Paper proposes a novel method to improve matrix completion with median loss for large datasets.
problem Matrix completion with absolute deviation loss for large-scale data.
method Proposes a refinement step using pseudo data to improve inefficient estimators of median matrix completion.
result Turns inefficient estimators into a rate (near-)optimal matrix completion procedure.
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…
A new method for fast and robust sparsity learning over networks.
problem Efficiently learning sparse models in decentralized networks.
method Decentralized surrogate median regression (deSMR) method.
result Linear convergence rate with a simple implementation.
This work shows that Gaussian is the only prior for optimal linear estimation in L1 loss.
problem Optimal linear estimation of a random variable from noisy observations under L1 fidelity criterion. method Analyzes the conditions under which the conditional median is a linear estimator and identifies the Gaussian distribution as the only prior that induces linearity.
result Gaussian is the only prior distribution that induces linearity in the conditional median for L1 loss. 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.
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 …
Proposes a new Huber loss combining absolute and quadratic properties.
problem Improving robustness in learning models.
method Introduces a generalized Huber loss with a log-exp transform and provides an efficient minimization algorithm.
result Shows that the new loss function can be minimized efficiently.
In large-scale distributed learning, security issues have become increasingly important. Particularly in a decentralized environment, some computing units may behave abnormally, or even exhibit Byzantine failures -- arbitrary and potentially adversarial behavior. In this paper, we develop distributed learning algorithm…
New method for neural networks to predict histogram data.
problem Lack of principled approach for histogram regression.
method Pinball loss applied to cumulative histogram.
result Accuracy similar to EMD with less computational cost.
Minimizing a convex risk function is the main step in many basic learning algorithms. We study protocols for convex optimization which provably leak very little about the individual data points that constitute the loss function. Specifically, we consider differentially private algorithms that operate in the local model…
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…
Introduces robust convex clustering with Median of Means for better data clustering.
problem Challenges in convex clustering with high-dimensional data and noise/outliers.
method Integrates convex clustering with Median of Means estimator for robustness and efficiency.
result Enhanced clustering performance on large-scale datasets compared to existing methods.
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.
The paper examines bounds for stop-loss payoffs using transformed random variables.
problem Bounding stop-loss payoffs for a difference of two random variables.
method Analyzes crossing points of cdfs of original and transformed random variables.
result Unique pairwise crossing points for mortality-linked securities under symmetric copulas.
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.
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…
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…
We show how to reduce the process of predicting general order statistics (and the median in particular) to solving classification. The accompanying theoretical statement shows that the regret of the classifier bounds the regret of the quantile regression under a quantile loss. We also test this reduction empirically ag…
Paper introduces robust deep learning method for handling random data corruption.
problem Random corruption in deep learning data due to limited quality of data.
method Inspired by median-of-means and Le Cam's principle, introduces a new approach.
result Demonstrates the approach performs well in practice and is a promising alternative to standard training methods.
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.
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.
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.
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.
This paper extends Median-of-Means to new learning problems involving pairwise comparisons.
problem Learning from pairwise comparisons in machine learning.
method Segmenting data into blocks, comparing pairs of decision rules, and declaring the winner based on majority performance.
result The Median-of-Means approach maintains robustness and performance under various sampling schemes.
Tukey median performance analyzed under TV corruptions.
problem Performance analysis of Tukey median under TV corruptions.
method Analysis of Tukey median and projection algorithm under TV corruptions.
result Breakdown point reduced to 1/4 under TV corruptions, compared to 1/3 under Huber's model.
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…
Clustering with fast algorithms large samples of high dimensional data is an important challenge in computational statistics. Borrowing ideas from MacQueen (1967) who introduced a sequential version of the k-means algorithm, a new class of recursive stochastic gradient algorithms designed for the k-medians loss cri…
Paper extends quantile factor analysis with probabilistic methods for better economic policy and financial condition prediction.
problem Improving accuracy in economic and financial condition prediction.
method Probabilistic quantile factor analysis with regularization and variational approximations.
result The probabilistic estimator outperforms a recent loss-based estimator in many cases.
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.
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.
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.
Empirical median performs well in estimating location with varying scales.
problem Estimating location with varying scales in data.
method Analysis of empirical median as an estimator.
result Matching upper and lower bounds on estimation error.
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,…,n} say, based on the observation of an input/explanatory r.v. X e.g. characteristics of the individual), when error is measured…
This work studies applications and generalizations of a simple estimation technique that provides exponential concentration under heavy-tailed distributions, assuming only bounded low-order moments. We show that the technique can be used for approximate minimization of smooth and strongly convex losses, and specificall…
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.
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.
Upper bound for Hausdorff distance between hyperbolic space and its medianization.
problem Calculating the Hausdorff distance between hyperbolic space and its medianization.
method Using de Sitter space to model finite-dimensional hyperbolic space and its medianization, calculating the Hausdorff distance.
result An upper bound for the Hausdorff distance between hyperbolic space and its medianization is calculated.
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 …
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 …
Study shows Roller compactification's median graph has limited asymptotic dimension.
problem Understanding the asymptotic dimension of Roller compactifications.
method Proved using finite dimensional CAT(0) cube complexes and Borel median graph.
result Borel asymptotic dimension is bounded by the complex's dimension.
Extends graph factor system to quasi-median graphs.
problem Constraint relaxation for combinatorial HHS machinery.
method Relaxing domain constraints on combinatorial HHS machinery and extending factor system to quasi-median graphs.
result Factor system applied to quasi-median graphs.
New method for estimating median and mean with high probability privacy.
problem Estimating median and mean with differential privacy.
method Propose, Test, Release (PTR) mechanism with concentration inequalities.
result First sub-Gaussian high probability bounds for differentially private median and mean estimation.
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.
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.