Ranking from Pairwise Comparisons in General Graphs and Graphs with Locality

This technical report studies the problem of ranking from pairwise comparisons in the classical Bradley-Terry-Luce (BTL) model, with a focus on score estimation. For general graphs, we show that, with sufficiently many samples, maximum likelihood estimation (MLE) achieves an entrywise estimation error matching the Cram\'er-Rao lower bound, which can be stated in terms of effective resistances; the key to our analysis is a connection between statistical estimation and iterative optimization by preconditioned gradient descent. We are also particularly interested in graphs with locality, where only nearby items can be connected by edges; our analysis identifies conditions under which locality does not hurt, i.e. comparing the scores between a pair of items that are far apart in the graph is nearly as easy as comparing a pair of nearby items. We further explore divide-and-conquer algorithms that can provably achieve similar guarantees even in the regime with the sparsest samples, while enjoying certain computational advantages. Numerical results validate our theory and confirm the efficacy of the proposed algorithms.

A Surface-Based Federated Chow Test Model for Integrating APOE Status, Tau Deposition Measure, and Hippocampal Surface Morphometry

Background: Alzheimer's Disease (AD) is the most common type of age-related dementia, affecting 6.2 million people aged 65 or older according to CDC data. It is commonly agreed that discovering an effective AD diagnosis biomarker could have enormous public health benefits, potentially preventing or delaying up to 40% of dementia cases. Tau neurofibrillary tangles are the primary driver of downstream neurodegeneration and subsequent cognitive impairment in AD, resulting in structural deformations such as hippocampal atrophy that can be observed in magnetic resonance imaging (MRI) scans. Objective: To build a surface-based model to 1) detect differences between APOE subgroups in patterns of tau deposition and hippocampal atrophy, and 2) use the extracted surface-based features to predict cognitive decline. Methods: Using data obtained from different institutions, we develop a surface-based federated Chow test model to study the synergistic effects of APOE, a previously reported significant risk factor of AD, and tau on hippocampal surface morphometry. Results: We illustrate that the APOE-specific morphometry features correlate with AD progression and better predict future AD conversion than other MRI biomarkers. For example, a strong association between atrophy and abnormal tau was identified in hippocampal subregion cornu ammonis 1 (CA1 subfield) and subiculum in e4 homozygote cohort. Conclusion: Our model allows for identifying MRI biomarkers for AD and cognitive decline prediction and may uncover a corner of the neural mechanism of the influence of APOE and tau deposition on hippocampal morphology.

Fast Computation of Optimal Transport via Entropy-Regularized Extragradient Methods

Efficient computation of the optimal transport distance between two distributions serves as an algorithm subroutine that empowers various applications. This paper develops a scalable first-order optimization-based method that computes optimal transport to within $\varepsilon$ additive accuracy with runtime $\widetilde{O}( n^2/\varepsilon)$, where $n$ denotes the dimension of the probability distributions of interest. Our algorithm achieves the state-of-the-art computational guarantees among all first-order methods, while exhibiting favorable numerical performance compared to classical algorithms like Sinkhorn and Greenkhorn. Underlying our algorithm designs are two key elements: (a) converting the original problem into a bilinear minimax problem over probability distributions; (b) exploiting the extragradient idea -- in conjunction with entropy regularization and adaptive learning rates -- to accelerate convergence.

Learning Mixtures of Linear Dynamical Systems

We study the problem of learning a mixture of multiple linear dynamical systems (LDSs) from unlabeled short sample trajectories, each generated by one of the LDS models. Despite the wide applicability of mixture models for time-series data, learning algorithms that come with end-to-end performance guarantees are largely absent from existing literature. There are multiple sources of technical challenges, including but not limited to (1) the presence of latent variables (i.e. the unknown labels of trajectories); (2) the possibility that the sample trajectories might have lengths much smaller than the dimension $d$ of the LDS models; and (3) the complicated temporal dependence inherent to time-series data. To tackle these challenges, we develop a two-stage meta-algorithm, which is guaranteed to efficiently recover each ground-truth LDS model up to error $\tilde{O}(\sqrt{d/T})$, where $T$ is the total sample size. We validate our theoretical studies with numerical experiments, confirming the efficacy of the proposed algorithm.

Learning Mixtures of Low-Rank Models

We study the problem of learning mixtures of low-rank models, i.e. reconstructing multiple low-rank matrices from unlabelled linear measurements of each. This problem enriches two widely studied settings -- low-rank matrix sensing and mixed linear regression -- by bringing latent variables (i.e. unknown labels) and structural priors (i.e. low-rank structures) into consideration. To cope with the non-convexity issues arising from unlabelled heterogeneous data and low-complexity structure, we develop a three-stage meta-algorithm that is guaranteed to recover the unknown matrices with near-optimal sample and computational complexities under Gaussian designs. In addition, the proposed algorithm is provably stable against random noise. We complement the theoretical studies with empirical evidence that confirms the efficacy of our algorithm.

Active Orthogonal Matching Pursuit for Sparse Subspace Clustering

Sparse Subspace Clustering (SSC) is a state-of-the-art method for clustering high-dimensional data points lying in a union of low-dimensional subspaces. However, while $\ell_1$ optimization-based SSC algorithms suffer from high computational complexity, other variants of SSC, such as Orthogonal Matching Pursuit-based SSC (OMP-SSC), lose clustering accuracy in pursuit of improving time efficiency. In this letter, we propose a novel Active OMP-SSC, which improves clustering accuracy of OMP-SSC by adaptively updating data points and randomly dropping data points in the OMP process, while still enjoying the low computational complexity of greedy pursuit algorithms. We provide heuristic analysis of our approach, and explain how these two active steps achieve a better tradeoff between connectivity and separation. Numerical results on both synthetic data and real-world data validate our analyses and show the advantages of the proposed active algorithm.