Lipschitz Metric Research Articles

Quantifying safety risks of deep neural networks

Safety concerns on the deep neural networks (DNNs) have been raised when they are applied to critical sectors. In this paper, we define safety risks by requesting the alignment of network’s decision with human perception. To enable a general methodology for quantifying safety risks, we define a generic safety property and instantiate it to express various safety risks. For the quantification of risks, we take the maximum radius of safe norm balls, in which no safety risk exists. The computation of the maximum safe radius is reduced to the computation of their respective Lipschitz metrics—the quantities to be computed. In addition to the known adversarial example, reachability example, and invariant example, in this paper, we identify a new class of risk—uncertainty example—on which humans can tell easily, but the network is unsure. We develop an algorithm, inspired by derivative-free optimization techniques and accelerated by tensor-based parallelization on GPUs, to support an efficient computation of the metrics. We perform evaluations on several benchmark neural networks, including ACSC-Xu, MNIST, CIFAR-10, and ImageNet networks. The experiments show that our method can achieve competitive performance on safety quantification in terms of the tightness and the efficiency of computation. Importantly, as a generic approach, our method can work with a broad class of safety risks and without restrictions on the structure of neural networks.

Convexity of balls in outer space

In this paper we study the convexity properties of geodesics and balls in Outer space equipped with the Lipschitz metric. We introduce a class of geodesics called balanced folding paths and show that, for every loop \alpha , the length of \alpha along a balanced folding path is not larger than the maximum of its lengths at the endpoints. This implies that out-going balls are weakly convex. We then show that these results are sharp by providing several counter examples.

Observability of the heat equation, geometric constants in control theory, and a conjecture of Luc Miller

This article is concerned in the first place with the short-time observability constant of the heat equation from a subdomain $\omega$ of a bounded domain $M$. The constant is of the form $e^{\frac{K}{T}}$, where $K$ depends only on the geometry of $M$ and $\omega$. Luc Miller (JDE, 2004) conjectured that $K$ is (universally) proportional to the square of the maximal distance from $\omega$ to a point of $M$. We show in particular geometries that $K$ may blow up like $|\log(r)|^2$ when $\omega$ is a ball of radius $r$, hence disproving the conjecture. We then prove in the general case the associated upper bound on this blowup. We also show that the conjecture is true for positive solutions of the heat equation. The proofs rely on the study of the maximal vanishing rate of (sums of) eigenfunctions. They also yield lower and upper bounds for other geometric constants appearing as tunneling constants or approximate control costs. As an intermediate step in the proofs, we provide a uniform Carleman estimate for Lipschitz metrics. The latter also implies uniform spectral inequalities and observability estimates for the heat equation in a bounded class of Lipschitz metrics, which are of independent interest.

Displacements of automorphisms of free groups I: Displacement functions, minpoints and train tracks

This is the first of two papers in which we investigate the properties of the displacement functions of automorphisms of free groups (more generally, free products) on Culler-Vogtmann Outer space and its simplicial bordification – the free splitting complex – with respect to the Lipschitz metric. The theory for irreducible automorphisms being well-developed, we concentrate on the reducible case. Since we deal with the bordification, we develop all the needed tools in the more general setting of deformation spaces, and their associated free splitting complexes. In the present paper we study the local properties of the displacement function. In particular, we study its convexity properties and the behaviour at bordification points, by geometrically characterising its continuity-points. We prove that the global-simplex-displacement spectrum of A u t ( F n ) Aut(F_n) is a well-ordered subset of R \mathbb {R} , this being helpful for algorithmic purposes. We introduce a weaker notion of train tracks, which we call partial train tracks (which coincides with the usual one for irreducible automorphisms) and we prove that, for any automorphism, points of minimal displacement – minpoints – coincide with the marked metric graphs that support partial train tracks. We show that any automorphism, reducible or not, has a partial train track (hence a minpoint) either in the outer space or its bordification. We show that, given an automorphism, any of its invariant free factors is seen in a partial train track map. In a subsequent paper we will prove that level sets of the displacement functions are connected, and we will apply that result to solve certain decision problems.

Continuous dependence on data under the Lipschitz metric for the rotation-Camassa-Holm equation

In this article, we consider the Lipschitz metric of conservative weak solutions for the rotation-Camassa-Holm equation. Based on defining a Finsler-type norm on the tangent space for solutions, we first establish the Lipschitz metric for smooth solutions, then by proving the generic regularity result, we extend this metric to general weak solutions.

A LIPSCHITZ METRIC FOR THE CAMASSA–HOLM EQUATION

We analyze stability of conservative solutions of the Cauchy problem on the line for the Camassa–Holm (CH) equation. Generically, the solutions of the CH equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this paper is the construction of a Lipschitz metric that compares two solutions of the CH equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

Shock Wave Interactions and the Riemann-Flat Condition: The Geometry Behind Metric Smoothing and the Existence of Locally Inertial Frames in General Relativity

We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether {\it regularity singularities} (points where the curvature is in $L^\infty$ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to $C^{1,1}$ by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to $C^{1,1}$ locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme.

The metric completion of Outer Space

We develop the theory of a metric completion of an asymmetric metric space. We characterize the points on the boundary of Outer Space that are in the metric completion of Outer Space with the Lipschitz metric. We prove that the simplicial completion, the subset of the completion consisting of simplicial tree actions, is homeomorphic to the free splitting complex. We use this to give a new proof of a theorem by Francaviglia and Martino that the isometry group of Outer Space is isomorphic to $${\text {Out}}(F_{n})$$ for $$n \ge 3$$ and to $$\text {PSL}(2,{{\mathbb {Z}}})$$ for $$n=2$$.

A Lipschitz metric for the Hunter–Saxton equation

We analyze stability of conservative solutions of the Cauchy problem on the line for the (integrated) Hunter–Saxton (HS) equation. Generically, the solutions of the HS equation develop singularities with steep gradients while preserving continuity of the solution itself. In order to obtain uniqueness, one is required to augment the equation itself by a measure that represents the associated energy, and the breakdown of the solution is associated with a complicated interplay where the measure becomes singular. The main result in this article is the construction of a Lipschitz metric that compares two solutions of the HS equation with the respective initial data. The Lipschitz metric is based on the use of the Wasserstein metric.

A dense geodesic ray in the Out$(F_r)$-quotient of reduced Outer Space

In [16] Masur proved the existence of a dense geodesic in the moduli space for a surface. We prove an analogue theorem for reduced Outer Space endowed with the Lipschitz metric. We also prove two results possibly of independent interest: we show Brun's unordered algorithm weakly converges and from this prove that the set of Perron–Frobenius eigenvectors of positive integer m \times m matrices is dense in the positive cone \mathbb R^m_+ (these matrices will in fact be the transition matrices of positive automorphisms). We give a proof in the appendix that not every point in the boundary of Outer Space is the limit of a flow line.

Lipschitz Metric for the Novikov Equation

We consider the Lipschitz continuous dependence of solutions for the Novikov equation with respect to the initial data. In particular, we construct a Finsler type optimal transport metric which renders the solution map Lipschitz continuous on bounded sets of $${H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}$$ , although it is not Lipschitz continuous under the natural Sobolev metric from an energy law due to the finite time gradient blowup. By an application of Thom’s transversality theorem, we also prove that when the initial data is in an open dense subset of $${H^1(\mathbb{R})\cap W^{1,4}(\mathbb{R})}$$ , the solution is piecewise smooth. This generic regularity result helps us extend the Lipschitz continuous metric to the general weak solutions. Our method of constructing the metric can be used to treat other kinds of quasi-linear equations, provided a good knowledge about the energy concentration.

Non-Regular Spacetime Geometry

PrefaceUntil recently, mathematical studies of Lorentzian causality theory assumed, explicitly or implicitly, smoothness of the metric. This changed a few years ago, after independent work by Fathi and Siconolfi, and by Chruściel and Grant. Since then many papers began to explore causality under weaker conditions such as C1,1 metrics, Lipschitz metrics, continuous metrics (and cone structures), or even topological settings. Indeed, new insights are needed when the metric is not twice-differentiable. Hence, manifolds for which the metric is not C2 might be called non-regular, though closer inspection reveals that there are not so many differences between the C2 and the C1,1 theories, so one can also refer to the non-regular theories as those whose metric has regularity below C1,1.In Riemannian signature geometers studied non-regular manifolds using comparison geometry and other approaches. Many of these techniques have yet to be exported to the Lorentzian signature since, of course, a number of difficulties have to be addressed. In spite of this, mathematical physicists study non-regular spaces with a strong motivation that stems from the desire of understanding the ultimate nature of spacetime. The idea is that at the quantum level and hence whenever the physical conditions become extreme (say gravitational collapse, or origin of the universe) the spacetime manifold should perhaps not be approximated by a smooth manifold. If this expectation is correct then an understanding of the role of differentiability conditions in general relativity might give hints to the very nature of gravity and spacetime at the quantum scale.

Lipschitz Metrics for a Class of Nonlinear Wave Equations

The nonlinear wave equation $${u_{tt}-c(u)(c(u)u_x)_x=0}$$ determines a flow of conservative solutions taking values in the space $${H^1(\mathbb{R})}$$ . However, this flow is not continuous with respect to the natural H 1 distance. The aim of this paper is to construct a new metric which renders the flow uniformly Lipschitz continuous on bounded subsets of $${H^1(\mathbb{R})}$$ . For this purpose, H 1 is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, one can carefully estimate how the weighted length grows in time. By the generic regularity result proved in [7], these piecewise regular paths are dense and can be used to construct a geodesic distance with the desired Lipschitz property.

A Lipschitz metric for conservative solutions of the two-component Hunter–Saxton system

We establish the existence of conservative solutions of the initial value problem of the two-component Hunter--Saxton system on the line. Furthermore we investigate the stability of these solutions by constructing a Lipschitz metric.

The horoboundary of outer space, and growth under random automorphisms

We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of $F_N$. We identify the set of Busemann points with the set of trees with dense orbits. We also investigate a few properties of the horoboundary of outer space for the backward Lipschitz metric, and show in particular that it is infinite-dimensional when $N\ge 3$. We then use our description of the horoboundary of outer space to derive an analogue of a theorem of Furstenberg--Kifer and Hennion for random products of outer automorphisms of $F_N$, that estimates possible growth rates of conjugacy classes of elements of $F_N$ under such products.

Lines of minima in outer space

We define lines of minima in the thick part of Outer space for the free group Fn with n>2 generators. We show that these lines of minima are contracting for the Lipschitz metric. Every fully irreducible outer automorphism of Fn defines such a line a minima. Now let G be a subgroup of the outer automorphism group of Fn which is not virtually abelian. We obtain as an immediate application that if G contains at least one fully irreducible element then for every p<1 the second bounded cohomology group with coefficients in lp(G) is infinite dimensional.

Every Lipschitz metric has -geodesics

We observe that the geodesic equation for any semi-Riemannian metric of regularity possesses -solutions in the sense of Filippov.

A positive mass theorem for Lipschitz metrics with small singular sets

We prove that the positive mass theorem applies to Lipschitz metrics as long as the singular set is low-dimensional, with no other conditions on the singular set. More precisely, let $g$ be an asymptotically flat Lipschitz metric on a smooth manifold $M^n$, such that $n<8$ or $M$ is spin. As long as $g$ has bounded $C^2$ norm and nonnegative scalar curvature on the complement of some singular set $S$ of Minkowski dimension less than $n/2$, the mass of $g$ must be nonnegative. We conjecture that the dimension of $S$ need only be less than $n-1$ for the result to hold. These results complement and contrast with earlier results of H. Bray, P. Miao, and Y. Shi and L.-F. Tam, where $S$ is a hypersurface.

The diameter of the thick part of moduli space and simultaneous Whitehead moves

Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the ϵ-thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log(g+pϵ). The same result also holds for the ϵ-thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrarily labeled tree with n labels using simultaneous Whitehead moves, where the number of steps is of order log(n). As a related combinatorial problem, we also compute, in the appendix of this paper, the asymptotic diameter of the moduli space of pants decompositions on S in the metric of elementary moves.

The isometry group of Outer Space

We prove analogues of Royden’s Theorem for the Lipschitz metrics of Outer Space, namely that Isom(CVn)=Out(Fn).