arXiv subjects

Robert McCann

Publications and source records attributed to Robert McCann.

9 recordsLinked to original sources

arXiv2026

Discontinuities in optimal transport between nonconvex domains

We develop a new approach to studying discontinuity sets of measure-preserving convex gradient maps between nonconvex domains. Such maps optimize the Euclidean distance squared. Our idea is to introduce a new `phantom' optimal transport problem, that transports the missing mass from the convex hull of the target to the `excess' Monge--Ampère measure of the minimal convex extension of the Brenier potential. We show that the closure of the discontinuity set of the original map contains the support of this phantom target. Moreover, inside a planar source domain these two sets actually coincide; on $n$-dimensional domains they coincide $\mathcal{H}^{n-1}$-a.e. The Hausdorff exponent $n-1$ is sharp in every dimension $n\geq3$. For planar polygonal targets, we establish a quantitative global length bound for the closure of the discontinuity set depending only on the target geometry, valid for bounded positive densities.

math.AP↗

arXiv2026

Causal Sobolev spaces and gradient flows

We develop a relaxation-based approach to Sobolev functions and Cheeger energies on metric spacetimes, in analogy with the construction used in positive signature. For an exponent $p<0$, we introduce a pre-$p$-Cheeger energy built from a local steepness constant on causal functions, and define the $p$-Cheeger energy as its relaxation in $L^0$; this yields, by construction, a convex and lower semicontinuous functional together with an associated maximal relaxed subslope. We establish natural calculus rules for the resulting Sobolev class. We then define the $p$-heat flow as the gradient flow of the $p$-Cheeger energy in $L^2$ and its generator, the $p$-d'Alembertian. Combining a Kuwada-type lemma with entropy growth estimates along the flow, we show that our notion of Sobolev class and relaxed subslope coincides with the Sobolev class and weak subslope introduced by Beran et al.

math.AP↗

arXiv2024

AI-Driven Guided Response for Security Operation Centers with Microsoft Copilot for Security

Security operation centers contend with a constant stream of security incidents, ranging from straightforward to highly complex. To address this, we developed Microsoft Copilot for Security Guided Response (CGR), an industry-scale ML architecture that guides security analysts across three key tasks -- (1) investigation, providing essential historical context by identifying similar incidents; (2) triaging to ascertain the nature of the incident -- whether it is a true positive, false positive, or benign positive; and (3) remediation, recommending tailored containment actions. CGR is integrated into the Microsoft Defender XDR product and deployed worldwide, generating millions of recommendations across thousands of customers. Our extensive evaluation, incorporating internal evaluation, collaboration with security experts, and customer feedback, demonstrates that CGR delivers high-quality recommendations across all three tasks. We provide a comprehensive overview of the CGR architecture, setting a precedent as the first cybersecurity company to openly discuss these capabilities in such depth. Additionally, we release GUIDE, the largest public collection of real-world security incidents, spanning 13M evidences across 1M incidents annotated with ground-truth triage labels by customer security analysts. This dataset represents the first large-scale cybersecurity resource of its kind, supporting the development and evaluation of guided response systems and beyond.

cs.LG↗

arXiv2022

ADMoE: Anomaly Detection with Mixture-of-Experts from Noisy Labels

Existing works on anomaly detection (AD) rely on clean labels from human annotators that are expensive to acquire in practice. In this work, we propose a method to leverage weak/noisy labels (e.g., risk scores generated by machine rules for detecting malware) that are cheaper to obtain for anomaly detection. Specifically, we propose ADMoE, the first framework for anomaly detection algorithms to learn from noisy labels. In a nutshell, ADMoE leverages mixture-of-experts (MoE) architecture to encourage specialized and scalable learning from multiple noisy sources. It captures the similarities among noisy labels by sharing most model parameters, while encouraging specialization by building "expert" sub-networks. To further juice out the signals from noisy labels, ADMoE uses them as input features to facilitate expert learning. Extensive results on eight datasets (including a proprietary enterprise security dataset) demonstrate the effectiveness of ADMoE, where it brings up to 34% performance improvement over not using it. Also, it outperforms a total of 13 leading baselines with equivalent network parameters and FLOPS. Notably, ADMoE is model-agnostic to enable any neural network-based detection methods to handle noisy labels, where we showcase its results on both multiple-layer perceptron (MLP) and the leading AD method DeepSAD.

cs.LG↗

arXiv2020

Independence of synthetic Curvature Dimension conditions on transport distance exponent

The celebrated Lott-Sturm-Villani theory of metric measure spaces furnishes synthetic notions of a Ricci curvature lower bound $K$ joint with an upper bound $N$ on the dimension. Their condition, called the Curvature-Dimension condition and denoted by $\mathrm{CD}(K,N)$, is formulated in terms of a modified displacement convexity of an entropy functional along $W_{2}$-Wasserstein geodesics. We show that the choice of the squared-distance function as transport cost does not influence the theory. By denoting with $\mathrm{CD}_{p}(K,N)$ the analogous condition but with the cost as the $p^{th}$ power of the distance, we show that $\mathrm{CD}_{p}(K,N)$ are all equivalent conditions for any $p>1$ -- at least in spaces whose geodesics do not branch. We show that the trait d'union between all the seemingly unrelated $\mathrm{CD}_{p}(K,N)$ conditions is the needle decomposition or localization technique associated to the $L^{1}$-optimal transport problem. We also establish the local-to-global property of $\mathrm{CD}_{p}(K,N)$ spaces.

math.DG↗

arXiv2017

Free discontinuities in optimal transport

We prove a nonsmooth implicit function theorem applicable to the zero set of the difference of convex functions. This theorem is explicit and global: it gives a formula representing this zero set as a difference of convex functions which holds throughout the entire domain of the original functions. As applications, we prove results on the stability of singularities of envelopes of semi-convex functions, and solutions to optimal transport problems under appropriate perturbations, along with global structure theorems on certain discontinuities arising in optimal transport maps for Ma-Trudinger-Wang costs. For targets whose components satisfy additional convexity, separation, multiplicity and affine independence assumptions we show these discontinuities occur on submanifolds of the appropriate codimension which are parameterized locally as differences of convex functions (DC, hence $C^2$ rectifiable), and --- depending on the precise assumptions --- $C^{1,α}$ smooth. In this case the highest codimension submanifolds consists of isolated points, each uniquely identified by the (affinely independent) components of the target to which it is transported.

math.AP↗

arXiv2016

Multidimensional matching

We present a general analysis of multidimensional matching problems with transferable utility, paying particular attention to the case in which the dimensions of heterogeneity on the two sides of the market are unequal. A particular emphasis is put on problems where agents on one side of the market are multidimensional and agents on the other side are uni-dimensional, we describe a general approach to solve such problems. Lastly, we analyze several examples, including an hedonic model with differentiated products, a marriage market model where wives are differentiated in income and fertility, and a competitive variation of the Rochet-Choné problem. In the latter example, we show that the bunching phenomena, observed by Rochet and Choné in the monopoly context, do not occur in the competitive context

econ.GN↗

arXiv2012

Higher order time asymptotics of fast diffusion in euclidean space: a dynamical systems approach

This paper quantifies the speed of convergence and higher- order asymptotics of fast diffusion dynamics on R^n to the Barenblatt (self similar) solution. Degeneracies in the parabolicity of this equation are cured by re-expressing the dynamics on a manifold with a cylindrical end, called the cigar. The nonlinear evolution becomes differentiable in Hoelder spaces on the cigar. The linearization of the dynamics is given by the Laplace-Beltrami operator plus a transport term (which can be suppressed by introducing appropriate weights into the function space norm), plus a finite-depth potential well with a universal profile. In the limiting case of the (linear) heat equation, the depth diverges, the number of eigenstates increases without bound, and the continuous spectrum recedes to infinity. We provide a detailed study of the linear and nonlinear problems in Hoelder spaces on the cigar, including a sharp boundedness estimate for the semi-group, and use this as a tool to obtain sharp convergence results toward the Barenblatt solution, and higher order asymptotics. In finer convergence results (after modding out symmetries of the problem), a subtle interplay between convergence rates and tail behavior is revealed. The difficulties involved in choosing the right functional spaces in which to carry out the analysis can be interpreted as genuine features of the equation rather than mere annoying technicalities.

math.AP↗

arXiv2010

Five lectures on optimal transportation: Geometry, regularity and applications

In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recent developments in the regularity theory for Monge-Ampere type equations. An application to microeconomics will also be described, which amounts to finding the equilibrium price distribution for a monopolist marketing a multidimensional line of products to a population of anonymous agents whose preferences are known only statistically.

math.AP↗