Adversarial consistency depends on the uniqueness of adversarial Bayes classifiers.
problem Consistency of adversarial surrogate losses is not guaranteed.
method Connected consistency of adversarial surrogate losses to the uniqueness of adversarial Bayes classifiers.
result A convex surrogate loss is statistically consistent for adversarial learning if and only if the adversarial Bayes classifier is unique.
Unique optimal strategy identified for state-dependent risk aversion.
problem Consistency of optimal portfolio choice for varying risk aversion.
method Analysis of state-dependent exponential utilities in arbitrage-free markets.
result Uniqueness of optimal strategy across any time horizon.
The paper simplifies complex jump-diffusion markets to complete models.
problem Pricing and hedging derivatives in incomplete jump-diffusion markets.
method Filtration reduction to a complete market, then consistency to original market.
result A unique equivalent martingale measure is obtained for pricing.
The paper introduces a sampling theory for graphons with a Poincaré inequality and proves consistency.
problem Sampling on large graphs is challenging due to their non-Euclidean nature.
method The paper introduces a signal sampling theory for graphons, proving a Poincaré inequality and showing consistency.
result Unique sampling sets for graphon signals are consistent across graph sequences.
The n-dimensional torus is uniquely characterized by specific harmonic forms.
problem Characterizing the n-dimensional torus via harmonic forms.
method Analyzing closed 1-forms on the torus to determine unique properties.
result The n-dimensional torus is the unique manifold supporting a linearly independent set of (n-1) closed 1-forms whose product determines a non-zero cohomological class.
In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Each surface diagram co…
The paper assesses conditions for the uniqueness of k-means clustering.
problem Conditions for the uniqueness of k-means clustering.
method Analyzes the choice of k and provides necessary and sufficient conditions for uniqueness.
result Determines the asymptotic distribution of the within cluster sum of squares (WCSS) and provides a bootstrap test for uniqueness.
Develops a new framework for causal models on cyclic graphs, solving unique solvability issues.
problem Challenges in specifying unique probability distributions for cyclic functional causal models.
method Introduces a new probability rule and graph-separation property (p-separation) for cyclic fCMs.
result Proves p-separation is sound and complete for all consistent cyclic fCMs, recovering d-separation for DAGs.
Study proves stability and uniqueness for a specific type of flow.
problem Volume-preserving mean curvature flow stability and uniqueness.
method New gradient flow calibrations for volume preservation, stability estimate in distributional solutions.
result Strong solutions are calibrated and stable under certain conditions.
The critical catenoid is uniquely determined by certain symmetries of its boundary.
problem Uniqueness of free boundary minimal annuli in a half-ball.
method Symmetry analysis and boundary conditions.
result An embedded free boundary minimal annulus with specific symmetries is congruent to the critical catenoid.
We consider a network in the Euclidean plane that consists of three distinct half-lines with common start points. From that network as initial condition, there exists a network that consists of three curves that all start at one point, where they form 120 degree angles, and expands homothetically under curve shortening…
This paper proves unique determination of certain non-quasi-Fuchsian manifolds by their end structure and bending lamination.
problem Unique determination of non-quasi-Fuchsian manifolds by their end structure and bending lamination.
method Analysis of the end structure (parabolic locus, ending laminations, conformal structures) and bending lamination.
result Non-quasi-Fuchsian manifolds are uniquely determined by their end structure and bending lamination.
We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.
We study existence and uniqueness of continuous-time stochastic Radner equilibria in an incomplete market model among a group of agents whose preference is characterized by cash invariant time-consistent monetary utilities. An assumption of "smallness" type is shown to be sufficient for existence and uniqueness. In par…
Robustifies elicitable functionals to handle small distribution misspecifications.
problem Determining uniquely optimal forecasts under distributional misspecification.
method Integrates statistical robustness into elicitable functionals using Kullback-Leibler divergence.
result Robust elicitable functionals admit unique solutions at the boundary of uncertainty regions.
Geodesics on polygons in a unit disk are studied with unique metric properties.
problem Characterizing geodesics on polygons within a unit disk.
method Defining a metric on polygons such that geodesics are curves in the family C.
result The constructed metric space is not isometric to any convex domain in R^2.
The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. F…
We relate ergodic-theoretic properties of a very small tree or lamination to the behavior of folding and unfolding paths in Outer space that approximate it, and we obtain a criterion for unique ergodicity in both cases. Our main result is that non-unique ergodicity gives rise to a transverse decomposition of the foldin…
The paper proves strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
problem Proving strong uniqueness and rectifiability of generalized cylindrical singularities in Ricci flow.
method Establishing a Lojasiewicz inequality for the pointed W-entropy in Ricci flow under the assumption of geometry near the base point being close to a generalized cylinder. result Proves strong uniqueness of generalized cylindrical tangent flows and shows that the subset of points with rectifiable Sqck(N) is horizontally parabolic. Inspired by Strotz's consistent planning strategy, we formulate the infinite horizon mean-variance stopping problem as a subgame perfect Nash equilibrium in order to determine time consistent strategies with no regret. Equilibria among stopping times or randomized stopping times may not exist. This motivates us to cons…
An axiomatic approach to signal reconstruction is formulated, involving a sample consistent set and a guiding set, describing desired reconstructions. New frame-less reconstruction methods are proposed, based on a novel concept of a reconstruction set, defined as a shortest pathway between the sample consistent set and…
This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, F…
This work develops a model to distinguish network and covariate information.
problem Identifying unique network and covariate information.
method Low-rank model with two-step estimation: spectral method followed by refinement.
result The method accurately recovers joint and individual components.
We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strate…
We prove that the unique least-perimeter way of partitioning the unit 2-dimensional disk into three regions of prescribed areas is by means of the standard graph consisting in three balanced constant geodesic curvature curves meeting themselves at 120 degrees, and reaching orthogonally the boundary of the disk.
New theory for clustering in geometric and adaptive settings.
problem Clustering in non-Euclidean spaces and adaptive parameters.
method Asymptotic theory for k-means and related methods. result Strong consistency and asymptotic limit theorems for various clustering procedures.
Unified framework connects two market-making models, revealing their underlying equivalence.
problem Independent calibration of two market-making frameworks (Avellaneda-Stoikov and Cartea-Jaimungal).
method Axiomatic approach to market preference functional, showing equivalence under specific conditions.
result Avellaneda-Stoikov and Cartea-Jaimungal frameworks are equivalent under certain conditions.
Large scale networks delineating collective dynamics often exhibit cascading failures across nodes leading to a system-wide collapse. Prominent examples of such phenomena would include collapse on financial and economic networks. Intertwined nature of the dynamics of nodes in such network makes it difficult to disentan…
Derives variance kernel for reaction boundary in financial models.
problem Separating components in financial volatility models.
method Operational-time variance kernel, damped Abel response kernel, closed asymptotic form.
result Operational variance has a closed asymptotic form involving various parameters.
Characterizes photon surfaces in static spacetimes, proving uniqueness.
problem Understanding photon surfaces in static spacetimes of arbitrary dimension.
method Complete characterization and new insights into spacetime geometry.
result Proves uniqueness of certain electrostatic spacetimes.
Derives operational-time variance kernel for reaction boundaries in financial markets.
problem Separating components in volatility models to better understand market dynamics.
method Derives a variance kernel for a latent-order-book reaction boundary, separating structural boundary cumulant, clock projection, and pricing-measure choice.
result Operational variance has a closed asymptotic form for long-memory forcing, with effective signed-forcing intensity and resilience.
The paper identifies a component of representations mapping modular group elements to isometries with unique fixed points.
problem Characterizing representations of the modular group into isometry groups.
method Analyzing the space of discrete faithful representations of the modular group into Isom(X) for X=SL3(R)/SO(3).
result The space of representations has a component homeomorphic to R^2 x [0,∞), parametrized by Pappus representations and containing Anosov representations.
Karigiannis discusses geometric flows of G2-structures, focusing on existence and uniqueness.
problem Existence and uniqueness of geometric flows of G2-structures.
method Introduced geometric structures, geometric flows, and discussed qualitative features. Focused on Ricci flow and DeTurck trick, then extended to G2-structures.
result Clarified conditions for short-time existence and uniqueness of G2-Laplacian flow.
The paper examines Nash equilibrium in GANs for stationary Gaussian processes.
problem Existence and uniqueness of Nash equilibrium in GANs for stationary Gaussian processes.
method Analyzes the existence of Nash equilibrium in GANs for stationary Gaussian processes, considering different discriminator families.
result The existence of Nash equilibrium depends on the discriminator family and symmetry properties of the generator family.
A unique hyperbolic metric is found for each spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
problem Finding a hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
method Constructing a strictly polyhedral hyperbolic metric on the 3-manifold such that the given spherical cone-metric is the induced dual metric on the boundary.
result The existence and uniqueness of a strictly polyhedral hyperbolic metric for a given spherical cone-metric on the boundary of a hyperbolizable 3-manifold.
New 3-manifolds created from 4-regular graphs with unique Eulerian cycles.
problem Creating compact 3-manifolds from specific graph structures. method Defining 3-manifolds via compatible Eulerian cycles in 4-regular graphs. result Each manifold in the class has a unique minimal ideal triangulation with n tetrahedra. We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor …
This work improves data reconstruction methods by ensuring unique solutions and refining optimization.
problem Ensuring unique solutions and optimizing reconstruction from KKT conditions.
method Discussion of sufficient conditions for unique solutions and introduction of sample splitting for optimization.
result Sample splitting improves reconstruction performance across various methods.
Interpreting predictions from tree ensemble methods such as gradient boosting machines and random forests is important, yet feature attribution for trees is often heuristic and not individualized for each prediction. Here we show that popular feature attribution methods are inconsistent, meaning they can lower a featur…
Novel weak solutions for volume-preserving mean curvature flow established.
problem Existence and uniqueness of solutions to volume-preserving mean curvature flow.
method Introducing varifold solutions coupled with phase volumes and new calibrations.
result Uniqueness of classical solutions among varifold solutions.
The study investigates the consistency of k-means clustering under finite expectation assumptions.
problem Consistency of k-means clustering under finite expectation assumptions. method Investigates the conditions under which k-means clustering is consistent, considering finite expectation instead of finite variance. result Inconsistency can arise due to extreme cluster imbalance, leading to some clusters having few points.
Measures consistency of tabular LLM predictions under fine-tuning multiplicity.
problem Conflicting predictions from fine-tuned tabular LLMs.
method Local stability measure in embedding space.
result Probabilistic guarantees on prediction consistency under multiplicity.
The paper explores game-theoretic alignment of LLMs with human preferences, finding limitations and conditions.
problem Aligning LLMs with human preferences using game theory.
method Systematic study of payoff choices in a two-player zero-sum game for desirable alignment properties.
result Impossibility of preference matching in game-theoretic LLM alignment under standard assumptions.
We study the concept of financial bubble in a market model endowed with a set of probability measures, typically mutually singular to each other. In this setting we introduce the notions of robust bubble and robust fundamental value in a consistent way with the existing literature in the case a unique prior exists. The…
A generalization of expectiles for d-dimensional multivariate distribution functions is introduced. The resulting geometric expectiles are unique solutions to a convex risk minimization problem and are given by d-dimensional vectors. They are well behaved under common data transformations and the corresponding sample v…
The consistency of Fréchet medians is proved for probability measures in proper metric spaces. In the context of Riemannian manifolds, assuming that the probability measure has more than a half mass lying in a convex ball and verifies some concentration conditions, the positions of its Fréchet medians are estimated. It…
We establish the Gaussian Double-Bubble Conjecture: the least Gaussian-weighted perimeter way to decompose Rn into three cells of prescribed (positive) Gaussian measure is to use a tripod-cluster, whose interfaces consist of three half-hyperplanes meeting along an (n−2)-dimensional plane at 120∘ …
Uniqueness of circle packings on certain translation surfaces is proven.
problem Proving the uniqueness of circle packings on specific translation surfaces.
method Using splitting bigons to characterize variations of circle packings.
result For certain circle packings on H(1,1) translation surfaces, there are only a finite number of ways the packing can vary without changing the contacts graph.