Whitney Stratification Research Articles

Whitney stratifications are conically smooth

The notion of conically smooth structure on a stratified space was introduced by Ayala, Francis and Tanaka. This is a very well behaved analogue of a differential structure in the context of stratified topological spaces, satisfying good properties such as the existence of resolutions of singularities and handlebody decompositions. In this paper we prove Ayala, Francis and Tanaka’s conjecture that any Whitney stratified space admits a canonical conically smooth structure. We thus establish a connection between the theory of conically smooth spaces and the classical examples of stratified spaces from differential topology.

Stochastic Subgradient Descent Escapes Active Strict Saddles on Weakly Convex Functions

In nonsmooth stochastic optimization, we establish the nonconvergence of the stochastic subgradient descent (SGD) to the critical points recently called active strict saddles by Davis and Drusvyatskiy. Such points lie on a manifold M, where the function f has a direction of second-order negative curvature. Off this manifold, the norm of the Clarke subdifferential of f is lower-bounded. We require two conditions on f. The first assumption is a Verdier stratification condition, which is a refinement of the popular Whitney stratification. It allows us to establish a strengthened version of the projection formula of Bolte et al. for Whitney stratifiable functions and which is of independent interest. The second assumption, termed the angle condition, allows us to control the distance of the iterates to M. When f is weakly convex, our assumptions are generic. Consequently, generically, in the class of definable weakly convex functions, SGD converges to a local minimizer. Funding: The work of Sholom Schechtman was supported by “Région Ile-de-France”.

Canonical stratification of definable Lie groupoids

Our aim is to precisely present a tame topology counterpart to canonical stratification of a Lie groupoid. We consider a definable Lie groupoid in semialgebraic, subanalytic, o-minimal over R, or more generally, Shiota's X-category. We show that there exists a canonical Whitney stratification of the Lie groupoid into definable strata which are invariant under the groupoid action. This is a generalization and refinement of results on real algebraic group action which J.N. Mather and V.A. Vassiliev independently stated with sketchy proofs. A crucial change to their proofs is to use Shiota's isotopy lemma and approximation theorem in the context of tame topology.

Lipschitz Analysis of Generalized Phase Retrievable Matrix Frames

The classical phase retrieval problem arises in contexts ranging from speech recognition to x-ray crystallography and quantum state tomography. The generalization to matrix frames is natural in the sense that it corresponds to quantum tomography of impure states. We provide computable global stability bounds for the quasi-linear analysis map $\beta$ and a path forward for understanding related problems in terms of the differential geometry of key spaces. In particular, we manifest a Whitney stratification of the positive semidefinite matrices of low rank which allows us to ``stratify'' the computation of the global stability bound. We show that for the impure state case no such global stability bounds can be obtained for the non-linear analysis map $\alpha$ with respect to certain natural distance metrics. Finally, our computation of the global lower Lipschitz constant for the $\beta$ analysis map provides novel conditions for a frame to be generalized phase retrievable.

Conormal Spaces and Whitney Stratifications

We describe a new algorithm for computing Whitney stratifications of complex projective varieties. The main ingredients are (a) an algebraic criterion, due to Lê and Teissier, which reformulates Whitney regularity in terms of conormal spaces and maps, and (b) a new interpretation of this conormal criterion via ideal saturations, which can be practically implemented on a computer. We show that this algorithm improves upon the existing state of the art by several orders of magnitude, even for relatively small input varieties. En route, we introduce related algorithms for efficiently stratifying affine varieties, flags on a given variety, and algebraic maps.

THE THOM CONDITION

In this note we explain the notion of the Thom condition for the Whitney stratifications of a complex analytic map. We give a question P. Deligne and indicate a possible way to answer it.

SOME GEOMETRIC PROPERTIES OF WHITNEY STRATIFICATIONS

We describe some old and new results about Whitney stratifications and state some open problems.

On the Continuity of the Tangent Cone to the Determinantal Variety

Tangent and normal cones play an important role in constrained optimization to describe admissible search directions and, in particular, to formulate optimality conditions. They notably appear in various recent algorithms for both smooth and nonsmooth low-rank optimization where the feasible set is the set $\mathbb {R}_{\le r}^{m \times n}$ of all m × n real matrices of rank at most r. In this paper, motivated by the convergence analysis of such algorithms, we study, by computing inner and outer limits, the continuity of the correspondence that maps each $X \in \mathbb {R}_{\le r}^{m \times n}$ to the tangent cone to $\mathbb {R}_{\le r}^{m \times n}$ at X. We also deduce results about the continuity of the corresponding normal cone correspondence. Finally, we show that our results include as a particular case the a-regularity of the Whitney stratification of $\mathbb {R}_{\le r}^{m \times n}$ following from the fact that this set is a real algebraic variety, called the real determinantal variety.

The orbit type stratification of the moduli space of Higgs bundles

The moduli space of Higgs bundles can be constructed as a quotient of an infinite-dimensional space and hence admits an orbit type decomposition. In this paper, we show that the orbit type decomposition is a complex Whitney stratification such that each stratum is a complex symplectic submanifold and hence admits a complex Poisson bracket. Moreover, these Poisson brackets glue to a Poisson bracket on the structure sheaf of the moduli space so that the moduli space is a stratified complex symplectic space.

A generalization of Milnor\u2019s formula

The Milnor number mu _f of a holomorphic function f :({mathbb {C}}^n,0) rightarrow ({mathbb {C}},0) with an isolated singularity has several different characterizations as, for example: 1) the number of critical points in a morsification of f, 2) the middle Betti number of its Milnor fiber M_f, 3) the degree of the differential {text {d}}f at the origin, and 4) the length of an analytic algebra due to Milnor’s formula mu _f = dim _{mathbb {C}}{mathcal {O}}_n/{text {Jac}}(f). Let (X,0) subset ({mathbb {C}}^n,0) be an arbitrarily singular reduced analytic space, endowed with its canonical Whitney stratification and let f :({mathbb {C}}^n,0) rightarrow ({mathbb {C}},0) be a holomorphic function whose restriction f|(X, 0) has an isolated singularity in the stratified sense. For each stratum {mathscr {S}}_alpha let mu _f(alpha ;X,0) be the number of critical points on {mathscr {S}}_alpha in a morsification of f|(X, 0). We show that the numbers mu _f(alpha ;X,0) generalize the classical Milnor number in all of the four characterizations above. To this end, we describe a homology decomposition of the Milnor fiber M_{f|(X,0)} in terms of the mu _f(alpha ;X,0) and introduce a new homological index which computes these numbers directly as a holomorphic Euler characteristic. We furthermore give an algorithm for this computation when the closure of the stratum is a hypersurface.

On the pseudo-manifold of quantum states

There are various statements in the physics literature about the stratification of quantum states, for example into orbits of a unitary group, and about generalized differentiable structures on it. Our aim is to clarify and make precise some of these statements. For A an arbitrary finite-dimensional C*-algebra and U(A) the group of unitary elements of A, we observe that the partition of the state space S(A) into U(A) orbits is not a decomposition and that the decomposition into orbit types is not a stratification (its pieces are not manifolds without boundary), while there is a natural Whitney stratification into matrices of fixed rank. For the latter, when A is a full matrix algebra, we give an explicit description of the pseudo-manifold structure (the conical neighborhood around any point). We then make some comments about the infinite-dimensional case.

Conservative set valued fields, automatic differentiation, stochastic gradient methods and deep learning

Modern problems in AI or in numerical analysis require nonsmooth approaches with a flexible calculus. We introduce generalized derivatives called conservative fields for which we develop a calculus and provide representation formulas. Functions having a conservative field are called path differentiable: convex, concave, Clarke regular and any semialgebraic Lipschitz continuous functions are path differentiable. Using Whitney stratification techniques for semialgebraic and definable sets, our model provides variational formulas for nonsmooth automatic differentiation oracles, as for instance the famous backpropagation algorithm in deep learning. Our differential model is applied to establish the convergence in values of nonsmooth stochastic gradient methods as they are implemented in practice.

On the rattleback dynamics

In this paper we present some relevant dynamical properties of an idealized conservative model of the rattleback, from the Poisson dynamics point of view. In the first half of the article, along with a dynamical study of the orbits, using a Hamilton–Poisson realization of the dynamical system, we provide a geometric characterization of the space of orbits in terms of Whitney stratifications associated to the image of the energy–Casimir mapping. In the second half of the article we provide an explicit method to stabilize asymptotically any arbitrary fixed orbit/cycle of the rattleback system and to keep unchanged the geometry of the model space.

Deformation retracts to intersections of Whitney stratifications

We give a counterexample to a conjecture of Eyral on the existence of deformation retracts to intersections of Whitney stratifications embedded in a smooth manifold. We then prove that the conjecture holds if the stratifications are definable in some o-minimal structure without assuming any regularity conditions. Moreover, we also show that the conjecture holds for Whitney stratifications if they intersect transversally.

Thom Isotopy Theorem for Nonproper Maps and Computation of Sets of Stratified Generalized Critical Values

Let Xsubset {mathbb {C}}^n be an affine variety and f:Xrightarrow {mathbb {C}}^m be the restriction to X of a polynomial map {mathbb {C}}^nrightarrow {mathbb {C}}^m. We construct an affine Whitney stratification of X. The set K(f) of stratified generalized critical values of f can also be computed. We show that K(f) is a nowhere dense subset of {mathbb {C}}^m which contains the set B(f) of bifurcation values of f by proving a version of the Thom isotopy lemma for nonproper polynomial maps on singular varieties.

Whitney stratifications and the continuity of local Lipschitz–Killing curvatures

In the paper we prove that the local Lipschitz–Killing curvatures of a definable set in a polynomially bounded o-minimal structure are continuous along the strata of a Whitney stratification. Moreover, if the stratification is (w)-regular the local Lipschitz–Killing curvatures are locally Lipschitz in any o-minimal structure.

Higher discriminants and the topology of algebraic maps

We show that the way in which Betti cohomology varies in a proper family of complex algebraic varieties is controlled by certain in the base. These discriminants are defined in terms of transversality conditions, which in the case of a morphism between smooth varieties can be checked by a tangent space calculation. They control the variation of cohomology in the following two senses: (1) the support of any summand of the pushforward of the IC sheaf along a projective map is a component of a higher discriminant, and (2) any component of the characteristic cycle of the proper pushforward of the constant function is a conormal variety to a component of a higher discriminant. The same would hold for the Whitney stratification of the family, but there are vastly fewer higher discriminants than Whitney strata. For example, in the case of the Hitchin fibration, the stratification by higher discriminants gives exactly the {\delta} stratification introduced by Ngo.

Upper Bounds for the Hausdorff Dimension and Stratification of an Invariant Set of an Evolution System on a Hilbert Manifold

We prove a generalization of the well-known Douady–Oesterle theorem on the upper bound for the Hausdorff dimension of an invariant set of a finite-dimensional mapping to the case of a smooth mapping generating a dynamical system on an infinite-dimensional Hilbert manifold. A similar estimate is given for the invariant set of a dynamical system generated by a differential equation on a Hilbert manifold. As an example, the well-known sine-Gordon equation is considered. In addition, we propose an algorithm for the Whitney stratification of semianalytic sets on finite-dimensional manifolds.

Non‐archimedean stratifications of tangent cones

AbstractWe study the impact of a kind of non‐archimedean stratifications (t‐stratifications) on tangent cones of definable sets in real closed fields. We prove that such stratifications induce stratifications of the same nature on the tangent cone of a definable set at a fixed point. As a consequence, the archimedean counterpart of a t‐stratification is shown to induce Whitney stratifications on the tangent cones of a semi‐algebraic set. Extensions of these results are proposed for real closed fields with further structure.

The Lê-Greuel formula for functions on analytic spaces

In this article we give an extension of the Lê-Greuel formula to the general setting of function germs $(f,g)$ defined on a complex analytic variety $X$ with arbitrary singular set, where $f = (f_1,\ldots,f_k): (X,\underline{0}) \to (\mathbb{C}^k,\underline{0})$ is generically a submersion with respect to some Whitney stratification on $X$. We assume further that the dimension of the zero set $V(f)$ is larger than 0, that $f$ has the Thom $a_f$-property with respect to this stratification, and $g: (X,\underline{0}) \to (\mathbb{C},0)$ has an isolated critical point in the stratified sense, both on $X$ and on $V(f)$.