Among techniques for high-dimensional linear regression, Sorted L-One Penalized Estimation (SLOPE) generalizes the LASSO via an adaptive $l_1$ regularization that applies heavier penalties to larger coefficients in the model. To achieve such adaptivity, SLOPE requires the specification of a complex hierarchy of penalties, i.e., a monotone penalty sequence in $R^p$, in contrast to a single penalty scalar for LASSO. Tuning this sequence when $p$ is large poses a challenge, as brute force search over a grid of values is computationally prohibitive. In this work, we study the 2-level SLOPE, an important subclass of SLOPE, with only three hyperparameters. We demonstrate both empirically and analytically that 2-level SLOPE not only preserves the advantages of general SLOPE -- such as improved mean squared error and overcoming the Donoho-Tanner power limit -- but also exhibits computational benefits by reducing the penalty hyperparameter space. In particular, we prove that 2-level SLOPE admits a sharp, theoretically tight characterization of the trade-off between true positive proportion (TPP) and false discovery proportion (FDP), contrasting with general SLOPE where only upper and lower bounds are known. Empirical evaluations further underscore the effectiveness of 2-level SLOPE in settings where predictors exhibit high correlation, when the noise is large, or when the underlying signal is not sparse. Our results suggest that 2-level SLOPE offers a robust, scalable alternative to both LASSO and general SLOPE, making it particularly suited for practical high-dimensional data analysis.

Sparse Principal Component Analysis (SPCA) is a fundamental technique for dimensionality reduction, and is NP-hard. In this paper, we introduce a randomized approximation algorithm for SPCA, which is based on the basic SDP relaxation. Our algorithm has an approximation ratio of at most the sparsity constant with high probability, if called enough times. Under a technical assumption, which is consistently satisfied in our numerical tests, the average approximation ratio is also bounded by $\mathcal{O}(\log{d})$, where $d$ is the number of features. We show that this technical assumption is satisfied if the SDP solution is low-rank, or has exponentially decaying eigenvalues. We then present a broad class of instances for which this technical assumption holds. We also demonstrate that in a covariance model, which generalizes the spiked Wishart model, our proposed algorithm achieves a near-optimal approximation ratio. We demonstrate the efficacy of our algorithm through numerical results on real-world datasets.

Accurate forecasting of energy demand and supply is critical for optimizing sustainable energy systems, yet it is challenged by the variability of renewable sources and dynamic consumption patterns. This paper introduces a neural framework that integrates continuous-time Neural Ordinary Differential Equations (Neural ODEs), graph attention, multi-resolution wavelet transformations, and adaptive learning of frequencies to address the issues of time series prediction. The model employs a robust ODE solver, using the Runge-Kutta method, paired with graph-based attention and residual connections to better understand both structural and temporal patterns. Through wavelet-based feature extraction and adaptive frequency modulation, it adeptly captures and models diverse, multi-scale temporal dynamics. When evaluated across seven diverse datasets: ETTh1, ETTh2, ETTm1, ETTm2 (electricity transformer temperature), and Waste, Solar, and Hydro (renewable energy), this architecture consistently outperforms state-of-the-art baselines in various forecasting metrics, proving its robustness in capturing complex temporal dependencies. Furthermore, the model enhances interpretability through SHAP analysis, making it suitable for sustainable energy applications.

Many dynamic processes such as telecommunication and transport networks can be described through discrete time series of graphs. Modelling the dynamics of such time series enables prediction of graph structure at future time steps, which can be used in applications such as detection of anomalies. Existing approaches for graph prediction have limitations such as assuming that the vertices do not to change between consecutive graphs. To address this, we propose to exploit time series prediction methods in combination with an adapted form of flux balance analysis (FBA), a linear programming method originating from biochemistry. FBA is adapted to incorporate various constraints applicable to the scenario of growing graphs. Empirical evaluations on synthetic datasets (constructed via Preferential Attachment model) and real datasets (UCI Message, HePH, Facebook, Bitcoin) demonstrate the efficacy of the proposed approach.

We consider stochastic approximation with block-coordinate stepsizes and propose adaptive stepsize rules that aim to minimize the expected distance from the next iterate to an optimal point. These stepsize rules employ online estimates of the second moment of the search direction along each block coordinate. The popular Adam algorithm can be interpreted as a particular heuristic for such estimation. By leveraging a simple conditional estimator, we derive a new method that obtains comparable performance as Adam but requires less memory and fewer hyper-parameters. We prove that this family of methods converges almost surely to a small neighborhood of the optimal point, and the radius of the neighborhood depends on the bias and variance of the second-moment estimator. Our analysis relies on a simple aiming condition that assumes neither convexity nor smoothness, thus has broad applicability.

Wavelet neural network (WNN), which learns an unknown nonlinear mapping from the data, has been widely used in signal processing, and time-series analysis. However, challenges in constructing accurate wavelet bases and high computational costs limit their application. This study introduces a constructive WNN that selects initial bases and trains functions by introducing new bases for predefined accuracy while reducing computational costs. For the first time, we analyze the frequency of unknown nonlinear functions and select appropriate initial wavelets based on their primary frequency components by estimating the energy of the spatial frequency component. This leads to a novel constructive framework consisting of a frequency estimator and a wavelet-basis increase mechanism to prioritize high-energy bases, significantly improving computational efficiency. The theoretical foundation defines the necessary time-frequency range for high-dimensional wavelets at a given accuracy. The framework's versatility is demonstrated through four examples: estimating unknown static mappings from offline data, combining two offline datasets, identifying time-varying mappings from time-series data, and capturing nonlinear dependencies in real time-series data. These examples showcase the framework's broad applicability and practicality. All the code will be released at https://github.com/dshuangdd/CWNN.

There is no much doubt that biotic interactions shape community assembly and ultimately the spatial co-variations between species. There is a hope that the signal of these biotic interactions can be observed and retrieved by investigating the spatial associations between species while accounting for the direct effects of the environment. By definition, biotic interactions can be both symmetric and asymmetric. Yet, most models that attempt to retrieve species associations from co-occurrence or co-abundance data internally assume symmetric relationships between species. Here, we propose and validate a machine-learning framework able to retrieve bidirectional associations by analyzing species community and environmental data. Our framework (1) models pairwise species associations as directed influences from a source to a target species, parameterized with two species-specific latent embeddings: the effect of the source species on the community, and the response of the target species to the community; and (2) jointly fits these associations within a multi-species conditional generative model with different modes of interactions between environmental drivers and biotic associations. Using both simulated and empirical data, we demonstrate the ability of our framework to recover known asymmetric and symmetric associations and highlight the properties of the learned association networks. By comparing our approach to other existing models such as joint species distribution models and probabilistic graphical models, we show its superior capacity at retrieving symmetric and asymmetric interactions. The framework is intuitive, modular and broadly applicable across various taxonomic groups.

Stochastic simulators exhibit intrinsic stochasticity due to unobservable, uncontrollable, or unmodeled input variables, resulting in random outputs even at fixed input conditions. Such simulators are common across various scientific disciplines; however, emulating their entire conditional probability distribution is challenging, as it is a task traditional deterministic surrogate modeling techniques are not designed for. Additionally, accurately characterizing the response distribution can require prohibitively large datasets, especially for computationally expensive high-fidelity (HF) simulators. When lower-fidelity (LF) stochastic simulators are available, they can enhance limited HF information within a multifidelity surrogate modeling (MFSM) framework. While MFSM techniques are well-established for deterministic settings, constructing multifidelity emulators to predict the full conditional response distribution of stochastic simulators remains a challenge. In this paper, we propose multifidelity generalized lambda models (MF-GLaMs) to efficiently emulate the conditional response distribution of HF stochastic simulators by exploiting data from LF stochastic simulators. Our approach builds upon the generalized lambda model (GLaM), which represents the conditional distribution at each input by a flexible, four-parameter generalized lambda distribution. MF-GLaMs are non-intrusive, requiring no access to the internal stochasticity of the simulators nor multiple replications of the same input values. We demonstrate the efficacy of MF-GLaM through synthetic examples of increasing complexity and a realistic earthquake application. Results show that MF-GLaMs can achieve improved accuracy at the same cost as single-fidelity GLaMs, or comparable performance at significantly reduced cost.

In epidemic modeling, outliers can distort parameter estimation and ultimately lead to misguided public health decisions. Although there are existing robust methods that can mitigate this distortion, the ability to simultaneously detect outliers is equally vital for identifying potential disease hotspots. In this work, we introduce a robust spatiotemporal generalized additive model (RST-GAM) to address this need. We accomplish this with a mean-shift parameter to quantify and adjust for the effects of outliers and rely on adaptive Lasso regularization to model the sparsity of outlying observations. We use univariate polynomial splines and bivariate penalized splines over triangulations to estimate the functional forms and a data-thinning approach for data-adaptive weight construction. We derive a scalable proximal algorithm to estimate model parameters by minimizing a convex negative log-quasi-likelihood function. Our algorithm uses adaptive step-sizes to ensure global convergence of the resulting iterate sequence. We establish error bounds and selection consistency for the estimated parameters and demonstrate our model's effectiveness through numerical studies under various outlier scenarios. Finally, we demonstrate the practical utility of RST-GAM by analyzing county-level COVID-19 infection data in the United States, highlighting its potential to inform public health decision-making.

Diffusion models have quickly become some of the most popular and powerful generative models for high-dimensional data. The key insight that enabled their development was the realization that access to the score -- the gradient of the log-density at different noise levels -- allows for sampling from data distributions by solving a reverse-time stochastic differential equation (SDE) via forward discretization, and that popular denoisers allow for unbiased estimators of this score. In this paper, we demonstrate that an alternative, backward discretization of these SDEs, using proximal maps in place of the score, leads to theoretical and practical benefits. We leverage recent results in proximal matching to learn proximal operators of the log-density and, with them, develop Proximal Diffusion Models (ProxDM). Theoretically, we prove that $\widetilde{O}(d/\sqrt{\varepsilon})$ steps suffice for the resulting discretization to generate an $\varepsilon$-accurate distribution w.r.t. the KL divergence. Empirically, we show that two variants of ProxDM achieve significantly faster convergence within just a few sampling steps compared to conventional score-matching methods.