Study on eigenvalue distribution of correlated time series, showing deformation of Marchenko-Pastur distribution.
problem Eigenvalue distribution of Wishart matrix with temporal correlation.
method Analysis of moments and convergence to deformed Marchenko-Pastur distribution for Gaussian process with temporal correlation.
result Eigenvalue distribution converges to deformed Marchenko-Pastur distribution with longer tail and higher peak.
The study of random walks on hyperbolic spaces and Teichmüller spaces, proving central limit theorems and geodesic tracking.
problem Analyzing random walks on hyperbolic and Teichmüller spaces.
method Proving central limit theorems and geodesic tracking using finite moments and logarithmic moments.
result Translation lengths of random isometries satisfy a central limit theorem if and only if the random walk has finite second moment.
In a recent paper [\textit{M. Cristelli, A. Zaccaria and L. Pietronero, Phys. Rev. E 85, 066108 (2012)}], Cristelli \textit{et al.} analysed relation between skewness and kurtosis for complex dynamical systems and identified two power-law regimes of non-Gaussianity, one of which scales with an exponent of 2 and the oth…
We prove that in metric measure spaces where the entropy functional is K-convex along every Wasserstein geodesic any optimal transport between two absolutely continuous measures with finite second moments lives on a non-branching set of geodesics. As a corollary we obtain that in these spaces there exists only one opti…
New method finds closest martingale to Brownian motion.
problem Finding optimal martingale interpolating marginals.
method Martingale Sinkhorn algorithm, iterative scheme.
result Algorithm yields Bass potential in arbitrary dimension.
This paper examines how data affects risk measures in uncertain distributions.
problem How does distributional ambiguity affect risk measures?
method Formulated and derived simpler dual problems for infinite and finite dimensional robust moment problems.
result Developed theory and conducted experiments in inventory control and portfolio management.
Independent component analysis (ICA) is the problem of efficiently recovering a matrix A∈Rn×n from i.i.d. observations of X=AS where S∈Rn is a random vector with mutually independent coordinates. This problem has been intensively studied, but all existing efficient algorithms w…
MOMENT selects and estimates mixed-effects models using moment identities.
problem Selecting and estimating random-effects covariance matrix and fixed-effects coefficients in multiresponse linear mixed-effects models.
method MOMENT is a stage-wise moment-based framework that reduces the random-effects selection problem to a smooth constrained convex optimization problem.
result MOMENT performs competitively and can outperform separate univariate analyses for correlated responses.
Study on SA with heavy-tailed and LRD noise, establishing finite-time bounds.
problem Analyzing stochastic approximation under heavy-tailed and LRD noise.
method Noise-averaging argument to regularize impact of non-classical noise.
result Established first finite-time moment bounds for SA under heavy-tailed and LRD noise.
We associate certain probability measures on R to geodesics in the space $\H_L$ of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H0(X,kL). We prove that the measures associated to the finite dimensional spaces converge weakly to t…
Unified framework for mean testing under truncation bias.
problem High-dimensional mean testing under arbitrary truncation.
method Characterizes fundamental limits and develops a simple second-order test.
result Unified framework connects finite-moment, sub-Gaussian, and median-regular structural regimes.
JME continually estimates data moments privately and accurately.
problem Private and accurate continual estimation of data moments.
method Uses matrix mechanism and joint sensitivity analysis.
result Improves accuracy in estimating mean and covariance with reduced noise.
PMT uses public data moments to make DP feasible for unbounded data.
problem Applying differential privacy to unbounded data distributions.
method Public-moment-guided Truncation (PMT) using second-moments from public data.
result PMT improves the accuracy and stability of DP models.
Corrected moment-based methods improve inference in topic model regression.
problem Inferential difficulties in topic model plug-in workflow for regression.
method Corrected spectral moment methods for LDA, response-weighted word moments.
result Direct identification of regression coefficients without estimating topic shares.
We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arxiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions…
Paper develops methods for statistical inference in SGD with infinite variance.
problem Challenges in statistical inference for SGD with infinite variance.
method Model-agnostic methodology based on weak convergence and subsampling calibration.
result Asymptotically valid confidence regions for SGD in both finite and infinite variance regimes.
Study examines robust regression in high dimensions with heavy-tailed data.
problem Analyzing robust regression in high-dimensional settings with heavy-tailed data.
method Sharp asymptotic characterisation of M-estimators and ridge regression in elliptical distributions.
result Ridge regression is optimal and universal for finite second moments but can decay faster without them.
Consider a random vector with finite second moments. If its precision matrix is an M-matrix, then all partial correlations are non-negative. If that random vector is additionally Gaussian, the corresponding Markov random field (GMRF) is called attractive. We study estimation of M-matrices taking the role of inverse sec…
The paper proves that Gaussian field critical points have finite moments.
problem Proving the finiteness of moments for Gaussian field critical points.
method General approach not specific to critical points, using Taylor polynomial non-degeneracy.
result The finiteness of moments of the number of critical points of Gaussian fields.
Second-order estimator improves continuous-time policy evaluation.
problem Estimating value surfaces from discrete data with time-inhomogeneous dynamics.
method Moment-matching coefficients for high-order generator regression.
result Second-order estimator consistently outperforms Bellman baseline.
Enhances DP linear regression using public data moments.
problem Limited utility of traditional DP methods in linear regression.
method Transform private data using public second-moment matrix for a better OLSE.
result Improved accuracy and robustness of OLSE in DP linear regression.
The paper proposes a method for constructing confidence sets that adapt to the cardinality of the smallest component of a mean vector.
problem Forming confidence sets for the smallest component of an unknown mean vector.
method Sample splitting and self-normalization approach to test each component for being the smallest, maintaining validity regardless of d and n. result The proposed tests achieve the local minimax separation rate and robust to heavy-tailed distributions.
We consider two stage estimation with a non-parametric first stage and a generalized method of moments second stage, in a simpler setting than (Chernozhukov et al. 2016). We give an alternative proof of the theorem given in (Chernozhukov et al. 2016) that orthogonal second stage moments, sample splitting and n1/4-…
New algorithm detects changes in heavy-tailed data streams.
problem Detecting changes in heavy-tailed data streams.
method Clipped Stochastic Gradient Descent (SGD) combined with union bound.
result First algorithm with finite-sample false-positive rate guarantees for heavy-tailed data.
A new memory-efficient Adam variant reduces second moments when feasible.
problem Memory constraints in training machine learning models.
method Signal-to-Noise Ratio (SNR) analysis to identify dimensions where second moments can be replaced by means.
result Memory-efficient Adam variant (SlimAdam) matches performance and stability of Adam while saving up to 98% of second moments.
A new method for estimating causal parameters from observables reduces the need for finite moment conditions.
problem Estimating causal parameters from observational data with unknown or infinite moment conditions.
method Variational Method of Moments (VMM) for a general class of estimators, including kernel and neural net-based methods.
result VMM estimators are consistent, asymptotically normal, and semiparametrically efficient.
Moment polytope of toric exponential families is a projection of a simplex.
problem Understanding the geometry of exponential families in finite sample spaces.
method Toric torification and projection of higher-dimensional simplices.
result Moment polytope is a projection of a higher-dimensional simplex.
Detection of power-law behavior and studies of scaling exponents uncover the characteristics of complexity in many real world phenomena. The complexity of financial markets has always presented challenging issues and provided interesting findings, such as the inverse cubic law in the tails of stock price fluctuation di…
Revisits Lee's Moment Formula, relaxing moment assumptions for implied volatility.
problem Implied volatility constraints under finite log-moments.
method Analyzes stock price martingale with finite log-moments, derives new bounds and proof.
result New bounds on implied volatility growth, relaxes moment assumptions.
New robust estimators achieve subgaussian bounds using VC-dimension.
problem Robust estimation of sparse and corrupted data.
method Use of VC-dimension to measure statistical complexity.
result First robust estimators for sparse estimation with subgaussian rate.
Paper provides unbiased spectral moment estimates from finite data.
problem Challenges in estimating spectral moments from limited data.
method Dynamic programming approach to estimate spectral moments of kernel integral operator.
result Demonstrates consistency with theoretical spectra and practical utility in neural networks.
We consider a class of finite Markov moment problems with arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the non-unique solution families. Moreover, we present a constructive algorithm to solve the moment problems numeric…
This study shows the moment-SOS hierarchy converges in polynomial optimization over product of spheres.
problem Minimizing multihomogeneous polynomials over product of spheres.
method Moment-SOS hierarchy, local optimality conditions, differential geometry, Morse theory.
result The moment-SOS hierarchy has finite convergence for generic multihomogeneous objective functions.
The paper connects moment maps to the stability of holomorphic fibrations.
problem Stability of holomorphic fibrations.
method Use of moment maps and K-stability criteria.
result Existence of optimal symplectic connections implies stability of fibrations.
We extend the classical Cox-Ross-Rubinstein binomial model in two ways. We first develop a binomial model with time-dependent parameters that equate all moments of the pricing tree increments with the corresponding moments of the increments of the limiting Itô price process. Second, we introduce a new trinomial model i…
This paper proposes a novel model of financial prices where: (i) prices are discrete; (ii) prices change in continuous time; (iii) a high proportion of price changes are reversed in a fraction of a second. Our model is analytically tractable and directly formulated in terms of the calendar time and price impact curve. …
Dynamic Boltzmann Machine (DyBM) has been shown highly efficient to predict time-series data. Gaussian DyBM is a DyBM that assumes the predicted data is generated by a Gaussian distribution whose first-order moment (mean) dynamically changes over time but its second-order moment (variance) is fixed. However, in many fi…
Wide neural networks learn features under μP, identifying weights and decomposing support.
problem Feature learning in wide neural networks under μP. method Proving mean-field limit, characterizing identifiability, sparse-dictionary decomposition, and feature-learning-error decomposition.
result The triple (w∗,Dorb∗,S∗) identifies the natural learning cell of the architecture-data pair (σ,ρ). New optimizer Eve uses examplewise gradients for better second-moment estimates.
problem Improving optimization methods for machine learning.
method Adaptive optimization with examplewise gradients.
result Eve optimizer slightly outperforms Adam on small scale benchmarks.
New result on group actions in CAT(0) spaces with vanishing escape rate.
problem Understanding group actions with vanishing escape rate on CAT(0) spaces.
method Equivariant μ-harmonic map proof. result Existence of a flat subspace invariant under the action of Γ. Investigates properties of moment maps and stratifications on Lie groups.
problem Understanding moment maps and stratifications on real reductive Lie groups.
method Functorial, algebraic approach to moment map and Kirwan-Ness stratification.
result Properties and properties of moment maps and stratifications established.
Kim-Milman flow map stable under regular target measures
problem Stability of Kim-Milman flow map under target measure variations
method Stability in relative entropy and 2-Wasserstein distance result Lipschitz stability up to logarithmic factor
Q-MMR evaluates policies using reweighted rewards and moment matching.
problem Off-policy evaluation in finite-horizon MDPs.
method Q-MMR learns scalar weights for data points via a moment matching objective against a value-function discriminator class.
result Data-dependent finite-sample guarantee with a dimension-free error bound.
Study finds the minimum number of finite Gaussian mixtures for best approximation.
problem Finding the minimum number of finite Gaussian mixtures for best approximation.
method Local moment matching for upper bound and spectral analysis for lower bound.
result Corrects a previous lower bound in the case of Gaussian mixing distributions.
AdamNX improves Adam's stability by adjusting its learning rate.
problem Adam's tendency to converge to non-flat minima in large-scale models.
method Proposes a novel exponential decay mechanism for Adam's second-order moment estimate.
result AdamNX outperforms Adam and its variants in stability and performance.
The paper addresses score-mismatched diffusion models and zero-shot conditional samplers.
problem Theoretical guarantees for score-mismatched diffusion models in zero-shot conditional sampling.
method Theoretical analysis of score-mismatched diffusion models and zero-shot conditional samplers.
result Theoretical performance guarantees with explicit dimensional dependencies for score-mismatched diffusion samplers.
New proof of Sobolev inequality with constraints on sphere.
problem Improving Sobolev inequality on sphere with constraints.
method Careful study of extremal problem on sphere.
result Explicit determination of constant in second order moments case.
Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.
problem Capturing higher-order tail behavior and dependence effects in risk measures.
method Second-order asymptotic expansions using extreme value theory and regular variation theory.
result Second-order approximations reduce approximation errors, especially at extreme confidence levels.