Ery Arias-Castro

Ery Arias-Castro, Elizabeth Coda

In this paper, we take an axiomatic approach to defining a population hierarchical clustering for piecewise constant densities, and in a similar manner to Lebesgue integration, extend this definition to more general densities. When the density satisfies some mild conditions, e.g., when it has connected support, is continuous, and vanishes only at infinity, or when the connected components of the density satisfy these conditions, our axiomatic definition results in Hartigan's definition of cluster tree.

Ery Arias-Castro, Wanli Qiao

We adapt concepts, methodology, and theory originally developed in the areas of multidimensional scaling and dimensionality reduction for multivariate data to the functional setting. We focus on classical scaling and Isomap -- prototypical methods that have played important roles in these area -- and showcase their use in the context of functional data analysis. In the process, we highlight the crucial role that the ambient metric plays.

Siddharth Vishwanath, Ery Arias-Castro

We establish the consistency of classical scaling under a broad class of noise models, encompassing many commonly studied cases in literature. Our approach requires only finite fourth moments of the noise, significantly weakening standard assumptions. We derive convergence rates for classical scaling and establish matching minimax lower bounds, demonstrating that classical scaling achieves minimax optimality in recovering the true configuration even when the input dissimilarities are corrupted by noise.

Siddharth Vishwanath, Ery Arias-Castro

We develop a formal statistical framework for classical multidimensional scaling (CMDS) applied to noisy dissimilarity data. We establish distributional convergence results for the embeddings produced by CMDS for various noise models, which enable the construction of \emph{bona~fide} uniform confidence sets for the latent configuration, up to rigid transformations. We further propose bootstrap procedures for constructing these confidence sets and provide theoretical guarantees for their validity. We find that the multiplier bootstrap adapts automatically to heteroscedastic noise such as multiplicative noise, while the empirical bootstrap seems to require homoscedasticity. Either form of bootstrap, when valid, is shown to substantially improve finite-sample accuracy. The empirical performance of the proposed methods is demonstrated through numerical experiments.

Elizabeth Coda, Ery Arias-Castro, Gal Mishne

Dimensionality reduction methods such as t-SNE and UMAP are popular methods for visualizing data with a potential (latent) clustered structure. They are known to group data points at the same time as they embed them, resulting in visualizations with well-separated clusters that preserve local information well. However, t-SNE and UMAP also tend to distort the global geometry of the underlying data. We propose a more transparent, modular approach consisting of first clustering the data, then embedding each cluster, and finally aligning the clusters to obtain a global embedding. We demonstrate this approach on several synthetic and real-world datasets and show that it is competitive with existing methods, while being much more transparent.

Ery Arias-Castro, Elizabeth Coda, Wanli Qiao

We present a method for graph clustering that is analogous with gradient ascent methods previously proposed for clustering points in space. We show that, when applied to a random geometric graph with data iid from some density with Morse regularity, the method is asymptotically consistent. Here, consistency is understood with respect to a density-level clustering defined by the partition of the support of the density induced by the basins of attraction of the density modes.

Ery Arias-Castro, Wanli Qiao

We consider several hill-climbing approaches to clustering as formulated by Fukunaga and Hostetler in the 1970's. We study both continuous-space and discrete-space (i.e., medoid) variants and establish their consistency.

Ery Arias-Castro, Wanli Qiao

Two important nonparametric approaches to clustering emerged in the 1970's: clustering by level sets or cluster tree as proposed by Hartigan, and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hosteler. In a recent paper, we argue the thesis that these two approaches are fundamentally the same by showing that the gradient flow provides a way to move along the cluster tree. In making a stronger case, we are confronted with the fact the cluster tree does not define a partition of the entire support of the underlying density, while the gradient flow does. In the present paper, we resolve this conundrum by proposing two ways of obtaining a partition from the cluster tree -- each one of them very natural in its own right -- and showing that both of them reduce to the partition given by the gradient flow under standard assumptions on the sampling density.

Ery Arias-Castro, Wanli Qiao

The paper establishes a strong correspondence between two important clustering approaches that emerged in the 1970's: clustering by level sets or cluster tree as proposed by Hartigan and clustering by gradient lines or gradient flow as proposed by Fukunaga and Hostetler. We do so by showing that we can move up the cluster tree by following the gradient ascent flow.

Ery Arias-Castro, Phong Alain Chau

We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction -- the tangential Delaunay complex -- for that purpose. We then turn to manifold learning and argue that a variant of Isomap where the distances are instead computed on a reconstructed surface is minimax optimal for the isometric variant of the problem.

Ery Arias-Castro, Adel Javanmard, Bruno Pelletier

One of the common tasks in unsupervised learning is dimensionality reduction, where the goal is to find meaningful low-dimensional structures hidden in high-dimensional data. Sometimes referred to as manifold learning, this problem is closely related to the problem of localization, which aims at embedding a weighted graph into a low-dimensional Euclidean space. Several methods have been proposed for localization, and also manifold learning. Nonetheless, the robustness property of most of them is little understood. In this paper, we obtain perturbation bounds for classical scaling and trilateration, which are then applied to derive performance bounds for Isomap, Landmark Isomap, and Maximum Variance Unfolding. A new perturbation bound for procrustes analysis plays a key role.

Ery Arias-Castro, Xiao Pu

Consider the problem of sparse clustering, where it is assumed that only a subset of the features are useful for clustering purposes. In the framework of the COSA method of Friedman and Meulman, subsequently improved in the form of the Sparse K-means method of Witten and Tibshirani, a natural and simpler hill-climbing approach is introduced. The new method is shown to be competitive with these two methods and others.

Ojash Neopane, Srinjoy Das, Ery Arias-Castro et al.

Restricted Boltzmann Machines and Deep Belief Networks have been successfully used in probabilistic generative model applications such as image occlusion removal, pattern completion and motion synthesis. Generative inference in such algorithms can be performed very efficiently on hardware using a Markov Chain Monte Carlo procedure called Gibbs sampling, where stochastic samples are drawn from noisy integrate and fire neurons implemented on neuromorphic substrates. Currently, no satisfactory metrics exist for evaluating the generative performance of such algorithms implemented on high-dimensional data for neuromorphic platforms. This paper demonstrates the application of nonparametric goodness-of-fit testing to both quantify the generative performance as well as provide decision-directed criteria for choosing the parameters of the neuromorphic Gibbs sampler and optimizing usage of hardware resources used during sampling.

Ery Arias-Castro, Nicolas Verzelen

We consider the problem of detecting a tight community in a sparse random network. This is formalized as testing for the existence of a dense random subgraph in a random graph. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph on $N$ vertices and with connection probability $p_0$; under the alternative, there is an unknown subgraph on $n$ vertices where the connection probability is p1 > p0. In Arias-Castro and Verzelen (2012), we focused on the asymptotically dense regime where p0 is large enough that np0>(n/N)^{o(1)}. We consider here the asymptotically sparse regime where p0 is small enough that np0<(n/N)^{c0} for some c0>0. As before, we derive information theoretic lower bounds, and also establish the performance of various tests. Compared to our previous work, the arguments for the lower bounds are based on the same technology, but are substantially more technical in the details; also, the methods we study are different: besides a variant of the scan statistic, we study other statistics such as the size of the largest connected component, the number of triangles, the eigengap of the adjacency matrix, etc. Our detection bounds are sharp, except in the Poisson regime where we were not able to fully characterize the constant arising in the bound.

Ery Arias-Castro, Nicolas Verzelen

We formalize the problem of detecting a community in a network into testing whether in a given (random) graph there is a subgraph that is unusually dense. We observe an undirected and unweighted graph on N nodes. Under the null hypothesis, the graph is a realization of an Erdös-Rényi graph with probability p0. Under the (composite) alternative, there is a subgraph of n nodes where the probability of connection is p1 > p0. We derive a detection lower bound for detecting such a subgraph in terms of N, n, p0, p1 and exhibit a test that achieves that lower bound. We do this both when p0 is known and unknown. We also consider the problem of testing in polynomial-time. As an aside, we consider the problem of detecting a clique, which is intimately related to the planted clique problem. Our focus in this paper is in the quasi-normal regime where n p0 is either bounded away from zero, or tends to zero slowly.

Ery Arias-Castro, Gilad Lerman, Teng Zhang

We propose a spectral clustering method based on local principal components analysis (PCA). After performing local PCA in selected neighborhoods, the algorithm builds a nearest neighbor graph weighted according to a discrepancy between the principal subspaces in the neighborhoods, and then applies spectral clustering. As opposed to standard spectral methods based solely on pairwise distances between points, our algorithm is able to resolve intersections. We establish theoretical guarantees for simpler variants within a prototypical mathematical framework for multi-manifold clustering, and evaluate our algorithm on various simulated data sets.

Ery Arias-Castro, Bruno Pelletier

Maximum Variance Unfolding is one of the main methods for (nonlinear) dimensionality reduction. We study its large sample limit, providing specific rates of convergence under standard assumptions. We find that it is consistent when the underlying submanifold is isometric to a convex subset, and we provide some simple examples where it fails to be consistent.

Joseph Salmon, Rebecca Willett, Ery Arias-Castro

This paper describes a simple image noise removal method which combines a preprocessing step with the Yaroslavsky filter for strong numerical, visual, and theoretical performance on a broad class of images. The framework developed is a two-stage approach. In the first stage the image is filtered with a classical denoising method (e.g., wavelet or curvelet thresholding). In the second stage a modification of the Yaroslavsky filter is performed on the original noisy image, where the weights of the filters are governed by pixel similarities in the denoised image from the first stage. Similar prefiltering ideas have proved effective previously in the literature, and this paper provides theoretical guarantees and important insight into why prefiltering can be effective. Empirically, this simple approach achieves very good performance for cartoon images, and can be computed much more quickly than current patch-based denoising algorithms.