On Comon’s and Strassen’s conjectures
Comon’s conjecture on the equality of the rank and the symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors are two of the most challenging and guiding problems in the area of tensor decomposition. We survey the main known results on these conjectures, and, under suitable bounds on the rank, we prove them, building on classical techniques used in the case of symmetric tensors, for mixed tensors. Finally, we improve the bound for Comon’s conjecture given by flattenings by producing new equations for secant varieties of Veronese and Segre varieties.
Key words and phrases:Strassen’s conjecture, Comon’s conjecture, Tensor decomposition, Waring decomposition
2010 Mathematics Subject Classification:Primary 15A69, 15A72, 11P05; Secondary 14N05, 15A69
Let be an irreducible and reduced non-degenerate variety. The rank with respect to of a point is the minimal integer such that lies in the linear span of distinct points of . In particular, if we have that .
Since the -secant variety of is the subvariety of obtained as the closure of the union of all -planes spanned by general points of , for a general point we have .
When the ambient projective space is a space parametrizing tensors we enter the area of tensor decomposition. A tensor rank decomposition expresses a tensor as a linear combination of simpler tensors. More precisely, given a tensor , lying in a given tensor space over a field , a tensor rank-1 decomposition of is an expression of the form
where the ’s are linearly independent rank one tensors, and . The rank of is the minimal positive integer such that admits such a decomposition.
Tensor decomposition problems come out naturally in many areas of mathematics and applied sciences. For instance, in signal processing, numerical linear algebra, computer vision, numerical analysis, neuroscience, graph analysis, control theory and electrical networks [KB09], [CM96], [CGLM08], [LO15], [MR13], [MR14], [BFFX17]. In pure mathematics tensor decomposition issues arise while studying the additive decompositions of a general tensor [Dol04], [DK93], [MM13], [Mas16], [RS00], [TZ11], [MMS18].
Comon’s conjecture [CGLM08], which states the equality of the rank and symmetric rank of a symmetric tensor, and Strassen’s conjecture on the additivity of the rank of tensors [Str73] are two of the most important and guiding problems in the area of tensor decomposition.
More precisely, Comon’s conjecture predicts that the rank of a homogeneous polynomial with respect to the Veronese variety is equal to its rank with respect to the Segre variety into which is diagonally embedded, that is .
Strassen’s conjecture was originally stated for triple tensors and then generalized to a number of different contexts. For instance, for homogeneous polynomials it says that if and are homogeneous polynomials in distinct sets of variables then .
In Sections 2 and 3, while surveying the state of the art on Comon’s and Strassen’s conjectures, we push a bit forward some standard techniques, based on catalecticant matrices and more generally on flattenings, to extend some results on these conjectures, known in the setting of Veronese and Segre varieties, for Segre-Veronese and Segre-Grassmann varieties that is to the context of mixed tensors.
In Section 4 we introduce a method to improve a classical result on Comon’s conjecture. By standard arguments involving catalecticant matrices it is not hard to prove that Comon’s conjecture holds for the general polynomial in of symmetric rank as soon as , see Proposition 2.2. We manage to improve this bound looking for equations for the -secant variety , not coming from catalecticant matrices, that are restrictions to the space of symmetric tensors of equations of the -secant variety . We will do so by embedding the space of degree polynomials into the space of degree polynomials by mapping to and then considering suitable catalecticant matrices of rather than those of itself.
Implementing this method in Macaulay2 we are able to prove for instance that Comon’s conjecture holds for the general cubic polynomial in variables of rank as long as . Note that for cubics the usual flattenings work for .
The second and third named authors are members of the Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni of the Istituto Nazionale di Alta Matematica "F. Severi" (GNSAGA-INDAM). We thank the referees for helping us to improve the exposition.
Let and be two -uples of positive integers. Set
Let be vector spaces of dimensions , and consider the product
The line bundle
induces an embedding
where . We call the image
a Segre-Veronese variety. It is a smooth variety of dimension and degree in .
When , is a Veronese variety. In this case we write for , and for the Veronese embedding. When , is a Segre variety. In this case we write for , and for the Segre embedding. Note that
Similarly, given a -uple of -vector spaces and -uple of positive integers we may consider the Segre-Plücker embedding
where . We call the image
a Segre-Grassmann variety.
Let be -vector spaces of finite dimension, and consider the tensor product with , . Then we may interpret a tensor
as a linear map . Clearly, if the rank of is at most then the rank of is at most as well. Indeed, a decomposition of as a linear combination of rank one tensors yields a linear subspace of , generated by the corresponding rank one tensors, containing . The matrix associated to the linear map is called an -flattening of .
In the case of mixed tensors we can consider the embedding
where , , with for any . In particular, if we may interpret a tensor as a degree homogeneous polynomial on . In this case the matrix associated to the linear map is nothing but the -th catalecticant matrix of , that is the matrix whose rows are the coefficient of the partial derivatives of order of .
Similarly, by considering the inclusion
where , , with for any , we get the so called skew-flattenings. We refer to [Lan12] for details on the subject.
The partial derivatives of an homogeneous polynomials are particular flattenings. The partial derivatives of a polynomial are homogeneous polynomials of degree spanning a linear space .
1.3. Rank and border rank
Let be an irreducible and reduced non-degenerate variety. We define the rank with respect to of a point as the minimal integer such that there exist points in linear general position with . Clearly, if we have that
The border rank of with respect to is the smallest integer such that is in the Zariski closure of the set of points such that . In particular .
Recall that given an irreducible and reduced non-degenerate variety , and a positive integer the -secant variety of is the subvariety of obtained as the Zariski closure of the union of all -planes spanned by general points of .
In other words is computed by the smallest secant variety containing .
Now, let be subvarieties of an irreducible projective variety , spanning two linear subspaces . Fix two points , and consider a point . Clearly
2. Comon’s conjecture
It is natural to ask under which assumptions (1.4) is indeed an equality. Consider the Segre-Veronese embedding with -vector spaces of dimension . Its composition with the diagonal embedding is the Veronese embedding of of degree . Let be the corresponding Veronese variety. We will denote by the linear span of in .
In the notations of Section 1.3 set and . For any symmetric tensor we may consider its symmetric rank and its rank as a mixed tensor. Comon’s conjecture predicts that in this particular setting the inequality (1.4) is indeed an equality [CGLM08].
Conjecture 1 (Comon’s).
Let be a symmetric tensor. Then .
Note that when Comon’s conjecture is true. Indeed, is cut out by the size minors of a general square matrix and is cut out by the size minors of a general symmetric matrix, that is .
Conjecture 1 has been proved in several special cases. For instance, when the symmetric rank is at most two [CGLM08], when the rank is less than or equal to the order [ZHQ16], for tensors belonging to tangential varieties to Veronese varieties [BB13], for tensors in [BL13], when the rank is at most the flattening rank plus one [Fri16], for the so called Coppersmith–Winograd tensors [LM17], for symmetric tensors in and also for symmetric tensors of symmetric rank at most seven in [Sei18].
On the other hand, a counter-example to Comon’s conjecture has recently been found by Y. Shitov [Shi17a]. The counter-example consists of a symmetric tensor in which can be written as a sum of rank one tensors but not as a sum of 903 symmetric rank one tensors. It is important to stress that for this tensor rank and border rank are quite different. Comon’s conjecture for border ranks is still completely open [Shi17a, Problem 25].
Even though it has been recently proven false in full generality, we believe that Comon’s conjecture is true for a general symmetric tensor, perhaps it is even true for those tensor for which .
In what follows we use simple arguments based on flattenings to give sufficient conditions for Comon’s conjecture, recovering a known result, and its skew-symmetric analogue.
The tensors such that for a given flattening form a proper closed subset of . Furthermore, the same result holds if we replace the Segre-Veronese variety with the Segre-Grassmann variety .
Let be a general point. Assume that . This condition forces the -flattening matrix to have rank at most . On the other hand, by [SU00, Proposition 4.1] these minors do not vanish on , and therefore define a proper closed subset of . In the Segre-Grassmann setting we argue in the same way by using skew-flattenings. ∎
[IK99] For any integer there exists an open subset such that for any the rank and the symmetric rank of coincide, that is
First of all, note that we always have . Furthermore, Section 1.1 yields that for any -flattening the inequality holds. Since is symmetric and its catalecticant matrices are particular flattenings we get that for any .
Now, for a general we have , and if , where , then Lemma 2.1 yields . Therefore, under these conditions we have the following chain of inequalities
and hence . ∎
Now, consider the Segre-Plücker embedding with -vector spaces of dimension . Its composition with the diagonal embedding is the Plücker embedding of with . Let be the corresponding Grassmannian and let us denote by its linear span in .
For any skew-symmetric tensor we may consider its skew rank that is its rank with respect to the Grassmannian , and its rank as a mixed tensor. Playing the same game as in Proposition 2.2 we have the following.
For any integer there exists an open subset such that for any the rank and the skew rank of coincide, that is
As before for any tensor we have . For any -skew-flattening we have . Furthermore, since is in particular a flattening also the inequality holds.
Now, for a general we have , and if , where , Lemma 2.1 yields , where is the skew-flattening corresponding to the partition of . Therefore, we deduce that
and hence . ∎
Propositions 2.2, 2.3 suggest that whenever we are able to write determinantal equations for secant varieties we are able to verify Comon’s conjecture. We conclude this section suggesting a possible way to improve the range where the general Comon’s conjecture holds giving a conjectural way to produce determinantal equations for some secant varieties.
Set , -times, , -times, and consider the corresponding Segre varieties , and Veronese varieties , . Fix the polynomial and let be the linear space spanned by the polynomials of the form , where is a polynomial of degree . This allow us to see . Note that polynomials of the form lie in the tangent space of at , and therefore .
Hence for a polynomial of degree we have . Our aim is to understand when the equality holds.
We may mimic the same construction for the Segre varieties and , and use determinantal equations for the secant varieties of to give determinantal equations of the secant varieties of and henceforth conclude Comon’s conjecture. In particular, as soon as is odd and , this produces new determinantal equations for and with . Therefore, this would give new cases in which the general Comon’s conjecture holds. Unfortunately, we are only able to successfully implement this procedure in very special cases, see Section 4.
3. Strassen’s conjecture
Another natural problem consists in giving hypotheses under which in (1.5) equality holds. Consider the triple Segre embedding , and let be the corresponding Segre variety. Now, take complementary subspaces , , , and let be the Segre varieties associated respectively to and .
In the notations of Section 1.3 set , and . Strassen’s conjecture states that the additivity of the rank holds for triple tensors, or in onther words that in this setting the inequality (1.5) is indeed an equality [Str73].
Conjecture 2 (Strassen’s).
In the above notation let be two tensors. Then .
Even though Conjecture 2 was originally stated in the context of triple tensors that is bilinear forms, with particular attention to the complexity of matrix multiplication, a number of generalizations are immediate. For instance, we could ask the same question for higher order tensors, symmetric tensors, mixed tensors and skew-symmetric tensors. It is also natural to ask for the analogue of Conjecture 2 for border rank. This has been answered negatively [Sch81].
Conjecture 2 and its analogues have been proven when either or has dimension at most two, when can be determined by the so called substitution method [LM17], when both for the rank and the border rank [BGL13], when are symmetric that is homogeneous polynomials in disjoint sets of variables, either is a power, or both and have two variables, or either or has small rank [CCC15], and also for other classes of homogeneous polynomials [CCO17], [Tei15].
As for Comon’s conjecture a counterexample to Strassen’s conjecture has recentely been given by Y. Shitov [Shi17b]. In this case Y. Shitov proved that over any infinite field there exist tensors such that the inequality in Conjecture 2 is strict.
In what follows we give sufficient conditions for Strassen’s conjecture, recovering a known result, and for its mixed and skew-symmetric analogues.
[IK99] Let be -vector spaces of dimensions , and consider . Let and be two homogeneous polynomials. If there exists an integer such that
Clearly, holds in general. On the other hand, our hypothesis yields
where the last inequality follows from Remark 1.2. ∎
The argument used in the proof of Proposition 3.1 works for general only if for the generic rank we have . For instance, when the generic rank is while the maximal dimension of the spaces spanned by partial derivatives is .
Let and be -vector spaces of dimension and respectively. Consider for every . Let and be two mixed tensors.
If for any there exists a pair with and -flattenings , as in (1.1) such that
Clearly, . On the other hand, our hypothesis yields
where denotes the -flattening of the mixed tensor . ∎
Arguing as in the proof of Proposition 3.3 with skew-symmetric flattenings we have an analogous statement in the Segre-Grassmann setting.
Let and be -vector spaces of dimension and respectively. Consider for every , and let and be two skew-symmetric tensors with .
If for any there exists a pair with and -skew-flattenings , as in (1.1) such that
4. On the rank of
In this section, building on Remark 2.4, we present new cases in which Comon’s conjecture holds. Recall, that for a smooth point , the -osculating space of at is roughly the smaller linear subspace locally approximating up to order at , and the -osculating variety of is defined as the closure of the union of all the osculating spaces
For any the osculating space of order at the point can be written as
Equivalently, is the space of homogeneous polynomials whose derivatives of order less than or equal to in the direction given by the linear form vanish. Note that and for any . Moreover, for any and we can embed a copy of into the osculating space by considering