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

Extensors and the Hilbert scheme

The Hilbert scheme $\mathbf{Hilb}^n_{p(t)}$ parametrizes closed subschemes and families of closed subschemes in the projective space $\mathbb{P}^n$ with a fixed Hilbert polynomial $p(t)$. It can be realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we consider schemes over a field $k$ of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian $\mathbf{Gr}^{N(r)}_{p(r)}$, where $N(r) = h^0(O_{\mathbb{P}^n}(r))$. Moreover, we exhibit explicit equations defining it in the Plücker coordinates of the Plücker embedding of $\mathbf{Gr}^{N(r)}_{p(r)}$. Our proof of existence does not need some of the classical tools used in previous proofs, as flattening stratifications and Gotzmann’s Persistence Theorem. The degree of our equations is $\deg p(t) + 2$, lower than the degree of the equations given by Iarrobino and Kleiman in 1999 and also lower (except for the case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels in 2004 after Bayer’s conjecture in 1982. The novelty of our approach mainly relies on the deeper attention to the intrinsic symmetries of the Hilbert scheme and on some results about Grassmannian based on the notion of extensors

Extensors and the Hilbert scheme

The Hilbert scheme $\mathbf{Hilb}_{p(t)}^{n}$ parametrizes closed subschemes and families of closed subschemes in the projective space $\mathbb{P}^n$ with a fixed Hilbert polynomial $p(t)$. It is classically realized as a closed subscheme of a Grassmannian or a product of Grassmannians. In this paper we consider schemes over a field $k$ of characteristic zero and we present a new proof of the existence of the Hilbert scheme as a subscheme of the Grassmannian $\mathbf{Gr}_{p(r)}^{N(r)}$, where $N(r)= h^0 (\mathcal{O}_{\mathbb{P}^n}(r))$. Moreover, we exhibit explicit equations defining it in the Pl\"ucker coordinates of the Pl\"ucker embedding of $\mathbf{Gr}_{p(r)}^{N(r)}$. Our proof of existence does not need some of the classical tools used in previous proofs, as flattening stratifications and Gotzmann's Persistence Theorem. The degree of our equations is $\text{deg} p(t)+2$, lower than the degree of the equations given by Iarrobino and Kleiman in 1999 and also lower (except for the case of hypersurfaces) than the degree of those proved by Haiman and Sturmfels in 2004 after Bayer's conjecture in 1982. The novelty of our approach mainly relies on the deeper attention to the intrinsic symmetries of the Hilbert scheme and on some results about Grassmannian based on the notion of extensors.Comment: Added equations of the Hilbert schemes of 2 points in the plane, 3-space, 4-space and of 3 points in the plane (a Macaulay2 file with the complete computation is available at http://tinyurl.com/EquationsHilbPoints-m2). Final version. To appear on Annali della Scuola Normale Superiore di Pisa - Classe di Scienz

General Tensor Decomposition, Moment Matrices and Applications

SubmittedInternational audienceThe tensor decomposition addressed in this paper may be seen as a generalisation of Singular Value Decomposition of matrices. We consider general multilinear and multihomogeneous tensors. We show how to reduce the problem to a truncated moment matrix problem and give a new criterion for flat extension of Quasi-Hankel matrices. We connect this criterion to the commutation characterisation of border bases. A new algorithm is described. It applies for general multihomogeneous tensors, extending the approach of J.J. Sylvester to binary forms. An example illustrates the algebraic operations involved in this approach and how the decomposition can be recovered from eigenvector computation

Schémas de Hilbert et décompositions de tenseurs

This thesis consists of two parts. The first one contains chapter 2 and 3 and is about the Hilbert scheme. These chapters correspond to joint works with M.E. Alonso and B. Mourrain : [3] and with P. Lella, B. Mourrain and M. Roggero : [10]. We are interested in the equations that define it as a closed sub-scheme of the Grassman- nian and especially their degree. We will give new global equations, more simple than those already known. Chapter 2 is about the case of constant Hilbert polynomial equal to μ. First, we will briefly recall the defini- μ tions and propositions related to the Hilbert functor associated to μ, denoted by HilbPn . Then we will prove that it is representable, we will use a local approach and build a covering of open representable sub-functors whose equations correspond to commutation relations that characterize border basis. The scheme that repre- The second part of this thesis is concerned with tensors decomposition, chapter 4. We will begin with the sym- metric case which corresponds to a joint work with P. Comon, B. Mourrain and E. Tsigaridas : [9]. We will extend the algorithm devised by Sylvester for the binary case. We will use a dual approach and give necessary and sufficient conditions for the existence of a decomposition of a given rank, using Hankel operators. We will deduce an algorithm for the symmetric case. Finally, we will conclude by studying the case of general tensors which corresponds to a joint work with A. Bernardi, P. Comon and B. Mourrain : [6]. In particular, we will prove how the formalism that as been used so far for the symmetric case, can be extended to solve the problem. μμn sents HilbPn is called the Hilbert scheme associated to μ and is denoted by Hilb (P ). Then, thanks to the theorems of Persistence and Regularity of Gotzmann, we will give a global description of Hilbμ(Pn). We will provide a set of homogeneous equations of degree 2 in the Plücker coordinates that characterizes Hilbμ(Pn) as a closed sub-scheme of the Grassmannian. We will finally conclude this chapter by studying the tangent plan of the Hilbert scheme. Chapter 3 deals with the general case of Hilbert scheme associated to a Hilbert polynomial P of degree d ≥ 0, denoted by HilbP (Pn). We will generalize chapter 2, giving homogeneous equations of degree d + 2 in the Plücker coordinates. The second part of this thesis is concerned with tensors decomposition, chapter 4. We will begin with the symmetric case which corresponds to a joint work with P. Comon, B. Mourrain and E. Tsigaridas : [9]. We will extend the algorithm devised by Sylvester for the binary case. We will use a dual approach and give necessary and sufficient conditions for the existence of a decomposition of a given rank, using Hankel operators. We will deduce an algorithm for the symmetric case. Finally, we will conclude by studying the case of general tensors which corresponds to a joint work with A. Bernardi, P. Comon and B. Mourrain : [6]. In particular, we will prove how the formalism that as been used so far for the symmetric case, can be extended to solve the problem.Cette thèse est constituée de deux parties. La première regroupe les chapitres 2 et 3 et traite du schéma de Hilbert. Ces chapitres correspondent respectivement à des travaux en collaboration avec M.E. Alonso et B. Mourrain : [3] et avec P. Lella, B. Mourrain et M. Roggero : [10]. Nous nous intéresserons aux équations qui le définissent comme sous-schéma fermé de la grassmannienne et plus précisément à leur degré. Nous fournirons ainsi de nouvelles équations globales, plus simples que celles qui existent déjà. Le chapitre 2 se concentre sur le cas des polynômes de Hilbert constants égaux à μ. Après avoir rappelé les définitions et propriétés élémen- μ taires du foncteur de Hilbert associé à μ, noté HilbPn , nous montrerons que celui-ci est représentable. Nous adopterons pour cela une approche locale et construirons un recouvrement ouvert de sous-foncteurs représen- tables, dont les équations correspondent aux relations de commutation qui caractérisent les bases de bord. Son représentant s'appelle le schéma de Hilbert associé à μ, noté Hilbμ(Pn). Nous fournirons ensuite, grâce aux théorèmes de Persistance et de Régularité de Gotzmann, une description globale de ce schéma. Nous donne- rons un système d'équations homogènes de degré 2 en les coordonnées de Plücker qui caractérise Hilbμ(Pn) comme sous-schéma fermé de la Grassmannienne. Nous conclurons ce chapitre par une étude du plan tangent au schéma de hilbert en exploitant l'approche locale et les relations de commutation précédemment introduites. Le chapitre 3 traite le cas général du schéma de Hilbert associé à un polynôme P de degré d ≥ 0, noté HilbP (Pn). Nous généraliserons le chapitre précédent en fournissant des équations globales homogènes de degré d + 2 en les coordonnées de Plücker. La deuxième partie de cette thèse concerne la décomposition de tenseurs, chapitre 4. Nous commencerons par étudier le cas symétrique, qui correspond à l'article [9] en collaboration avec P. Comon, B. Mourrain et E. Tsi- garidas. Nous étendrons pour cela l'algorithme de Sylvester proposé pour le cas binaire. Nous utiliserons une approche duale et fournirons des conditions nécessaires et suffisantes pour l'existence d'une décomposition de rang donné, en utilisant les opérateurs de Hankel. Nous en déduirons un algorithme pour le cas symétrique. Nous aborderons aussi la question de l'unicité de la décomposition minimale. Enfin, nous conclurons en étu- diant le cas des tenseurs généraux qui correspond à un article en collaboration avec A. Bernardi, P. Comon et B. Mourrain : [6]. Nous montrerons en particulier comment le formalisme introduit pour le cas symétrique peut s'adapter pour résoudre le problème

Genomic profiling reveals that transient adipogenic activation is a hallmark of mouse models of skeletal muscle regeneration.

The marbling of skeletal muscle by ectopic adipose tissue is a hallmark of many muscle diseases, including sarcopenia and muscular dystrophies, and generally associates with impaired muscle regeneration. Although the etiology and the molecular mechanisms of ectopic adipogenesis are poorly understood, fatty regeneration can be modeled in mice using glycerol-induced muscle damage. Using comprehensive molecular and histological profiling, we compared glycerol-induced fatty regeneration to the classical cardiotoxin (CTX)-induced regeneration model previously believed to lack an adipogenic response in muscle. Surprisingly, ectopic adipogenesis was detected in both models, but was stronger and more persistent in response to glycerol. Importantly, extensive differential transcriptomic profiling demonstrated that glycerol induces a stronger inflammatory response, and promotes adipogenic regulatory networks while reducing fatty acid β-oxidation. Altogether, these results provide a comprehensive repository of gene expression changes during the time course of two muscle regeneration models, and strongly suggest that adipogenic commitment is a hallmark of muscle regeneration, which can lead to ectopic adipocyte accumulation in response to specific physiopathological challenge

Genomic Profiling Reveals That Transient Adipogenic Activation Is a Hallmark of Mouse Models of Skeletal Muscle Regeneration

<div><p>The marbling of skeletal muscle by ectopic adipose tissue is a hallmark of many muscle diseases, including sarcopenia and muscular dystrophies, and generally associates with impaired muscle regeneration. Although the etiology and the molecular mechanisms of ectopic adipogenesis are poorly understood, fatty regeneration can be modeled in mice using glycerol-induced muscle damage. Using comprehensive molecular and histological profiling, we compared glycerol-induced fatty regeneration to the classical cardiotoxin (CTX)-induced regeneration model previously believed to lack an adipogenic response in muscle. Surprisingly, ectopic adipogenesis was detected in both models, but was stronger and more persistent in response to glycerol. Importantly, extensive differential transcriptomic profiling demonstrated that glycerol induces a stronger inflammatory response and promotes adipogenic regulatory networks while reducing fatty acid β-oxidation. Altogether, these results provide a comprehensive mapping of gene expression changes during the time course of two muscle regeneration models, and strongly suggest that adipogenic commitment is a hallmark of muscle regeneration, which can lead to ectopic adipocyte accumulation in response to specific physio-pathological challenges.</p></div

Adipogenesis and β-oxidation are differentially regulated in muscle after glycerol or CTX injection.

<p>qPCR analysis of the mRNA levels of different adipogenic (A), or in fatty-acid oxidation (B) regulators. Data are expressed as mean ± s.e.m., n = 5–6/group. * p-value <0.05 <i>vs</i>. control, # p-value <0.05 in Glycerol <i>vs</i>. CTX at same time points. Acadm, acyl-CoA dehydrogenase medium; Acs/l, acyl-CoA synthesase short-/long-chain; Acss, Acetyl-coenzyme A synthetase; Acox, Acyl-coenzyme A oxidase, Palmitoyl; C/EBP: CCAAT/ Enhancer binding protein; Cpt, carnitine palmitoyltransferase; Hadh, hydroxyacyl-CoA dehydrogenase; PPAR, peroxisome proliferator activated receptor.</p

Gene set enrichment mapping of glycerol- <i>vs.</i> CTX-injected muscle.

<p>Gene set enrichment analysis was performed on glycerol-injected compared to CTX-injected muscles 3 and 7 days after injection, and clustered according to gene set ontology. The size of nodes is proportional to the number of genes contained in the gene set. Red nodes: gene sets upregulated in glycerol <i>vs</i>. CTX model, blue nodes: gene sest downregulated in glycerol <i>vs</i>. CTX model, green bar: link between two gene sets sharing regulated genes.</p

Ectopic adipogenesis occurs in both glycerol- and CTX-induced muscle regeneration.

<p>(A) qPCR analysis of the mRNA level of the platelet-derived growth factor receptor alpha (PDGFRα). (B) Cryosections were performed at the mid-belly part of TA and subjected to H&E and perilipin staining at each time points after injection. Representative perilipin (green) /DAPI (blue) fluorescent stainings at 21 dpi are shown next to an H&E staining of the same region. Scale bars, 50 μm. (C), Quantitative analysis of perilipin expression assessed by counting and measuring the area of all perilipin expressing cells per section. Data are expressed as mean ± s.e.m., n = 5–6/group. * p-value <0.05 <i>vs</i>. control, # p-value <0.05 in Glycerol <i>vs</i>. CTX at same time points.</p