Marina Meila

Nikolaos Evangelou, Felix Dietrich, Eliodoro Chiavazzo et al.

We introduce a data-driven approach to building reduced dynamical models through manifold learning; the reduced latent space is discovered using Diffusion Maps (a manifold learning technique) on time series data. A second round of Diffusion Maps on those latent coordinates allows the approximation of the reduced dynamical models. This second round enables mapping the latent space coordinates back to the full ambient space (what is called lifting); it also enables the approximation of full state functions of interest in terms of the reduced coordinates. In our work, we develop and test three different reduced numerical simulation methodologies, either through pre-tabulation in the latent space and integration on the fly or by going back and forth between the ambient space and the latent space. The data-driven latent space simulation results, based on the three different approaches, are validated through (a) the latent space observation of the full simulation through the Nyström Extension formula, or through (b) lifting the reduced trajectory back to the full ambient space, via Latent Harmonics. Latent space modeling often involves additional regularization to favor certain properties of the space over others, and the mapping back to the ambient space is then constructed mostly independently from these properties; here, we use the same data-driven approach to construct the latent space and then map back to the ambient space.

Hanyu Zhang, Samson Koelle, Marina Meila

We propose a paradigm for interpretable Manifold Learning for scientific data analysis, whereby we parametrize a manifold with $d$ smooth functions from a scientist-provided dictionary of meaningful, domain-related functions. When such a parametrization exists, we provide an algorithm for finding it based on sparse non-linear regression in the manifold tangent bundle, bypassing more standard manifold learning algorithms. We also discuss conditions for the existence of such parameterizations in function space and for successful recovery from finite samples. We demonstrate our method with experimental results from a real scientific domain.

Hanyu Zhang, Marina Meila

We quantify the parameter stability of a spherical Gaussian Mixture Model (sGMM) under small perturbations in distribution space. Namely, we derive the first explicit bound to show that for a mixture of spherical Gaussian $P$ (sGMM) in a pre-defined model class, all other sGMM close to $P$ in this model class in total variation distance has a small parameter distance to $P$. Further, this upper bound only depends on $P$. The motivation for this work lies in providing guarantees for fitting Gaussian mixtures; with this aim in mind, all the constants involved are well defined and distribution free conditions for fitting mixtures of spherical Gaussians. Our results tighten considerably the existing computable bounds, and asymptotically match the known sharp thresholds for this problem.

Luke Evans, Octavian-Vlad Murad, Lars Dingeldein et al.

Simulation-based inference provides a powerful framework for cryo-electron microscopy, employing neural networks in methods like CryoSBI to infer biomolecular conformations via learned latent representations. This latent space represents a rich opportunity, encoding valuable information about the physical system and the inference process. Harnessing this potential hinges on understanding the underlying geometric structure of these representations. We investigate this structure by applying manifold learning techniques to CryoSBI representations of hemagglutinin (simulated and experimental). We reveal that these high-dimensional data inherently populate low-dimensional, smooth manifolds, with simulated data effectively covering the experimental counterpart. By characterizing the manifold's geometry using Diffusion Maps and identifying its principal axes of variation via coordinate interpretation methods, we establish a direct link between the latent structure and key physical parameters. Discovering this intrinsic low-dimensionality and interpretable geometric organization not only validates the CryoSBI approach but enables us to learn more from the data structure and provides opportunities for improving future inference strategies by exploiting this revealed manifold geometry.

Samson Koelle, Marina Meila

Isometry pursuit is a convex algorithm for identifying orthonormal column-submatrices of wide matrices. It consists of a novel normalization method followed by multitask basis pursuit. Applied to Jacobians of putative coordinate functions, it helps identity isometric embeddings from within interpretable dictionaries. We provide theoretical and experimental results justifying this method. For problems involving coordinate selection and diversification, it offers a synergistic alternative to greedy and brute force search.

Yali Wan, Marina Meila

Finding communities in networks is a problem that remains difficult, in spite of the amount of attention it has recently received. The Stochastic Block-Model (SBM) is a generative model for graphs with "communities" for which, because of its simplicity, the theoretical understanding has advanced fast in recent years. In particular, there have been various results showing that simple versions of spectral clustering using the Normalized Laplacian of the graph can recover the communities almost perfectly with high probability. Here we show that essentially the same algorithm used for the SBM and for its extension called Degree-Corrected SBM, works on a wider class of Block-Models, which we call Preference Frame Models, with essentially the same guarantees. Moreover, the parametrization we introduce clearly exhibits the free parameters needed to specify this class of models, and results in bounds that expose with more clarity the parameters that control the recovery error in this model class.

Marina Meila

Meila (2018) introduces an optimization based method called the Sublevel Set method, to guarantee that a clustering is nearly optimal and "approximately correct" without relying on any assumptions about the distribution that generated the data. This paper extends the Sublevel Set method to the cost-based hierarchical clustering paradigm proposed by Dasgupta (2016).

Yali Wan, Marina Meila

In this paper, we propose a perturbation framework to measure the robustness of graph properties. Although there are already perturbation methods proposed to tackle this problem, they are limited by the fact that the strength of the perturbation cannot be well controlled. We firstly provide a perturbation framework on graphs by introducing weights on the nodes, of which the magnitude of perturbation can be easily controlled through the variance of the weights. Meanwhile, the topology of the graphs are also preserved to avoid uncontrollable strength in the perturbation. We then extend the measure of robustness in the robust statistics literature to the graph properties.

Samson Koelle, Hanyu Zhang, Marina Meila et al.

Manifold embedding algorithms map high-dimensional data down to coordinates in a much lower-dimensional space. One of the aims of dimension reduction is to find intrinsic coordinates that describe the data manifold. The coordinates returned by the embedding algorithm are abstract, and finding their physical or domain-related meaning is not formalized and often left to domain experts. This paper studies the problem of recovering the meaning of the new low-dimensional representation in an automatic, principled fashion. We propose a method to explain embedding coordinates of a manifold as non-linear compositions of functions from a user-defined dictionary. We show that this problem can be set up as a sparse linear Group Lasso recovery problem, find sufficient recovery conditions, and demonstrate its effectiveness on data.

James McQueen, Marina Meila, Jacob VanderPlas et al.

Manifold Learning is a class of algorithms seeking a low-dimensional non-linear representation of high-dimensional data. Thus manifold learning algorithms are, at least in theory, most applicable to high-dimensional data and sample sizes to enable accurate estimation of the manifold. Despite this, most existing manifold learning implementations are not particularly scalable. Here we present a Python package that implements a variety of manifold learning algorithms in a modular and scalable fashion, using fast approximate neighbors searches and fast sparse eigendecompositions. The package incorporates theoretical advances in manifold learning, such as the unbiased Laplacian estimator and the estimation of the embedding distortion by the Riemannian metric method. In benchmarks, even on a single-core desktop computer, our code embeds millions of data points in minutes, and takes just 200 minutes to embed the main sample of galaxy spectra from the Sloan Digital Sky Survey --- consisting of 0.6 million samples in 3750-dimensions --- a task which has not previously been possible.

Zaeem Hussain, Marina Meila

This report discusses two new indices for comparing clusterings of a set of points. The motivation for looking at new ways for comparing clusterings stems from the fact that the existing clustering indices are based on set cardinality alone and do not consider the positions of data points. The new indices, namely, the Random Walk index (RWI) and Variation of Information with Neighbors (VIN), are both inspired by the clustering metric Variation of Information (VI). VI possesses some interesting theoretical properties which are also desirable in a metric for comparing clusterings. We define our indices and discuss some of their explored properties which appear relevant for a clustering index. We also include the results of these indices on clusterings of some example data sets.

Dominique Perrault-Joncas, Marina Meila

We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set the bandwidth by optimizing the Laplacian's ability to preserve the geometry of the data. Experiments show that this principled approach is effective and robust.

Dominique Perrault-Joncas, Marina Meila

This paper considers the problem of embedding directed graphs in Euclidean space while retaining directional information. We model a directed graph as a finite set of observations from a diffusion on a manifold endowed with a vector field. This is the first generative model of its kind for directed graphs. We introduce a graph embedding algorithm that estimates all three features of this model: the low-dimensional embedding of the manifold, the data density and the vector field. In the process, we also obtain new theoretical results on the limits of "Laplacian type" matrices derived from directed graphs. The application of our method to both artificially constructed and real data highlights its strengths.

Hoyt Koepke, Marina Meila

We present an exploration of the rich theoretical connections between several classes of regularized models, network flows, and recent results in submodular function theory. This work unifies key aspects of these problems under a common theory, leading to novel methods for working with several important models of interest in statistics, machine learning and computer vision. In Part 1, we review the concepts of network flows and submodular function optimization theory foundational to our results. We then examine the connections between network flows and the minimum-norm algorithm from submodular optimization, extending and improving several current results. This leads to a concise representation of the structure of a large class of pairwise regularized models important in machine learning, statistics and computer vision. In Part 2, we describe the full regularization path of a class of penalized regression problems with dependent variables that includes the graph-guided LASSO and total variation constrained models. This description also motivates a practical algorithm. This allows us to efficiently find the regularization path of the discretized version of TV penalized models. Ultimately, our new algorithms scale up to high-dimensional problems with millions of variables.

Dominique Perraul-Joncas, Marina Meila

In recent years, manifold learning has become increasingly popular as a tool for performing non-linear dimensionality reduction. This has led to the development of numerous algorithms of varying degrees of complexity that aim to recover man ifold geometry using either local or global features of the data. Building on the Laplacian Eigenmap and Diffusionmaps framework, we propose a new paradigm that offers a guarantee, under reasonable assumptions, that any manifo ld learning algorithm will preserve the geometry of a data set. Our approach is based on augmenting the output of embedding algorithms with geometric informatio n embodied in the Riemannian metric of the manifold. We provide an algorithm for estimating the Riemannian metric from data and demonstrate possible application s of our approach in a variety of examples.

Marina Meila, David Heckerman

We examine methods for clustering in high dimensions. In the first part of the paper, we perform an experimental comparison between three batch clustering algorithms: the Expectation-Maximization (EM) algorithm, a winner take all version of the EM algorithm reminiscent of the K-means algorithm, and model-based hierarchical agglomerative clustering. We learn naive-Bayes models with a hidden root node, using high-dimensional discrete-variable data sets (both real and synthetic). We find that the EM algorithm significantly outperforms the other methods, and proceed to investigate the effect of various initialization schemes on the final solution produced by the EM algorithm. The initializations that we consider are (1) parameters sampled from an uninformative prior, (2) random perturbations of the marginal distribution of the data, and (3) the output of hierarchical agglomerative clustering. Although the methods are substantially different, they lead to learned models that are strikingly similar in quality.

Marina Meila, Tommi S. Jaakkola

In this paper we present decomposable priors, a family of priors over structure and parameters of tree belief nets for which Bayesian learning with complete observations is tractable, in the sense that the posterior is also decomposable and can be completely determined analytically in polynomial time. This follows from two main results: First, we show that factored distributions over spanning trees in a graph can be integrated in closed form. Second, we examine priors over tree parameters and show that a set of assumptions similar to (Heckerman and al. 1995) constrain the tree parameter priors to be a compactly parameterized product of Dirichlet distributions. Beside allowing for exact Bayesian learning, these results permit us to formulate a new class of tractable latent variable models in which the likelihood of a data point is computed through an ensemble average over tree structures.

Susan Shortreed, Marina Meila

In spectral clustering and spectral image segmentation, the data is partioned starting from a given matrix of pairwise similarities S. the matrix S is constructed by hand, or learned on a separate training set. In this paper we show how to achieve spectral clustering in unsupervised mode. Our algorithm starts with a set of observed pairwise features, which are possible components of an unknown, parametric similarity function. This function is learned iteratively, at the same time as the clustering of the data. The algorithm shows promosing results on synthetic and real data.

Marina Meila, Kapil Phadnis, Arthur Patterson et al.

We analyze the generalized Mallows model, a popular exponential model over rankings. Estimating the central (or consensus) ranking from data is NP-hard. We obtain the following new results: (1) We show that search methods can estimate both the central ranking pi0 and the model parameters theta exactly. The search is n! in the worst case, but is tractable when the true distribution is concentrated around its mode; (2) We show that the generalized Mallows model is jointly exponential in (pi0; theta), and introduce the conjugate prior for this model class; (3) The sufficient statistics are the pairwise marginal probabilities that item i is preferred to item j. Preliminary experiments confirm the theoretical predictions and compare the new algorithm and existing heuristics.

Marina Meila, Le Bao

This paper presents a natural extension of stagewise ranking to the the case of infinitely many items. We introduce the infinite generalized Mallows model (IGM), describe its properties and give procedures to estimate it from data. For estimation of multimodal distributions we introduce the Exponential-Blurring-Mean-Shift nonparametric clustering algorithm. The experiments highlight the properties of the new model and demonstrate that infinite models can be simple, elegant and practical.

Marina Meila, Harr Chen

We present a Dirichlet process mixture model over discrete incomplete rankings and study two Gibbs sampling inference techniques for estimating posterior clusterings. The first approach uses a slice sampling subcomponent for estimating cluster parameters. The second approach marginalizes out several cluster parameters by taking advantage of approximations to the conditional posteriors. We empirically demonstrate (1) the effectiveness of this approximation for improving convergence, (2) the benefits of the Dirichlet process model over alternative clustering techniques for ranked data, and (3) the applicability of the approach to exploring large realworld ranking datasets.