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Best rank-k approximations for tensors: generalizing Eckart–Young

Best rank-k approximations for tensors: generalizing Eckart–Young jan.draisma@math.unibe.ch Mathematisches Institut, Given a tensor f in a Euclidean tensor space, we are interested in the critical points of Universität Bern, Sidlerstrasse 5, the distance function from f to the set of tensors of rank at most k, which we call the 3012 Bern, Switzerland The first author was partially critical rank-at-most-k tensors for f.When f is a matrix, the critical rank-one matrices for supported by the NWO Vici grant f correspond to the singular pairs of f . The critical rank-one tensors for f lie in a linear entitled Stabilisation in Algebra subspace H , the critical space of f . Our main result is that, for any k, the critical and Geometry. The second author is member of rank-at-most-k tensors for a sufficiently general f also lie in the critical space H .Thisis INDAM-GNSAGA. the part of Eckart–Young Theorem that generalizes from matrices to tensors. Moreover, Full list of author information is we show that when the tensor format satisfies the triangle inequalities, the critical available at the end of the article space H is spanned by the complex critical rank-one tensors. Since f itself belongs to H ,wededucethatalso f itself is a linear combination of its critical rank-one tensors. Keywords: Tensor, Eckart–Young Theorem, Best rank-k approximation Mathematics Subject Classification: 15A69, 15A18, 14M17, 14P05 1 Introduction The celebrated Eckart–Young Theorem says that, for a real m × n-matrix A with m ≤ n and for an integer k ≤ m,amatrix B of rank at most k nearest to A is obtained from A as follows: Compute the singular value decomposition A = U V , where U, V are orthogonal matrices and where  = diag(σ , ... , σ ) is the “diagonal” m × n- 1 m matrix with the singular values σ ≥ ··· ≥ σ ≥ 0 on its main diagonal, and set 1 m B := Udiag(σ , ... , σ , 0, ... , 0)V . Such a best rank-k approximation is unique if and 1 k only if σ >σ , and for us “nearest” refers to the Frobenius norm (but in fact, the result k k+1 holds for arbitrary O × O -invariant norms [12]). m n For higher-order tensors, an analogous approach for finding best rank-k approximations fails in general [18]. It succeeds, with respect to the Frobenius norm, for orthogonally decomposable tensors [1,18], but this is a very low-dimensional real-algebraic variety in the space of all tensors. In this paper, we will establish versions of the Eckart–Young Theorem and the Spectral Theorem that do hold for general tensors. To motivate this theorem, consider matrices once again, and assume that the σ are distinct and positive. A statement generalizing the Eckart–Young Theorem says that we obtain all critical points of the distance function d (B):=||A − B|| on the manifold of rank-k matrices by setting any m −k of the singular values equal to zero [3], so as to obtain amatrix © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,–: vol 27 Page 2 of 13 Draisma et al. Res Math Sci (2018) 5:27 B := Udiag(0, ... , 0, σ , 0, ... , 0, σ , 0, ... , 0, σ , 0, ...)V i ,...,i i i i 1 k 1 2 k for any ordered k-tuple i < ··· < i in {1, ... ,m}. We will call these critical points critical rank-k matrices for A. In particular, the critical rank-one matrices are B , ... ,B ,and we 1 m draw attention to the fact that for each k ≥ 1 and each k-tuple i < ··· < i the critical rank-k matrix B lies in the linear span of B , ... ,B . Moreover, this linear span has a i ,...,i 1 m 1 k T T direct description in terms of A: It consists of all matrices B such that both A B and AB are symmetric matrices. Taking cue from this observation, we will associate a critical space H to a tensor f show that H contains the critical rank-at-most-k tensors for f for each value of k (see below for a definition), and that H is spanned by the critical rank-one tensors for f .Wewill establish these results for sufficiently general partially symmetric tensors, and we work over the base field C rather than R. d n +1 d n +1 1 1 p p Theorem 1.1 Let f be a sufficiently general tensor in S C ⊗ ··· ⊗ S C .Then, for each natural number k, the critical rank-at-most-k tensors for f lie in the critical space H . Moreover, if for each  with d = 1 the triangle inequality n ≤ n holds, f   i i= n +1 then codim H = and H is spanned by the critical rank-one tensors for f . In f f particular, f itself lies in the linear span of the critical rank-one tensors for f . We record the following two corollaries over the real numbers. Corollary 1.2 If n , ... ,n satisfy the triangle inequality n ≤ n for each  = 1 p  i i= n +1 1, ... , p, then for a sufficiently general tensor f ∈ R and any natural number k, i=1 any real tensor of real rank at most k closest to f in the Frobenius norm lies in the linear span of the complex critical rank-one tensors for f . In particular, f itself lies in the linear span of the complex critical rank-one tensors for f . d n+1 Corollary 1.3 For a sufficiently general symmetric tensor f ∈ S R and any natural number k, any real symmetric tensor of real symmetric rank at most k closest to f in the Frobenius norm lies in the linear span of the complex critical symmetric rank-one tensors for f . In particular, f itself lies in the linear span of the complex critical rank-one tensors for f . In the case of symmetric tensors, these critical rank-one tensors correspond to the so- called eigenvectors of f [11], while in the case of ordinary tensors, they correspond to singular vector tuples [10]. In the case n = 1 of binary forms, Corollary 1.3 was proved in [16]. The two corollaries above can be regarded as generalizations of the Eckart–Young Theorem and the Spectral Theorem from matrices to tensors. Several remarks are in order here. First, we complexify d to the quadratic polynomial d (u):= (u − f |u − f ), where (.|.) is the standard complex bilinear form on the space of tensors (and not a Hermitian form). The point of doing this is that, unlike for matrices, the critical points of this function on low-rank tensors are in general not real anymore, even if f is real. Accordingly, the critical space H , while defined by linear equations over R if f is real, is taken to be the space of complex solutions to those equations. Second, we denote the dimensions by n + 1 rather than n since we will be using methods from projective algebraic geometry where the formulas look more appealing in terms of the projective dimension n than in the affine dimension n + 1. An example of this phenomenon is the Draisma et al. Res Math Sci (2018) 5:27 Page 3 of 13 27 triangle inequalities in the theorem, which hold if and only if the variety dual to the Segre– n n 1 p Veronese embedding of the product P ×···×P via degrees d , ... ,d isahypersurface 1 p [7, Corollary 5.11]. The remainder of this paper is organized as follows. In Sect. 2, we define the critical space H for a partially symmetric tensor f and prove that the critical rank-at-most-k tensors for f lie in H . Then, in Sect. 3, we use vector bundle techniques to compute the dimension of the space spanned by the critical rank-one tensors for f and to show that this space equals H . In Sect. 4, we combine these ingredients to establish the results above. 2 The critical space of a tensor 2.1 Partially symmetric tensors and their ranks Let p ∈ Z ,let V , ... ,V be complex vector spaces, and let d , ... ,d ∈ Z .Let S V ≥1 1 p 1 p ≥1 be the dth symmetric power of V . We will study tensors in the space d d 1 p T := S V ⊗ ··· ⊗ S V . 1 p So for p = 1, T is a symmetric power of V , which is canonically isomorphic with the space of symmetric, d -way n ×···×n -tensors. On the other hand, when all d are equal p 1 1 i to 1, then T is a space of p-way ordinary tensors. We will write [p]:={1, ... ,p}. Inside T,let X be the set of all tensors of the form d p v ⊗ ··· ⊗ v (v ∈ V ,  ∈ [p]). Then, X is a closed subvariety of T known as the affine cone over the Segre–Veronese n n 1 p embedding of P × ··· × P of degrees (d , ... ,d ). Let kX denote the set of sums of k 1 p elements of X. An arbitrary element t of T lies in kX for some k, and the minimal such k is called the rank of t [9, Definition 5.2.1.1]. For p = 1, this is the symmetric or Waring rank, and if all d are 1, this notion is ordinary tensor rank. We write Sec (X) for the Zariski (or Euclidean) closure of kX in T. For real tensors, a few modifications are needed. The real rank of a real tensor t is the minimum k such that t = λ x with λ ∈ R and x ∈ X (it is enough to allow i i i i R i=1 3 3 λ =±1). For example (e + ie ) + (e − ie ) has complex rank 2 and real rank 3. Real i 1 2 1 2 rank is subtle for low-rank approximation of tensors. A classical example of de Silva and Lim [2] shows that for almost all 2 × 2 × 2-tensors of real rank 3 (like the above one) does not exist a closest tensor of real rank 2, while such phenomena may happen only for measure zero subsets of the set of complex tensors of given rank. 2.2 Symmetric bilinear forms and pairings If V, W are complex vector spaces with symmetric bilinear forms (.|.), and if d ∈ Z , then ≥0 S V and V ⊗ W carry unique symmetric bilinear forms, also denoted (.|.), that satisfy d d  d (v |v ):= (v|v ) and (v ⊗ w|v ⊗ w ):= (v|v )(w|w ). The first of these equalities implies (v ... v |v ) = (v |v ) 1 d i i=1 27 Page 4 of 13 Draisma et al. Res Math Sci (2018) 5:27 and more in general v ··· v |v ··· v = v |v . 1 d i d π(i) d! π ∈S i=1 We now fix nondegenerate symmetric bilinear forms on each V ,  ∈ [p]. Then, iterating these constructions, we obtain a canonical bilinear form on T. Using the bilinear forms on V and W , we can also build more general bilinear maps whose outputs are vectors or tensors rather than scalars. We will call these bilinear maps pairings anddenotethemby[.|.]. Of particular relevance to us is the skew-symmetric d d pairing S V × S V → V determined by d d d−1 [v |w ]:= (v|w) v ∧ w, which implies ⎛ ⎞ ⎝ ⎠ [v ... v |w ] = (v |w) v ∧ w 1 d i i i ∈[d] i=i and more in general ⎛ ⎞ ⎝ ⎠ [v ··· v |w ··· w ] = (v |w ) v ∧ w , 1 1 i d d π(i) i j d · d! i ,j ∈[d] π:[d]\i →[d]\j i=i where π runs over all bijections [d] \ i → [d] \ j . Remark 2.1 In the case of binary forms (dim V = 2 and arbitrary d), the pairing [f |g] coincides (up to scalar multiples) with (f |D(g)), where D(g) = g y − g x;see [16]. Note x y the skew-symmetry property (f |D(g)) =− (g |D(f )). On the other hand, in the case of symmetric matrices (d = 2 and arbitrary V ), the pairing [f |g] coincides (up to scalar multiples) with the bracket fg − gf . Building on this construction, for each  ∈ [p] we define a pairing [.|.] : T ×T → V by ⎛ ⎞ ⎝ ⎠ [f ⊗ ··· ⊗ f |g ⊗ ··· ⊗ g ] := (f |g ) [f |g ],f ,g ∈ S V , (1) 1 p 1 p  i i   i i i i= which we will use to define the critical space. Remark 2.2 In the case of matrices T = V ⊗ V , the pairing [f, g] coincides (up to scalar 1 2 1 t t t t multiples) with fg − gf , while [f, g] is (up to scalar multiples) f g − g f . 2.3 Basis, orthogonal basis and monomials If B is a basis of V , then the degree-d monomials in the elements of B form a basis of S V . Such a basis is orthogonal if B is orthogonal. Hence, if we fix bases (respectively, orthog- onal bases) of V , ... ,V , then by taking tensor products we obtain a basis (respectively, 1 p orthogonal basis) of T, whose elements we will call monomials of degree D := d . =1 We will use the word gcd of two such monomials x, y for the highest-degree monomial z such that both x and y can be obtained from z by multiplying z with suitable monomials. Draisma et al. Res Math Sci (2018) 5:27 Page 5 of 13 27 Example 2.3 If p = 3and V = V = V = C with the standard bilinear form, and 1 2 3 2 2 3 d = d = 3and d = 2, then the gcd of (e e ) ⊗(e e e ) ⊗(e )and (e e e ) ⊗(e ) ⊗(e e ) 1 2 3 2 1 2 3 1 2 3 2 3 1 1 3 equals (e e ) ⊗ (e ) ⊗ (1). 1 2 3 Lemma 2.4 For two monomials f = f ⊗ ··· ⊗ f ,g = g ⊗ ··· ⊗ g in T relative to the 1 p 1 p same orthogonal bases of V , ... ,V and for  ∈ [p] we have [f |g] = 0 unless f = g for 1 p  i i all i =  and h := gcd(f ,g ) has degree d − 1; in this case gcd(f, g) has degree D − 1 and [u|v] ∈ C (f /h) ∧ (g /h). This is immediate from the definition of the pairing in (1). 2.4 Critical rank-one tensors Let f ∈ T. Then, the critical points of the distance function d : x → (f −x|f −x)on X are by definition those x ∈ X \{0} for which f − x is perpendicular to the tangent space T X to X at x; we write this as f − x ⊥ T X. We call these tensors the critical rank-one tensors for f . For sufficiently general f , each of these critical rank-one tensors is non-isotropic, i.e., satisfies (x|x) = 0(see[4, Lemma 4.2], in next Proposition 2.6 we will prove a slightly more general fact). We will establish a bilinear characterization of these critical rank-one tensors for f . First, we describe the tangent space of X at a point x in more detail. For this, write x = v ⊗ ··· ⊗ v . (2) Hence, we may extend each v to a basis of V .Wethenobtainan x-adapted basis of T consisting of monomials. If moreover x is non-isotropic, we have (v |v ) = 0 and we may extend each v to an orthogonal basis. We then obtain an x-adapted orthogonal basis of T. Lemma 2.5 Let x ∈ Xas in (2). (1) Then, relative to any x-adapted basis, T X is spanned by all degree-D monomials whose gcd with x has degree at least D − 1. (2) Assume moreover that x is non-isotropic. Then, relative to any x-adapted orthogonal basis, (T X) is spanned by all degree-D monomials whose gcd with x has degree at most D − 2. Proof Part (1) follows by applying the Leibniz rule to the parameterization (2)of X;part (2) is a straightforward consequence. Proposition 2.6 Let f ∈ T and let x ∈ X be non-isotropic. Then, the following two statements are equivalent: (1) some (nonzero) scalar multiple of x is a critical rank-one tensor for f and (2) a unique (nonzero) scalar multiple of x is a critical rank-one tensor for f ; and they imply the following statement: (3) for each  ∈ [p], [f |x] ∈ V is zero. Moreover, if f is sufficiently general, then every nonzero x ∈ X satisfying (3) is non-isotropic and satisfies (1) and (2). 27 Page 6 of 13 Draisma et al. Res Math Sci (2018) 5:27 The pairing in item (3) is the pairing from (1). Proof For the equivalence of the first two statements, we note that if cx, c x with c, c = 0 are critical rank-one tensors for f , then T X = T  X = T X and f − cx ⊥ T X and cx c x x x f − c x ⊥ T X.Since x ∈ T X, we find that (c − c )x ⊥ x, and using that x is non-isotropic x x we find that c = c . To prove that (1) implies (3), write x as in (2) and extend each v to an orthogonal basis of V ,soastoobtainan x-adapted orthogonal basis of T. Now assume that f − cx ⊥ T X. Then, by Lemma 2.5, f − cx is a linear combination of degree-D monomials whose gcds with x have degrees at most D − 2. Hence by Lemma 2.4,[x|f − cx] = 0. By the skew- symmetry, [x|x] = 0, so [x|f ] = 0. d p For the last statement, consider an x = v ⊗ ··· ⊗ v ∈ X where, say, v , ... ,v p 1 a with a > 0 are isotropic but the remaining factors are not. Extend each v , > a to an orthogonal basis of V , and for v with  ≤ a find an isotropic w ∈ V with (v |w ) = 1 and extend v ,w with an orthogonal basis of the orthogonal complement of v ,w  to a basis of V . In the corresponding (non-orthogonal) monomial basis of T, the monomials y with [y|x] = 0 for  ≤ a are those of the form d d −1 d p 1  d a+1 w ⊗ ··· ⊗ w u ⊗ ··· ⊗ w ⊗ v ⊗ ··· ⊗ v , 1  a a+1 where u is a basis vector of V that is distinct from v but possibly equal to w .These monomials all satisfy [y|x] = 0 for i = . Similarly, the monomials y with [y|x] = 0 for > a are those of the form d d −1 d d p 1 d a+1 l w ⊗ ··· ⊗ w ⊗ v ⊗ ··· ⊗ v u ⊗ ··· ⊗ v l p 1 a+1 l with u a basis vector of V distinct from v ; they, too, satisfy [y|x] = 0 for i = .The remaining monomials span the space of f swith[x|f ] = 0 for all ; this space therefore has dimension dim T − (n + ··· + n ), 1 p and it does not change when we scale x. Since the isotropic projective points x∈ PT form a subvariety of positive codimension in the (n + ··· + n )-dimensional projective 1 p variety PX, the locus of all f for which there is a nonzero isotropic x ∈ X with [f |x] = 0 for all  has dimension less than dim T. Now assume that f is sufficiently general and let x ∈ X \{0} satisfy [x|f ] = 0 for all .By the above, x is non-isotropic. Suppose that f ,expandedonthe x-adapted orthogonal basis, contains a monomial y whose gcd with x has degree exactly D −1. If y agrees with x except d −1 in the factor S V where it equals v u ,thenin[x|f ] , expanded on the standard basis of V relative to the chosen basis of V , the term v ∧ u has a nonzero coefficient. Hence, [x|f ] is nonzero, a contradiction. Therefore, f contains only monomials whose gcds with x have degrees at most D − 2, and possibly the monomial x itself. Then, f − cx ⊥ T X for a unique constant c.By generality of f , it does not lie in (T X) for any x ∈ X \{0} (the union of these orthogonal complements is the cone over the variety dual to the projective variety defined by X,and of positive codimension). Hence, c = 0, and cx is a critical rank-one tensor for f . Remark 2.7 The implication (1) =⇒ (3) in Proposition 2.6 holds without the assumption of non-isotropy of x. This follows from the fact that the ED correspondence {(x, f ) ∈ X × V | x is critical for f } Draisma et al. Res Math Sci (2018) 5:27 Page 7 of 13 27 is a irreducible variety (see [3, §4 and Lemma 2.1]) and the nonempty open part in it where x is non-isotropic lies in the variety defined by [f |x] = 0 ∀ ∈ [p]byProposition 2.6. 2.5 The critical space In view of Proposition 2.6, we introduce the following notion. Definition 2.8 For a tensor f ∈ T, the critical space H ⊆ Tof f is defined as H :={g ∈ T | [f |g] = 0 for all  ∈ [p]}. Remark 2.9 By the skew-symmetry, it follows immediately that f ∈ H . Remark 2.10 In the case of binary forms (dim V = 2), H is the hyperplane orthogonal to D(f )[16]. In the case of ordinary tensors, H was first defined in [15] where it was called singular space, but in view of the results in this paper we feel that critical space isabetter name. Proposition 2.6 establishes that the non-isotropical critical rank-one tensors all lie inside H ; hence for a sufficiently general f , all critical rank-one tensors lie in H . In the next f f subsection, we will establish an analogous statement for higher ranks. p 2 Note that the number of linear conditions for g to lie in H is at most dim V = =1 p n +1 —the linear conditions in the definition may not all be linearly independent. =1 2 In Proposition 3.6 we will see that, assuming the triangle inequalities from Theorem 1.1 and assuming that f is sufficiently general, equality holds. 2.6 Higher rank We will now establish a generalization of Proposition 2.6 to higher-rank tensors. Definition 2.11 Let f ∈ T and let k be any nonnegative integer. A critical rank-at-most-k tensor for f is a tensor g ∈ kX such that f − g ⊥ T Sec (X). g k Note that by [4, Lemma 4.2], all the critical rank-at-most-k tensors for a sufficiently general f ∈ T are smooth points of Sec (X) and can be written as a sum of k non- isotropic rank-one tensors. Moreover, if we assume that k is at most the generic rank of tensors in T, then these critical tensors to a sufficiently general f have rank equal to k.If k is at least the generic rank of tensors in T, then the only critical rank-at-most-k tensor for a sufficiently general f is f itself. Proposition 2.12 Let f ∈ T be sufficiently general and let k be a nonnegative integer. Then, all the critical rank-at-most-k tensors for f lie in the critical space H . Proof Let g be a critical rank-at-most-k tensor. By generality of f , g can be written as x + ··· + x with each x ∈ X non-isotropic. Then, T Sec X ⊇ T X, so that for 1 i g x k k i=1 i each i ∈ [k]wehave f − g ⊥ T X. By Lemma 2.5 this means that, in the x -adapted x i orthogonal basis, f − g is a linear combination of monomials whose gcds with x have degrees at most D − 2. Hence, by Lemma 2.4,[f − g |x ] = 0 for all  = 1, ... ,p.We conclude that, for each , [f − g |g] = [f − g |x ] = 0, i=1 27 Page 8 of 13 Draisma et al. Res Math Sci (2018) 5:27 and therefore [f |g] = [f − g |g] + [g |g] = 0 + 0, where in the last step we used that [.|.] is skew-symmetric. Hence, g ∈ H . In the next section, we compute the dimension of the space spanned by the critical rank-one tensors for a general tensor and show that this space equals H . 3 The scheme of critical rank-one tensors 3.1 Critical rank-one tensors as the zero locus of a vector bundle section Let f ∈ T = S V be a tensor. We assume that p ≥ 2, d ≥ 1, and dim V = =1 n + 1 ≥ 1 for all . We adapt the notation of [15, Section 5.1] to our current setting. Consider the Segre–Veronese variety PX = PV × ... × PV embedded with 1 p O(d , ... ,d )in PT;so PX is the projective variety associated with the affine cone X ⊆ T. 1 p Let π : PX → PV be the projection on the th factor and set N := dim PX = n + ... + n . For each  ∈ [p]let Q be the quotient bundle on PV , whose fiber 1 p over a point v is V /v. From these quotient bundles, we construct the following vector bundles on PX: E := E where E := π Q ⊗ O(d , ... ,d ,d − 1,d , ... ,d ). l l  1 −1  +1 p =1 Note that E has rank N. The fiber of E over a point v := (v , ... , v ) ∈ PX 1 p consists of polynomial maps v → V /v  that are multi-homogeneous of i=1 multi-degree (d , ... ,d − 1, ... ,d ). The tensor f yields a global section of E which 1  p over the point v is the map sending (c v , ... ,c v ) to the natural pairing of f with 1 1 p p d d −1 d 1  p (c v ) ··· (c v ) ··· (c v ) —a vector in V —taken modulo v . Combining these 1 1   p p p sections, f yields a global section s of E.ByProposition 2.6, for f sufficiently general, the tensor x := v ⊗ ··· ⊗ v is a nonzero scalar multiple of a critical rank-one tensor for f if and only if the point (v , ... , v ) is in the zero locus Z of the section s .In[5], 1 p f f this is used to compute the cardinality of Z for f sufficiently general as the top Chern class of E. Our current task is different: we want to show that, if we assume the triangle inequalities of Theorem 1.1 and that f is sufficiently general, the linear span Z  equals the projectivized critical space PH ; this is the second part of Theorem 1.1. 3.2 Bott’s formulas and a consequence Our central tool will be the following formulas for the cohomology of vector bundles over projective spaces [13]. Recall that  (k)isthe O(k)-twisted sheaf of differential r-forms on P . Lemma 3.1 (Bott’s formulas) For q,n,r ∈ Z and k ∈ Z,wehave ≥0 k+n−r k−1 if q = 0 ≤ r ≤ n and k > r, k r 1if0 ≤ q = r ≤ n and k = 0, q n r h P ,  (k) = −k+r −k−1 ⎪ if q = n ≥ r ≥ 0and k < r − n, and −k n−r 0 otherwise. A consequence featuring the triangle inequalities of Theorem 1.1 is the following. Draisma et al. Res Math Sci (2018) 5:27 Page 9 of 13 27 Lemma 3.2 Suppose that n ≤ n holds for all  with d = 1.Let k ≥ 2 be an i= integer, q , ... ,q be nonnegative integers with q < kand r , ... ,r be nonnegative 1 p  1 p =1 integers with r = k. Then, =1 H PV ,  (−d (k − 1) + 2r ) = 0. PV =1 Proof Assume that all factors in the tensor product are nonzero. First, if all of the factors were nonzero by virtue of the second and third line in Bott’s formulas, then we would have q ≥ r for all , and hence k > q ≥ r = k,a contradiction. So some factor is nonzero by virtue of the first line in Bott’s formulas; without loss of generality this is the first factor. Hence we have q = 0 ≤ r ≤ n and −d (k−1)+2r > r . 1 1 1 1 1 1 This last inequality reads r > d (k − 1). Combining this with r = k and the fact that 1 1 d is a positive integer, we find that r = k, d = 1, and r = 0 for > 1. In particular, 1 1 1 there are no > 1 for which the first line in Bott’s formulas applies. For any > 1, if the second line applies, then 0 = r = q =−d (k − 1) + 2r , which contradicts that both d and k − 1 are positive. Hence, the third line applies for all > 1, and in particular we have q = n . But then p p n ≥ r = k > q = n , 1 1 l l l=1 l=2 which together with d = 1 contradicts the triangle inequality in the lemma. 3.3 Vanishing cohomology The vanishing result in this subsection uses Lemma 3.2 and the following version of Künneth’s formula. Lemma 3.3 (Künneth’s formula) For vector bundles G on PV for  = 1, ... ,p and a nonnegative integer q we have q ∗ q H PX, π G H (PV , G ), q +...+q =q 1 p where the sum is over all p-tuples of nonnegative integers summing to q. Lemma 3.4 Suppose that n ≤ n holds for all  such that d = 1.Let k ≥ 2 be an i= integer and q ∈{0, ... ,k − 1}. Then, we have ⎛ ⎛ ⎞ ⎞ q ∗ ⎝ ⎝ ⎠ ⎠ H PX, E ⊗ O(d , ... ,d ) = 0. 1 p Proof First, ∗ ∗ ∗ E = π Q ⊗ O(−d , ... , −d , −(d − 1), −d , ... , −d ). 1 −1  +1 p =1 A well-known formula for kth wedge power of a direct sum yields k r ∗ ∗ ∗ E = (π Q ⊗ O(−d , ... , −(d − 1), ... , −d )). 1  p r +...+r =k 1 p 27 Page 10 of 13 Draisma et al. Res Math Sci (2018) 5:27 r r r ∗ 1 1 r Using (F ⊗ O(ω)) = ( F)(rω), Q =  (1), and ( (1)) =  (r), we obtain ∗ ∗ E = π  (r ) ⊗ O(−r d , ... , −r (d − 1), ... , −r d . 1    p PV r +...+r =k 1 p Twisting by O(d , ... ,d ), regrouping in each projection, and using r = k we find: 1 p ⎛ ⎞ ∗ ∗ ⎝ ⎠ E ⊗ O(d , ... ,d ) = π  (−d (k − 1) + 2r ) . 1 p PV r +...+r =k 1 p To compute H of each summand we apply Künneth’s formula, and obtain subsummands which are exactly of the form in Lemma 3.2, hence zero. 3.4 Comparing PH and Z f f Assume that f is sufficiently general in T. By the first subsection of this section and by Proposition 2.6, Z is contained in the projectivized critical space PH , hence so is Z . f f f Our goal now is to show that Z  is equal to PH and to compute its dimension. Both f f of these goals are achieved through the following lemma. The section s of E yields a homomorphism E → O of sheaves whose image is contained in the ideal sheaf I of the zero locus of s . Lemma 3.5 Assume that for each  ∈ [p] we have n ≤ n and that f is sufficiently i= general. Then, the induced homomorphism E ⊗ O(d , ... ,d ) → I ⊗ O(d , ... ,d ) 1 p Z 1 p induces an isomorphism at the level of global sections. The following proof can be shortened considerably using spectral sequences, but we found it more informative in its current form. To make the formulas more transparent, q q we write H (.) instead of H (PX, .) everywhere. Proof To establish the desired isomorphism 0 ∗ 0 H (E ⊗ O(d , ... ,d )) H (I ⊗ O(d , ... ,d )) 1 p Z 1 p we use the following Koszul complex (see, e.g., [8, Chapter III,Proposition 7.10A]): N +1 N 2 ∗ ∗ ∗ ∗ 0 = E → E → ··· → E → E → I → 0. k k+1 ∗ ∗ Letting F be the quotient of E by the image of E , this yields the short exact sequence 0 → F → E → I → 0. 2 Z Tensoring with O(d , ... ,d ) yields the short exact sequence 1 p 0 → F ⊗ O(d , ... ,d ) → E ⊗ O(d , ... ,d ) → I ⊗ O(d , ... ,d ) → 0, 2 1 p 1 p Z 1 p and this gives a long exact sequence in cohomology beginning with 0 0 ∗ 0 0 → H (F ⊗ O(d , ... ,d )) → H (E ⊗ O(d , ... ,d )) → H (I ⊗ O(d , ... ,d )) 2 1 p 1 p Z 1 p → H (F ⊗ O(d , ... ,d )) → 2 1 p So to obtain the desired isomorphism we want that H (F ⊗ O(d , ... ,d )) = 0 for q = 0, 1. 2 1 p Draisma et al. Res Math Sci (2018) 5:27 Page 11 of 13 27 For each k = 2, ... ,N, we have the short exact sequence 0 → F → E → F → 0 k+1 k which yields the long exact sequence ⎛ ⎞ k−2 ∗ k−2 ⎝ ⎠ → H E ⊗ O(d , ... ,d ) → H F ⊗ O(d , ... ,d ) 1 p 1 p ⎛ ⎞ k−1 k−1 ∗ ⎝ ⎠ → H (F ⊗ O(d , ... ,d )) → H E ⊗ O(d , ... ,d ) k+1 1 p 1 p k−1 k → H F ⊗ O(d , ... ,d ) → H F ⊗ O(d , ... ,d ) → k 1 p k+1 1 p Using Lemma 3.4, the two leftmost spaces are zero, so that k−2 k−1 H (F ⊗ O(d , ... ,d )) H (F ⊗ O(d , ... ,d )) and k 1 p k+1 1 p k−1 k H (F ⊗ O(d , ... ,d )) ⊆ H (F ⊗ O(d , ... ,d )). k 1 p k+1 1 p Hence, using that F = 0, we find that N +1 0 N −1 ∼ ∼ H (F ⊗ O(d , ... ,d )) ··· H (F ⊗ O(d , ... ,d )) = 0and = = 2 1 p N +1 1 p 1 N H (F ⊗ O(d , ... ,d )) ⊆ ··· ⊆ H (F ⊗ O(d , ... ,d )) = 0, 2 1 p N +1 1 p as desired. Proposition 3.6 Suppose that for each  ∈ [p] we have n ≤ n and that f is i= n +1 sufficiently general. Then, Z = PH and codim H = . f f T f Proof Since PX is embedded by O(d , ... ,d ), the space of linear forms on T vanishing 1 p on Z is H (I ⊗ O(d , ... ,d )). By Lemma 3.5, this space is isomorphic to Z 1 p 0 ∗ 0 ∗ ∗ H (E ⊗ O(d , ... ,d )) = H π Q ⊗ O(0, ... , 1, ... , 0) 1 p 0 1 0 1 = H (π ( (2))) = H PV ,  (2) , PV PV n +1 which by the first line in Bott’s formulas has dimension . This means that n +1 codim Z = , so the second statement in the proposition follows from PT the first statement. To establish the first statement, we spell out the map 0 ∗ 0 1 0 H (PV , Q ⊗ O(1)) = H (PV ,  (2)) → H (I ⊗ O(d , ... ,d )) Z 1 p PV f in greater detail. The space on the left is canonically ( V ) , and an element ξ in this space is mapped to the linear form T → C,g → ξ([f |g] ). As  varies, these are precisely the linear forms that cut out H . This proves that PH =Z . f f f 27 Page 12 of 13 Draisma et al. Res Math Sci (2018) 5:27 Remark 3.7 In general, for the equality Z = PH we only need that the linear equations f f cutting out PH also cut out Z , i.e., we only need that the linear map in Lemma 3.5 is f f surjective. One might wonder whether this surjectivity remains true when the triangle inequalities fail. In the case of (n + 1) × (n + 1)-matrices, it does indeed—there we 1 2 already knew the critical rank-one approximations span the critical space—but for p = 3 and 2 × 2 × 4-tensors (so that n = 3 > 1 + 1 = n + n ) the space Z  has dimension 3 1 2 6 while computer experiments suggest that the space PH has dimension 7 , hence the surjectivity fails. Still, in these experiments, f itself seems to lie in the span of Z .Thisleads to the open problem whether our analogue of the Spectral Theorem and the Eckart–Young Theorem persists when the triangle inequalities fail. 4 Proofs of the main results Proof of Theorem 1.1 The first statement is Proposition 2.12; the second and third state- ment are Proposition 3.6. The last statement follows from Remark 2.9. Proof of Corollaries 1.2 and 1.3. If g is a real tensor of real rank at most k closest to f , then one can write it as x + ··· + x with x , ... ,x real points of X. In particular, all of these 1 k 1 k points are non-isotropic, and the argument of Proposition 2.12 applies. Hence, g lies in H . Now the result follows from Proposition 3.6. The argument applies, in particular, to k equal to the rank of f , which gives the last statement of the corollaries. Note that, if f is any real tensor, then any real tensor of real rank at most k closest to f lies in H by the argument above. Only for the conclusion that it lies in the span of the complex critical rank-one tensors of f do we use that f is sufficiently general. We do not know whether this generality is really needed. Also note that we do not shed new light on the question of when for sufficiently general f there exists a closest real tensor of rank at most k. For an update on the complex case, see [17]. Author details 1 2 Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands, Dipartimento di Matematica e Informatica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Florence, Italy, Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Bulevar Louis Pasteur, 31, 29010 Málaga, Spain. Received: 30 November 2017 Accepted: 9 May 2018 Published online: 23 May 2018 References 1. Boralevi, A., Draisma, J., Horobe¸t, E., Robeva, E.: Orthogonal and unitary tensor decomposition from an algebraic perspective. Isr. J. Math. 222, 223–260 (2017) 2. de Silva, V., Lim, L.-H.: Tensor rank and the ill-posedness of the best low-rank approximation problem. SIAM J. Matrix Anal. Appl. 30, 1084–1127 (2008) 3. Draisma, J., Horobe¸t, E., Ottaviani, G., Sturmfels, B., Thomas, R.: The Euclidean distance degree of an algebraic variety. Found. Comput. Math. 16(1), 99–149 (2016) 4. Drusvyatskiy, D., Lee, H.-L., Ottaviani, G., Thomas, R.: The Euclidean distance degree of orthogonally invariant matrix varieties. Isr. J. Math 221, 291–316 (2017) 5. Friedland, S., Ottaviani, G.: The number of singular vector tuples and uniqueness of best rank one approximation of tensors. Found. Comput. Math. 14(6), 1209–1242 (2014) 6. Friedland, S., Tammali, V.: Low-rank approximation of tensors. In: Benner, P., Bollhöfer, M., Kressner, D., Mehl, C., Stykel, T. (eds.) Numerical Algebra, Matrix Theory, Differential-Algebraic Equations and Control Theory. Springer, Cham (2015) 7. Gelfand, I., Kapranov, M., Zelevinsky, A.: Discriminants, Resultants and Multidimensional Determinants. Birkhäuser, Boston (1994) 8. Hartshorne, R.: Algebraic Geometry, Graduate Texts in Mathematics 52. Springer, New York (1977) 9. Landsberg, J.M.: Tensors: Geometry and Applications, Volume 128 of Graduate Studies in Mathematics. American Mathematical Society (AMS), Providence (2012) Draisma et al. Res Math Sci (2018) 5:27 Page 13 of 13 27 10. Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of IEEE International Workshop on Computing Advances in Multi-sensor Adaptive Processing (CAMSAP 2005), pp. 129–132 11. Maccioni, M.: The number of real eigenvectors of a real polynomial, to appear in Bollettino dell’Unione Matematica Italiana. arXiv:1606.04737 12. Mirsky, L.: Symmetric gauge functions and unitarily invariant norms. Q. J. Math. Oxf. II Ser. 11, 50–59 (1960) 13. Okonek, C., Schneider, M., Spindler, H.: Vector Bundles on Complex Projective Spaces, Progress in Mathematics, vol. 3. Birkhäuser, Boston (1980) 14. Oeding, L., Ottaviani, G.: Eigenvectors of tensors and algorithms for waring decomposition. J. Symb. Comput. 54, 9–35 (2013) 15. Ottaviani, G., Paoletti, R.: A geometric perspective on the singular value decomposition. Rend. Istit. Mat. Univ. Trieste 47, 107–125 (2015) 16. Ottaviani, G., Tocino, A.: Best rank k approximation for binary forms. Collect. Math. 69, 163–171 (2018) 17. Qi, Y., Michałek, M., Lim, L.H.: Complex tensors almost always have best low-rank approximations. preprint. arXiv:1711.11269 18. Vannieuwenhoven, N., Nicaise, J., Vandebril, R., Meerbergen, K.: On generic nonexistence of the Schmidt–Eckart– Young decomposition for complex tensors. SIAM J. Matrix Anal. 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Best rank-k approximations for tensors: generalizing Eckart–Young

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Springer Journals
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Copyright © 2018 by The Author(s)
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Mathematics; Mathematics, general; Applications of Mathematics; Computational Mathematics and Numerical Analysis
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Abstract

jan.draisma@math.unibe.ch Mathematisches Institut, Given a tensor f in a Euclidean tensor space, we are interested in the critical points of Universität Bern, Sidlerstrasse 5, the distance function from f to the set of tensors of rank at most k, which we call the 3012 Bern, Switzerland The first author was partially critical rank-at-most-k tensors for f.When f is a matrix, the critical rank-one matrices for supported by the NWO Vici grant f correspond to the singular pairs of f . The critical rank-one tensors for f lie in a linear entitled Stabilisation in Algebra subspace H , the critical space of f . Our main result is that, for any k, the critical and Geometry. The second author is member of rank-at-most-k tensors for a sufficiently general f also lie in the critical space H .Thisis INDAM-GNSAGA. the part of Eckart–Young Theorem that generalizes from matrices to tensors. Moreover, Full list of author information is we show that when the tensor format satisfies the triangle inequalities, the critical available at the end of the article space H is spanned by the complex critical rank-one tensors. Since f itself belongs to H ,wededucethatalso f itself is a linear combination of its critical rank-one tensors. Keywords: Tensor, Eckart–Young Theorem, Best rank-k approximation Mathematics Subject Classification: 15A69, 15A18, 14M17, 14P05 1 Introduction The celebrated Eckart–Young Theorem says that, for a real m × n-matrix A with m ≤ n and for an integer k ≤ m,amatrix B of rank at most k nearest to A is obtained from A as follows: Compute the singular value decomposition A = U V , where U, V are orthogonal matrices and where  = diag(σ , ... , σ ) is the “diagonal” m × n- 1 m matrix with the singular values σ ≥ ··· ≥ σ ≥ 0 on its main diagonal, and set 1 m B := Udiag(σ , ... , σ , 0, ... , 0)V . Such a best rank-k approximation is unique if and 1 k only if σ >σ , and for us “nearest” refers to the Frobenius norm (but in fact, the result k k+1 holds for arbitrary O × O -invariant norms [12]). m n For higher-order tensors, an analogous approach for finding best rank-k approximations fails in general [18]. It succeeds, with respect to the Frobenius norm, for orthogonally decomposable tensors [1,18], but this is a very low-dimensional real-algebraic variety in the space of all tensors. In this paper, we will establish versions of the Eckart–Young Theorem and the Spectral Theorem that do hold for general tensors. To motivate this theorem, consider matrices once again, and assume that the σ are distinct and positive. A statement generalizing the Eckart–Young Theorem says that we obtain all critical points of the distance function d (B):=||A − B|| on the manifold of rank-k matrices by setting any m −k of the singular values equal to zero [3], so as to obtain amatrix © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 0123456789().,–: vol 27 Page 2 of 13 Draisma et al. Res Math Sci (2018) 5:27 B := Udiag(0, ... , 0, σ , 0, ... , 0, σ , 0, ... , 0, σ , 0, ...)V i ,...,i i i i 1 k 1 2 k for any ordered k-tuple i < ··· < i in {1, ... ,m}. We will call these critical points critical rank-k matrices for A. In particular, the critical rank-one matrices are B , ... ,B ,and we 1 m draw attention to the fact that for each k ≥ 1 and each k-tuple i < ··· < i the critical rank-k matrix B lies in the linear span of B , ... ,B . Moreover, this linear span has a i ,...,i 1 m 1 k T T direct description in terms of A: It consists of all matrices B such that both A B and AB are symmetric matrices. Taking cue from this observation, we will associate a critical space H to a tensor f show that H contains the critical rank-at-most-k tensors for f for each value of k (see below for a definition), and that H is spanned by the critical rank-one tensors for f .Wewill establish these results for sufficiently general partially symmetric tensors, and we work over the base field C rather than R. d n +1 d n +1 1 1 p p Theorem 1.1 Let f be a sufficiently general tensor in S C ⊗ ··· ⊗ S C .Then, for each natural number k, the critical rank-at-most-k tensors for f lie in the critical space H . Moreover, if for each  with d = 1 the triangle inequality n ≤ n holds, f   i i= n +1 then codim H = and H is spanned by the critical rank-one tensors for f . In f f particular, f itself lies in the linear span of the critical rank-one tensors for f . We record the following two corollaries over the real numbers. Corollary 1.2 If n , ... ,n satisfy the triangle inequality n ≤ n for each  = 1 p  i i= n +1 1, ... , p, then for a sufficiently general tensor f ∈ R and any natural number k, i=1 any real tensor of real rank at most k closest to f in the Frobenius norm lies in the linear span of the complex critical rank-one tensors for f . In particular, f itself lies in the linear span of the complex critical rank-one tensors for f . d n+1 Corollary 1.3 For a sufficiently general symmetric tensor f ∈ S R and any natural number k, any real symmetric tensor of real symmetric rank at most k closest to f in the Frobenius norm lies in the linear span of the complex critical symmetric rank-one tensors for f . In particular, f itself lies in the linear span of the complex critical rank-one tensors for f . In the case of symmetric tensors, these critical rank-one tensors correspond to the so- called eigenvectors of f [11], while in the case of ordinary tensors, they correspond to singular vector tuples [10]. In the case n = 1 of binary forms, Corollary 1.3 was proved in [16]. The two corollaries above can be regarded as generalizations of the Eckart–Young Theorem and the Spectral Theorem from matrices to tensors. Several remarks are in order here. First, we complexify d to the quadratic polynomial d (u):= (u − f |u − f ), where (.|.) is the standard complex bilinear form on the space of tensors (and not a Hermitian form). The point of doing this is that, unlike for matrices, the critical points of this function on low-rank tensors are in general not real anymore, even if f is real. Accordingly, the critical space H , while defined by linear equations over R if f is real, is taken to be the space of complex solutions to those equations. Second, we denote the dimensions by n + 1 rather than n since we will be using methods from projective algebraic geometry where the formulas look more appealing in terms of the projective dimension n than in the affine dimension n + 1. An example of this phenomenon is the Draisma et al. Res Math Sci (2018) 5:27 Page 3 of 13 27 triangle inequalities in the theorem, which hold if and only if the variety dual to the Segre– n n 1 p Veronese embedding of the product P ×···×P via degrees d , ... ,d isahypersurface 1 p [7, Corollary 5.11]. The remainder of this paper is organized as follows. In Sect. 2, we define the critical space H for a partially symmetric tensor f and prove that the critical rank-at-most-k tensors for f lie in H . Then, in Sect. 3, we use vector bundle techniques to compute the dimension of the space spanned by the critical rank-one tensors for f and to show that this space equals H . In Sect. 4, we combine these ingredients to establish the results above. 2 The critical space of a tensor 2.1 Partially symmetric tensors and their ranks Let p ∈ Z ,let V , ... ,V be complex vector spaces, and let d , ... ,d ∈ Z .Let S V ≥1 1 p 1 p ≥1 be the dth symmetric power of V . We will study tensors in the space d d 1 p T := S V ⊗ ··· ⊗ S V . 1 p So for p = 1, T is a symmetric power of V , which is canonically isomorphic with the space of symmetric, d -way n ×···×n -tensors. On the other hand, when all d are equal p 1 1 i to 1, then T is a space of p-way ordinary tensors. We will write [p]:={1, ... ,p}. Inside T,let X be the set of all tensors of the form d p v ⊗ ··· ⊗ v (v ∈ V ,  ∈ [p]). Then, X is a closed subvariety of T known as the affine cone over the Segre–Veronese n n 1 p embedding of P × ··· × P of degrees (d , ... ,d ). Let kX denote the set of sums of k 1 p elements of X. An arbitrary element t of T lies in kX for some k, and the minimal such k is called the rank of t [9, Definition 5.2.1.1]. For p = 1, this is the symmetric or Waring rank, and if all d are 1, this notion is ordinary tensor rank. We write Sec (X) for the Zariski (or Euclidean) closure of kX in T. For real tensors, a few modifications are needed. The real rank of a real tensor t is the minimum k such that t = λ x with λ ∈ R and x ∈ X (it is enough to allow i i i i R i=1 3 3 λ =±1). For example (e + ie ) + (e − ie ) has complex rank 2 and real rank 3. Real i 1 2 1 2 rank is subtle for low-rank approximation of tensors. A classical example of de Silva and Lim [2] shows that for almost all 2 × 2 × 2-tensors of real rank 3 (like the above one) does not exist a closest tensor of real rank 2, while such phenomena may happen only for measure zero subsets of the set of complex tensors of given rank. 2.2 Symmetric bilinear forms and pairings If V, W are complex vector spaces with symmetric bilinear forms (.|.), and if d ∈ Z , then ≥0 S V and V ⊗ W carry unique symmetric bilinear forms, also denoted (.|.), that satisfy d d  d (v |v ):= (v|v ) and (v ⊗ w|v ⊗ w ):= (v|v )(w|w ). The first of these equalities implies (v ... v |v ) = (v |v ) 1 d i i=1 27 Page 4 of 13 Draisma et al. Res Math Sci (2018) 5:27 and more in general v ··· v |v ··· v = v |v . 1 d i d π(i) d! π ∈S i=1 We now fix nondegenerate symmetric bilinear forms on each V ,  ∈ [p]. Then, iterating these constructions, we obtain a canonical bilinear form on T. Using the bilinear forms on V and W , we can also build more general bilinear maps whose outputs are vectors or tensors rather than scalars. We will call these bilinear maps pairings anddenotethemby[.|.]. Of particular relevance to us is the skew-symmetric d d pairing S V × S V → V determined by d d d−1 [v |w ]:= (v|w) v ∧ w, which implies ⎛ ⎞ ⎝ ⎠ [v ... v |w ] = (v |w) v ∧ w 1 d i i i ∈[d] i=i and more in general ⎛ ⎞ ⎝ ⎠ [v ··· v |w ··· w ] = (v |w ) v ∧ w , 1 1 i d d π(i) i j d · d! i ,j ∈[d] π:[d]\i →[d]\j i=i where π runs over all bijections [d] \ i → [d] \ j . Remark 2.1 In the case of binary forms (dim V = 2 and arbitrary d), the pairing [f |g] coincides (up to scalar multiples) with (f |D(g)), where D(g) = g y − g x;see [16]. Note x y the skew-symmetry property (f |D(g)) =− (g |D(f )). On the other hand, in the case of symmetric matrices (d = 2 and arbitrary V ), the pairing [f |g] coincides (up to scalar multiples) with the bracket fg − gf . Building on this construction, for each  ∈ [p] we define a pairing [.|.] : T ×T → V by ⎛ ⎞ ⎝ ⎠ [f ⊗ ··· ⊗ f |g ⊗ ··· ⊗ g ] := (f |g ) [f |g ],f ,g ∈ S V , (1) 1 p 1 p  i i   i i i i= which we will use to define the critical space. Remark 2.2 In the case of matrices T = V ⊗ V , the pairing [f, g] coincides (up to scalar 1 2 1 t t t t multiples) with fg − gf , while [f, g] is (up to scalar multiples) f g − g f . 2.3 Basis, orthogonal basis and monomials If B is a basis of V , then the degree-d monomials in the elements of B form a basis of S V . Such a basis is orthogonal if B is orthogonal. Hence, if we fix bases (respectively, orthog- onal bases) of V , ... ,V , then by taking tensor products we obtain a basis (respectively, 1 p orthogonal basis) of T, whose elements we will call monomials of degree D := d . =1 We will use the word gcd of two such monomials x, y for the highest-degree monomial z such that both x and y can be obtained from z by multiplying z with suitable monomials. Draisma et al. Res Math Sci (2018) 5:27 Page 5 of 13 27 Example 2.3 If p = 3and V = V = V = C with the standard bilinear form, and 1 2 3 2 2 3 d = d = 3and d = 2, then the gcd of (e e ) ⊗(e e e ) ⊗(e )and (e e e ) ⊗(e ) ⊗(e e ) 1 2 3 2 1 2 3 1 2 3 2 3 1 1 3 equals (e e ) ⊗ (e ) ⊗ (1). 1 2 3 Lemma 2.4 For two monomials f = f ⊗ ··· ⊗ f ,g = g ⊗ ··· ⊗ g in T relative to the 1 p 1 p same orthogonal bases of V , ... ,V and for  ∈ [p] we have [f |g] = 0 unless f = g for 1 p  i i all i =  and h := gcd(f ,g ) has degree d − 1; in this case gcd(f, g) has degree D − 1 and [u|v] ∈ C (f /h) ∧ (g /h). This is immediate from the definition of the pairing in (1). 2.4 Critical rank-one tensors Let f ∈ T. Then, the critical points of the distance function d : x → (f −x|f −x)on X are by definition those x ∈ X \{0} for which f − x is perpendicular to the tangent space T X to X at x; we write this as f − x ⊥ T X. We call these tensors the critical rank-one tensors for f . For sufficiently general f , each of these critical rank-one tensors is non-isotropic, i.e., satisfies (x|x) = 0(see[4, Lemma 4.2], in next Proposition 2.6 we will prove a slightly more general fact). We will establish a bilinear characterization of these critical rank-one tensors for f . First, we describe the tangent space of X at a point x in more detail. For this, write x = v ⊗ ··· ⊗ v . (2) Hence, we may extend each v to a basis of V .Wethenobtainan x-adapted basis of T consisting of monomials. If moreover x is non-isotropic, we have (v |v ) = 0 and we may extend each v to an orthogonal basis. We then obtain an x-adapted orthogonal basis of T. Lemma 2.5 Let x ∈ Xas in (2). (1) Then, relative to any x-adapted basis, T X is spanned by all degree-D monomials whose gcd with x has degree at least D − 1. (2) Assume moreover that x is non-isotropic. Then, relative to any x-adapted orthogonal basis, (T X) is spanned by all degree-D monomials whose gcd with x has degree at most D − 2. Proof Part (1) follows by applying the Leibniz rule to the parameterization (2)of X;part (2) is a straightforward consequence. Proposition 2.6 Let f ∈ T and let x ∈ X be non-isotropic. Then, the following two statements are equivalent: (1) some (nonzero) scalar multiple of x is a critical rank-one tensor for f and (2) a unique (nonzero) scalar multiple of x is a critical rank-one tensor for f ; and they imply the following statement: (3) for each  ∈ [p], [f |x] ∈ V is zero. Moreover, if f is sufficiently general, then every nonzero x ∈ X satisfying (3) is non-isotropic and satisfies (1) and (2). 27 Page 6 of 13 Draisma et al. Res Math Sci (2018) 5:27 The pairing in item (3) is the pairing from (1). Proof For the equivalence of the first two statements, we note that if cx, c x with c, c = 0 are critical rank-one tensors for f , then T X = T  X = T X and f − cx ⊥ T X and cx c x x x f − c x ⊥ T X.Since x ∈ T X, we find that (c − c )x ⊥ x, and using that x is non-isotropic x x we find that c = c . To prove that (1) implies (3), write x as in (2) and extend each v to an orthogonal basis of V ,soastoobtainan x-adapted orthogonal basis of T. Now assume that f − cx ⊥ T X. Then, by Lemma 2.5, f − cx is a linear combination of degree-D monomials whose gcds with x have degrees at most D − 2. Hence by Lemma 2.4,[x|f − cx] = 0. By the skew- symmetry, [x|x] = 0, so [x|f ] = 0. d p For the last statement, consider an x = v ⊗ ··· ⊗ v ∈ X where, say, v , ... ,v p 1 a with a > 0 are isotropic but the remaining factors are not. Extend each v , > a to an orthogonal basis of V , and for v with  ≤ a find an isotropic w ∈ V with (v |w ) = 1 and extend v ,w with an orthogonal basis of the orthogonal complement of v ,w  to a basis of V . In the corresponding (non-orthogonal) monomial basis of T, the monomials y with [y|x] = 0 for  ≤ a are those of the form d d −1 d p 1  d a+1 w ⊗ ··· ⊗ w u ⊗ ··· ⊗ w ⊗ v ⊗ ··· ⊗ v , 1  a a+1 where u is a basis vector of V that is distinct from v but possibly equal to w .These monomials all satisfy [y|x] = 0 for i = . Similarly, the monomials y with [y|x] = 0 for > a are those of the form d d −1 d d p 1 d a+1 l w ⊗ ··· ⊗ w ⊗ v ⊗ ··· ⊗ v u ⊗ ··· ⊗ v l p 1 a+1 l with u a basis vector of V distinct from v ; they, too, satisfy [y|x] = 0 for i = .The remaining monomials span the space of f swith[x|f ] = 0 for all ; this space therefore has dimension dim T − (n + ··· + n ), 1 p and it does not change when we scale x. Since the isotropic projective points x∈ PT form a subvariety of positive codimension in the (n + ··· + n )-dimensional projective 1 p variety PX, the locus of all f for which there is a nonzero isotropic x ∈ X with [f |x] = 0 for all  has dimension less than dim T. Now assume that f is sufficiently general and let x ∈ X \{0} satisfy [x|f ] = 0 for all .By the above, x is non-isotropic. Suppose that f ,expandedonthe x-adapted orthogonal basis, contains a monomial y whose gcd with x has degree exactly D −1. If y agrees with x except d −1 in the factor S V where it equals v u ,thenin[x|f ] , expanded on the standard basis of V relative to the chosen basis of V , the term v ∧ u has a nonzero coefficient. Hence, [x|f ] is nonzero, a contradiction. Therefore, f contains only monomials whose gcds with x have degrees at most D − 2, and possibly the monomial x itself. Then, f − cx ⊥ T X for a unique constant c.By generality of f , it does not lie in (T X) for any x ∈ X \{0} (the union of these orthogonal complements is the cone over the variety dual to the projective variety defined by X,and of positive codimension). Hence, c = 0, and cx is a critical rank-one tensor for f . Remark 2.7 The implication (1) =⇒ (3) in Proposition 2.6 holds without the assumption of non-isotropy of x. This follows from the fact that the ED correspondence {(x, f ) ∈ X × V | x is critical for f } Draisma et al. Res Math Sci (2018) 5:27 Page 7 of 13 27 is a irreducible variety (see [3, §4 and Lemma 2.1]) and the nonempty open part in it where x is non-isotropic lies in the variety defined by [f |x] = 0 ∀ ∈ [p]byProposition 2.6. 2.5 The critical space In view of Proposition 2.6, we introduce the following notion. Definition 2.8 For a tensor f ∈ T, the critical space H ⊆ Tof f is defined as H :={g ∈ T | [f |g] = 0 for all  ∈ [p]}. Remark 2.9 By the skew-symmetry, it follows immediately that f ∈ H . Remark 2.10 In the case of binary forms (dim V = 2), H is the hyperplane orthogonal to D(f )[16]. In the case of ordinary tensors, H was first defined in [15] where it was called singular space, but in view of the results in this paper we feel that critical space isabetter name. Proposition 2.6 establishes that the non-isotropical critical rank-one tensors all lie inside H ; hence for a sufficiently general f , all critical rank-one tensors lie in H . In the next f f subsection, we will establish an analogous statement for higher ranks. p 2 Note that the number of linear conditions for g to lie in H is at most dim V = =1 p n +1 —the linear conditions in the definition may not all be linearly independent. =1 2 In Proposition 3.6 we will see that, assuming the triangle inequalities from Theorem 1.1 and assuming that f is sufficiently general, equality holds. 2.6 Higher rank We will now establish a generalization of Proposition 2.6 to higher-rank tensors. Definition 2.11 Let f ∈ T and let k be any nonnegative integer. A critical rank-at-most-k tensor for f is a tensor g ∈ kX such that f − g ⊥ T Sec (X). g k Note that by [4, Lemma 4.2], all the critical rank-at-most-k tensors for a sufficiently general f ∈ T are smooth points of Sec (X) and can be written as a sum of k non- isotropic rank-one tensors. Moreover, if we assume that k is at most the generic rank of tensors in T, then these critical tensors to a sufficiently general f have rank equal to k.If k is at least the generic rank of tensors in T, then the only critical rank-at-most-k tensor for a sufficiently general f is f itself. Proposition 2.12 Let f ∈ T be sufficiently general and let k be a nonnegative integer. Then, all the critical rank-at-most-k tensors for f lie in the critical space H . Proof Let g be a critical rank-at-most-k tensor. By generality of f , g can be written as x + ··· + x with each x ∈ X non-isotropic. Then, T Sec X ⊇ T X, so that for 1 i g x k k i=1 i each i ∈ [k]wehave f − g ⊥ T X. By Lemma 2.5 this means that, in the x -adapted x i orthogonal basis, f − g is a linear combination of monomials whose gcds with x have degrees at most D − 2. Hence, by Lemma 2.4,[f − g |x ] = 0 for all  = 1, ... ,p.We conclude that, for each , [f − g |g] = [f − g |x ] = 0, i=1 27 Page 8 of 13 Draisma et al. Res Math Sci (2018) 5:27 and therefore [f |g] = [f − g |g] + [g |g] = 0 + 0, where in the last step we used that [.|.] is skew-symmetric. Hence, g ∈ H . In the next section, we compute the dimension of the space spanned by the critical rank-one tensors for a general tensor and show that this space equals H . 3 The scheme of critical rank-one tensors 3.1 Critical rank-one tensors as the zero locus of a vector bundle section Let f ∈ T = S V be a tensor. We assume that p ≥ 2, d ≥ 1, and dim V = =1 n + 1 ≥ 1 for all . We adapt the notation of [15, Section 5.1] to our current setting. Consider the Segre–Veronese variety PX = PV × ... × PV embedded with 1 p O(d , ... ,d )in PT;so PX is the projective variety associated with the affine cone X ⊆ T. 1 p Let π : PX → PV be the projection on the th factor and set N := dim PX = n + ... + n . For each  ∈ [p]let Q be the quotient bundle on PV , whose fiber 1 p over a point v is V /v. From these quotient bundles, we construct the following vector bundles on PX: E := E where E := π Q ⊗ O(d , ... ,d ,d − 1,d , ... ,d ). l l  1 −1  +1 p =1 Note that E has rank N. The fiber of E over a point v := (v , ... , v ) ∈ PX 1 p consists of polynomial maps v → V /v  that are multi-homogeneous of i=1 multi-degree (d , ... ,d − 1, ... ,d ). The tensor f yields a global section of E which 1  p over the point v is the map sending (c v , ... ,c v ) to the natural pairing of f with 1 1 p p d d −1 d 1  p (c v ) ··· (c v ) ··· (c v ) —a vector in V —taken modulo v . Combining these 1 1   p p p sections, f yields a global section s of E.ByProposition 2.6, for f sufficiently general, the tensor x := v ⊗ ··· ⊗ v is a nonzero scalar multiple of a critical rank-one tensor for f if and only if the point (v , ... , v ) is in the zero locus Z of the section s .In[5], 1 p f f this is used to compute the cardinality of Z for f sufficiently general as the top Chern class of E. Our current task is different: we want to show that, if we assume the triangle inequalities of Theorem 1.1 and that f is sufficiently general, the linear span Z  equals the projectivized critical space PH ; this is the second part of Theorem 1.1. 3.2 Bott’s formulas and a consequence Our central tool will be the following formulas for the cohomology of vector bundles over projective spaces [13]. Recall that  (k)isthe O(k)-twisted sheaf of differential r-forms on P . Lemma 3.1 (Bott’s formulas) For q,n,r ∈ Z and k ∈ Z,wehave ≥0 k+n−r k−1 if q = 0 ≤ r ≤ n and k > r, k r 1if0 ≤ q = r ≤ n and k = 0, q n r h P ,  (k) = −k+r −k−1 ⎪ if q = n ≥ r ≥ 0and k < r − n, and −k n−r 0 otherwise. A consequence featuring the triangle inequalities of Theorem 1.1 is the following. Draisma et al. Res Math Sci (2018) 5:27 Page 9 of 13 27 Lemma 3.2 Suppose that n ≤ n holds for all  with d = 1.Let k ≥ 2 be an i= integer, q , ... ,q be nonnegative integers with q < kand r , ... ,r be nonnegative 1 p  1 p =1 integers with r = k. Then, =1 H PV ,  (−d (k − 1) + 2r ) = 0. PV =1 Proof Assume that all factors in the tensor product are nonzero. First, if all of the factors were nonzero by virtue of the second and third line in Bott’s formulas, then we would have q ≥ r for all , and hence k > q ≥ r = k,a contradiction. So some factor is nonzero by virtue of the first line in Bott’s formulas; without loss of generality this is the first factor. Hence we have q = 0 ≤ r ≤ n and −d (k−1)+2r > r . 1 1 1 1 1 1 This last inequality reads r > d (k − 1). Combining this with r = k and the fact that 1 1 d is a positive integer, we find that r = k, d = 1, and r = 0 for > 1. In particular, 1 1 1 there are no > 1 for which the first line in Bott’s formulas applies. For any > 1, if the second line applies, then 0 = r = q =−d (k − 1) + 2r , which contradicts that both d and k − 1 are positive. Hence, the third line applies for all > 1, and in particular we have q = n . But then p p n ≥ r = k > q = n , 1 1 l l l=1 l=2 which together with d = 1 contradicts the triangle inequality in the lemma. 3.3 Vanishing cohomology The vanishing result in this subsection uses Lemma 3.2 and the following version of Künneth’s formula. Lemma 3.3 (Künneth’s formula) For vector bundles G on PV for  = 1, ... ,p and a nonnegative integer q we have q ∗ q H PX, π G H (PV , G ), q +...+q =q 1 p where the sum is over all p-tuples of nonnegative integers summing to q. Lemma 3.4 Suppose that n ≤ n holds for all  such that d = 1.Let k ≥ 2 be an i= integer and q ∈{0, ... ,k − 1}. Then, we have ⎛ ⎛ ⎞ ⎞ q ∗ ⎝ ⎝ ⎠ ⎠ H PX, E ⊗ O(d , ... ,d ) = 0. 1 p Proof First, ∗ ∗ ∗ E = π Q ⊗ O(−d , ... , −d , −(d − 1), −d , ... , −d ). 1 −1  +1 p =1 A well-known formula for kth wedge power of a direct sum yields k r ∗ ∗ ∗ E = (π Q ⊗ O(−d , ... , −(d − 1), ... , −d )). 1  p r +...+r =k 1 p 27 Page 10 of 13 Draisma et al. Res Math Sci (2018) 5:27 r r r ∗ 1 1 r Using (F ⊗ O(ω)) = ( F)(rω), Q =  (1), and ( (1)) =  (r), we obtain ∗ ∗ E = π  (r ) ⊗ O(−r d , ... , −r (d − 1), ... , −r d . 1    p PV r +...+r =k 1 p Twisting by O(d , ... ,d ), regrouping in each projection, and using r = k we find: 1 p ⎛ ⎞ ∗ ∗ ⎝ ⎠ E ⊗ O(d , ... ,d ) = π  (−d (k − 1) + 2r ) . 1 p PV r +...+r =k 1 p To compute H of each summand we apply Künneth’s formula, and obtain subsummands which are exactly of the form in Lemma 3.2, hence zero. 3.4 Comparing PH and Z f f Assume that f is sufficiently general in T. By the first subsection of this section and by Proposition 2.6, Z is contained in the projectivized critical space PH , hence so is Z . f f f Our goal now is to show that Z  is equal to PH and to compute its dimension. Both f f of these goals are achieved through the following lemma. The section s of E yields a homomorphism E → O of sheaves whose image is contained in the ideal sheaf I of the zero locus of s . Lemma 3.5 Assume that for each  ∈ [p] we have n ≤ n and that f is sufficiently i= general. Then, the induced homomorphism E ⊗ O(d , ... ,d ) → I ⊗ O(d , ... ,d ) 1 p Z 1 p induces an isomorphism at the level of global sections. The following proof can be shortened considerably using spectral sequences, but we found it more informative in its current form. To make the formulas more transparent, q q we write H (.) instead of H (PX, .) everywhere. Proof To establish the desired isomorphism 0 ∗ 0 H (E ⊗ O(d , ... ,d )) H (I ⊗ O(d , ... ,d )) 1 p Z 1 p we use the following Koszul complex (see, e.g., [8, Chapter III,Proposition 7.10A]): N +1 N 2 ∗ ∗ ∗ ∗ 0 = E → E → ··· → E → E → I → 0. k k+1 ∗ ∗ Letting F be the quotient of E by the image of E , this yields the short exact sequence 0 → F → E → I → 0. 2 Z Tensoring with O(d , ... ,d ) yields the short exact sequence 1 p 0 → F ⊗ O(d , ... ,d ) → E ⊗ O(d , ... ,d ) → I ⊗ O(d , ... ,d ) → 0, 2 1 p 1 p Z 1 p and this gives a long exact sequence in cohomology beginning with 0 0 ∗ 0 0 → H (F ⊗ O(d , ... ,d )) → H (E ⊗ O(d , ... ,d )) → H (I ⊗ O(d , ... ,d )) 2 1 p 1 p Z 1 p → H (F ⊗ O(d , ... ,d )) → 2 1 p So to obtain the desired isomorphism we want that H (F ⊗ O(d , ... ,d )) = 0 for q = 0, 1. 2 1 p Draisma et al. Res Math Sci (2018) 5:27 Page 11 of 13 27 For each k = 2, ... ,N, we have the short exact sequence 0 → F → E → F → 0 k+1 k which yields the long exact sequence ⎛ ⎞ k−2 ∗ k−2 ⎝ ⎠ → H E ⊗ O(d , ... ,d ) → H F ⊗ O(d , ... ,d ) 1 p 1 p ⎛ ⎞ k−1 k−1 ∗ ⎝ ⎠ → H (F ⊗ O(d , ... ,d )) → H E ⊗ O(d , ... ,d ) k+1 1 p 1 p k−1 k → H F ⊗ O(d , ... ,d ) → H F ⊗ O(d , ... ,d ) → k 1 p k+1 1 p Using Lemma 3.4, the two leftmost spaces are zero, so that k−2 k−1 H (F ⊗ O(d , ... ,d )) H (F ⊗ O(d , ... ,d )) and k 1 p k+1 1 p k−1 k H (F ⊗ O(d , ... ,d )) ⊆ H (F ⊗ O(d , ... ,d )). k 1 p k+1 1 p Hence, using that F = 0, we find that N +1 0 N −1 ∼ ∼ H (F ⊗ O(d , ... ,d )) ··· H (F ⊗ O(d , ... ,d )) = 0and = = 2 1 p N +1 1 p 1 N H (F ⊗ O(d , ... ,d )) ⊆ ··· ⊆ H (F ⊗ O(d , ... ,d )) = 0, 2 1 p N +1 1 p as desired. Proposition 3.6 Suppose that for each  ∈ [p] we have n ≤ n and that f is i= n +1 sufficiently general. Then, Z = PH and codim H = . f f T f Proof Since PX is embedded by O(d , ... ,d ), the space of linear forms on T vanishing 1 p on Z is H (I ⊗ O(d , ... ,d )). By Lemma 3.5, this space is isomorphic to Z 1 p 0 ∗ 0 ∗ ∗ H (E ⊗ O(d , ... ,d )) = H π Q ⊗ O(0, ... , 1, ... , 0) 1 p 0 1 0 1 = H (π ( (2))) = H PV ,  (2) , PV PV n +1 which by the first line in Bott’s formulas has dimension . This means that n +1 codim Z = , so the second statement in the proposition follows from PT the first statement. To establish the first statement, we spell out the map 0 ∗ 0 1 0 H (PV , Q ⊗ O(1)) = H (PV ,  (2)) → H (I ⊗ O(d , ... ,d )) Z 1 p PV f in greater detail. The space on the left is canonically ( V ) , and an element ξ in this space is mapped to the linear form T → C,g → ξ([f |g] ). As  varies, these are precisely the linear forms that cut out H . This proves that PH =Z . f f f 27 Page 12 of 13 Draisma et al. Res Math Sci (2018) 5:27 Remark 3.7 In general, for the equality Z = PH we only need that the linear equations f f cutting out PH also cut out Z , i.e., we only need that the linear map in Lemma 3.5 is f f surjective. One might wonder whether this surjectivity remains true when the triangle inequalities fail. In the case of (n + 1) × (n + 1)-matrices, it does indeed—there we 1 2 already knew the critical rank-one approximations span the critical space—but for p = 3 and 2 × 2 × 4-tensors (so that n = 3 > 1 + 1 = n + n ) the space Z  has dimension 3 1 2 6 while computer experiments suggest that the space PH has dimension 7 , hence the surjectivity fails. Still, in these experiments, f itself seems to lie in the span of Z .Thisleads to the open problem whether our analogue of the Spectral Theorem and the Eckart–Young Theorem persists when the triangle inequalities fail. 4 Proofs of the main results Proof of Theorem 1.1 The first statement is Proposition 2.12; the second and third state- ment are Proposition 3.6. The last statement follows from Remark 2.9. Proof of Corollaries 1.2 and 1.3. If g is a real tensor of real rank at most k closest to f , then one can write it as x + ··· + x with x , ... ,x real points of X. In particular, all of these 1 k 1 k points are non-isotropic, and the argument of Proposition 2.12 applies. Hence, g lies in H . Now the result follows from Proposition 3.6. The argument applies, in particular, to k equal to the rank of f , which gives the last statement of the corollaries. Note that, if f is any real tensor, then any real tensor of real rank at most k closest to f lies in H by the argument above. Only for the conclusion that it lies in the span of the complex critical rank-one tensors of f do we use that f is sufficiently general. We do not know whether this generality is really needed. Also note that we do not shed new light on the question of when for sufficiently general f there exists a closest real tensor of rank at most k. For an update on the complex case, see [17]. Author details 1 2 Mathematisches Institut, Universität Bern, Sidlerstrasse 5, 3012 Bern, Switzerland, Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, The Netherlands, Dipartimento di Matematica e Informatica U. Dini, Università di Firenze, viale Morgagni 67/A, 50134 Florence, Italy, Departamento de Álgebra, Geometría y Topología, Universidad de Málaga, Bulevar Louis Pasteur, 31, 29010 Málaga, Spain. Received: 30 November 2017 Accepted: 9 May 2018 Published online: 23 May 2018 References 1. 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