Median-of-means sampling outperforms mean-of-means for large sample sizes in numerical integration.
problem Improving numerical integration accuracy in high dimensions.
method Median-of-means sampling compared to mean-of-means using RQMC methods.
result Median-of-means sampling is superior for large sample sizes, while mean-of-means is better for smaller sample sizes.
New graph properties inherited by Frechet mean and median.
problem Characterizing the average of graph-valued samples.
method Analysis of Frechet mean and median graphs.
result Edge density is hereditary in Frechet mean and median graphs.
Evolutionary algorithms (EAs) are a sort of nature-inspired metaheuristics, which have wide applications in various practical optimization problems. In these problems, objective evaluations are usually inaccurate, because noise is almost inevitable in real world, and it is a crucial issue to weaken the negative effect …
This paper introduces online algorithms to estimate robust geometric median in large data streams.
problem Detecting outliers in large data sets using robust statistical measures.
method Online stochastic Newton methods for estimating the geometric median.
result Rates of convergence for online estimation of the geometric median.
In kernel methods, the median heuristic has been widely used as a way of setting the bandwidth of RBF kernels. While its empirical performances make it a safe choice under many circumstances, there is little theoretical understanding of why this is the case. Our aim in this paper is to advance our understanding of the …
New estimator for symmetric kernel expectations, robust to missing data.
problem Efficient estimation of symmetric kernel expectations with missing data.
method Median-of-Incomplete-U-Statistics (MIU) estimator.
result Established finite-sample concentration rate for MIU.
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…
Sharp threshold found for Frechet mean of inhomogeneous graphs.
problem Finding the Frechet mean of inhomogeneous Erdos-Renyi random graphs.
method Thresholding the expected adjacency matrix of the ensemble.
result The Frechet mean graph of inhomogeneous Erdos-Renyi random graphs exhibits a sharp threshold.
A new depth measure and median defined on Hadamard manifolds.
problem Statistical depth and median on Hadamard manifolds.
method Horospherical depth and Busemann median defined using renormalized distance functions.
result The Busemann median exists for every Borel probability measure on Hadamard manifolds.
Paper proposes a robust method for federated ICA with geometric median aggregation.
problem Federated ICA with permutation ambiguity in client estimations.
method Geometric median aggregation with k-means clustering to resolve permutation ambiguity.
result The method provably remains effective in highly heterogeneous scenarios.
Anchor-TS uses median anchoring to improve online decision-making from offline data with distribution shift.
problem Improving online decision-making from offline data with distribution shift.
method Sample-Mean Anchored Thompson Sampling (Anchor-TS) with median anchoring.
result Anchor-TS safely leverages offline data to accelerate online learning and reduces regret.
Three privacy-preserving methods for median regression are proposed.
problem Protecting individual privacy in median regression analysis.
method Three privacy-preserving methods: finite smoothing, iterative, and greedy coordinate descent.
result Numerical results show varying performance across different sample sizes.
A method for estimating the median of gradients in stochastic optimization.
problem Robust gradient estimation in stochastic optimization for various applications.
method Stochastic Proximal Point Method for median gradient estimation.
result The proposed method can converge even under heavy-tailed, state-dependent noise.
Improved private geometric median estimation with nearly-linear time complexity.
problem Estimating the geometric median of a dataset while maintaining privacy.
method Improved algorithm using subsampling and geometric aggregation, achieving nearly-linear runtime.
result Achieves the same approximation quality as previous methods but with nearly-linear runtime.
This paper investigates the phase retrieval problem, which aims to recover a signal from the magnitudes of its linear measurements. We develop statistically and computationally efficient algorithms for the situation when the measurements are corrupted by sparse outliers that can take arbitrary values. We propose a nove…
Clustering is a fundamental problem in unsupervised learning, and has been studied widely both as a problem of learning mixture models and as an optimization problem. In this paper, we study clustering with respect the emph{k-median} objective function, a natural formulation of clustering in which we attempt to minimiz…
We analyze the performance of the Tukey median estimator under total variation (TV) distance corruptions. Previous results show that under Huber's additive corruption model, the breakdown point is 1/3 for high-dimensional halfspace-symmetric distributions. We show that under TV corruptions, the breakdown point reduces …
Replicable clustering algorithms for k-medians, k-means, and k-centers are proposed.
problem Designing clustering algorithms that produce the same partition on repeated runs under the same distribution.
method Utilizing approximation routines for combinatorial clustering problems in a black-box manner.
result Replicable algorithms for statistical k-medians, k-means, and k-centers with specified approximation and sample complexities. New method improves mean estimation for heavy-tailed data.
problem Estimating mean of heavy-tailed distributions.
method Median-of-Means (MoM) with symmetrization technique.
result Improved sample complexity bound for mean estimation.
Medix uses the median to detect outliers from unlabeled data for robust OOD detection.
problem Challenges in using unlabeled data for OOD detection due to mixed InD and OOD samples.
method Introduces Medix, a framework using the median operation to identify outliers from unlabeled data.
result Empirical results show Medix outperforms existing methods in open-world settings.
Unified meta algorithms estimate various distribution functionals in infinite-armed bandits.
problem Estimating various distribution functionals in infinite-armed bandits.
method Unified meta algorithms for offline and online settings, achieving optimal sample complexities.
result Online estimation offers significant advantage for certain distribution functionals.
This paper proposes methods to compute differentially private confidence intervals for the median.
problem Ensuring privacy in statistical inference for the median.
method Directly estimating interval bounds for the median under differential privacy constraints.
result The proposed methods provide valid differentially private confidence intervals for the median.
We study the sample-based k-median clustering objective under a sequential setting without substitutions. In this setting, an i.i.d. sequence of examples is observed. An example can be selected as a center only immediately after it is observed, and it cannot be substituted later. The goal is to select a set of centers …
Develops privacy-preserving multivariate median estimation methods.
problem Lack of rigorous privacy guarantees for robust multivariate location estimation.
method Novel finite-sample performance guarantees for differentially private multivariate depth-based medians.
result Sharp performance guarantees for multivariate depth-based medians under differential privacy.
We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of CAT(0) cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper median spaces of infinite rank. Our methods also work for general median algebra…
Paper introduces MoM-KDE for robust density estimation robust to anomalous data.
problem Density estimation robustness to anomalous data.
method Combines Kernel Density Estimation and Median-of-Means principle.
result Achieves competitive results with lower computational complexity compared to other robust estimators.
New concept of coarse medians for higher rank symmetric spaces.
problem Understanding medians in higher rank symmetric spaces.
method Introducing coarse r-median spaces and proving their existence. result Existence of coarse higher medians on divisible and quasi-homogeneous convex domains.
Unique median structures found in hyperbolic spaces.
problem Uniqueness of median structures in hyperbolic spaces.
method Analyzing product of hyperbolic spaces and properties of relative hyperbolicity.
result Non-hyperbolic pants graphs can have unique median structures.
Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.
problem Understanding median algebra structures on Euclidean spaces and manifolds.
method Showed local CAT(0) cubulation for median structures on ER homology manifolds.
result Median structures on ER homology manifolds have a local CAT(0) cubulation structure.
K-bMOM robustly clusters data with outliers, improving on Lloyd-type methods.
problem Outliers in datasets disrupt traditional clustering algorithms.
method Lloyd-type iterations with robust median-of-means estimates.
result K-bMOM outperforms existing robust K-means methods.
We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theor…
We show that uniform lattices of isometries of products of real hyperbolic spaces act properly discontinuously and cocompactly on a median space. For lattices in products of at least two factors, this is the strongest degree of compatibility possible with the median geometry. Our theorem is also relevant for potential …
Convex cores found for group actions on median spaces.
problem Understanding group actions on median spaces without metric or topology.
method Introduced convex cores for actions on finite-rank median algebras.
result Actions on median spaces have nonempty convex cores.
MFRDE uses medians of forest estimators to robustly estimate densities in noisy data.
problem Robust density estimation in the presence of outliers.
method MFRDE uses pointwise median operation on forest density estimators fitted on subsampled datasets.
result MFRDE achieves robustness against all outliers while maintaining accuracy for density estimation.
We introduce and begin to explore the mean and median of finite sets of shapes represented as integral currents. The median can be computed efficiently in practice, and we focus most of our theoretical and computational attention on medians. We consider questions on the existence and regularity of medians. While the me…
Paper shows MoM is optimal under adversarial contamination for certain distributions.
problem Optimality of MoM under adversarial contamination.
method Upper and lower bounds for MoM's error under adversarial contamination.
result MoM is (minimax) optimal for distributions with finite variance and infinite variance with finite absolute moments.
Algorithm identifies Pareto set in bandits with contaminated feedback.
problem Identifying Pareto set in multi-objective bandits with adversarial contamination.
method Sample median-based multi-objective adaptive elimination algorithm.
result Sample complexity bound that depends on contamination probability.
Efficiently private clustering algorithms with tight approximation ratios.
problem Differentially private clustering of various types.
method Efficient algorithms achieving tight approximation ratios for clustering problems.
result Achieves approximation ratios similar to non-private algorithms with small additive errors.
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…
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.
This paper studies active learning in the context of robust statistics. Specifically, we propose a variant of the Best Arm Identification problem for \emph{contaminated bandits}, where each arm pull has probability ε of generating a sample from an arbitrary contamination distribution instead of the true und…
From a young age humans learn to use grammatical principles to hierarchically combine words into sentences. Action grammars is the parallel idea, that there is an underlying set of rules (a "grammar") that govern how we hierarchically combine actions to form new, more complex actions. We introduce the Action Grammar Re…
MCE reduces embedding instability in nonlinear dimensionality reduction.
problem Embedding instability caused by random initialization.
method Median of multiple embeddings (MCE) based on large deviation theory.
result MCE achieves consistency at an exponential rate and effectively mitigates instability.
The study finds that maximizing median returns is the only viable strategy in portfolio selection.
problem Difficulties in studying optimal portfolio strategies due to discontinuity and time inconsistency in maximizing median and quantile returns.
method Used intra-personal equilibrium approach to analyze portfolio selection under median and quantile maximization.
result Median maximization is the only viable strategy, with no investment in risky assets for other quantiles.
For massive data sets, efficient computation commonly relies on distributed algorithms that store and process subsets of the data on different machines, minimizing communication costs. Our focus is on regression and classification problems involving many features. A variety of distributed algorithms have been proposed …
In high dimensions, the mean and geometric median are nearly identical.
problem Understanding the relationship between mean and geometric median in high-dimensional spaces.
method Analytical derivation and simulation of the distance between mean and geometric median.
result The distance between mean and geometric median vanishes with dimensionality in high dimensions.
New subspace prototype flag median improves clustering on noisy data.
problem Finding robust prototypes for datasets of images and videos.
method Proposes flag median and introduces FlagIRLS algorithm for its calculation.
result Flag median is robust to outliers and improves cluster purity.
Improved median of means estimator with tighter bounds.
problem Improving the efficiency and reliability of median of means estimator.
method Modification of the median of means estimator with sub-Gaussian deviation bounds.
result Achieves nearly optimal constants under minimal assumptions.