arXiv subjects

Luca Chiantini

Publications and source records attributed to Luca Chiantini.

At least 37 records · Page 2Linked to original sources

arXiv2020

On the Terracini locus of projective varieties

We introduce and study properties of the Terracini locus of projective varieties X, which is the locus of finite subsets S of X such that 2S fails to impose independent conditions to a linear system L. Terracini loci are relevant in the study of interpolation problems over double points in special position, but they also enter naturally in the study of special loci contained in secant varieties to projective varieties. We find some criteria which exclude that a set S belongs to the Terracini locus. Furthermore, in the case where X is a Veronese variety, we bound the dimension of the Terracini locus and we determine examples in which the locus has codimension 1 in the symmetric product of X.

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arXiv2020

On secant defective varieties, in particular of dimension 4

In this paper we prove some general results on secant defective varieties. Then we focus on the 4--dimensional case and we give the full classification of secant defective 4--folds. This paper has been inspired by classical work by G. Scorza,

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arXiv2020

Sets of points which project to complete intersections

The motivating problem addressed by this paper is to describe those non-degenerate sets of points $Z$ in $\mathbb P^3$ whose general projection to a general plane is a complete intersection of curves in that plane. One large class of such $Z$ is what we call $(m,n)$-grids. We relate this problem to the {\em unexpected cone property} ${\mathcal C}(d)$, a special case of the unexpected hypersurfaces which have been the focus of much recent research. After an analysis of ${\mathcal C}(d)$ for small $d$, we show that a non-degenerate set of $9$ points has a general projection that is the complete intersection of two cubics if and only if the points form a $(3,3)$-grid. However, in an appendix we describe a set of $24$ points that are not a grid but nevertheless have the projection property. These points arise from the $F_4$ root system. Furthermore, from this example we find subsets of $20$, $16$ and $12$ points with the same feature.

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arXiv2020

Minimality and uniqueness for decompositions of specific ternary forms

The paper deals with the computation of the rank and the identifiability of a specific ternary form. Often, one knows some short Waring decomposition of a given form, and the problem is to determine whether the decomposition is minimal and unique. We show how the analysis of the Hilbert-Burch matrix of the set of points representing the decomposition can solve this problem in the case of ternary forms. Moreover, when the decomposition is not unique, we show how the procedure of liaison can provide alternative, maybe shorter, decompositions. We give an explicit algorithm that tests our criterion of minimality for the case of ternary forms of degree $9$. This is the first numerical case in which a new phaenomenon appears: the span of $18$ general powers of linear forms contains points of (subgeneric) rank $18$, but it also contains points whose rank is $17$, due to the existence of a second shorter decomposition which is completely different from the given one.

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arXiv2020

On the identifiability of ternary forms

We describe a new method to determine the minimality and identifiability of a Waring decomposition $A$ of a specific form (symmetric tensor) $T$ in three variables. Our method, which is based on the Hilbert function of $A$, can distinguish between forms in the span of the Veronese image of $A$, which in general contains both identifiable and not identifiable points, depending on the choice of coefficients in the decomposition. This makes our method applicable for all values of the length $r$ of the decomposition, from $2$ up to the generic rank, a range which was not achievable before. Though the method in principle can handle all cases of specific ternary forms, we introduce and describe it in details for forms of degree $8$.

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arXiv2020

Catalecticant intersections and confinement of decompositions of forms

We introduce the notion of confinement of decompositions for forms or vector of forms. The confinement, when it holds, lowers the number of parameters that one needs to consider, in order to find all the possible decompositions of a given set of data. With the technique of confinement, we obtain here two results. First, we give a new, shorter proof of a result by London (\cite{London90}) that $3$ general plane cubics have $2$ simultaneous Waring decompositions of rank $6$. Then we compute, with the software Bertini, that $4$ general plane quartics have $18$ different decompositions of rank $10$ (a result which was not known before).

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arXiv2019

Identifiability for a class of symmetric tensors

We use methods of algebraic geometry to find new, effective methods for detecting the identifiability of symmetric tensors. In particular, for ternary symmetric tensors T of degree 7, we use the analysis of the Hilbert function of a finite projective set, and the Cayley-Bacharach property, to prove that, when the Kruskal's rank of a decomposition of T are maximal (a condition which holds outside a Zariski closed set of measure 0), then the tensor T is identifiable, i.e. the decomposition is unique, even if the rank lies beyond the range of application of both the Kruskal's and the reshaped Kruskal's criteria.

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arXiv2018

Hilbert functions and tensor analysis

We show how well known tools of algebraic geometry for the study of finite sets can be fruitfully applied to the study of Waring decompositions of symmetric tensors (forms). We mainly focus on the uniqueness of a given decomposition (the identifiability problem), and show how, in some cases, one can effectively determine the uniqueness even in some range in which the Kruskal's criterion does not apply.

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arXiv2017

Identifiability beyond Kruskal's bound for symmetric tensors of degree 4

We show how methods of algebraic geometry can produce criteria for the identifiability of specific tensors that reach beyond the range of applicability of the celebrated Kruskal criterion. More specifically, we deal with the symmetric identifiability of symmetric tensors in Sym$^4(\mathbb{C}^{n+1})$, i.e., quartic hypersurfaces in a projective space $\mathbb{P}^n$, that have a decomposition in 2n+1 summands of rank 1. This is the first case where the reshaped Kruskal criterion no longer applies. We present an effective algorithm, based on efficient linear algebra computations, that checks if the given decomposition is minimal and unique. The criterion is based on the application of advanced geometric tools, like Castelnuovo's lemma for the existence of rational normal curves passing through a finite set of points, and the Cayley-Bacharach condition on the postulation of finite sets. In order to apply these tools to our situation, we prove a reformulation of these results, hereby extending classical results such as Castelnuovo's lemma and the analysis of Geramita, Kreuzer, and Robbiano, "Cayley-Bacharach schemes and their canonical modules", Trans. Amer. Math. Soc. 339:443-452, 1993.

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arXiv2017

On the dimension of contact loci and the identifiability of tensors

Let $X\subset \mathbb{P}^r$ be an integral and non-degenerate variety. Set $n:= \dim (X)$. We prove that if the $(k+n-1)$-secant variety of $X$ has (the expected) dimension $(k+n-1)(n+1)-1<r$ and $X$ is not uniruled by lines, then $X$ is not $k$-weakly defective and hence the $k$-secant variety satisfies identifiability, i.e. a general element of it is in the linear span of a unique $S\subset X$ with $\sharp (S) =k$. We apply this result to many Segre-Veronese varieties and to the identifiability of Gaussian mixtures $G_{1,d}$. If $X$ is the Segre embedding of a multiprojective space we prove identifiability for the $k$-secant variety (assuming that the $(k+n-1)$-secant variety has dimension $(k+n-1)(n+1)-1<r$, this is a known result in many cases), beating several bounds on the identifiability of tensors.

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arXiv2017

Polynomials and the exponent of matrix multiplication

We define tensors, corresponding to cubic polynomials, which have the same exponent $ω$ as the matrix multiplication tensor. In particular, we study the symmetrized matrix multiplication tensor $sM_n$ defined on an $n\times n$ matrix $A$ by $sM_n(A)=trace(A^3)$. The use of polynomials enables the introduction of additional techniques from algebraic geometry in the study of the matrix multiplication exponent $ω$.

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arXiv2017

Bounds on the tensor rank

We give a sufficient criterion for a lower bound of the cactus rank of a tensor. Then we refine that criterion in order to be able to give an explicit sufficient condition for a non-redundant decomposition of a tensor to be minimal and unique.

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arXiv2017

Waring-like decompositions of polynomials - 1

Let $F$ be a homogeneous form of degree $d$ in $n$ variables. A Waring decomposition of $F$ is a way to express $F$ as a sum of $d^{th}$ powers of linear forms. In this paper we consider the decompositions of a form as a sum of expressions, each of which is a fixed monomial evaluated at linear forms.

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arXiv2016

The geometry of rank decompositions of matrix multiplication I: 2x2 matrices

This is the first in a series of papers on rank decompositions of the matrix multiplication tensor. In this paper we: establish general facts about rank decompositions of tensors, describe potential ways to search for new matrix multiplication decompositions, give a geometric proof of the theorem of Burichenko's theorem establishing the symmetry group of Strassen's algorithm, and present two particularly nice subfamilies in the Strassen family of decompositions.

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arXiv2016

Real identifiability vs complex identifiability

Let $T$ be a real tensor of (real) rank $r$. $T$ is 'identifiable' when it has a unique decomposition in terms of rank $1$ tensors. There are cases in which the identifiability fails over the complex field, for general tensors of rank $r$. This behavior is quite peculiar when the rank $r$ is submaximal. Often, the failure is due to the existence of an elliptic normal curve through general points of the corresponding Segre, Veronese or Grassmann variety. We prove the existence of nonempty euclidean open subsets of some variety of tensors of rank $r$, whose elements have several decompositions over $\mathbb C$, but only one of them is formed by real summands. Thus, in the open sets, tensors are not identifiable over $\mathbb C$, but are identifiable over $\mathbb R$. We also provide examples of non trivial euclidean open subsets in a whole space of symmetric tensors (of degree $7$ and $8$ in three variables) and of almost unbalanced tensors Segre Product ($\mathbb P^2\times \mathbb P^4\times \mathbb P^9$) whose elements have typical real rank equal to the complex rank, and are identifiable over $\mathbb R$, but not over $\mathbb C$. On the contrary, we provide examples of tensors of given real rank, for which real identifiability cannot hold in non-trivial open subsets.

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arXiv2016

Effective criteria for specific identifiability of tensors and forms

In applications where the tensor rank decomposition arises, one often relies on its identifiability properties for interpreting the individual rank-$1$ terms appearing in the decomposition. Several criteria for identifiability have been proposed in the literature, however few results exist on how frequently they are satisfied. We propose to call a criterion effective if it is satisfied on a dense, open subset of the smallest semi-algebraic set enclosing the set of rank-$r$ tensors. We analyze the effectiveness of Kruskal's criterion when it is combined with reshaping. It is proved that this criterion is effective for both real and complex tensors in its entire range of applicability, which is usually much smaller than the smallest typical rank. Our proof explains when reshaping-based algorithms for computing tensor rank decompositions may be expected to recover the decomposition. Specializing the analysis to symmetric tensors or forms reveals that the reshaped Kruskal criterion may even be effective up to the smallest typical rank for some third, fourth and sixth order symmetric tensors of small dimension as well as for binary forms of degree at least three. We extended this result to $4 \times 4 \times 4 \times 4$ symmetric tensors by analyzing the Hilbert function, resulting in a criterion for symmetric identifiability that is effective up to symmetric rank $8$, which is optimal.

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arXiv2016

On generic identifiability of symmetric tensors of subgeneric rank

We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional cases arise, all of which were known in the literature. Our original contribution regards the case of cubics ($d=3$), while for $d\ge 4$ we rely on known results on weak defectivity by Ballico, Ciliberto, Chiantini, and Mella.

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arXiv2015

Symmetric tensors: rank and Strassen's conjecture

In this paper we introduce the notion of linear computability as a method of finding the Waring rank of forms. We use this notion to find infinitely many new examples which satisfy Strassen's Conjecture.

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