The Data Singular and the Data Isotropic Loci for Affine Cones

The generic number of critical points of the Euclidean distance function from a data point to a variety is called the Euclidean distance degree. The two special loci of the data points where the number of critical points is smaller then the ED degree are called the Euclidean Distance Data Singular Locus and the Euclidean Distance Data Isotropic Locus. In this article we present connections between these two special loci of an affine cone and its dual cone.


Introduction
Models in science are o en expressed as real solution sets of systems of polynomial equations, namely real algebraic varieties. One of the most fundamental optimization problems that can be formulated on such sets is the following: Given a real algebraic variety and given a general data point of the ambient space, minimize the Euclidean distance from the given data point to the variety.
To solve this problem algebraically, we examine the critical points of the squared Euclidean distance function. The number of such critical points is an important complexity parameter for both numerical and exact algorithms, [5,13] for nding the optimal solution to the distance minimization problem and is called the Euclidean Distance Degree (or ED degree). This optimization problem arises in a wide range of applications, such as low-rank approximations (Example 8), control theory (Example 10), formation control (Example 13), algebraic statistics (Example 14), and multiview geometry (Example 15).
For a general data point u, the number of complex critical points is constant, while the number of real critical points is typically not constant for all general u. For example, if one of the critical points has a multiplicity, then the number of real critical points typically changes. This locus is called the EDdiscriminant (or classically focal locus) [2,3,6,9,14].
In this article, we want to discuss the locus (di erent from the ED discriminant) of exceptional data points u for which the number of complex critical points is smaller than the ED degree. We consider three cases in which we can have a di erent number of critical points than expected. The rst reason is because a critical point may wander o into the singular locus of the variety. The study of this special locus was proposed by Bernd Sturmfels, rst examples were developed [4] and it was named ED data singular locus. In a similar fashion, the second case is when a critical point becomes isotropic with respect to the Euclidean inner product (i.e., it has norm zero); this locus will be called ED data isotropic locus. In these two cases, the number of critical points is smaller than the ED degree. Finally, a data point can have in nitely many critical points, but this phenomenon is apparently recorded by the ED discriminant, so we do not deal with it in this article. A classical example would be that there are in nitely many critical rank 2 approximations of a matrix with two identical singular values.
In this article, we aim to describe the data singular and the data isotropic loci of a ne cones.

The special Loci of data points
To nd the critical points algebraically, we consider X to be a variety in C n and we examine all complex critical points of the complexi ed distance, induced by the standard symmetric bilinear form, with x ∈ X reg , where X reg denotes the locus of regular points of X, so we only allow those critical points that are nonsingular. Since the ED degree is additive over the components of a variety from now on, we assume that X ⊆ C n is an irreducible algebraic variety of codimension c with de ning radical ideal I. If x ∈ X reg is a critical point of d u , then the following holds: u − x ⊥ T x X. This latter condition can be formulated as x ∈ X reg is a critical point of d u if and only if all the (c + 1) × (c + 1) minors of the matrix: where Jac x (I) is the Jacobian of I at the point x.
We de ne the ED-correspondence to be the closure of the set of all pairs (u, x), such that x ∈ X reg is critical to d u , and we denote it by E X ⊆ C n u × C n x . In other words, E X is the closure of: We have two natural projection maps π 1 : E X → C n u sending (u, x) to u and π 2 : E X → C n x sending (u, x) to x. Let SingX denotes the singular locus of X, that is, the set of all points of x ∈ X such that all the c × c minors of Jac x (I) vanish.
So for a given data point u, the cardinality of the ber of π 1 over u, π −1 1 (u), measures the number of critical points.
We want to discuss the locus of exceptional data points u at which the number of complex critical points is di erent from the ED degree. As mentioned in the introduction, we consider three cases in which we can have di erent number of critical points than expected. The rst one is because a critical point may wander o into SingX because of the closure appearing in the de nition of E X . This locus is called the ED data singular locus.

Data singular locus
We use the precise de nition of the ED data singular locus [4], that is, the Zariski closure of the set: . We denote the ED data singular locus of an algebraic variety X by DS(X) (abbreviating "data singular" locus) and we aim to describe the data singular locus of a ne cones. We de ne X * the dual variety to X to be the Zariski closure of the set: y ∈ C n | ∃x ∈ X reg : y ⊥ T x X . More precisely, we view X * as subset of C n through the standard symmetric bilinear form on C n . Our main result in this section is the following theorem. Theorem 1. Let X ⊆ C n be an irreducible a ne cone that is not a linear space. Then the following two inclusions hold: Proof. First, we prove inclusion (1) for a dense subset of X * . For this, take u ∈ X * , such that there exists a regular point x r ∈ X reg , such that u ⊥ T x r X, that is, all the (c + 1) × (c + 1) minors of u Jac xr (I) vanish, where c is the codimension of X and Jac x r (I) is the Jacobian of the (radical) ideal I of X at the point x r . We denote an arbitrary (c + 1) × (c + 1) minor of this matrix by u Jac xr (I) (c+1) . We claim that (u + λx r , λx r ) ∈ E X for all real λ ≥ 0. We have that if f ∈ I is homogeneous of degree d, then ∇f (λx) = λ d ∇f (x). So if x r is a regular point, then λx r is also regular, for any λ > 0. Moreover, we get that for any (c + 1) × (c + 1) minor: where N is the sum of degrees of the de ning polynomials of I, which appear in the particular (c + 1) × (c + 1) minor. So (u + λx r , λx r ) ∈ E X for all real λ > 0. But then taking the limit when λ goes to zero, we get that is Zariski closed (hence closed wrt. Euclidean topology as well) and since 0 ∈ SingX. Indeed, for every x ∈ X, the line {λ·x} is in the tangent space to 0, so T 0 X is equal to the linear span of X, which has a greater dimension than X if and only if X is not a linear space, and hence 0 ∈ SingX. So then u = π 1 ((u, 0)) ∈ DS(X).
For the proof of (2), take an element (u, x 0 ) ∈ E X ∩π −1 2 (SingX). Then, this point can be approximated by a sequence in the part of E X over X reg . That is, there exists a sequence δ i → 0 in C n and x i → x 0 with all the x i ∈ X reg , such that By the ED Duality Theorem for a ne cones [3, Theorem 5.2], we get that (u + δ i ) − x i ∈ X * , for all i. Now taking the limit, when i goes to in nity, we get that u − x 0 ∈ X * , since X * is closed (hence closed wrt. Euclidean topology as well). Finally, this means that u ∈ x 0 + X * ⊆ SingX + X * .
Note that the condition in the theorem that X is not a linear space is necessary to prove the theorem. Otherwise, if X is a linear subspace of C n , then it has a nonempty dual (its orthogonal complement with respect to the inner product), but its singular locus is empty, hence its data singular locus is empty as well.

Data isotropic locus
A second possibility for a data point u to have smaller number of critical points than expected is by letting one of the critical points become isotropic. Let us denote by Q = {x ∈ C n : n i=1 x 2 i = 0} the isotropic quadric with respect to the standard symmetric bilinear form. Draisma et al. de ne the ED degree of a projective variety in P n−1 to be the ED degree of the corresponding a ne cone in C n [3]. Moreover, given a data point u, the critical points to these two objects are in a one-to-one correspondence, given that none of the critical points lies in the isotropic quadric [3, Lemma 2.8]. In particular, the role of Q shows that the computation of ED degree is a metric problem. This is the reason that even though in the de nition of the a ne E X we keep the isotropic critical points, when we pass to projective varieties we will exclude the isotropic points. This way the data isotropic locus represents the locus of data points which have a di erent number of critical points when X is considered as an a ne cone compared to when X is considered as a projective variety. More precisely, we de ne the ED data isotropic locus to be the Zariski closure of the set: . We denote the ED data isotropic locus of an algebraic variety X by DI(X) (abbreviating "data isotropic" locus). We have the following theorem for the ED data isotropic locus of a ne cones.

Theorem 2.
Let X ⊆ C n be an irreducible a ne cone. Then the following two inclusions hold: where X * denotes the dual variety to X.
Proof. The proof follows the lines of the proof of Theorem 1, keeping in mind that 0 ∈ X is always an isotropic point.
In the following two sections, we will give examples to show that both inclusions appearing in Theorems 1 and 2 can be strict and/or equalities.

Examples of the ED data singular locus
In this section, we present several useful examples concerning the ED data singular locus of an a ne cone. Before we get to the examples, we present how one can computationally determine the objects we are working with. We illustrate the main algorithms with code in Macaulay2 [7]. For an a ne cone X ⊆ C n , of codimension c with de ning radical ideal I, one can determine its dual X * using the following code [12, Algorithm 5.1].
Example 3 (Computing the dual variety). We present the algorithm for the real a ne cone X ⊆ C 3 de ned by the homogeneous equation The output reveals that X * is the zero locus of the polynomial f * = 4x 3 1 − 27x 2 2 x 3 .
Following the de nition of the data singular locus, the next example contains an algorithm for calculating the ideal of it.
Example 4 (Computing the data singular locus). We present the algorithm for the real a ne cone X ⊆ C 3 de ned by the homogeneous equation From the output, we see that the data singular locus is the zero set of the polynomial x 1 (4x 3 1 − 27x 2 2 x 3 ). Now, we arrived at the point to present a sequence of interesting varieties and the corresponding duals and data singular loci. The rst example is the one we used for presenting the algorithms previously. In this example, both inclusions (1) and (2) are strict, as it will be seen.

Example 5 (Cuspidal cubic cone). Let X ⊆ C 3 be the real variety de ned by the homogeneous equation
Since it is an a ne cone, it has a dual X * , which is de ned by the dual equation f * = 4x 3 1 − 27x 2 2 x 3 . For the data singular locus, we get that DS(X) is the zero locus of the polynomial . So we can see that X * is even a component of DS(X). Moreover, X * + SingX is something much larger and not equal to DS(X). For example, the point: (3, 2, 1) + (0, 0, 1) ∈ X * + SingX, but is not on DS(X). Figure 1 shows X in blue and X * in green and DS(X) is the union of the greencolored X * and the additional surface in red.
The next example shows that both inclusions (1) and (2) can in fact be equalities. More generally, we have the following corollary to Theorem 1. Proof. The rst part follows directly from the claim of Theorem 1. The "moreover" part is classical [12, Exercise 5.14].
Example 7 (Cone over ellipse). Let X ⊆ C 3 the cone over an ellipse, de ned by the homogeneous equation f = x 2 1 +4x 2 2 −9x 2 3 . The singular locus SingX only contains 0, so as a consequence of Theorem 1,  we have that DS(X) equals the dual variety X * , de ned by the dual equation f * = x 2 1 + x 2 2 /4 − x 2 3 /9. Figure 2 shows X in blue and X * in green.
The next example concerned the well-known and much used determinantal varieties. We will see that for this variety, inclusion (1) is strict and inclusion (2) is an equality.

Example 8 (Determinantal varieties). Denote by M ≤r
n×m the variety of n × m matrices (suppose n ≤ m) of rank at most r. It is classical that the singular locus is the variety M ≤r−1 n×m . We have that the dual variety is exactly M ≤n−r n×m [6, Chapter 1, Proposition 4.11]. So applying Theorem 1, we get that M ≤n−r n×m ⊆ DS(M ≤r n×m ) ⊆ M ≤n−r n×m + M ≤r−1 n×m = M ≤n−1 n×m . So for rank-one matrices (r = 1), we get that DS(M ≤1 n×m ) = M ≤n−1 n×m , which is not a surprise based on Corollary 6, since M ≤1 n×m is smooth, except at 0. But something more is true for general r. We claim that the upper bound for the inclusions is always attained. For this, we have the following proposition. Proposition 9. The ED data singular locus of the determinantal variety M ≤r n×m is equal to M ≤n−1 n×m , for all 1 ≤ r ≤ n − 1.
Proof. An n × m matrix U lies in the interior of DS(M ≤r n×m ) if and only if it has a singular critical point. All the critical points of U look like T 1 · Diag(0, 0, . . . , σ i 1 , 0, . . . , 0, σ i r , 0, . . . , 0) · T 2 , where the singular value decomposition of U is equal to U = T 1 · Diag(σ 1 , . . . , σ n ) · T 2 , with σ 1 ≥ · · · ≥ σ n singular values and T 1 , T 2 orthogonal matrices of size n × n and m × m [3, Example 2.3]. Such a critical point is singular if and only if it has rank at most r − 1, which can only happen if one of the singular values σ i 1 , . . . , σ i r is zero. So there exists a singular critical point to U if and only if there is a zero singular value of U, which can only happen if U has a rank defect. Hence all the (n − 1) × (n − 1) minors are zero, that is, U ∈ M ≤n−1 n×m . Now since M ≤n−1 n×m is Zariski closed, we have the desired equality.
The next example shows that X * is a subvariety of DS(X) but not necessarily a component of it.
Example 10 (Hurwitz determinant). In control theory, to check whether a given polynomial is stable, one builds up the so-called Hurwitz matrix H n and checks if every leading principal minor of H n is positive. Take n = 4, then the 4th Hurwitz matrix looks like: The ratio Ŵ 4 = det(H 4 )/x 5 is a homogeneous polynomial and is called the Hurwitz determinant for n = 4 [3, Example 3.5].
We have thus seen examples of varieties with: both inclusions in Theorem 1 being strict, both inclusions in Theorem 1 being equalities and the second inclusion being an equality, while the rst one is strict. It is natural to ask if there are examples where the rst inclusion is an equality, while the second one is strict. The author could not nd such an example, so the following question arises.

Problem 11.
Find an a ne cone X, such that X * = DS ⊂ X * + Sing(X) or prove that there is no such X.

Examples of the ED data isotropic locus
In this section, we present several application-oriented examples concerning the ED data isotropic locus of an a ne cone. We begin with presenting how can one computationally determine the data isotropic locus of a variety.
Example 12 (Computing the data isotropic locus). We present the algorithm for the a ne cone de ned by f = x 1 x 6 − x 2 x 5 + x 3 x 4 , representing the Grassmannian of planes in 4-space.
From the output, we learn that DI(X) is the zero locus of the polynomial x 1 x 6 − x 2 x 5 + x 3 x 4 , so we get that the data isotropic locus is equal to the dual variety which in this case equals the variety.
The next example shows that the data isotropic locus can be equal to the dual and strictly contained in X * + (X ∩ Q).
Example 13 (Cayley-Menger variety). Let X denote the variety in C 3 with parametric representation: Based on [1] and on [3,Example 3.7], the points in X record the squared distances among three interacting agents with coordinates z 1 , z 2 , and z 3 on the line R. The prime ideal of X is given by the determinant of the Cayley-Menger matrix So X is de ned by the irreducible polynomial: A er running the computations, one can see that the data isotropic locus equals the dual variety, which is de ned by f * = x 1 x 2 + x 1 x 3 + x 2 x 3 ( Figure 3). And it does not equal X * + (Q ∩ X), for example, because the point (1, 0, 0) + (0, 1, i) ∈ X * + (Q ∩ X), but it does not lie on DI(X).
The next example shows that both inclusions from Theorem 2 can be strict. Example 14 (Cayley's cubic). Let X be de ned by f = x 3 1 − x 1 x 2 2 − x 1 x 2 3 + 2x 2 x 3 x 4 − x 1 x 2 4 , the 3 × 3 symmetric determinant in C 4 . This hypersurface is sometimes called the Cayley's cubic surface and receives much attention in the study of elliptopes and exponential varieties in algebraic statistics ( [12,Example 5.44], [11, Example 1.1], [10]). Its dual variety is the quartic Steiner surface de ned by f * = x 2 2 x 2 3 − 2x 1 x 2 x 3 x 4 + x 2 2 x 2 4 + x 2 3 x 2 4 . A er running the computations, one nds that the data isotropic locus is the union: So it is clearly not equal to the dual variety. And it is not equal to X * + (Q ∩ X) either, because, for example, the point: but it is not in DI(X) Our next example shows that the second inclusion in Theorem 2 can be equality and moreover, it can give the whole space.
Example 15 (Special essential variety). Essential matrices play an important role in multiview geometry [8]. The connections between the ED degree theory and multiview geometry were investigated [3,Example 3.3]. The set of essential matrices is called the essential variety and is de ned as follows: It is a codimension 3 variety of degree 10. We are interested in the data isotropic locus of this variety, but for computational reasons, we will take a linear section of it and we will only consider the symmetric, constant diagonal essential matrices, which we will call the special essential variety and will denote by SE. More precisely, we de ne SE to be: Since this variety is not irreducible, we will perform our computations, componentwise. When running the computations, one will nd that the data isotropic locus is the whole space. Indeed, one can observe that SE is inside the isotropic quadric Q, so every critical point is isotropic. We have that DI(X) = X * + (X ∩ Q) = X * + X = C 4 .
Moreover, DI(X) is not equal to the dual variety, since X * is a proper variety de ned by f * = (x 2 2 + x 2 4 )(x 2 2 + x 2 3 )(x 2 3 + x 2 4 ). Moreover, clearly, the dual is not a component of DI(X).
In the last example, the reader can see that both inclusions from Theorem 2 can be equalities.
Example 16 (Line through the origin). In what follows let X be the line through the origin in C 3 de ned by the vanishing of the polynomials x 1 + 2x 2 + 3x 3 and 4x 1 + 5x 2 + 6x 3 . Then, we get that X intersects the quadric Q only in the point 0, so by Theorem 2, we immediately get that X * = DI(X) = X * + {0}, and the dual is the orthogonal complement of X, so it is de ned by x 1 − 2x 2 + x 3 .