Spectral methods have emerged as a simple yet surprisingly effective approach for extracting information from massive, noisy and incomplete data. In a nutshell, spectral methods refer to a collection of algorithms built upon the eigenvalues (resp. singular values) and eigenvectors (resp. singular vectors) of some properly designed matrices constructed from data. A diverse array of applications have been found in machine learning, data science, and signal processing. Due to their simplicity and effectiveness, spectral methods are not only used as a stand-alone estimator, but also frequently employed to initialize other more sophisticated algorithms to improve performance. 
While the studies of spectral methods can be traced back to classical matrix perturbation theory and methods of moments, the past decade has witnessed tremendous theoretical advances in demystifying their efficacy through the lens of statistical modeling, with the aid of non-asymptotic random matrix theory. This monograph aims to present a systematic, comprehensive, yet accessible introduction to spectral methods from a modern statistical perspective, highlighting their algorithmic implications in diverse large-scale applications. In particular, our exposition gravitates around several central questions that span various applications: how to characterize the sample efficiency of spectral methods in reaching a target level of statistical accuracy, and how to assess their stability in the face of random noise, missing data, and adversarial corruptions? In addition to conventional $\ell_2$ perturbation analysis, we present a systematic $\ell_{\infty}$ and $\ell_{2,\infty}$ perturbation theory for eigenspace and singular subspaces, which has only recently become available owing to a powerful "leave-one-out" analysis framework.

Spectral Methods for Data Science: A Statistical Perspective

Reconstructing the 3D model of a physical object typically requires us to align the depth scans obtained from different camera poses into the same coordinate system. Solutions to this global alignment problem usually proceed in two steps. The first step estimates relative transformations between pairs of scans using an off-the-shelf technique. Due to limited information presented between pairs of scans, the resulting relative transformations are generally noisy. The second step then jointly optimizes the relative transformations among all input depth scans. A natural constraint used in this step is the cycle-consistency constraint, which allows us to prune incorrect relative transformations by detecting inconsistent cycles. The performance of such approaches, however, heavily relies on the quality of the input relative transformations. Instead of merely using the relative transformations as the input to perform transformation synchronization, we propose to use a neural network to learn the weights associated with each relative transformation. Our approach alternates between transformation synchronization using weighted relative transformations and predicting new weights of the input relative transformations using a neural network. We demonstrate the usefulness of this approach across a wide range of datasets.

/pdf/learning-transformation-synchronization-1d7o25kqad.pdf

Learning Transformation Synchronization

We present an entirely new geometric and probabilistic approach to synchronization of correspondences across multiple sets of objects or images. In particular, we present two algorithms: (1) Birkhoff-Riemannian L-BFGS for optimizing the relaxed version of the combinatorially intractable cycle consistency loss in a principled manner, (2) Birkhoff-Riemannian Langevin Monte Carlo for generating samples on the Birkhoff Polytope and estimating the confidence of the found solutions. To this end, we first introduce the very recently developed Riemannian geometry of the Birkhoff Polytope. Next, we introduce a new probabilistic synchronization model in the form of a Markov Random Field (MRF). Finally, based on the first order retraction operators, we formulate our problem as simulating a stochastic differential equation and devise new integrators. We show on both synthetic and real datasets that we achieve high quality multi-graph matching results with faster convergence and reliable confidence/uncertainty estimates.

/pdf/probabilistic-permutation-synchronization-using-the-28yzg72gcx.pdf

Probabilistic Permutation Synchronization Using the Riemannian Structure of the Birkhoff Polytope

Unmanned aerial vehicles (UAVs) are widely used platforms to carry data capturing sensors for various applications. The reason for this success can be found in many aspects: the high maneuverability of the UAVs, the capability of performing autonomous data acquisition, flying at different heights, and the possibility to reach almost any vantage point. The selection of appropriate viewpoints and planning the optimum trajectories of UAVs is an emerging topic that aims at increasing the automation, efficiency and reliability of the data capturing process to achieve a dataset with desired quality. On the other hand, 3D reconstruction using the data captured by UAVs is also attracting attention in research and industry. This review paper investigates a wide range of model-free and model-based algorithms for viewpoint and path planning for 3D reconstruction of large-scale objects. The analyzed approaches are limited to those that employ a single-UAV as a data capturing platform for outdoor 3D reconstruction purposes. In addition to discussing the evaluation strategies, this paper also highlights the innovations and limitations of the investigated approaches. It concludes with a critical analysis of the existing challenges and future research perspectives. 

/pdf/a-review-on-viewpoints-and-path-planning-for-uav-based-3-d-gnkr47qg.pdf

A Review on Viewpoints and Path Planning for UAV-Based 3-D Reconstruction

We introduce a new paradigm, `measure synchronization', for synchronizing graphs with measure-valued edges. We formulate this problem as maximization of the cycle-consistency in the space of probability measures over relative rotations. In particular, we aim at estimating marginal distributions of absolute orientations by synchronizing the `conditional' ones, which are defined on the Riemannian manifold of quaternions. Such graph optimization on distributions-on-manifolds enables a natural treatment of multimodal hypotheses, ambiguities and uncertainties arising in many computer vision applications such as SLAM, SfM, and object pose estimation. We first formally define the problem as a generalization of the classical rotation graph synchronization, where in our case the vertices denote probability measures over rotations. We then measure the quality of the synchronization by using Sinkhorn divergences, which reduces to other popular metrics such as Wasserstein distance or the maximum mean discrepancy as limit cases. We propose a nonparametric Riemannian particle optimization approach to solve the problem. Even though the problem is non-convex, by drawing a connection to the recently proposed sparse optimization methods, we show that the proposed algorithm converges to the global optimum in a special case of the problem under certain conditions. Our qualitative and quantitative experiments show the validity of our approach and we bring in new perspectives to the study of synchronization.

/pdf/synchronizing-probability-measures-on-rotations-via-optimal-koyu0h5g3l.pdf

Synchronizing Probability Measures on Rotations via Optimal Transport

Most existing techniques in map computation (e.g., in the form of feature or dense correspondences) assume that the underlying map between an object pair is unique. This assumption, however, easily breaks when visual objects possess self-symmetries. In this paper, we study the problem of jointly optimizing symmetry groups and pair-wise maps among a collection of symmetric objects. We introduce a lifting map representation for encoding both symmetry groups and maps between symmetry groups. Based on this representation, we introduce a computational framework for joint symmetry and map synchronization. Experimental results show that this approach outperforms state-of-the-art approaches for symmetry detection from a single object as well as joint map optimization among an object collection.

/pdf/joint-map-and-symmetry-synchronization-1tfgfvwed7.pdf

Joint Map and Symmetry Synchronization

In this paper, we introduce the problem of K-best transformation synchronization for the purpose of multiple scan matching. Given noisy pair-wise transformations computed between a subset of depth scan pairs, K-best transformation synchronization seeks to output multiple consistent relative transformations. This problem naturally arises in many geometry reconstruction applications, where the underlying object possesses self-symmetry. For approximately symmetric or even non-symmetric objects, K-best solutions offer an intermediate presentation for recovering the underlying single-best solution. We introduce a simple yet robust iterative algorithm for K-best transformation synchronization, which alternates between transformation propagation and transformation clustering. We present theoretical guarantees on the robust and exact recoveries of our algorithm. Experimental results demonstrate the advantage of our approach against state-of-the-art transformation synchronization techniques on both synthetic and real datasets.

/pdf/k-best-transformation-synchronization-3fw9h5hj34.pdf

K-Best Transformation Synchronization

Efficient 3D space sampling to represent an underlying 3D object/scene is essential for 3D vision, robotics, and beyond. A standard approach is to explicitly sample a dense collection of views and formulate it as a view selection problem, or, more generally, a set cover problem. In this paper, we introduce a novel approach that avoids dense view sampling. The key idea is to learn a view prediction network and a trainable aggregation module that takes the predicted views as input and outputs an approximation of their generic scores (e.g., surface coverage, viewing angle from surface normals). This methodology allows us to turn the set cover problem (or multi-view representation optimization) into a continuous optimization problem. We then explain how to effectively solve the induced optimization problem using continuation, i.e., aggregating a hierarchy of smoothed scoring modules. Experimental results show that our approach arrives at similar or better solutions with about 10 x speed up in running time, comparing with the standard methods.

/pdf/learning-view-selection-for-3d-scenes-415e5reupn.pdf

Learning View Selection for 3D Scenes

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Joint Map and Symmetry Synchronization

K-Best Transformation Synchronization

Learning View Selection for 3D Scenes