LI Cheng-hui

Chenghui Li, Rishi Sonthalia, Nicolas Garcia Trillos

There is a large variety of machine learning methodologies that are based on the extraction of spectral geometric information from data. However, the implementations of many of these methods often depend on traditional eigensolvers, which present limitations when applied in practical online big data scenarios. To address some of these challenges, researchers have proposed different strategies for training neural networks as alternatives to traditional eigensolvers, with one such approach known as Spectral Neural Network (SNN). In this paper, we investigate key theoretical aspects of SNN. First, we present quantitative insights into the tradeoff between the number of neurons and the amount of spectral geometric information a neural network learns. Second, we initiate a theoretical exploration of the optimization landscape of SNN's objective to shed light on the training dynamics of SNN. Unlike typical studies of convergence to global solutions of NN training dynamics, SNN presents an additional complexity due to its non-convex ambient loss function.

Jelena Diakonikolas, Chenghui Li, Swati Padmanabhan et al.

Nonnegative (linear) least square problems are a fundamental class of problems that is well-studied in statistical learning and for which solvers have been implemented in many of the standard programming languages used within the machine learning community. The existing off-the-shelf solvers view the non-negativity constraint in these problems as an obstacle and, compared to unconstrained least squares, perform additional effort to address it. However, in many of the typical applications, the data itself is nonnegative as well, and we show that the nonnegativity in this case makes the problem easier. In particular, while the oracle complexity of unconstrained least squares problems necessarily scales with one of the data matrix constants (typically the spectral norm) and these problems are solved to additive error, we show that nonnegative least squares problems with nonnegative data are solvable to multiplicative error and with complexity that is independent of any matrix constants. The algorithm we introduce is accelerated and based on a primal-dual perspective. We further show how to provably obtain linear convergence using adaptive restart coupled with our method and demonstrate its effectiveness on large-scale data via numerical experiments.

Nicolás García Trillos, Chenghui Li, Raghavendra Venkatraman

We study the problem of estimating eigenpairs of elliptic differential operators from samples of a distribution $ρ$ supported on a manifold $M$. The operators discussed in the paper are relevant in unsupervised learning and in particular are obtained by taking suitable scaling limits of widely used graph Laplacians over data clouds. We study the minimax risk for this eigenpair estimation problem and explore the rates of approximation that can be achieved by commonly used graph Laplacians built from random data. More concretely, assuming that $ρ$ belongs to a certain family of distributions with controlled second derivatives, and assuming that the $d$-dimensional manifold $M$ where $ρ$ is supported has bounded geometry, we prove that the statistical minimax rate for approximating eigenvalues and eigenvectors in the $H^1(M)$-sense is $n^{-2/(d+4)}$, a rate that matches the minimax rate for a closely related density estimation problem. We then revisit the literature studying Laplacians over proximity graphs in the large data limit and prove that, under slightly stronger regularity assumptions on the data generating model, eigenpairs of graph Laplacians induce manifold agnostic estimators with an error of approximation that, up to logarithmic corrections, matches our lower bounds. Our analysis allows us to expand the existing literature on graph-based learning in at least two significant ways: 1) we consider stronger norms to measure the error of approximation than the ones that had been analyzed in the past; 2) our rates of convergence are uniform over a family of smooth distributions and do not just apply to densities with special symmetries, and, as a consequence of our lower bounds, are essentially sharp when the connectivity of the graph is sufficiently high.

Chenghui Li, Nicolás García Trillos, Housen Li et al.

Given i.i.d.\ samples $X_n =\{ x_1, \dots, x_n \}$ from a distribution supported on a low dimensional manifold ${M}$ embedded in Eucliden space, we consider the graph Laplacian operator $Δ_n$ associated to an $\varepsilon$-proximity graph over $X_n$ and study the asymptotic fluctuations of its eigenvalues around their means. In particular, letting $\hatλ_l^\varepsilon$ denote the $l$-th eigenvalue of $Δ_n$, and under suitable assumptions on the data generating model and on the rate of decay of $\varepsilon$, we prove that $\sqrt{n } (\hatλ_{l}^\varepsilon - \mathbb{E}[\hatλ_{l}^\varepsilon] )$ is asymptotically Gaussian with a variance that we can explicitly characterize. A formal argument allows us to interpret this asymptotic variance as the dissipation of a gradient flow of a suitable energy with respect to the Fisher-Rao geometry. This geometric interpretation allows us to give, in turn, a statistical interpretation of the asymptotic variance in terms of a Cramer-Rao lower bound for the estimation of the eigenvalues of certain weighted Laplace-Beltrami operator. The latter interpretation suggests a form of asymptotic statistical efficiency for the eigenvalues of the graph Laplacian. We also present CLTs for multiple eigenvalues and through several numerical experiments explore the validity of our results when some of the assumptions that we make in our theoretical analysis are relaxed.

Chenghui Li, A. Martina Neuman

Contrastive learning leverages data augmentation to develop feature representation without relying on large labeled datasets. However, despite its empirical success, the theoretical foundations of contrastive learning remain incomplete, with many essential guarantees left unaddressed, particularly the realizability assumption concerning neural approximability of an optimal spectral contrastive loss solution. In this work, we overcome these limitations by analyzing pointwise and spectral consistency of the augmentation graph Laplacian. We establish that, under specific conditions for data generation and graph connectivity, as the augmented dataset size increases, the augmentation graph Laplacian converges to a weighted Laplace-Beltrami operator on the natural data manifold. These consistency results ensure that the graph Laplacian spectrum effectively captures the manifold geometry. Consequently, they give way to a robust framework for establishing neural approximability, directly resolving the realizability assumption in a current paradigm.

Nicolas Garcia Trillos, Pengfei He, Chenghui Li

In this work we study statistical properties of graph-based algorithms for multi-manifold clustering (MMC). In MMC the goal is to retrieve the multi-manifold structure underlying a given Euclidean data set when this one is assumed to be obtained by sampling a distribution on a union of manifolds $\mathcal{M} = \mathcal{M}_1 \cup\dots \cup \mathcal{M}_N$ that may intersect with each other and that may have different dimensions. We investigate sufficient conditions that similarity graphs on data sets must satisfy in order for their corresponding graph Laplacians to capture the right geometric information to solve the MMC problem. Precisely, we provide high probability error bounds for the spectral approximation of a tensorized Laplacian on $\mathcal{M}$ with a suitable graph Laplacian built from the observations; the recovered tensorized Laplacian contains all geometric information of all the individual underlying manifolds. We provide an example of a family of similarity graphs, which we call annular proximity graphs with angle constraints, satisfying these sufficient conditions. We contrast our family of graphs with other constructions in the literature based on the alignment of tangent planes. Extensive numerical experiments expand the insights that our theory provides on the MMC problem.