Symmetric tensor decomposition

We present an algorithm for decomposing a symmetric tensor, of dimension n and order d as a sum of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for binary forms. We recall the correspondence between the decomposition of a homogeneous polynomial in n variables of total degree d as a sum of powers of linear forms (Waring's problem), incidence properties on secant varieties of the Veronese Variety and the representation of linear forms as a linear combination of evaluations at distinct points. Then we reformulate Sylvester's approach from the dual point of view. Exploiting this duality, we propose necessary and sufficient conditions for the existence of such a decomposition of a given rank, using the properties of Hankel (and quasi-Hankel) matrices, derived from multivariate polynomials and normal form computations. This leads to the resolution of polynomial equations of small degree in non-generic cases. We propose a new algorithm for symmetric tensor decomposition, based on this characterization and on linear algebra computations with these Hankel matrices. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved global convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the rank

Constructing IGA-suitable planar parameterization from complex CAD boundary by domain partition and global/local optimization

In this paper, we propose a general framework for constructing IGA-suitable planar B-spline parameterizations from given complex CAD boundaries consisting of a set of B-spline curves. Instead of forming the computational domain by a simple boundary, planar domains with high genus and more complex boundary curves are considered. Firstly, some pre-processing operations including B\'ezier extraction and subdivision are performed on each boundary curve in order to generate a high-quality planar parameterization; then a robust planar domain partition framework is proposed to construct high-quality patch-meshing results with few singularities from the discrete boundary formed by connecting the end points of the resulting boundary segments. After the topology information generation of quadrilateral decomposition, the optimal placement of interior B\'ezier curves corresponding to the interior edges of the quadrangulation is constructed by a global optimization method to achieve a patch-partition with high quality. Finally, after the imposition of C1=G1-continuity constraints on the interface of neighboring B\'ezier patches with respect to each quad in the quadrangulation, the high-quality B\'ezier patch parameterization is obtained by a C1-constrained local optimization method to achieve uniform and orthogonal iso-parametric structures while keeping the continuity conditions between patches. The efficiency and robustness of the proposed method are demonstrated by several examples which are compared to results obtained by the skeleton-based parameterization approach

On the asymptotic and practical complexity of solving bivariate systems over the reals

This paper is concerned with exact real solving of well-constrained, bivariate polynomial systems. The main problem is to isolate all common real roots in rational rectangles, and to determine their intersection multiplicities. We present three algorithms and analyze their asymptotic bit complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based method, and \sOB(N^{12}) for two subresultant-based methods: this notation ignores polylogarithmic factors, where $N$ bounds the degree and the bitsize of the polynomials. The previous record bound was \sOB(N^{14}). Our main tool is signed subresultant sequences. We exploit recent advances on the complexity of univariate root isolation, and extend them to sign evaluation of bivariate polynomials over two algebraic numbers, and real root counting for polynomials over an extension field. Our algorithms apply to the problem of simultaneous inequalities; they also compute the topology of real plane algebraic curves in \sOB(N^{12}), whereas the previous bound was \sOB(N^{14}). All algorithms have been implemented in MAPLE, in conjunction with numeric filtering. We compare them against FGB/RS, system solvers from SYNAPS, and MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is among the most robust, and its runtimes are comparable, or within a small constant factor, with respect to the C/C++ libraries. Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure

A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI

International audienceIn this paper we extract the geometric characteristics from an antipodally symmetric spherical function (ASSF), which can be de- scribed equivalently in the spherical harmonic (SH) basis, in the symmet- ric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. All three bases span the same vector space and are bijective when the rank of the SH series equals the order of the ST and equals the degree of the HP. We show, therefore, how it is possible to extract the maxima and minima of an ASSF by computing the stationary points of a constrained HP. In Diffusion MRI, the Orientation Distribution Function (ODF), repre- sents a state of the art reconstruction method whose maxima are aligned with the dominant fiber bundles. It is, therefore, important to be able to correctly estimate these maxima to detect the fiber directions. The ODF is an ASSF. To illustrate the potential of our method, we take up the example of the ODF, and extract its maxima to detect the fiber directions. Thanks to our method we are able to extract the maxima without limiting our search to a discrete set of values on the sphere, but by searching the maxima of a continuous function. Our method is also general, not dependent on the ODF, and the framework we present can be applied to any ASSF described in one of the three bases

Real Algebraic Numbers: Complexity Analysis and Experimentations

International audienceWe present algorithmic, complexity and implementation results concerning real root isolation of a polynomial of degree $d$, with integer coefficients of bit size $\le\tau$, using Sturm (-Habicht) sequences and the Bernstein subdivision solver. In particular, we unify and simplify the analysis of both methods and we give an asymptotic complexity bound of \sOB( d^4 \tau^2). This matches the best known bounds. Moreover, we generalize this to cover the non-squarefree polynomials and show that within the same complexity we can also compute the multiplicities of the roots. We also consider algorithms for sign evaluation, comparison of real algebraic numbers and simultaneous inequalities (SI) and we improve the known bounds at least by a factor of $d$. Finally, we present our C++ implementation in Synaps and experiments on various data sets

Symmetric tensor decomposition

International audienceWe present an algorithm for decomposing a symmetric tensor of dimension n and order d as a sum of of rank-1 symmetric tensors, extending the algorithm of Sylvester devised in 1886 for symmetric tensors of dimension 2. We exploit the known fact that every symmetric tensor is equivalently represented by a homogeneous polynomial in n variables of total degree d. Thus the decomposition corresponds to a sum of powers of linear forms. The impact of this contribution is two-fold. First it permits an efficient computation of the decomposition of any tensor of sub-generic rank, as opposed to widely used iterative algorithms with unproved convergence (e.g. Alternate Least Squares or gradient descents). Second, it gives tools for understanding uniqueness conditions, and for detecting the tensor rank

A polynomial based approach to extract the maxima of an antipodally symmetric spherical function and its application to extract fiber directions from the Orientation Distribution Function in Diffusion MRI

International audienceIn this paper we extract the geometric characteristics from an antipodally symmetric spherical function (ASSF), which can be de- scribed equivalently in the spherical harmonic (SH) basis, in the symmet- ric tensor (ST) basis constrained to the sphere, and in the homogeneous polynomial (HP) basis constrained to the sphere. All three bases span the same vector space and are bijective when the rank of the SH series equals the order of the ST and equals the degree of the HP. We show, therefore, how it is possible to extract the maxima and minima of an ASSF by computing the stationary points of a constrained HP. In Diffusion MRI, the Orientation Distribution Function (ODF), repre- sents a state of the art reconstruction method whose maxima are aligned with the dominant fiber bundles. It is, therefore, important to be able to correctly estimate these maxima to detect the fiber directions. The ODF is an ASSF. To illustrate the potential of our method, we take up the example of the ODF, and extract its maxima to detect the fiber directions. Thanks to our method we are able to extract the maxima without limiting our search to a discrete set of values on the sphere, but by searching the maxima of a continuous function. Our method is also general, not dependent on the ODF, and the framework we present can be applied to any ASSF described in one of the three bases

RDR2 Partially Antagonizes the Production of RDR6-Dependent siRNA in Sense Transgene-Mediated PTGS

Background: RNA-DEPENDENT RNA POLYMERASE6 (RDR6) and SUPPRESSOR of GENE SILENCING 3 (SGS3) are required for DNA methylation and post-transcriptional gene silencing (PTGS) mediated by 21-nt siRNAs produced by sense transgenes (S-PTGS). In contrast, RDR2, but not RDR6, is required for DNA methylation and TGS mediated by 24-nt siRNAs, and for cellto-cell spreading of IR-PTGS mediated by 21-nt siRNAs produced by inverted repeat transgenes under the control of a phloem-specific promoter. Principal Findings: In this study, we examined the role of RDR2 and RDR6 in S-PTGS. Unlike RDR6, RDR2 is not required for DNA methylation of transgenes subjected to S-PTGS. RDR6 is essential for the production of siRNAs by transgenes subjected to S-PTGS, but RDR2 also contributes to the production of transgene siRNAs when RDR6 is present because rdr2 mutations reduce transgene siRNA accumulation. However, the siRNAs produced via RDR2 likely are counteractive in wildtype plants because impairement of RDR2 increases S-PTGS efficiency at a transgenic locus that triggers limited silencing, and accelerates S-PTGS at a transgenic locus that triggers efficient silencing. Conclusions/Significance: These results suggest that RDR2 and RDR6 compete for RNA substrates produced by transgenes subjected to S-PTGS. RDR2 partially antagonizes RDR6 because RDR2 action likely results in the production of counteractiv

Involvement of RDR6 in short-range intercellular RNA silencing in Nicotiana benthamiana

In plants, non-cell autonomous RNA silencing spreads between cells and over long distances. Recent work has revealed insight on the genetic and molecular components essential for cell-to-cell movement of RNA silencing in Arabidopsis. Using a local RNA silencing assay, we report on a distinct mechanism that may govern the short-range (6–10 cell) trafficking of virus-induced RNA silencing from epidermal to neighbouring palisade and spongy parenchyma cells in Nicotiana benthamiana. This process involves a previously unrecognised function of the RNA-dependent RNA polymerase 6 (RDR6) gene. Our data suggest that plants may have evolved distinct genetic controls in intercellular RNA silencing among different types of cells

RNA silencing can explain chlorotic infection patterns on plant leaves

<p>Abstract</p> <p>Background</p> <p>RNA silencing has been implicated in virus symptom development in plants. One common infection symptom in plants is the formation of chlorotic tissue in leaves. Chlorotic and healthy tissue co-occur on a single leaf and form patterns. It has been shown that virus levels in chlorotic tissue are high, while they are low in healthy tissue. Additionally, the presence of siRNAs is confined to the chlorotic spots and the boundaries between healthy and infected tissue. These results strongly indicate that the interaction between virus growth and RNA silencing plays a role in the formation of infection patterns on leaves. However, how RNA silencing leads to the intricate patterns is not known.</p> <p>Results</p> <p>Here we elucidate the mechanisms leading to infection patterns and the conditions which lead to the various patterns observed. We present a modeling approach in which we combine intra- and inter-cellular dynamics of RNA silencing and viral growth. We observe that, due to the spread of viruses and the RNA silencing response, parts of the tissue become infected while other parts remain healthy. As is observed in experiments high virus levels coincide with high levels of siRNAs, and siRNAs are also present in the boundaries between infected and healthy tissue. We study how single- and double-stranded cleavage by Dicer and amplification by RNA-dependent RNA polymerase can affect the patterns formed.</p> <p>Conclusion</p> <p>This work shows that RNA silencing and virus growth within a cell, and the local spread of virions and siRNAs between cells can explain the heterogeneous spread of virus in leaf tissue, and therewith the observed infection patterns in plants.</p