arXiv research

Improved private geometric median estimation with nearly-linear time complexity.

This paper introduces online algorithms to estimate robust geometric median in large data streams.

In high dimensions, the mean and geometric median are nearly identical.

Paper proposes a robust method for federated ICA with geometric median aggregation.

Paper shows how to use geometric median for robust SGD in high dimensions.

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 …

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 …

The study finds that maximizing median returns is the only viable strategy in portfolio selection.

Improves clustering interpretability with decision trees.

Develops privacy-preserving multivariate median estimation methods.

We develop a coreset for robust geometric median, reducing size dependency on outliers.

This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the only risk measures that satisfy a set of economic axioms for the Choquet expected …

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…

Paper addresses robust federated linear bandits against Byzantine attacks.

In this paper, we define the geometric median of a probability measure on a Riemannian manifold, give its characterization and a natural condition to ensure its uniqueness. In order to calculate the median in practical cases, we also propose a subgradient algorithm and prove its convergence as well as estimating the er…

Stochastic gradient descent's long-term fluctuations are described by a diffusion limit.

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 …

Boosting framework for vector-valued prediction with geometric stability.

This work achieves exponential concentration in heavy-tailed data over CAT(κ) spaces using the Fréchet median.

A new method for fast and robust sparsity learning over networks.

This study presents a long-term alternative formula for stock price variation described by a geometric Brownian motion on the basis of median instead of mean or expected values. The proposed method is motivated by the observation made in remote fields, where optimality of bet-hedging or diversification strategies is ex…

Byrd-SAGA reduces variance to robustify SGD against Byzantine attacks.

Algorithm identifies Pareto set in bandits with contaminated feedback.

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ 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…

This paper provides new algorithms for distributed clustering for two popular center-based objectives, k-median and k-means. These algorithms have provable guarantees and improve communication complexity over existing approaches. Following a classic approach in clustering by \cite{har2004coresets}, we reduce the proble…

MCE reduces embedding instability in nonlinear dimensionality reduction.

New concept of coarse medians for higher rank symmetric spaces.

Unique median structures found in hyperbolic spaces.

Study on median algebra structures on Euclidean spaces and manifolds with local CAT(0) cubulation.

Robust score matching improves parameter estimation in contaminated data.

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…

Distributed model training is vulnerable to byzantine system failures and adversarial compute nodes, i.e., nodes that use malicious updates to corrupt the global model stored at a parameter server (PS). To guarantee some form of robustness, recent work suggests using variants of the geometric median as an aggregation r…

AsylADMM improves gossip-based learning for non-smooth objectives.

Convex cores found for group actions on median spaces.

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…

Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…

Novel algorithm resists Byzantine attacks in federated learning for PCA and LRCS.

Proposes a robust clustering method using the Median-of-Means estimator.

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…

In a recent work, [19] studied the following "fair" variants of classical clustering problems such as $k$-means and $k$-median: given a set of $n$ data points in $\mathbb{R}^d$ and a binary type associated to each data point, the goal is to cluster the points while ensuring that the proportion of each type in each clus…

A method for estimating the median of gradients in stochastic optimization.

New graph properties inherited by Frechet mean and median.

Tukey median performance analyzed under TV corruptions.

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…

New subspace prototype flag median improves clustering on noisy data.

Improved median of means estimator with tighter bounds.

Empirical median performs well in estimating location with varying scales.

This article is devoted to the problem of predicting the value taken by a random permutation $Σ$, describing the preferences of an individual over a set of numbered items $\{1,\; \ldots,\; n\}$ say, based on the observation of an input/explanatory r.v. $X$ e.g. characteristics of the individual), when error is measured…