The paper studies properties of optimal metrics associated to curves on surfaces.
problem Investigating properties of optimal metrics associated to curves on surfaces.
method Starting from a filling curve and a separating curve, constructing a two integer parameter family of curves and deriving coarse length bounds and qualitative properties of their associated optimal metrics.
result There are infinitely many pairs of filling curves with distinct inf invariants but the same self-intersection number.
Study risk sharing among agents with varying risk preferences.
problem Risk sharing among agents with heterogeneous risk measures.
method Derive explicit solutions for inf-convolution and counter-monotonic inf-convolution under varying risk seeking.
result Explicit solutions for inf-convolution and counter-monotonic inf-convolution can be represented by a generalization of distortion risk measures.
We define a new differential invariant a compact manifold by VM(M)=infgVc(M,[g]), where Vc(M,[g]) is the conformal volume of M for the conformal class [g], and prove that it is uniformly bounded above. The main motivation is that this bound provides a upper bound of the Friedlander-Nadirashvili…
Establishes relationships between prudence and stability properties of risk functionals.
problem Stability properties of risk functionals
method General relationships and preservation of prudence under cash-additive hulls and inf-convolutions
result General methods for constructing prudent risk measures
Algorithmic solutions to the conjugacy problem in the braid groups B_n were given by Elrifai-Morton in 1994 and by the authors in 1998. Both solutions yield two conjugacy class invariants which are known as `inf' and `sup'. A problem which was left unsolved in both papers was the number m of times one must `cycle' (res…
INF-clip optimizes heavy-tailed MAB problems with improved performance.
problem Optimizing multi-armed bandit problems with heavy-tailed rewards.
method INF-clip algorithm for adversarial and stochastic heavy-tailed MAB settings.
result INF-clip is optimal for linear and non-linear heavy-tailed stochastic MAB problems.
In this paper we will discuss the optimal risk transfer problems when risk measures are generated by G-expectations, and we present the relationship between inf-convolution of G-expectations and the inf-convolution of drivers G.
This paper tightens the law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.
problem Developing nonasymptotic concentration bounds for empirical KL_inf with optimal constants and rates.
method Presenting a tight law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.
result A tight law of the iterated logarithm for empirical KL_inf, applicable to unbounded data.
Inequality found for a specific equation on 5D manifolds.
problem Finding an inequality for a Yamabe type equation on 5D manifolds.
method Analyzing a Yamabe type equation in 5D manifolds.
result An inequality of type sup x inf found for the Yamabe type equation in 5D.
SurvLIME-Inf simplifies explanation of survival models using a linear programming approach.
problem Explain complex survival models using simple linear programming.
method Uses L∞-norm for feature importance and explains black-box models. result SurvLIME-Inf outperforms SurvLIME in small training set scenarios.
Simplified proof for Tsallis-INF algorithm without conjugate functions.
problem Deriving a best-of-both-worlds guarantee for Tsallis-INF.
method Modern tools from online convex optimization, avoiding conjugate functions.
result A slimmer proof with simplified constants.
In this paper we analyze a dynamic recursive extension of the (static) notion of a deviation measure and its properties. We study distribution invariant deviation measures and show that the only dynamic deviation measure which is law invariant and recursive is the variance. We also solve the problem of optimal risk-sha…
Extends inf-convolution to countable risk measures for risk sharing.
problem Limited inf-convolution theory to finite sets of risk measures.
method Extends inf-convolution to countable sets, investigates properties and results.
result Generalizes known properties and results to countable case.
We prove an a priori estimate of type sup*inf on Riemannian manifold of dimension 3 (not necessarily compact).
Study uncovers new phase transitions in asymmetric causal inference scenarios.
problem Understanding typical phase transitions in asymmetric causal inference.
method Combining Causal inference (C-inf) and Low-rank recovery (LRR) with Random duality - Free probability theory (RDT-FPT).
result Discovering a doubling low-rankness phenomenon in asymmetric scenarios.
In this paper, we explore several Fatou-type properties of risk measures. The paper continues to reveal that the strong Fatou property, which was introduced in [17], seems to be most suitable to ensure nice dual representations of risk measures. Our main result asserts that every quasiconvex law-invariant functional on…
Study risk sharing with Lambda VaR under diverse beliefs.
problem Risk sharing among agents with different beliefs.
method Use Lambda Value-at-Risk as preference, analyze under heterogeneous beliefs.
result Explicit formulas for risk sharing under various belief scenarios.
We give some a priori estimates of type sup*inf for Yamabe and prescribed scalar curvature type equations on Riemannian manifolds of dimension >2. The product sup*inf is caracteristic of those equations, like the usual Harnack inequalities for non negative harmonic functions. First, we have a lower bound for sup*inf fo…
Let Mn⊂Rn+1 be the graph of a C2-real valued function defined in a closed ball of Rn. In this work, we obtain upper bounds for infM∣H∣ and infM∣R∣, where H and R are, respectively, the mean curvature and the scalar curvature of Mn, generalizing estimates given by Heinz i…
Optimizes convex functions in finite vs infinite dimensions, revealing slow convergence rates.
problem Analyzing gradient flows in finite and infinite-dimensional Hilbert spaces.
method Proves convergence rates and optimality conditions for gradient flows and related methods.
result Gradient flow convergence rates in finite dimensions are slower than in infinite dimensions, with optimal rates achievable in Hilbert spaces.
Improved regret bounds for Tsallis-INF in adversarial bandits and corruptions.
problem Adversarial bandits and corruptions in multiarmed bandit problems.
method Improved regret bounds for Tsallis-INF algorithm.
result Achieves $\mathcal{O}\left(\left(\sum_{i
eq i^*} \frac{1}{Δ_i}
ight)\log_+\left(\frac{(K-1)T}{\left(\sum_{i
eq i^*} \frac{1}{Δ_i}
ight)^2}
ight)+\sqrt{C\left(\sum_{i
eq i^*}\frac{1}{Δ_i}
ight)\log_+\left(\frac{(K-1)T}{C\sum_{i
eq i^*}\frac{1}{Δ_i}}
ight)}
ight)$ regret bound.
This paper studies moduli spaces of statistical structures on Lie groups.
problem Understanding statistical structures on Lie groups.
method Introduced and studied moduli spaces for left-invariant statistical structures on Lie groups.
result Moduli spaces of left-invariant Riemannian metrics are singletons for certain Lie groups.
New bounds for sequential tests under power-one error levels.
problem Determining stopping times for sequential tests with power-one error levels.
method Proved two lower bounds for stopping times under specific conditions.
result Upper and lower bounds for sequential tests are shown to be tight.
We connect Causal inference and low-rank recovery via RDT and free probability theory.
problem Determining the applicability of causal inference via low-rank recovery.
method Random Duality Theory, free probability theory, and mathematical rigor.
result Exact closed-form worst case phase transitions for causal inference.
In this paper, we study a family of non-convex and possibly non-smooth inf-projection minimization problems, where the target objective function is equal to minimization of a joint function over another variable. This problem include difference of convex (DC) functions and a family of bi-convex functions as special cas…
We study the existence of optimal actions in a zero-sum game infτsupPEP[Xτ] between a stopper and a controller choosing a probability measure. This includes the optimal stopping problem infτE(Xτ) for a class of sublinear expectations E(⋅) such as the G-expectation. We show that …
We show how an operation of inf-convolution can be used to approximate convex functions with C1 smooth convex functions on Riemannian manifolds with nonpositive curvature (in a manner that not only is explicit but also preserves some other properties of the original functions, such as ordering, symmetries, infima …
New spectral Dehn function characterizes word-hyperbolic groups.
problem Characterizing word-hyperbolic groups using spectral properties.
method Introducing spectral Dehn functions and proving inequalities relating them to Dehn functions.
result A spectral Dehn function characterizes word-hyperbolic groups.
We prove sharp blow up rates of solutions of higher order conformally invariant equations in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This is an extension of the celebrated theorem of Caffarelli-Gidas-Spruck for the second order Yamabe…
The paper explores numerical characteristics of compact Riemannian manifolds and proves inequalities.
problem Analyzing numerical characteristics of compact Riemannian manifolds.
method Proving inequalities involving scalar curvature, Ricci curvature, and sectional curvature.
result Proven inequalities for the curvature of compact Riemannian manifolds.
We show how Lasry-Lions's result on regularization of functions defined on Rn or on Hilbert spaces by sup-inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds M of bounded sectional curvature. More specifically, among other things we show that…
New examples of non-homeomorphic foliation leaves found.
problem Finding non-homeomorphic foliation leaves in manifolds.
method Examples of 5-manifolds and foliations of 6-manifolds.
result Examples of non-homeomorphic foliation leaves in C1 and C∞ foliations. The paper finds the optimal wealth growth rate in betting games.
problem Optimizing wealth growth in Kelly betting games against arbitrary hypotheses.
method Analyzes the growth rate using KL divergence and proves it equals a specific limit.
result The optimal wealth growth rate is characterized and proven to be achievable.
Let M be a compact manifold with a spin structure χand a Riemannian metric g. Let λ_g^2 be the smallest eigenvalue of the square of the Dirac operator with respect to g and χ. The τ-invariant is defined as τ(M,χ):= sup inf \sqrt{λ_g^2} Vol(M,g)^{1/n} where the supremum runs over the set of all conformal classes on M, a…
Let M be a compact manifold with a metric g and with a fixed spin structure χ. Let λ_1+(g) be the first non-negative eigenvalue of the Dirac operator on (M,g,χ). We set τ(M,χ):=supinfλ_1+(g) where the infimum runs over all metrics g of volume 1 in a conformal class [g_0] on M and where the…
Optimal tests for composite nulls achieve the KL inf lower bound.
problem Designing optimal tests for composite null hypotheses.
method Constructive schemes based on universal e-processes.
result Optimal tests match the KL inf lower bound as α → 0.
Lewis and Mordecki have computed the Wiener-Hopf factorization of a Lévy process whose restriction on ]0,+∞[ of their Lévy measure has a rational Laplace transform. That allows to compute the distribution of (Xt,inf0≤s≤tXs). For the same class of Lévy processes, we compute the distribution of $ (…
Kurdyka-Lojasiewicz (KL) exponent plays an important role in estimating the convergence rate of many contemporary first-order methods. In particular, a KL exponent of 21 for a suitable potential function is related to local linear convergence. Nevertheless, KL exponent is in general extremely hard to estimate. I…
Given a spacelike 2-surface Σ in a spacetime N and a constant future timelike unit vector T0 in R3,1, we derive upper and lower estimates of Wang-Yau quasilocal energy E(Σ,X,T0) for a given isometric embedding X of Σ into a flat 3-slice in R3,1. The quantity E(Σ,X,T0) itself depends …
Let (M,g) be a compact Riemannian manifold of dimension n≥3. For a metric g on M, we let $\la_2(g)$ be the second eigenvalue of the Yamabe operator $L_g:= \frac{4(n-1)}{n-2} Δ_g + \scal_g$. Then, the second Yamabe invariant is defined as $$ \si_2(M) \definedas \sup \inf_{h \in [g]} \la_2(h) \Vol(M,h)^{2/n}.…
Inf-FS selects features by graph paths, ranking them for infinite feature sets.
problem Feature selection in large datasets with relevance and redundancy.
method Graph-based feature selection with infinite paths, evaluating feature subsets using matrix power series and Markov chains.
result Inf-FS outperforms other methods in various feature selection scenarios.
Paper proves a spinorial version of Aubin's estimate for the Yamabe problem.
problem Solving the Yamabe problem for spin manifolds.
method Spinorial approach using the Dirac operator and conformal metrics.
result Shows a spinorial analogue to Aubin's estimate for the Yamabe problem.
New connections found on zero-mean multivariate normal distributions.
problem Characterizing statistical connections on zero-mean multivariate normal distributions.
method Investigating invariant conjugate symmetric statistical connections on the submanifold of zero-mean multivariate normal distributions.
result Invariant connections on zero-mean multivariate normal distributions are not uniquely characterized by invariance under the general linear group action.
A risk-neutral method is always used to price and hedge contingent claims in complete market, but another method based on utility maximization or risk minimization is wildly used in more general case. One can find all kinds of special risk measure in literature. In this paper, instead of using market modified risk meas…
Study of Steklov eigenvalues on degenerating conformal classes.
problem Understanding Steklov eigenvalues on surfaces with boundaries.
method Precise formula for the limit of Steklov eigenvalues on degenerating conformal classes.
result The limit of Steklov eigenvalues equals 2πk for surfaces with boundaries. The paper revisits the σk-Yamabe problem and proves the existence of a conformal metric with constant σ2-scalar curvature.
problem Finding a conformal metric with constant σk-scalar curvature on closed manifolds. method Analyzing the σ2-Yamabe constant and proving its achievability under certain conditions. result The σ2-Yamabe constant is achieved by a conformal metric, solving the σ2-Yamabe problem on manifolds with positive Yamabe constant. Let M,N be two smooth compact hypersurfaces of Rn which bound strictly convex domains equipped with two absolutely continuous measures μ and ν (with respect to the volume measures of M and N). We consider the optimal transportation from μ to ν for the quadratic cost. Let $(φ:m \to \mathbb{R},ψ…
On a filtered probability space (Ω,F,P,F=(Ft)t=0,…,T), we consider stopper-stopper games $\overline V:=\inf_{\Rho\in\bT^{ii}}\sup_{τ\in\T}\E[U(\Rho(τ),τ)]$ and $\underline V:=\sup_{\Tau\in\bT^i}\inf_{ρ\in\T}\E[U(\Rho(τ),τ)]$ in discrete time, where U(s,t) is $\mathcal{F}_{s\vee…