arXiv research

LocalKMeans parallelizes Lloyd's algorithm for distributed data.

Clustering is a fundamental problem in statistics and machine learning. Lloyd's algorithm, proposed in 1957, is still possibly the most widely used clustering algorithm in practice due to its simplicity and empirical performance. However, there has been little theoretical investigation on the statistical and computatio…

Fair k-means algorithm ensures equitable costs for different groups.

Research simulates Lloyd's of London's specialty insurance market dynamics.

The Lloyd-Max algorithm is a classical approach to perform K-means clustering. Unfortunately, its cost becomes prohibitive as the training dataset grows large. We propose a compressive version of K-means (CKM), that estimates cluster centers from a sketch, i.e. from a drastically compressed representation of the traini…

K-means fails catastrophically in high dimensions, Hartigan's avoids it.

K-bMOM robustly clusters data with outliers, improving on Lloyd-type methods.

Paper establishes universal lower bounds and optimal rates for clustering sub-exponential mixture models.

Using a trimming approach, we investigate a k-means type method based on Bregman divergences for clustering data possibly corrupted with clutter noise. The main interest of Bregman divergences is that the standard Lloyd algorithm adapts to these distortion measures, and they are well-suited for clustering data sampled …

The paper provides convergence bounds for approximating a distribution using point clouds.

New algorithms cluster nodes in SBM graphs faster and more accurately.

We consider $K$-means clustering in networked environments (e.g., internet of things (IoT) and sensor networks) where data is inherently distributed across nodes and processing power at each node may be limited. We consider a clustering algorithm referred to as networked $K$-means, or $NK$-means, which relies only on l…

A new method for robust product Markovian quantization overcomes numerical instabilities.

We propose a novel method to accelerate Lloyd's algorithm for K-Means clustering. Unlike previous acceleration approaches that reduce computational cost per iterations or improve initialization, our approach is focused on reducing the number of iterations required for convergence. This is achieved by treating the assig…

Paper proposes efficient methods for high-order clustering in tensor block models.

A method uses Wasserstein clustering to simplify financial data analysis.

K-means -- and the celebrated Lloyd algorithm -- is more than the clustering method it was originally designed to be. It has indeed proven pivotal to help increase the speed of many machine learning and data analysis techniques such as indexing, nearest-neighbor search and prediction, data compression; its beneficial u…

Optimizes clustering in Gaussian mixtures with varying covariance matrices.

A new heuristic LM algorithm improves $k$-segmentation accuracy with less computation.

In this paper, we study the problem of learning a mixture of Gaussians with streaming data: given a stream of $N$ points in $d$ dimensions generated by an unknown mixture of $k$ spherical Gaussians, the goal is to estimate the model parameters using a single pass over the data stream. We analyze a streaming version of …

K-means fails in high dimensions with noise and few samples.

We propose a general modeling and algorithmic framework for discrete structure recovery that can be applied to a wide range of problems. Under this framework, we are able to study the recovery of clustering labels, ranks of players, signs of regression coefficients, cyclic shifts, and even group elements from a unified…

The generalization properties of Gaussian processes depend heavily on the choice of kernel, and this choice remains a dark art. We present the Neural Kernel Network (NKN), a flexible family of kernels represented by a neural network. The NKN architecture is based on the composition rules for kernels, so that each unit …

K-means clustering improved for robustness to outliers and distribution shifts.

Quantum computers can optimize foreign exchange reserves management.

New method speeds up k-means clustering for large k by improving nearest-neighbor search.

Recursive Marginal Quantization (RMQ) allows fast approximation of solutions to stochastic differential equations in one-dimension. When applied to two factor models, RMQ is inefficient due to the fact that the optimization problem is usually performed using stochastic methods, e.g., Lloyd's algorithm or Competitive Le…

New clustering method for exponential family data.

We show that $k$-means (Lloyd's algorithm) is obtained as a special case when truncated variational EM approximations are applied to Gaussian Mixture Models (GMM) with isotropic Gaussians. In contrast to the standard way to relate $k$-means and GMMs, the provided derivation shows that it is not required to consider Gau…

Energy statistics was proposed by Sz\' ekely in the 80's inspired by Newton's gravitational potential in classical mechanics and it provides a model-free hypothesis test for equality of distributions. In its original form, energy statistics was formulated in Euclidean spaces. More recently, it was generalized to metric…

Epanechnikov Mean Shift is a simple yet empirically very effective algorithm for clustering. It localizes the centroids of data clusters via estimating modes of the probability distribution that generates the data points, using the `optimal' Epanechnikov kernel density estimator. However, since the procedure involves n…

Cluster-wise linear regression (CLR), a clustering problem intertwined with regression, is to find clusters of entities such that the overall sum of squared errors from regressions performed over these clusters is minimized, where each cluster may have different variances. We generalize the CLR problem by allowing each…

We study a recently proposed large-scale distributed learning paradigm, namely Federated Learning, where the worker machines are end users' own devices. Statistical and computational challenges arise in Federated Learning particularly in the presence of heterogeneous data distribution (i.e., data points on different de…

IVF k-means algorithm improves performance on large sparse data sets.

New algorithm improves clustering and quantization using MMD.

Potential games, originally introduced in the early 1990's by Lloyd Shapley, the 2012 Nobel Laureate in Economics, and his colleague Dov Monderer, are a very important class of models in game theory. They have special properties such as the existence of Nash equilibria in pure strategies. This note introduces graphical…

One-bit clustering method for two-component sub-Gaussian mixture models

This paper proposes an original approach to cluster multi-component data sets, including an estimation of the number of clusters. From the construction of a minimal spanning tree with Prim's algorithm, and the assumption that the vertices are approximately distributed according to a Poisson distribution, the number of …

Kernelized convex clustering handles non-linear and non-convex data.

We describe $k$-MLE, a fast and efficient local search algorithm for learning finite statistical mixtures of exponential families such as Gaussian mixture models. Mixture models are traditionally learned using the expectation-maximization (EM) soft clustering technique that monotonically increases the incomplete (expec…

Finding the optimal $k$-means clustering is NP-hard in general and many heuristics have been designed for minimizing monotonically the $k$-means objective. We first show how to extend Lloyd's batched relocation heuristic and Hartigan's single-point relocation heuristic to take into account empty-cluster and single-poin…

Paper optimizes clustering for multi-layer networks and discrete mixtures.

FWC creates fair synthetic samples for machine learning tasks.

Fuzzy clustering with similarity queries improves efficiency and accuracy.

Unified framework for robust clustering under various dissimilarity measures.

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

The paper introduces Absolute Shapley Value to handle negative contributions in machine learning model training.

One iteration of standard $k$-means (i.e., Lloyd's algorithm) or standard EM for Gaussian mixture models (GMMs) scales linearly with the number of clusters $C$, data points $N$, and data dimensionality $D$. In this study, we explore whether one iteration of $k$-means or EM for GMMs can scale sublinearly with $C$ at run…