Study random walks on groups with superlinear divergent geodesics.
problem Existence of superlinear divergent geodesics in groups.
method Developed theory of superlinear divergence and applied Gouëzel's pivoting technique.
result Established a central limit theorem for random walks on groups with superlinear divergent geodesics.
We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A(G) with connected defining graph G. We use this to determine when two points in an asymptotic cone of A(G) are separated by a cut-point. As an application, we show that if G does not d…
New quasi-Newton method guarantees global superlinear convergence.
problem Global convergence and superlinear convergence of quasi-Newton methods.
method Hybrid proximal extragradient method with online learning for Hessian approximation.
result First globally convergent quasi-Newton method with explicit superlinear convergence rate.
Study on divergence and thickness for Coxeter groups, generalizing previous work.
problem Characterizing and bounding divergence and thickness for Coxeter groups.
method Characterization of linear divergence, introduction of hypergraph index, new construction of Coxeter systems.
result Upper bounds on divergence and thickness for Coxeter groups, conjectured to be equalities.
Optimal transport with f-divergence regularization using generalized Sinkhorn algorithm.
problem Optimal transport with f-divergence regularization. method Generalized Sinkhorn algorithm for solving optimal transport problems with various f-divergences. result Strong duality holds, optimums are attained, and convergence to an optimal solution is guaranteed under certain conditions.
Study shows income inequality increases with city size, affecting only the wealthiest deciles.
problem Understanding income inequality in urban areas.
method Urban scaling analysis of total income scaling in population percentiles.
result Income in the poorest decile does not increase with city size, while the wealthiest deciles show superlinear scaling.
EM algorithm converges in KL divergence for exponential families via mirror descent.
problem Lack of understanding of EM's non-asymptotic convergence properties.
method Viewing EM as a mirror descent algorithm, showing convergence rates in KL divergence.
result KL divergence rates for EM in exponential families, invariant to parametrization.
Diffusion models converge linearly to complex data manifolds.
problem Sampling from high-dimensional complex data distributions.
method Score-matching generative models with novel integration scheme.
result Linear convergence in KL divergence to intrinsic dimension d. We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.
Paper improves a method for fast global and local convergence in optimization.
problem Slow global convergence in optimization methods with noisy Hessian estimates.
method Stochastic Newton Proximal Extragradient method using HPE framework.
result Faster global linear rate and superlinear convergence in fewer iterations.
New averaging technique speeds up Newton method convergence.
problem Superlinear convergence of stochastic Newton methods with noisy Hessians.
method Hessian averaging to reduce noise and maintain superlinear convergence.
result Hessian averaging achieves superlinear convergence with a non-asymptotic rate.
New method achieves superlinear convergence rate with limited memory.
problem Achieving superlinear convergence rate in quasi-Newton methods with limited memory.
method Limited-memory Greedy BFGS (LG-BFGS) method with displacement aggregation and basis vector selection.
result Explicit non-asymptotic superlinear convergence rate demonstrated.
New bounds show diffusion models converge nearly linearly in data dimension.
problem Improving convergence bounds for diffusion models.
method Refined discretization of reverse SDE using stochastic localization.
result Linear convergence in data dimension with logarithmic factors.
Paper proposes new Langevin samplers for sampling from log-concave distributions with superlinear gradient growth.
problem Sampling from log-concave distributions with superlinear gradient growth.
method Proposes two novel discretizations of kinetic Langevin SDEs, showing contractivity and log-Sobolev inequality.
result Establishes non-asymptotic bounds in 2-Wasserstein distance between sampled distributions and target measures.
Improved stability for matrix recovery from rank-one measurements.
problem Phase retrieval problem of recovering rank-one positive semidefinite matrices.
method Developed a smoothing Newton method based on Bures-Wasserstein gradient descent.
result Superlinear convergence with rigorous guarantees and stable implementation.
New algorithm samples superlinearly growing log-gradient distributions.
problem Sampling from distributions with superlinearly growing log-gradient.
method Proposes a novel taming Langevin-based scheme called sTULA.
result Derives non-asymptotic convergence bounds in KL, TV, and W2 distances.
New methods optimize functions faster with less gradient accuracy needed.
problem Optimizing complex functions with limited gradient accuracy.
method Hessian averaging and adaptive gradient sampling methods.
result Improved convergence rates for various function types.
Memory-constrained algorithms need superlinear memory for efficient convex optimization.
problem Efficiently minimizing convex functions with limited memory.
method Analyzing first-order algorithms with superlinear memory constraints.
result Superlinear memory is necessary for optimal performance in convex optimization.
New algorithm TUSLA improves learning of non-convex neural networks.
problem Optimizing non-convex loss functions in neural networks with superlinear gradient growth.
method Tamed Unadjusted Stochastic Langevin Algorithm (TUSLA) based on SGLD with taming technology.
result Finite-time guarantees for TUSLA to find approximate minimizers of empirical and population risks.
Sublinear functionals of random variables are known as sublinear expectations; they are convex homogeneous functionals on infinite-dimensional linear spaces. We extend this concept for set-valued functionals defined on measurable set-valued functions (which form a nonlinear space), equivalently, on random closed sets. …
New method improves smoothness of minimizing currents near singular points.
problem Improving smoothness of minimizing currents near singular points.
method New method to estimate the full singular set of the foliation by minimizers and proof of superlinear decay of closeness.
result Generic smoothness of minimizers improved to n−9−εn for n≥11. KSS method converges and recovers correct clustering under certain conditions.
problem Subspace clustering for semi-randomly sampled data.
method Local convergence analysis and recovery guarantee for KSS method.
result KSS method converges superlinearly and finds correct clustering within loglog N iterations.
Paper proposes a new method to efficiently incorporate curvature information in stochastic optimization.
problem Minimizing nonconvex functions with limited curvature information.
method Structured stochastic quasi-Newton method using partial Hessian information.
result Global convergence to stationary point and local superlinear convergence rate established.
OSGM uses online learning to adapt stepsize for faster convergence.
problem Improving convergence rates of first-order methods.
method OSGM combines online learning and feedback functions to adjust stepsize.
result OSGM achieves convergence rates asymptotically no worse than optimal.
RELTA-SGLD stabilizes nonconvex SGLD updates with a lighter taming scheme.
problem Stabilizing superlinear stochastic-gradient updates in nonconvex optimization.
method Threshold-based taming with relative-growth principle for stability.
result Polynomial moment stability and first-order stationary accuracy in nonconvex SGLD.
In a continuous-time model with multiple assets described by càdlàg processes, this paper characterizes superhedging prices, absence of arbitrage, and utility maximizing strategies, under general frictions that make execution prices arbitrarily unfavorable for high trading intensity. Such frictions induce a duality bet…
The paper develops efficient estimators for semi-parametric binary models in distributed computing.
problem Estimation and inference challenges in large-scale data under non-smooth objective functions.
method Proposes one-shot and multi-round divide-and-conquer estimators with adaptive kernel smoothing to relax constraints and achieve superlinear optimization error.
result Establishes quadratic convergence up to optimal statistical error rate and handles dataset heterogeneity and high-dimensional sparse parameters.
We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation ρ between the driving Brownian motions of the …
Study PL bordism theories with quantitative bounds on filling simplices.
problem Understanding PL bordism theories with geometric constraints.
method Quantitative analysis of PL manifolds and exotic theories.
result Bounding the number of simplices in fillings of cycles.
We study an optimal liquidation problem under the ambiguity with respect to price impact parameters. Our main results show that the value function and the optimal trading strategy can be characterized by the solution to a semi-linear PDE with superlinear gradient, monotone generator and singular terminal value. We also…
State of the art methods in astronomical image reconstruction rely on the resolution of a regularized or constrained optimization problem. Solving this problem can be computationally intensive and usually leads to a quadratic or at least superlinear complexity w.r.t. the number of pixels in the image. We investigate in…
We generalize the notion of cusp excursion of geodesic rays by introducing for any k≥1 the kth excursion in the cusps of a hyperbolic N-manifold of finite volume. We show that on one hand, this excursion is at most linear for geodesics that are generic with respect to the hitting measure of a random walk.…
Study on nonlinear elliptic equations with variable exponents, proving existence and multiplicity of solutions.
problem Existence and multiplicity of solutions for Dirichlet boundary value problems involving (p(m),q(m))-equation. method Proved using the mountain pass theorem and Fountain theorem with Cerami sequences.
result Existence and multiplicity of solutions for (p(m),q(m))-equation. In this paper, we introduce new classes of divergences by extending the definitions of the Bregman divergence and the skew Jensen divergence. These new divergence classes (g-Bregman divergence and skew g-Jensen divergence) satisfy some properties similar to the Bregman or skew Jensen divergence. We show these g-diverge…
Let a A be the 1-skeleton of a triangulated topological annulus. We establish bounds on the combinatorial modulus of a refinement A′, formed by attaching new vertices and edges to A, that depend only on the refinement and not on the structure of A itself. This immediately applies to showing that a disk triangul…
Divergence functions play a key role as to measure the discrepancy between two points in the field of machine learning, statistics and signal processing. Well-known divergences are the Bregman divergences, the Jensen divergences and the f-divergences. In this paper, we show that the symmetric Bregman divergence can be …
The techniques and analysis presented in this thesis provide new methods to solve optimization problems posed on Riemannian manifolds. These methods are applied to the subspace tracking problem found in adaptive signal processing and adaptive control. A new point of view is offered for the constrained optimization prob…
NR retraction approximates geodesics on submanifolds efficiently.
problem Efficiently approximating geodesics on submanifolds for practical algorithms.
method Introducing Newton retraction (NR) as a class of retractions on submanifolds induced by a foliation of the ambient manifold.
result NR is more stable and computationally cheaper than oblique projection, with superlinear convergence regions.
Study explores relationship between Hölder and FDPD divergences.
problem Understanding the relationship between Hölder and FDPD divergences.
method Intersection and generalization of divergence families, proving nonnegativity, deriving inequalities.
result Established ξ-Hölder divergence and derived inequalities. Unified representation of density-power-based divergences simplifies estimation to M-estimation.
problem Outliers in density estimation.
method Define a norm-based Bregman density power divergence (NB-DPD) that reduces to M-estimation.
result NB-DPD connects and generalizes existing divergences, highlighting robustness properties.
Artificial Neural Networks (ANNs) have received increasing attention in recent years with applications that span a wide range of disciplines including vital domains such as medicine, network security and autonomous transportation. However, neural network architectures are becoming increasingly complex and with an incre…
This paper improves active learning by using robust divergences for committee disagreement.
problem Active learning with high measurement costs.
method Query by committee with Bregman divergence (including Kullback-Leibler divergence as a special case).
result The proposed method is more robust and performs as well as or better than conventional methods.
Research examines correlations of complex logarithms of lattice points, showing level repulsion and Poissonian behavior.
problem Analyzing correlations of complex logarithms of lattice points.
method Proving existence of pair correlation functions and examining behavior at various scalings.
result Level repulsion observed at linear scaling, Poissonian behavior at sublinear scalings.
We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general …
The study examines correlations of logarithms of integers at different scalings.
problem Analyzing pair correlations of logarithms of integers at various scalings.
method Examined correlations of logarithms of positive integers at different scalings, proving the existence of pair correlation functions.
result Level repulsion at linear scaling, total loss of mass at superlinear scalings, and Poissonian behavior at sublinear scalings.
We define two non-linear operations with random (not necessarily closed) sets in Banach space: the conditional core and the conditional convex hull. While the first is sublinear, the second one is superlinear (in the reverse set inclusion ordering). Furthermore, we introduce the generalised conditional expectation of r…
New divergence measures improve KL approximation.
problem Improving KL divergence approximation without AC condition.
method Introduced α-geodesical skew divergence. result Properties of α-geodesical skew divergence studied. A new algorithm estimates mean adaptively to covariance, faster and more flexible than existing methods.
problem Estimating mean of a distribution with unknown covariance efficiently and privately.
method Adaptive differentially private algorithm with optimal convergence rates and near-linear sample complexity.
result Achieves optimal rates of convergence with respect to the Mahalanobis norm ∣∣⋅∣∣Σ.