Cruz-Iarobbino and Buczynska-Wisniewski. Inspired by the zonotopal algebras of Holtz and Ron, our study emphasizes explicit computations, and offers a new approach to Hilbert functions of fat points.

math.AG↗

arXiv2005

An explicit formulation for two dimensional vector partition functions

Based on discrete truncated powers, the beautiful Popoviciu's formulation for restricted integer partition function is generalized. An explicit formulation for two dimensional multivariate truncated power functions is presented. Therefore, a simplified explicit formulation for two dimensional vector partition functions is given. Moreover, the generalized Frobenius problem is also discussed.

math.CO↗