Uniform convergence of adversarially robust classifiers

In this paper we present a computationally efficient algorithm utilizing a fully or seminonlocal graph Laplacian for solving a wide range of learning problems in binary data classification and image processing. In their recent work [Multiscale Model. Simul., 10 (2012), pp. 1090--1118], Bertozzi and Flenner introduced a graph-based diffuse interface model utilizing the Ginzburg--Landau functional for solving problems in data classification. Here, we propose an adaptation of the classic numerical Merriman--Bence--Osher (MBO) scheme for minimizing graph-based diffuse interface functionals, like those originally proposed by Bertozzi and Flenner. We also make use of fast numerical solvers for finding eigenvalues and eigenvectors of the graph Laplacian. Various computational examples are presented to demonstrate the performance of our algorithm, which is successful on images with texture and repetitive structure due to its nonlocal nature. The results show that our method is multiple times more efficient than other well-known nonlocal models.

On the convergence of almost minimal sets for the Hausdorff and varifold topologies

We establish strong, new connections between convex sets and geometric measure theory. We use geometric measure theory to improve several standard theorems from the theory of convex sets, which have found wide application in fields such as functional analysis, economics, optimization, and control theory. For example, we prove that a closed subsetKKofRn\mathbb {R}^{n}with non-empty interior is convex if and only if it has locally finite perimeter inRn\mathbb {R}^{n}and has a supporting hyperplane through each point of its reduced boundary. This refines the standard result that such a setKKis convex if and only if it has a supporting hyperplane through each point of its topological boundary, which may be much larger than the reduced boundary. Thus, the reduced boundary from geometric measure theory contains all the convexity information for such a setKK. We similarly refine a standard separation theorem, as well as a representation theorem for convex sets. We then extend all of our results to other notions of boundary from the literature and deduce the corresponding classical results from convex analysis as special cases.

A hybrid sampling algorithm combining M-SMOTE and ENN based on Random forest for medical imbalanced data.

The structural features that protrude above or below a soft matter interface are well-known to be related to interfacially mediated chemical reactivity and transport processes. It is a challenge to develop a robust algorithm for identifying these organized surface structures, as the morphology can be highly varied and they may exist on top of an interface containing significant interfacial roughness. A new algorithm that employs concepts from geometric measure theory, algebraic topology, and optimization, is developed to identify candidate structures at a soft matter surface, and then using a probabilistic approach, to rank their likelihood of being a complex structural feature. The algorithm is tested for a surfactant laden water/oil interface, where it is robust to identifying protrusions responsible for water transport against a set identified by visual inspection. To our knowledge, this is the first example of applying geometric measure theory to analyze the properties of a chemical/materials science system.

The spherical maximal operator $$Af(x) = \mathop {sup}\limits_{t > 0} \left| {{A_t}f(x)} \right| = \mathop {sup}\limits_{t > 0} \left| \int{f(x - ty)d\sigma (y)} \right|$$ where σ is the surface measure on the unit sphere, is a classical object that appears in a variety of contexts in harmonic analysis, geometric measure theory, partial differential equation and geometric combinatorics. We establish Lp bounds for the Stein spherical maximal operator in the setting of compactly supported Borel measures μ, ν satisfying natural local size assumptions $$\mu (B(x,r)) \leqslant C{r {{s_\mu }}},v(B(x,r)) \leqslant C{r {{s_v}}}$$ . Taking the supremum over all t > 0 is not in general possible for reasons that are fundamental to the fractal setting, but we can obtain single scale (t ∈ [a, b] ⊂ (0,∞)) results. The range of possible Lp exponents is, in general, a bounded open interval where the upper endpoint is closely tied with the local smoothing estimates for Fourier Integral Operators. In the process, we establish L2(μ) → L2(ν) bounds for the convolution operator Tλf(x) = λ * (fμ), where λ is a tempered distribution satisfying a suitable Fourier decay condition. More generally, we establish a transference mechanism which yields Lp(μ) → Lp(ν) bounds for a large class of operators satisfying suitable Lp-Sobolev bounds. This allows us to effectively study the dimension of a blowup set ({x: Tf(x) = ∞}) for a wide class of operators, including the solution operator for the classical wave equation. Some of the results established in this paper have already been used to study a variety of Falconer type problems in geometric measure theory.

Curvature measures are an important tool in geometric measure theory and other fields of mathematics for describing the geometry of sets in Euclidean space. But the ‘classical’ concepts of curvature are not directly applicable to fractal sets. We try to bridge this gap between geometric measure theory and fractal geometry by introducing a notion of curvature for fractals. For compact sets F ⊆ R d (e.g. fractals), for which classical geometric characteristics such as curvatures or Euler characteristic are not available, we study these notions for their -parallel sets Fe := {x ∈ R d : inf y∈F k x − yk ≤ }, instead, expecting that their limiting behaviour as → 0 does provide information about the structure of the initial set F. In particular, we investigate the limiting behaviour of the total curvatures (or intrinsic volumes) Ck(Fe), k = 0, . . . , d, as well as weak limits of the corresponding curvature measures Ck(Fe, � ) as → 0. This leads to the notions of fractal curvature and fractal curvature measure, respectively. The well known Minkowski content appears in this concept as one of the fractal curvatures. For certain classes of self-similar sets, results on the existence of (averaged) fractal curvatures are presented. These limits can be calculated explicitly and are in a certain sense ‘invariants’ of the sets, which may help to distinguish and classify fractals. Based on these results also the fractal curvature measures of these sets are characterized. As a special case and a significant refinement of known results, a local characterization of the Minkowski content is given.

We analyze a previously unexplored generalization of the scalar total variation to vector-valued functions, which is motivated by geometric measure theory. A complete mathematical characterization is given, which proves important invariance properties as well as existence of solutions of the vectorial ROF model. As an important feature, there exists a dual formulation for the proposed vectorial total variation, which leads to a fast and stable minimization algorithm. The main difference to previous approaches with similar properties is that we penalize across a common edge direction for all channels, which is a major theoretical advantage. Experiments show that this leads to a significantly better restoration of color edges in practice.

In his curriculum vitae, John Hawkes lists his research interests as geometric measure theory, probability (Lévy processes), and potential theory (probabilistic). In fact, he made significant contributions to all three areas, and there are strong relationships between them. He used both geometric measure theory and potential theory as tools for his study of the trajectories of particular Lévy processes, but in many cases he needed to develop the tool before it was ready to be used. We will summarise his research later, but first we discuss what is known of his life history.

Chapter 24 - Geometric Measure Theory: Selected Concepts, Results and Problems

In this paper we study stationary graphs for functionals of geometric nature defined on currents or varifolds. The point of view we adopt is the one of differential inclusions, introduced in this context in the recent papers (De Lellis et al. in Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335; Tione in Minimal graphs and differential inclusions. Commun Part Differ Equ 7:1–33, 2021). In particular, given a polyconvex integrand f, we define a set of matrices C_f that allows us to rewrite the stationarity condition for a graph with multiplicity as a differential inclusion. Then we prove that if f is assumed to be non-negative, then in C_f there is no T'_N configuration, thus recovering the main result of De Lellis et al. (Geometric measure theory and differential inclusions, 2019. arXiv:1910.00335) as a corollary. Finally, we show that if the hypothesis of non-negativity is dropped, one can not only find T'_N configurations in C_f, but it is also possible to construct via convex integration a very degenerate stationary point with multiplicity.

In this thesis we focus on different problems in the Calculus of Variations and Geometric Measure Theory, with the common peculiarity of dealing with anisotropic energies. We can group them in two big topics: 1. The anisotropic Plateau problem: Recently in [37], De Lellis, Maggi and Ghiraldin have proposed a direct approach to the isotropic Plateau problem in codimension one, based on the “elementary” theory of Radon measures and on a deep result of Preiss concerning rectifiable measures. In the joint works [44],[38],[43] we extend the results of [37] espectively to any codimension, to the anisotropic setting in codimension one and to the anisotropic setting in any codimension. For the latter result, we exploit the anisotropic counterpart of Allard’s rectifiability Theorem, [2], which we prove in [42]. It asserts that every d-varifold in Rn with locally bounded anisotropic first variation is d-rectifiable when restricted to the set of points in Rn with positive lower d-dimensional density. In particular we identify a necessary and sufficient condition on the Lagrangian for the validity of the Allard type rectifiability result. We are also able to prove that in codimension one this condition is equivalent to the strict convexity of the integrand with respect to the tangent plane. In the paper [45], we apply the main theorem of [42] to the minimization of anisotropic energies in classes of rectifiable varifolds. We prove that the limit of a minimizing sequence of varifolds with density uniformly bounded from below is rectifiable. Moreover, with the further assumption that all the elements of the minimizing sequence are integral varifolds with uniformly locally bounded anisotropic first variation, we show that the limiting varifold is also integral. 2. Stability in branched transport: Models involving branched structures are employed to describe several supply-demand systems such as the structure of the nerves of a leaf, the system of roots of a tree and the nervous or cardiovascular systems. Given a flow (traffic path) that transports a given measure \(μ^−\) onto a target measure \(μ^+\), along a 1-dimensional network, the transportation cost per unit length is supposed in these models to be proportional to a concave power α ∈ (0,1) of the intensity of the flow. The transportation cost is called α-mass. In the paper [27] we address an open problem in the book [15] and we improve the stability for optimal traffic paths in the Euclidean space \(R^n\) with respect to variations of the given measures (\(μ^−\), \(μ^+\)), which was known up to now only for α > 1− \(\frac 1n\). We prove it for exponents α > 1− \(\frac{1}{n−1}\) (in particular, for every α ∈ (0,1) when n = 2), for a fairly large class of measures (\μ^+\) and (\μ^−\). The α- mass is a particular case of more general energies induced by even, subadditive, and lower semicontinuous functions H : R → [0,∞) satisfying H (0) = 0. In the paper [28], we prove that the lower semicontinuous envelope of these energy functionals defined on polyhedral chains coincides on rectifiable currents with the H -mass.

Chapter 1 - Geometric Measure Theory

CHAPTER 1 - Geometric Measure Theory

Highly non-ideal solutions are ever-present within chemistry, physics, and materials science – and are characterized by many-body effects across length and timescale. Understanding, and predicting, many-body correlations in the condensed phase is a grand challenge for the modeling and simulation community. Yet within the data science community, a large suite of tools exist for elucidating complex, correlating, relationships amongst variables. Molecular modeling and simulation data is in fact well-suited for study by methods that include the topology of graphs, point cloud data, and recent advances in applied mathematics methods that investigate surfaces like sublevel set persistent homology and geometric measure theory. We adapt, develop, and apply these tools to study highly non-ideal solutions and their interfaces, with examples drawn from separations science. The new physical insight derived from these methods is paving the way for bespoke liquid/liquid interfaces that optimize transport characteristics for purification and synthesis.