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A Morse Theoretic Approach to Non-Isolated Singularities and Applications to Optimization

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A Morse Theoretic Approach to Non-Isolated Singularities and Applications to Optimization A MORSE THEORETIC APPROACH TO NON-ISOLATED SINGULARITIES AND APPLICATIONS TO OPTIMIZATION LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG Abstract. Let X be a complex affine variety in CN , and let f : CN ! C be a polyno- mial function whose restriction to X is nonconstant. For g : CN ! C a general linear function, we study the limiting behavior of the critical points of the one-parameter family of ft := f − tg as t ! 0. Our main result gives an expression of this limit in terms of critical sets of the restrictions of g to the singular strata of (X; f). We apply this result in the context of optimization problems. For example, we consider nearest point problems (e.g., Euclidean distance degrees) for affine varieties and a possibly nongeneric data point. 1. Introduction The motivation for this work is to study nearest point problems for algebraic models and Euclidean distance degrees. For example, given a circle and a point P outside its center, there is a unique point on the circle which is closest to P , as seen in Figure1. However, if P is taken to be the center, then every point on the circle is a closest point. Our aim is to understand such a special (non-generic) behavior on (arbitrary) algebraic varieties by a limiting procedure on a set of critical points. In terms of applied algebraic geometry, our results can be understood as describing what happens when genericity assumptions of statements on Euclidean distance degrees are removed (see Section 5.3). In optimization, our results state what happens as we take a regularization term to zero. Before stating the general result, we start with the following simple, but enlightening example. Let X = CN , and let f : CN ! C be a polynomial function with isolated critical points P1;:::;Pl. In this case, the singularity behavior of the function f at each Pi is governed by the Milnor number of f at Pi (see [23]), which we denote by ni. In particular, f is a holomorphic Morse function (that is, it has only non-degenerate isolated critical points) if and only if each Milnor number ni is 1. Fix a general linear N N function g : C ! C. Then ft := f − tg is a holomorphic Morse function on C for all but finitely many t 2 C. The limit of the critical locus of ft has the following behavior as t goes to 0. In a small neighborhood Ui of Pi, there are ni non-degenerate critical points of ft for nonzero t with small absolute value. As t approaches zero, these critical points collide together to Pi. This process is the Morsification of f, which is a well-known result in singularity theory (see [5, Appendix]). Date: February 2, 2020. 2010 Mathematics Subject Classification. 13P25, 57R20, 90C26. Key words and phrases. Euclidean distance degree, Euler characteristic, local Euler obstruction func- tion, optimal solution, stationary point, maximum likelihood, objective function. 1 2 LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG Figure 1. Each red curves is the set of real points of an algebraic variety X and a purple point on the curve is a critical point of the distance function with respect to U. As U moves along the dotted path, the critical points move along the purple arcs. Left: When U is at the origin, every point on the circle X is equidistant from U. Right: As U approaches the origin, the three critical points move along the cardioid curve X and two of them come together. The green curve denotes the ED discriminant and when U is in the shaded region there is only one real critical point on the smooth locus of X. In general, we allow X to be a possibly singular subvariety of CN , and we allow f to be any polynomial whose restriction to X is nonconstant. If g is a general linear function, then ft := f − tg is a holomorphic Morse function on the smooth locus Xreg of X for all but finitely many t 2 C. We are interested in the limiting behavior of the set of critical points of ftjXreg as t approaches zero. In order to formulate our main result, let X ⊂ CN be a complex affine variety and let f : CN ! C be a polynomial function whose restriction to X is nonconstant. Consider S a stratification X = i2I Xi of X into smooth locally closed subvarieties such that the p ∗ N Lagrangian cycles of the perverse vanishing cycle functors Φf−c([TX C ]) are \locally constant along Xi" for all values of c 2 C and all i 2 I. Such a stratification of X can be obtained explicitly as follows. As it will be explained in Section 2.2, there exists a constructible complex F on CN with support on X, whose characteristic cycle is ∗ N exactly the conormal space TX C . We regard F as a constructible complex on X. The restriction fjX has only finitely many critical values in the stratified sense (see, e.g., [6, Definition 4.2.7]), and for each such critical value c 2 C of fjX the (perverse) vanishing p b b cycle functor Φf−c : Dc(X) ! Dc(X \ ff = cg) is constructible and supported in the stratified singular locus of f (see, e.g., [6, Proposition 4.2.8]). Choose a stratification p of X \ ff = cg into smooth locally closed subvarieties with respect to which Φf−c(F ) 3 is constructible. The required stratification of X is then obtained by collecting all the strata in X \ ff = cg for each critical value c of fjX , together with a Whitney stratification of the complement of these critical fibers in X. Once such a stratification S X = i2I Xi of X is fixed, we have the following equality of Lagrangian cycles X p ∗ N X ∗ N (1) Φf−c([T ]) = ni[T ] X C Xi C c2C i2I for ni 2 Z≥0. Notice that the sum on the left-hand side of (1) is a finite sum, since p ∗ N fjX has only finitely many critical values in the stratified sense, and Φf−c([TX C ]) = 0 when c is not a critical value. Moreover, it follows from work of Massey (see Theorem 3.5) that the coefficients ni are nonnegative. By the characteristic cycle functor (see (9)), equation (1) amounts to express (up to signs), for each critical value c of f, the constructible function 'f−c(EuX ) in terms of the basis of local Euler obstruction functions Eu corresponding to closures of strata in Xi f = c. Here, EuX denote the local Euler obstruction function introduced by MacPherson in [16]. In general, an explicit calculation of the coefficients ni is difficult (see Example 5.6). However, when fjX has simple singularities, the vanishing cycle on the left-hand side of (1) can be computed by hand, as the following examples show. Example 1.1. Suppose that X is smooth and fjX has isolated critical points P1;:::;Pl. Then we can take the stratification X0 = X n fP1;:::;Plg and Xi = fPig: The corresponding coefficients ni in (1) are computed directly as n0 = 0, and ni is equal to the Milnor number of fjX at Pi, that is the length of the Artinian algebra @f @f OX;P =h ;:::; i, where OX;P is the germ of holomorphic functions on X at Pi and i @z1 @zd i z1; : : : ; zd 2 OX;Pi are the local coordinates. Example 1.2. Let X ⊂ CN be a possibly singular complex affine variety. Let f : CN ! C be a polynomial function whose restriction to X is nonconstant and has only isolated critical points P1;:::;Pl in the stratified sense. Then formula (1), written in the language of constructible functions (see Section 2.3), becomes: l X dim X X (2) − Φf−c((−1) EuX ) = ni EuPi : c2C i=1 Fix i 2 f1; : : : ; lg and apply the equality of constructible functions in (2) to the point Pi to get: dim X−1 (3) ni = (−1) Φf−f(Pi)(EuX )(Pi): Of course, if X is smooth, then EuX = 1X and, via (12), (14) and (18), ni becomes the Milnor number of f at Pi, as already mentioned in Example 1.1. N Let g : C ! C be a general linear function, and write as above ft := f − tg. Our main result is the following: 4 LAURENTIU G. MAXIM, JOSE ISRAEL RODRIGUEZ, AND BOTONG WANG Theorem 1.3. The limit of the critical points of ft satisfies X (4) lim Crit(ftjX ) = ni · Crit(gjX ) t!0 reg i i2I where the symbol Crit denotes the set of critical points and the numbers ni are determined by formula (1). Remark 1.4. The limit in (4) is defined in Subsection 2.1. It is always well-defined in our setting, and it does not count the points going to infinity. Example 1.5. As in Example 1.1, suppose that X is smooth and fjX has only isolated critical points P1;:::;Pl. Then Theorem 1.3 specializes to the well-known Morsification result that X lim Crit(ft) = niPi t!0 1≤i≤l where ni is the Milnor number of fjX at Pi. Moreover, consider the situation of Example 1.2 of a possibly singular affine variety X, with f having only isolated stratified singular points P1;:::;Pl on X. Theorem 1.3 specializes in this case to X lim Crit(ft) = niPi; t!0 1≤i≤l with ni computed as in formula (3). Remark 1.6.
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