We propose a new algorithm for the problem of recovering data that adheres to multiple, heterogeneous low-dimensional structures from linear observations. Focusing on data matrices that are simultaneously row-sparse and low-rank, we propose and analyze an iteratively reweighted least squares (IRLS) algorithm that is able to leverage both structures. In particular, it optimizes a combination of non-convex surrogates for row-sparsity and rank, a balancing of which is built into the algorithm. We prove locally quadratic convergence of the iterates to a simultaneously structured data matrix in a regime of minimal sample complexity (up to constants and a logarithmic factor), which is known to be impossible for a combination of convex surrogates. In experiments, we show that the IRLS method exhibits favorable empirical convergence, identifying simultaneously row-sparse and low-rank matrices from fewer measurements than state-of-the-art methods.
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A package query returns a package - a multiset of tuples - that maximizes or minimizes a linear objective function subject to linear constraints, thereby enabling in-database decision support. Prior work has established the equivalence of package queries to Integer Linear Programs (ILPs) and developed the SketchRefine algorithm for package query processing. While this algorithm was an important first step toward supporting prescriptive analytics scalably inside a relational database, it struggles when the data size grows beyond a few hundred million tuples or when the constraints become very tight. In this paper, we present Progressive Shading, a novel algorithm for processing package queries that can scale efficiently to billions of tuples and gracefully handle tight constraints. Progressive Shading solves a sequence of optimization problems over a hierarchy of relations, each resulting from an ever-finer partitioning of the original tuples into homogeneous groups until the original relation is obtained. This strategy avoids the premature discarding of high-quality tuples that can occur with SketchRefine. Our novel partitioning scheme, Dynamic Low Variance, can handle very large relations with multiple attributes and can dynamically adapt to both concentrated and spread-out sets of attribute values, provably outperforming traditional partitioning schemes such as KD-tree. We further optimize our system by replacing our off-the-shelf optimization software with customized ILP and LP solvers, called Dual Reducer and Parallel Dual Simplex respectively, that are highly accurate and orders of magnitude faster.
Understanding and adequately assessing the difference between a true and a learnt causal graphs is crucial for causal inference under interventions. As an extension to the graph-based structural Hamming distance and structural intervention distance, we propose a novel continuous-measured metric that considers the underlying data in addition to the graph structure for its calculation of the difference between a true and a learnt causal graph. The distance is based on embedding intervention distributions over each pair of nodes as conditional mean embeddings into reproducing kernel Hilbert spaces and estimating their difference by the maximum (conditional) mean discrepancy. We show theoretical results which we validate with numerical experiments on synthetic data.
Three-dimensional (3D) reconstruction of head Computed Tomography (CT) images elucidates the intricate spatial relationships of tissue structures, thereby assisting in accurate diagnosis. Nonetheless, securing an optimal head CT scan without deviation is challenging in clinical settings, owing to poor positioning by technicians, patient's physical constraints, or CT scanner tilt angle restrictions. Manual formatting and reconstruction not only introduce subjectivity but also strain time and labor resources. To address these issues, we propose an efficient automatic head CT images 3D reconstruction method, improving accuracy and repeatability, as well as diminishing manual intervention. Our approach employs a deep learning-based object detection algorithm, identifying and evaluating orbitomeatal line landmarks to automatically reformat the images prior to reconstruction. Given the dearth of existing evaluations of object detection algorithms in the context of head CT images, we compared ten methods from both theoretical and experimental perspectives. By exploring their precision, efficiency, and robustness, we singled out the lightweight YOLOv8 as the aptest algorithm for our task, with an mAP of 92.91% and impressive robustness against class imbalance. Our qualitative evaluation of standardized reconstruction results demonstrates the clinical practicability and validity of our method.
Persistent homology is a methodology central to topological data analysis that extracts and summarizes the topological features within a dataset as a persistence diagram; it has recently gained much popularity from its myriad successful applications to many domains. However, its algebraic construction induces a metric space of persistence diagrams with a highly complex geometry. In this paper, we prove convergence of the $k$-means clustering algorithm on persistence diagram space and establish theoretical properties of the solution to the optimization problem in the Karush--Kuhn--Tucker framework. Additionally, we perform numerical experiments on various representations of persistent homology, including embeddings of persistence diagrams as well as diagrams themselves and their generalizations as persistence measures; we find that clustering performance directly on persistence diagrams and measures outperform their vectorized representations.
This note addresses the question of optimally estimating a linear functional of an object acquired through linear observations corrupted by random noise, where optimality pertains to a worst-case setting tied to a symmetric, convex, and closed model set containing the object. It complements the article "Statistical Estimation and Optimal Recovery" published in the Annals of Statistics in 1994. There, Donoho showed (among other things) that, for Gaussian noise, linear maps provide near-optimal estimation schemes relatively to a performance measure relevant in Statistical Estimation. Here, we advocate for a different performance measure arguably more relevant in Optimal Recovery. We show that, relatively to this new measure, linear maps still provide near-optimal estimation schemes even if the noise is merely log-concave. Our arguments, which make a connection to the deterministic noise situation and bypass properties specific to the Gaussian case, offer an alternative to parts of Donoho's proof.
The crossed random-effects model is widely used in applied statistics, finding applications in various fields such as longitudinal studies, e-commerce, and recommender systems, among others. However, these models encounter scalability challenges, as the computational time grows disproportionately with the number of data points, typically following a cubic root relationship $(N^{(3/2)}$ or worse) with $N$. Our inspiration for addressing this issue comes from observing the recommender system employed by an online clothing retailer. Our dataset comprises over 700,000 clients, 5,000 items, and 5,000,000 measurements. When applying the maximum likelihood approach to fit crossed random effects, computational inefficiency becomes a significant concern, limiting the applicability of this approach in large-scale settings. To tackle the scalability issues, previous research by Ghosh et al. (2022a) and Ghosh et al. (2022b) has explored linear and logistic regression models utilizing fixed-effect features based on client and item variables, while incorporating random intercept terms for clients and items. In this study, we present a more generalized version of the problem, allowing random effect sizes/slopes. This extension enables us to capture the variability in effect size among both clients and items. Importantly, we have developed a scalable solution to address the aforementioned problem and have empirically demonstrated the consistency of our estimates. Specifically, as the number of data points increases, our estimates converge towards the true parameters. To validate our approach, we implement the proposed algorithm using Stitch Fix data.
Neural point estimators are neural networks that map data to parameter point estimates. They are fast, likelihood free and, due to their amortised nature, amenable to fast bootstrap-based uncertainty quantification. In this paper, we aim to increase the awareness of statisticians to this relatively new inferential tool, and to facilitate its adoption by providing user-friendly open-source software. We also give attention to the ubiquitous problem of making inference from replicated data, which we address in the neural setting using permutation-invariant neural networks. Through extensive simulation studies we show that these neural point estimators can quickly and optimally (in a Bayes sense) estimate parameters in weakly-identified and highly-parameterised models with relative ease. We demonstrate their applicability through an analysis of extreme sea-surface temperature in the Red Sea where, after training, we obtain parameter estimates and bootstrap-based confidence intervals from hundreds of spatial fields in a fraction of a second.
The vertex cover problem is a fundamental and widely studied combinatorial optimization problem. It is known that its standard linear programming relaxation is integral for bipartite graphs and half-integral for general graphs. As a consequence, the natural rounding algorithm based on this relaxation computes an optimal solution for bipartite graphs and a $2$-approximation for general graphs. This raises the question of whether one can interpolate the rounding curve of the standard linear programming relaxation in a beyond the worst-case manner, depending on how close the graph is to being bipartite. In this paper, we consider a simple rounding algorithm that exploits the knowledge of an induced bipartite subgraph to attain improved approximation ratios. Equivalently, we suppose that we work with a pair $(G, S)$, consisting of a graph with an odd cycle transversal. If $S$ is a stable set, we prove a tight approximation ratio of $1 + 1/\rho$, where $2\rho -1$ denotes the odd girth (i.e., length of the shortest odd cycle) of the contracted graph $\tilde{G} := G /S$ and satisfies $\rho \in [2,\infty]$. If $S$ is an arbitrary set, we prove a tight approximation ratio of $\left(1+1/\rho \right) (1 - \alpha) + 2 \alpha$, where $\alpha \in [0,1]$ is a natural parameter measuring the quality of the set $S$. The technique used to prove tight improved approximation ratios relies on a structural analysis of the contracted graph $\tilde{G}$. Tightness is shown by constructing classes of weight functions matching the obtained upper bounds. As a byproduct of the structural analysis, we obtain improved tight bounds on the integrality gap and the fractional chromatic number of 3-colorable graphs. We also discuss algorithmic applications in order to find good odd cycle transversals and show optimality of the analysis.
We study the reverse shortest path problem on disk graphs in the plane. In this problem we consider the proximity graph of a set of $n$ disks in the plane of arbitrary radii: In this graph two disks are connected if the distance between them is at most some threshold parameter $r$. The case of intersection graphs is a special case with $r=0$. We give an algorithm that, given a target length $k$, computes the smallest value of $r$ for which there is a path of length at most $k$ between some given pair of disks in the proximity graph. Our algorithm runs in $O^*(n^{5/4})$ randomized expected time, which improves to $O^*(n^{6/5})$ for unit disk graphs, where all the disks have the same radius. Our technique is robust and can be applied to many variants of the problem. One significant variant is the case of weighted proximity graphs, where edges are assigned real weights equal to the distance between the disks or between their centers, and $k$ is replaced by a target weight $w$; that is, we seek a path whose length is at most $w$. In other variants, we want to optimize a parameter different from $r$, such as a scale factor of the radii of the disks. The main technique for the decision version of the problem (determining whether the graph with a given $r$ has the desired property) is based on efficient implementations of BFS (for the unweighted case) and of Dijkstra's algorithm (for the weighted case), using efficient data structures for maintaining the bichromatic closest pair for certain bicliques and several distance functions. The optimization problem is then solved by combining the resulting decision procedure with enhanced variants of the interval shrinking and bifurcation technique of [4].
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
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