The matrix sensing problem is an important low-rank optimization problem that has found a wide range of applications, such as matrix completion, phase synchornization/retrieval, robust PCA, and power system state estimation. In this work, we focus on the general matrix sensing problem with linear measurements that are corrupted by random noise. We investigate the scenario where the search rank $r$ is equal to the true rank $r^*$ of the unknown ground truth (the exact parametrized case), as well as the scenario where $r$ is greater than $r^*$ (the overparametrized case). We quantify the role of the restricted isometry property (RIP) in shaping the landscape of the non-convex factorized formulation and assisting with the success of local search algorithms. First, we develop a global guarantee on the maximum distance between an arbitrary local minimizer of the non-convex problem and the ground truth under the assumption that the RIP constant is smaller than $1/(1+\sqrt{r^*/r})$. We then present a local guarantee for problems with an arbitrary RIP constant, which states that any local minimizer is either considerably close to the ground truth or far away from it. More importantly, we prove that this noisy, overparametrized problem exhibits the strict saddle property, which leads to the global convergence of perturbed gradient descent algorithm in polynomial time. The results of this work provide a comprehensive understanding of the geometric landscape of the matrix sensing problem in the noisy and overparametrized regime.
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Cholesky factorization is a widely used method for solving linear systems involving symmetric, positive-definite matrices, and can be an attractive choice in applications where a high degree of numerical stability is needed. One such application is numerical optimization, where direct methods for solving linear systems are widely used and often a significant performance bottleneck. An example where this is the case, and the specific type of optimization problem motivating this work, is radiation therapy treatment planning, where numerical optimization is used to create individual treatment plans for patients. To address this bottleneck, we propose a task-based multi-threaded method for Cholesky factorization of banded matrices with medium-sized bands. We implement our algorithm using OpenMP tasks and compare our performance with state-of-the-art libraries such as Intel MKL. Our performance measurements show a performance that is on par or better than Intel MKL (up to ~26%) for a wide range of matrix bandwidths on two different Intel CPU systems.
We study how to reduce the reconfiguration time in hybrid optical-electrical Datacenter Networks (DCNs). With a layer of Optical Circuit Switches (OCSes), hybrid optical-electrical DCNs can reconfigure its logical topologies to better match the on-going traffic patterns, but the reconfiguration time may directly affect the benefits of reconfigurability. The reconfiguration time consists of the topology solver running time and the network convergence time after triggering reconfiguration. However, existing topology solvers either incur high algorithmic complexity or fail to minimize the reconfiguration overhead. In this paper, we propose a novel algorithm that combines the ideas of bipartition and Minimum Cost Flow (MCF) to reduce the overall reconfiguration time. For the first time, we formulate the topology solving problem as a MCF problem with piecewise-linear cost, which strikes a better balance between solver complexity and solution optimality. Our evaluation shows that our algorithm can significantly reduce the network convergence time while consuming less topology solver running time, making its overall performance superior to existing algorithms. Our code and test cases are available at a public repository.
In this paper, we extend the Discrete Empirical Interpolation Method (DEIM) to the third-order tensor case based on the t-product and use it to select important/ significant lateral and horizontal slices/features. The proposed Tubal DEIM (TDEIM) is investigated both theoretically and numerically. The experimental results show that the TDEIM can provide more accurate approximations than the existing methods. An application of the proposed method to the supervised classification task is also presented.
Calibrating the extrinsic parameters of sensory devices is crucial for fusing multi-modal data. Recently, event cameras have emerged as a promising type of neuromorphic sensors, with many potential applications in fields such as mobile robotics and autonomous driving. When combined with LiDAR, they can provide more comprehensive information about the surrounding environment. Nonetheless, due to the distinctive representation of event cameras compared to traditional frame-based cameras, calibrating them with LiDAR presents a significant challenge. In this paper, we propose a novel method to calibrate the extrinsic parameters between a dyad of an event camera and a LiDAR without the need for a calibration board or other equipment. Our approach takes advantage of the fact that when an event camera is in motion, changes in reflectivity and geometric edges in the environment trigger numerous events, which can also be captured by LiDAR. Our proposed method leverages the edges extracted from events and point clouds and correlates them to estimate extrinsic parameters. Experimental results demonstrate that our proposed method is highly robust and effective in various scenes.
Most existing studies on linear bandits focus on the one-dimensional characterization of the overall system. While being representative, this formulation may fail to model applications with high-dimensional but favorable structures, such as the low-rank tensor representation for recommender systems. To address this limitation, this work studies a general tensor bandits model, where actions and system parameters are represented by tensors as opposed to vectors, and we particularly focus on the case that the unknown system tensor is low-rank. A novel bandit algorithm, coined TOFU (Tensor Optimism in the Face of Uncertainty), is developed. TOFU first leverages flexible tensor regression techniques to estimate low-dimensional subspaces associated with the system tensor. These estimates are then utilized to convert the original problem to a new one with norm constraints on its system parameters. Lastly, a norm-constrained bandit subroutine is adopted by TOFU, which utilizes these constraints to avoid exploring the entire high-dimensional parameter space. Theoretical analyses show that TOFU improves the best-known regret upper bound by a multiplicative factor that grows exponentially in the system order. A novel performance lower bound is also established, which further corroborates the efficiency of TOFU.
In modern computer experiment applications, one often encounters the situation where various models of a physical system are considered, each implemented as a simulator on a computer. An important question in such a setting is determining the best simulator, or the best combination of simulators, to use for prediction and inference. Bayesian model averaging (BMA) and stacking are two statistical approaches used to account for model uncertainty by aggregating a set of predictions through a simple linear combination or weighted average. Bayesian model mixing (BMM) extends these ideas to capture the localized behavior of each simulator by defining input-dependent weights. One possibility is to define the relationship between inputs and the weight functions using a flexible non-parametric model that learns the local strengths and weaknesses of each simulator. This paper proposes a BMM model based on Bayesian Additive Regression Trees (BART). The proposed methodology is applied to combine predictions from Effective Field Theories (EFTs) associated with a motivating nuclear physics application.
We study the connections between sorting and the binary search tree (BST) model, with an aim towards showing that the fields are connected more deeply than is currently appreciated. While any BST can be used to sort by inserting the keys one-by-one, this is a very limited relationship and importantly says nothing about parallel sorting. We show what we believe to be the first formal relationship between the BST model and sorting. Namely, we show that a large class of sorting algorithms, which includes mergesort, quicksort, insertion sort, and almost every instance-optimal sorting algorithm, are equivalent in cost to offline BST algorithms. Our main theoretical tool is the geometric interpretation of the BST model introduced by Demaine et al., which finds an equivalence between searches on a BST and point sets in the plane satisfying a certain property. To give an example of the utility of our approach, we introduce the log-interleave bound, a measure of the information-theoretic complexity of a permutation $\pi$, which is within a $\lg \lg n$ multiplicative factor of a known lower bound in the BST model; we also devise a parallel sorting algorithm with polylogarithmic span that sorts a permutation $\pi$ using comparisons proportional to its log-interleave bound. Our aforementioned result on sorting and offline BST algorithms can be used to show existence of an offline BST algorithm whose cost is within a constant factor of the log-interleave bound of any permutation $\pi$.
This paper categorizes the parameterized complexity of the algorithmic problems Perfect Phylogeny and Triangulating Colored Graphs. We show that they are complete for the parameterized complexity class XALP using a reduction from Tree-chained Multicolor Independent Set and a proof of membership. We introduce the problem Triangulating Multicolored Graphs as a stepping stone and prove XALP-completeness for this problem as well. We also show that, assuming the Exponential Time Hypothesis, there exists no algorithm that solves any of these problems in time $f(k) n^{o(k)}$, where $n$ is the input size, $k$ the parameter, and $f$ any computable function.
We present self-supervised geometric perception (SGP), the first general framework to learn a feature descriptor for correspondence matching without any ground-truth geometric model labels (e.g., camera poses, rigid transformations). Our first contribution is to formulate geometric perception as an optimization problem that jointly optimizes the feature descriptor and the geometric models given a large corpus of visual measurements (e.g., images, point clouds). Under this optimization formulation, we show that two important streams of research in vision, namely robust model fitting and deep feature learning, correspond to optimizing one block of the unknown variables while fixing the other block. This analysis naturally leads to our second contribution -- the SGP algorithm that performs alternating minimization to solve the joint optimization. SGP iteratively executes two meta-algorithms: a teacher that performs robust model fitting given learned features to generate geometric pseudo-labels, and a student that performs deep feature learning under noisy supervision of the pseudo-labels. As a third contribution, we apply SGP to two perception problems on large-scale real datasets, namely relative camera pose estimation on MegaDepth and point cloud registration on 3DMatch. We demonstrate that SGP achieves state-of-the-art performance that is on-par or superior to the supervised oracles trained using ground-truth labels.
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
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