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Optimization under uncertainty is important in many applications, particularly to inform policy and decision making in areas such as public health. A key source of uncertainty arises from the incorporation of environmental variables as inputs into computational models or simulators. Such variables represent uncontrollable features of the optimization problem and reliable decision making must account for the uncertainty they propagate to the simulator outputs. Often, multiple, competing objectives are defined from these outputs such that the final optimal decision is a compromise between different goals. Here, we present emulation-based optimization methodology for such problems that extends expected quantile improvement (EQI) to address multi-objective optimization. Focusing on the practically important case of two objectives, we use a sequential design strategy to identify the Pareto front of optimal solutions. Uncertainty from the environmental variables is integrated out using Monte Carlo samples from the simulator. Interrogation of the expected output from the simulator is facilitated by use of (Gaussian process) emulators. The methodology is demonstrated on an optimization problem from public health involving the dispersion of anthrax spores across a spatial terrain. Environmental variables include meteorological features that impact the dispersion, and the methodology identifies the Pareto front even when there is considerable input uncertainty.

We show that the known list-decoding algorithms for univariate multiplicity and folded Reed-Solomon (FRS) codes can be made to run in nearly-linear time. This yields, to our knowledge, the first known family of codes that can be decoded in nearly linear time, even as they approach the list decoding capacity. Univariate multiplicity codes and FRS codes are natural variants of Reed-Solomon codes that were discovered and studied for their applications to list-decoding. It is known that for every $\epsilon >0$, and rate $R \in (0,1)$, there exist explicit families of these codes that have rate $R$ and can be list-decoded from a $(1-R-\epsilon)$ fraction of errors with constant list size in polynomial time (Guruswami & Wang (IEEE Trans. Inform. Theory, 2013) and Kopparty, Ron-Zewi, Saraf & Wootters (SIAM J. Comput. 2023)). In this work, we present randomized algorithms that perform the above tasks in nearly linear time. Our algorithms have two main components. The first builds upon the lattice-based approach of Alekhnovich (IEEE Trans. Inf. Theory 2005), who designed a nearly linear time list-decoding algorithm for Reed-Solomon codes approaching the Johnson radius. As part of the second component, we design nearly-linear time algorithms for two natural algebraic problems. The first algorithm solves linear differential equations of the form $Q\left(x, f(x), \frac{df}{dx}, \dots,\frac{d^m f}{dx^m}\right) \equiv 0$ where $Q$ has the form $Q(x,y_0,\dots,y_m) = \tilde{Q}(x) + \sum_{i = 0}^m Q_i(x)\cdot y_i$. The second solves functional equations of the form $Q\left(x, f(x), f(\gamma x), \dots,f(\gamma^m x)\right) \equiv 0$ where $\gamma$ is a high-order field element. These algorithms can be viewed as generalizations of classical algorithms of Sieveking (Computing 1972) and Kung (Numer. Math. 1974) for computing the modular inverse of a power series, and might be of independent interest.

Constant weight codes (CWCs) and constant composition codes (CCCs) are two important classes of codes that have been studied extensively in both combinatorics and coding theory for nearly sixty years. In this paper we show that for {\it all} fixed odd distances, there exist near-optimal CWCs and CCCs asymptotically achieving the classic Johnson-type upper bounds. Let $A_q(n,w,d)$ denote the maximum size of $q$-ary CWCs of length $n$ with constant weight $w$ and minimum distance $d$. One of our main results shows that for {\it all} fixed $q,w$ and odd $d$, one has $\lim_{n\rightarrow\infty}\frac{A_q(n,d,w)}{\binom{n}{t}}=\frac{(q-1)^t}{\binom{w}{t}}$, where $t=\frac{2w-d+1}{2}$. This implies the existence of near-optimal generalized Steiner systems originally introduced by Etzion, and can be viewed as a counterpart of a celebrated result of R\"odl on the existence of near-optimal Steiner systems. Note that prior to our work, very little is known about $A_q(n,w,d)$ for $q\ge 3$. A similar result is proved for the maximum size of CCCs. We provide different proofs for our two main results, based on two strengthenings of the well-known Frankl-R\"odl-Pippenger theorem on the existence of near-optimal matchings in hypergraphs: the first proof follows by Kahn's linear programming variation of the above theorem, and the second follows by the recent independent work of Delcour-Postle, and Glock-Joos-Kim-K\"uhn-Lichev on the existence of near-optimal matchings avoiding certain forbidden configurations. We also present several intriguing open questions for future research.

In modern computational materials science, deep learning has shown the capability to predict interatomic potentials, thereby supporting and accelerating conventional simulations. However, existing models typically sacrifice either accuracy or efficiency. Moreover, lightweight models are highly demanded for offering simulating systems on a considerably larger scale at reduced computational costs. A century ago, Felix Bloch demonstrated how leveraging the equivariance of the translation operation on a crystal lattice (with geometric symmetry) could significantly reduce the computational cost of determining wavefunctions and accurately calculate material properties. Here, we introduce a lightweight equivariant interaction graph neural network (LEIGNN) that can enable accurate and efficient interatomic potential and force predictions in crystals. Rather than relying on higher-order representations, LEIGNN employs a scalar-vector dual representation to encode equivariant features. By extracting both local and global structures from vector representations and learning geometric symmetry information, our model remains lightweight while ensuring prediction accuracy and robustness through the equivariance. Our results show that LEIGNN consistently outperforms the prediction performance of the representative baselines and achieves significant efficiency across diverse datasets, which include catalysts, molecules, and organic isomers. Finally, to further validate the predicted interatomic potentials from our model, we conduct classical molecular dynamics (MD) and ab initio MD simulation across various systems, including solid, liquid, and gas. It is found that LEIGNN can achieve the accuracy of ab initio MD and retain the computational efficiency of classical MD across all examined systems, demonstrating its accuracy, efficiency, and universality.

Among semiparametric regression models, partially linear additive models provide a useful tool to include additive nonparametric components as well as a parametric component, when explaining the relationship between the response and a set of explanatory variables. This paper concerns such models under sparsity assumptions for the covariates included in the linear component. Sparse covariates are frequent in regression problems where the task of variable selection is usually of interest. As in other settings, outliers either in the residuals or in the covariates involved in the linear component have a harmful effect. To simultaneously achieve model selection for the parametric component of the model and resistance to outliers, we combine preliminary robust estimators of the additive component, robust linear $MM-$regression estimators with a penalty such as SCAD on the coefficients in the parametric part. Under mild assumptions, consistency results and rates of convergence for the proposed estimators are derived. A Monte Carlo study is carried out to compare, under different models and contamination schemes, the performance of the robust proposal with its classical counterpart. The obtained results show the advantage of using the robust approach. Through the analysis of a real data set, we also illustrate the benefits of the proposed procedure.

The proximal gradient method is a generic technique introduced to tackle the non-smoothness in optimization problems, wherein the objective function is expressed as the sum of a differentiable convex part and a non-differentiable regularization term. Such problems with tensor format are of interest in many fields of applied mathematics such as image and video processing. Our goal in this paper is to address the solution of such problems with a more general form of the regularization term. An adapted iterative proximal gradient method is introduced for this purpose. Due to the slowness of the proposed algorithm, we use new tensor extrapolation methods to enhance its convergence. Numerical experiments on color image deblurring are conducted to illustrate the efficiency of our approach.

In this paper we develop a linear expectile hidden Markov model for the analysis of cryptocurrency time series in a risk management framework. The methodology proposed allows to focus on extreme returns and describe their temporal evolution by introducing in the model time-dependent coefficients evolving according to a latent discrete homogeneous Markov chain. As it is often used in the expectile literature, estimation of the model parameters is based on the asymmetric normal distribution. Maximum likelihood estimates are obtained via an Expectation-Maximization algorithm using efficient M-step update formulas for all parameters. We evaluate the introduced method with both artificial data under several experimental settings and real data investigating the relationship between daily Bitcoin returns and major world market indices.

In this paper, we tackle structure learning of Directed Acyclic Graphs (DAGs), with the idea of exploiting available prior knowledge of the domain at hand to guide the search of the best structure. In particular, we assume to know the topological ordering of variables in addition to the given data. We study a new algorithm for learning the structure of DAGs, proving its theoretical consistence in the limit of infinite observations. Furthermore, we experimentally compare the proposed algorithm to a number of popular competitors, in order to study its behavior in finite samples.

Alon and Shapira proved that every monotone class (closed under taking subgraphs) of undirected graphs is strongly testable, that is, under the promise that a given graph is either in the class or $\varepsilon$-far from it, there is a test using a constant number of samples (depending on $\varepsilon$ only) that rejects every graph not in the class with probability at least one half, and always accepts a graph in the class. However, their bound on the number of samples is quite large, since they heavily rely on Szemer\'edi's regularity lemma. We study the case of posets and show that every monotone class of posets is easily testable, that is, a polynomial number of samples is sufficient. We achieve this via proving a polynomial removal lemma for posets. We give a simple classification: for every monotone class of posets there is an $h$ such that the class is indistinguishable (every large enough poset in one class is $\varepsilon$-close to a poset in the other class) from the class of posets free of the chain $C_h$. This allows to test every monotone class of posets using $O(\varepsilon^{-1})$ samples. The test has two-sided error, but it is almost complete: the probability to refute a poset in the class is polynomially small in the size of the poset. The analogous results hold for comparability graphs, too.

Continuous queries over data streams may suffer from blocking operations and/or unbound wait, which may delay answers until some relevant input arrives through the data stream. These delays may turn answers, when they arrive, obsolete to users who sometimes have to make decisions with no help whatsoever. Therefore, it can be useful to provide hypothetical answers - "given the current information, it is possible that X will become true at time t" - instead of no information at all. In this paper we present a semantics for queries and corresponding answers that covers such hypothetical answers, together with an online algorithm for updating the set of facts that are consistent with the currently available information.