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In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where the hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.

We consider the estimation of two-sample integral functionals, of the type that occur naturally, for example, when the object of interest is a divergence between unknown probability densities. Our first main result is that, in wide generality, a weighted nearest neighbour estimator is efficient, in the sense of achieving the local asymptotic minimax lower bound. Moreover, we also prove a corresponding central limit theorem, which facilitates the construction of asymptotically valid confidence intervals for the functional, having asymptotically minimal width. One interesting consequence of our results is the discovery that, for certain functionals, the worst-case performance of our estimator may improve on that of the natural `oracle' estimator, which is given access to the values of the unknown densities at the observations.

The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is the most prominent multi-objective evolutionary algorithm for real-world applications. While it performs evidently well on bi-objective optimization problems, empirical studies suggest that it is less effective when applied to problems with more than two objectives. A recent mathematical runtime analysis confirmed this observation by proving the NGSA-II for an exponential number of iterations misses a constant factor of the Pareto front of the simple 3-objective OneMinMax problem. In this work, we provide the first mathematical runtime analysis of the NSGA-III, a refinement of the NSGA-II aimed at better handling more than two objectives. We prove that the NSGA-III with sufficiently many reference points -- a small constant factor more than the size of the Pareto front, as suggested for this algorithm -- computes the complete Pareto front of the 3-objective OneMinMax benchmark in an expected number of O(n log n) iterations. This result holds for all population sizes (that are at least the size of the Pareto front). It shows a drastic advantage of the NSGA-III over the NSGA-II on this benchmark. The mathematical arguments used here and in previous work on the NSGA-II suggest that similar findings are likely for other benchmarks with three or more objectives.

Distributed graph coloring is one of the most extensively studied problems in distributed computing. There is a canonical family of distributed graph coloring algorithms known as the locally-iterative coloring algorithms, first formalized in the seminal work of [Szegedy and Vishwanathan, STOC'93]. In such algorithms, every vertex iteratively updates its own color according to a predetermined function of the current coloring of its local neighborhood. Due to the simplicity and naturalness of its framework, locally-iterative coloring algorithms are of great significance both in theory and practice. In this paper, we give a locally-iterative $(\Delta+1)$-coloring algorithm with $O(\Delta^{3/4}\log\Delta)+\log^*n$ running time. This is the first locally-iterative $(\Delta+1)$-coloring algorithm with sublinear-in-$\Delta$ running time, and answers the main open question raised in a recent breakthrough [Barenboim, Elkin, and Goldberg, JACM'21]. A key component of our algorithm is a locally-iterative procedure that transforms an $O(\Delta^2)$-coloring to a $(\Delta+O(\Delta^{3/4}\log\Delta))$-coloring in $o(\Delta)$ time. Inside this procedure we work on special proper colorings that encode (arb)defective colorings, and reduce the number of used colors quadratically in a locally-iterative fashion. As a main application of our result, we also give a self-stabilizing distributed algorithm for $(\Delta+1)$-coloring with $O(\Delta^{3/4}\log\Delta)+\log^*n$ stabilization time. To the best of our knowledge, this is the first self-stabilizing algorithm for $(\Delta+1)$-coloring with sublinear-in-$\Delta$ stabilization time.

Sparse model identification enables nonlinear dynamical system discovery from data. However, the control of false discoveries for sparse model identification is challenging, especially in the low-data and high-noise limit. In this paper, we perform a theoretical study on ensemble sparse model discovery, which shows empirical success in terms of accuracy and robustness to noise. In particular, we analyse the bootstrapping-based sequential thresholding least-squares estimator. We show that this bootstrapping-based ensembling technique can perform a provably correct variable selection procedure with an exponential convergence rate of the error rate. In addition, we show that the ensemble sparse model discovery method can perform computationally efficient uncertainty estimation, compared to expensive Bayesian uncertainty quantification methods via MCMC. We demonstrate the convergence properties and connection to uncertainty quantification in various numerical studies on synthetic sparse linear regression and sparse model discovery. The experiments on sparse linear regression support that the bootstrapping-based sequential thresholding least-squares method has better performance for sparse variable selection compared to LASSO, thresholding least-squares, and bootstrapping-based LASSO. In the sparse model discovery experiment, we show that the bootstrapping-based sequential thresholding least-squares method can provide valid uncertainty quantification, converging to a delta measure centered around the true value with increased sample sizes. Finally, we highlight the improved robustness to hyperparameter selection under shifting noise and sparsity levels of the bootstrapping-based sequential thresholding least-squares method compared to other sparse regression methods.

Peridynamic (PD) theory is significant and promising in engineering and materials science; however, it imposes challenges owing to the enormous computational cost caused by its nonlocality. Our main contribution, which overcomes the restrictions of the existing fast method, is a general computational framework for the linear bond-based peridynamic models based on the meshfree method, called the matrix-structure-based fast method (MSBFM), which is suitable for the general case, including 2D/3D problems, and static/dynamic issues, as well as problems with general boundary conditions, in particular, problems with crack propagation. Consequently, we provide a general calculation flow chart. The proposed computational framework is practical and easily embedded into the existing computational algorithm. With this framework, the computational cost is reduced from $O(N^2)$ to $O(N\log N)$, and the storage request is reduced from $O(N^2)$ to $O(N)$, where N is the degree of freedom. Finally, the vast reduction of the computational and memory requirement is verified by numerical examples.

Dynamical models described by ordinary differential equations (ODEs) are a fundamental tool in the sciences and engineering. Exact reduction aims at producing a lower-dimensional model in which each macro-variable can be directly related to the original variables, and it is thus a natural step towards the model's formal analysis and mechanistic understanding. We present an algorithm which, given a polynomial ODE model, computes a longest possible chain of exact linear reductions of the model such that each reduction refines the previous one, thus giving a user control of the level of detail preserved by the reduction. This significantly generalizes over the existing approaches which compute only the reduction of the lowest dimension subject to an approach-specific constraint. The algorithm reduces finding exact linear reductions to a question about representations of finite-dimensional algebras. We provide an implementation of the algorithm, demonstrate its performance on a set of benchmarks, and illustrate the applicability via case studies. Our implementation is freely available at //github.com/x3042/ExactODEReduction.jl

The geometric optimisation of crystal structures is a procedure widely used in Chemistry that changes the geometrical placement of the particles inside a structure. It is called structural relaxation and constitutes a local minimization problem with a non-convex objective function whose domain complexity increases along with the number of particles involved. In this work we study the performance of the two most popular first order optimisation methods, Gradient Descent and Conjugate Gradient, in structural relaxation. The respective pseudocodes can be found in Section 6. Although frequently employed, there is a lack of their study in this context from an algorithmic point of view. In order to accurately define the problem, we provide a thorough derivation of all necessary formulae related to the crystal structure energy function and the function's differentiation. We run each algorithm in combination with a constant step size, which provides a benchmark for the methods' analysis and direct comparison. We also design dynamic step size rules and study how these improve the two algorithms' performance. Our results show that there is a trade-off between convergence rate and the possibility of an experiment to succeed, hence we construct a function to assign utility to each method based on our respective preference. The function is built according to a recently introduced model of preference indication concerning algorithms with deadline and their run time. Finally, building on all our insights from the experimental results, we provide algorithmic recipes that best correspond to each of the presented preferences and select one recipe as the optimal for equally weighted preferences.

Modern health care systems are conducting continuous, automated surveillance of the electronic medical record (EMR) to identify adverse events with increasing frequency; however, many events such as sepsis do not have elucidated prodromes (i.e., event chains) that can be used to identify and intercept the adverse event early in its course. Currently, there does not exist reliable framework for discovering or describing causal chains that precede adverse hospital events. Clinically relevant and interpretable results require a framework that can (1) infer temporal interactions across multiple patient features found in EMR data (e.g., labs, vital signs, etc.) and (2) can identify patterns that precede and are specific to an impending adverse event (e.g., sepsis). In this work, we propose a linear multivariate Hawkes process model, coupled with ReLU link function, to recover a Granger Causal (GC) graph with both exciting and inhibiting effects. We develop a scalable two-phase gradient-based method to maximize a surrogate-likelihood and estimate the problem parameters, which is shown to be effective via extensive numerical simulation. Our method is subsequently extended to a data set of patients admitted to an academic level 1 trauma center located in Atalanta, GA, where the estimated GC graph identifies several highly interpretable chains that precede sepsis. Here, we demonstrate the effectiveness of our approach in learning a GC graph over Sepsis Associated Derangements (SADs), but it can be generalized to other applications with similar requirements.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.