We consider the Gaussian kernel density estimator with bandwidth $\beta^{-\frac12}$ of $n$ iid Gaussian samples. Using the Kac-Rice formula and an Edgeworth expansion, we prove that the expected number of modes on the real line scales as $\Theta(\sqrt{\beta\log\beta})$ as $\beta,n\to\infty$ provided $n^c\lesssim \beta\lesssim n^{2-c}$ for some constant $c>0$. An impetus behind this investigation is to determine the number of clusters to which Transformers are drawn in a metastable state.

For many graph-related problems, it can be essential to have a set of structurally diverse graphs. For instance, such graphs can be used for testing graph algorithms or their neural approximations. However, to the best of our knowledge, the problem of generating structurally diverse graphs has not been explored in the literature. In this paper, we fill this gap. First, we discuss how to define diversity for a set of graphs, why this task is non-trivial, and how one can choose a proper diversity measure. Then, for a given diversity measure, we propose and compare several algorithms optimizing it: we consider approaches based on standard random graph models, local graph optimization, genetic algorithms, and neural generative models. We show that it is possible to significantly improve diversity over basic random graph generators. Additionally, our analysis of generated graphs allows us to better understand the properties of graph distances: depending on which diversity measure is used for optimization, the obtained graphs may possess very different structural properties which gives a better understanding of the graph distance underlying the diversity measure.

Despite the widespread use of the data augmentation (DA) algorithm, the theoretical understanding of its convergence behavior remains incomplete. We prove the first non-asymptotic polynomial upper bounds on mixing times of three important DA algorithms: DA algorithm for Bayesian Probit regression (Albert and Chib, 1993, ProbitDA), Bayesian Logit regression (Polson, Scott, and Windle, 2013, LogitDA), and Bayesian Lasso regression (Park and Casella, 2008, Rajaratnam et al., 2015, LassoDA). Concretely, we demonstrate that with $\eta$-warm start, parameter dimension $d$, and sample size $n$, the ProbitDA and LogitDA require $\mathcal{O}\left(nd\log \left(\frac{\log \eta}{\epsilon}\right)\right)$ steps to obtain samples with at most $\epsilon$ TV error, whereas the LassoDA requires $\mathcal{O}\left(d^2(d\log d +n \log n)^2 \log \left(\frac{\eta}{\epsilon}\right)\right)$ steps. The results are generally applicable to settings with large $n$ and large $d$, including settings with highly imbalanced response data in the Probit and Logit regression. The proofs are based on the Markov chain conductance and isoperimetric inequalities. Assuming that data are independently generated from either a bounded, sub-Gaussian, or log-concave distribution, we improve the guarantees for ProbitDA and LogitDA to $\tilde{\mathcal{O}}(n+d)$ with high probability, and compare it with the best known guarantees of Langevin Monte Carlo and Metropolis Adjusted Langevin Algorithm. We also discuss the mixing times of the three algorithms under feasible initialization.

Spatiotemporal dynamics pervade the natural sciences, from the morphogen dynamics underlying patterning in animal pigmentation to the protein waves controlling cell division. A central challenge lies in understanding how controllable parameters induce qualitative changes in system behavior called bifurcations. This endeavor is made particularly difficult in realistic settings where governing partial differential equations (PDEs) are unknown and data is limited and noisy. To address this challenge, we propose TRENDy (Temporal Regression of Effective Nonlinear Dynamics), an equation-free approach to learning low-dimensional, predictive models of spatiotemporal dynamics. Following classical work in spatial coarse-graining, TRENDy first maps input data to a low-dimensional space of effective dynamics via a cascade of multiscale filtering operations. Our key insight is the recognition that these effective dynamics can be fit by a neural ordinary differential equation (NODE) having the same parameter space as the input PDE. The preceding filtering operations strongly regularize the phase space of the NODE, making TRENDy significantly more robust to noise compared to existing methods. We train TRENDy to predict the effective dynamics of synthetic and real data representing dynamics from across the physical and life sciences. We then demonstrate how our framework can automatically locate both Turing and Hopf bifurcations in unseen regions of parameter space. We finally apply our method to the analysis of spatial patterning of the ocellated lizard through development. We found that TRENDy's effective state not only accurately predicts spatial changes over time but also identifies distinct pattern features unique to different anatomical regions, highlighting the potential influence of surface geometry on reaction-diffusion mechanisms and their role in driving spatially varying pattern dynamics.

We study a system with finitely many groups of multi-action bandit processes, each of which is a Markov decision process (MDP) with finite state and action spaces and potentially different transition matrices when taking different actions. The bandit processes of the same group share the same state and action spaces and, given the same action that is taken, the same transition matrix. All the bandit processes across various groups are subject to multiple weakly coupled constraints over their state and action variables. Unlike the past studies that focused on the offline case, we consider the online case without assuming full knowledge of transition matrices and reward functions a priori and propose an effective scheme that enables simultaneous learning and control. We prove the convergence of the relevant processes in both the timeline and the number of the bandit processes, referred to as the convergence in the time and the magnitude dimensions. Moreover, we prove that the relevant processes converge exponentially fast in the magnitude dimension, leading to exponentially diminishing performance deviation between the proposed online algorithms and offline optimality. Jing Fu is with Department of Electrical and Electronic Engineering, School of Engineering, STEM College, RMIT University, Australia (e-mail: jing.fu@rmit.edu.au). Bill Moran is with Department of Electrical and Electronic Engineering, the University of Melbourne, VIC 3010, Australia (e-mail:wmoran@unimelb.edu.au).

Complex networked systems driven by latent inputs are common in fields like neuroscience, finance, and engineering. A key inference problem here is to learn edge connectivity from node outputs (potentials). We focus on systems governed by steady-state linear conservation laws: $X_t = {L^{\ast}}Y_{t}$, where $X_t, Y_t \in \mathbb{R}^p$ denote inputs and potentials, respectively, and the sparsity pattern of the $p \times p$ Laplacian $L^{\ast}$ encodes the edge structure. Assuming $X_t$ to be a wide-sense stationary stochastic process with a known spectral density matrix, we learn the support of $L^{\ast}$ from temporally correlated samples of $Y_t$ via an $\ell_1$-regularized Whittle's maximum likelihood estimator (MLE). The regularization is particularly useful for learning large-scale networks in the high-dimensional setting where the network size $p$ significantly exceeds the number of samples $n$. We show that the MLE problem is strictly convex, admitting a unique solution. Under a novel mutual incoherence condition and certain sufficient conditions on $(n, p, d)$, we show that the ML estimate recovers the sparsity pattern of $L^\ast$ with high probability, where $d$ is the maximum degree of the graph underlying $L^{\ast}$. We provide recovery guarantees for $L^\ast$ in element-wise maximum, Frobenius, and operator norms. Finally, we complement our theoretical results with several simulation studies on synthetic and benchmark datasets, including engineered systems (power and water networks), and real-world datasets from neural systems (such as the human brain).

This paper introduces the Parsimonious Dynamic Mode Decomposition (parsDMD), a novel algorithm designed to automatically select an optimally sparse subset of dynamic modes for both spatiotemporal and purely temporal data. By incorporating time-delay embedding and leveraging Orthogonal Matching Pursuit (OMP), parsDMD ensures robustness against noise and effectively handles complex, nonlinear dynamics. The algorithm is validated on a diverse range of datasets, including standing wave signals, identifying hidden dynamics, fluid dynamics simulations (flow past a cylinder and transonic buffet), and atmospheric sea-surface temperature (SST) data. ParsDMD addresses a significant limitation of the traditional sparsity-promoting DMD (spDMD), which requires manual tuning of sparsity parameters through a rigorous trial-and-error process to balance between single-mode and all-mode solutions. In contrast, parsDMD autonomously determines the optimally sparse subset of modes without user intervention, while maintaining minimal computational complexity. Comparative analyses demonstrate that parsDMD consistently outperforms spDMD by providing more accurate mode identification and effective reconstruction in noisy environments. These advantages render parsDMD an effective tool for real-time diagnostics, forecasting, and reduced-order model construction across various disciplines.

We introduce an algorithm for efficiently representing convolution with zero-padding and stride as a sparse transformation matrix, applied to a vectorized input through sparse matrix-vector multiplication (SpMV). We provide a theoretical contribution with an explicit expression for the number of non-zero multiplications in convolutions with stride and padding, offering insight into the potential for leveraging sparsity in convolution operations. A proof-of-concept implementation is presented in Python, demonstrating the performance of our method on both CPU and GPU architectures. This work contributes to the broader exploration of sparse matrix techniques in convolutional algorithms, with a particular focus on leveraging matrix multiplications for parallelization. Our findings lay the groundwork for future advancements in exploiting sparsity to improve the efficiency of convolution operations in fields such as machine learning and signal processing.

Recurrent stochastic configuration networks (RSCNs) have shown great potential in modelling nonlinear dynamic systems with uncertainties. This paper presents an RSCN with hybrid regularization to enhance both the learning capacity and generalization performance of the network. Given a set of temporal data, the well-known least absolute shrinkage and selection operator (LASSO) is employed to identify the significant order variables. Subsequently, an improved RSCN with L2 regularization is introduced to approximate the residuals between the output of the target plant and the LASSO model. The output weights are updated in real-time through a projection algorithm, facilitating a rapid response to dynamic changes within the system. A theoretical analysis of the universal approximation property is provided, contributing to the understanding of the network's effectiveness in representing various complex nonlinear functions. Experimental results from a nonlinear system identification problem and two industrial predictive tasks demonstrate that the proposed method outperforms other models across all testing datasets.

Non-maximum suppression (NMS) is an indispensable post-processing step in object detection. With the continuous optimization of network models, NMS has become the ``last mile'' to enhance the efficiency of object detection. This paper systematically analyzes NMS from a graph theory perspective for the first time, revealing its intrinsic structure. Consequently, we propose two optimization methods, namely QSI-NMS and BOE-NMS. The former is a fast recursive divide-and-conquer algorithm with negligible mAP loss, and its extended version (eQSI-NMS) achieves optimal complexity of $\mathcal{O}(n\log n)$. The latter, concentrating on the locality of NMS, achieves an optimization at a constant level without an mAP loss penalty. Moreover, to facilitate rapid evaluation of NMS methods for researchers, we introduce NMS-Bench, the first benchmark designed to comprehensively assess various NMS methods. Taking the YOLOv8-N model on MS COCO 2017 as the benchmark setup, our method QSI-NMS provides $6.2\times$ speed of original NMS on the benchmark, with a $0.1\%$ decrease in mAP. The optimal eQSI-NMS, with only a $0.3\%$ mAP decrease, achieves $10.7\times$ speed. Meanwhile, BOE-NMS exhibits $5.1\times$ speed with no compromise in mAP.