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Solving multiscale diffusion problems is often computationally expensive due to the spatial and temporal discretization challenges arising from high-contrast coefficients. To address this issue, a partially explicit temporal splitting scheme is proposed. By appropriately constructing multiscale spaces, the spatial multiscale property is effectively managed, and it has been demonstrated that the temporal step size is independent of the contrast. To enhance simulation speed, we propose a parallel algorithm for the multiscale flow problem that leverages the partially explicit temporal splitting scheme. The idea is first to evolve the partially explicit system using a coarse time step size, then correct the solution on each coarse time interval with a fine propagator, for which we consider both the sequential solver and all-at-once solver. This procedure is then performed iteratively till convergence. We analyze the stability and convergence of the proposed algorithm. The numerical experiments demonstrate that the proposed algorithm achieves high numerical accuracy for high-contrast problems and converges in a relatively small number of iterations. The number of iterations stays stable as the number of coarse intervals increases, thus significantly improving computational efficiency through parallel processing.

Vintage factor analysis is one important type of factor analysis that aims to first find a low-dimensional representation of the original data, and then to seek a rotation such that the rotated low-dimensional representation is scientifically meaningful. The most widely used vintage factor analysis is the Principal Component Analysis (PCA) followed by the varimax rotation. Despite its popularity, little theoretical guarantee can be provided to date mainly because varimax rotation requires to solve a non-convex optimization over the set of orthogonal matrices. In this paper, we propose a deflation varimax procedure that solves each row of an orthogonal matrix sequentially. In addition to its net computational gain and flexibility, we are able to fully establish theoretical guarantees for the proposed procedure in a broader context. Adopting this new deflation varimax as the second step after PCA, we further analyze this two step procedure under a general class of factor models. Our results show that it estimates the factor loading matrix in the minimax optimal rate when the signal-to-noise-ratio (SNR) is moderate or large. In the low SNR regime, we offer possible improvement over using PCA and the deflation varimax when the additive noise under the factor model is structured. The modified procedure is shown to be minimax optimal in all SNR regimes. Our theory is valid for finite sample and allows the number of the latent factors to grow with the sample size as well as the ambient dimension to grow with, or even exceed, the sample size. Extensive simulation and real data analysis further corroborate our theoretical findings.

We give a complete expansion, at any accuracy order, for the iterated convolution of a complex valued integrable sequence in one space dimension. The remainders are estimated sharply with generalized Gaussian bounds. The result applies in probability theory for random walks as well as in numerical analysis for studying the large time behavior of numerical schemes.

Virtual environments provide a rich and controlled setting for collecting detailed data on human behavior, offering unique opportunities for predicting human trajectories in dynamic scenes. However, most existing approaches have overlooked the potential of these environments, focusing instead on static contexts without considering userspecific factors. Employing the CREATTIVE3D dataset, our work models trajectories recorded in virtual reality (VR) scenes for diverse situations including road-crossing tasks with user interactions and simulated visual impairments. We propose Diverse Context VR Human Motion Prediction (DiVR), a cross-modal transformer based on the Perceiver architecture that integrates both static and dynamic scene context using a heterogeneous graph convolution network. We conduct extensive experiments comparing DiVR against existing architectures including MLP, LSTM, and transformers with gaze and point cloud context. Additionally, we also stress test our model's generalizability across different users, tasks, and scenes. Results show that DiVR achieves higher accuracy and adaptability compared to other models and to static graphs. This work highlights the advantages of using VR datasets for context-aware human trajectory modeling, with potential applications in enhancing user experiences in the metaverse. Our source code is publicly available at //gitlab.inria.fr/ffrancog/creattive3d-divr-model.

These last few years, image decomposition algorithms have been proposed to split an image into two parts: the structures and the textures. These algorithms are not adapted to the case of noisy images because the textures are corrupted by noise. In this paper, we propose a new model which decomposes an image into three parts (structures, textures and noise) based on a local regularization scheme. We compare our results with the recent work of Aujol and Chambolle. We finish by giving another model which combines the advantages of the two previous ones.

In longitudinal data analysis, observation points of repeated measurements over time often vary among subjects except in well-designed experimental studies. Additionally, measurements for each subject are typically obtained at only a few time points. From such sparsely observed data, identifying underlying cluster structures can be challenging. This paper proposes a fast and simple clustering method that generalizes the classical $k$-means method to identify cluster centers in sparsely observed data. The proposed method employs the basis function expansion to model the cluster centers, providing an effective way to estimate cluster centers from fragmented data. We establish the statistical consistency of the proposed method, as with the classical $k$-means method. Through numerical experiments, we demonstrate that the proposed method performs competitively with, or even outperforms, existing clustering methods. Moreover, the proposed method offers significant gains in computational efficiency due to its simplicity. Applying the proposed method to real-world data illustrates its effectiveness in identifying cluster structures in sparsely observed data.

We address the problem of identifying functional interactions among stochastic neurons with variable-length memory from their spiking activity. The neuronal network is modeled by a stochastic system of interacting point processes with variable-length memory. Each chain describes the activity of a single neuron, indicating whether it spikes at a given time. One neuron's influence on another can be either excitatory or inhibitory. To identify the existence and nature of an interaction between a neuron and its postsynaptic counterpart, we propose a model selection procedure based on the observation of the spike activity of a finite set of neurons over a finite time. The proposed procedure is also based on the maximum likelihood estimator for the synaptic weight matrix of the network neuronal model. In this sense, we prove the consistency of the maximum likelihood estimator followed by a proof of the consistency of the neighborhood interaction estimation procedure. The effectiveness of the proposed model selection procedure is demonstrated using simulated data, which validates the underlying theory. The method is also applied to analyze spike train data recorded from hippocampal neurons in rats during a visual attention task, where a computational model reconstructs the spiking activity and the results reveal interesting and biologically relevant information.

This work establishes novel, optimum mixing bounds for the Glauber dynamics on the Hard-core and Ising models using the powerful Spectral Independence method from [Anari, Liu, Oveis-Gharan: FOCS 2020]. These bounds are expressed in terms of the local connective constant of the underlying graph $G$. This is a notion of effective degree of $G$ on a local scale. Our results have some interesting consequences for bounded degree graphs: (a) They include the max-degree bounds as a special case, (b) They improve on the running time of the FPTAS considered in [Sinclair, Srivastava, Stefankonic and Yin: PTRF 2017], (c) They allow us to obtain mixing bounds in terms of the spectral radius of the adjacency matrix and improve on results in [Hayes: FOCS 2006], (d) They also allow us to refine the celebrated connection between the hardness of approximate counting and the phase transitions from statistical physics. We obtain our mixing bounds by utilising the $k$-non-backtracking matrix $H_{G,k}$. This is a very interesting, alas technically intricate, object to work with. We upper bound the spectral radius of the pairwise influence matrix $I^{\Lambda,\tau}_{G}$ by means of the 2-norm of $H_{G,k}$. To our knowledge, obtaining mixing bound using $H_{G,k}$ has not been considered before in the literature.

Synthetic active collectives, composed of many nonliving individuals capable of cooperative changes in group shape and dynamics, hold promise for practical applications and for the elucidation of guiding principles of natural collectives. However, the design of collective robotic systems that operate effectively without intelligence or complex control at either the individual or group level is challenging. We investigate how simple steric interaction constraints between active individuals produce a versatile active system with promising functionality. Here we introduce the link-bot: a V-shape-based, single-stranded chain composed of active bots whose dynamics are defined by its geometric link constraints, allowing it to possess scale- and processing-free programmable collective behaviors. A variety of emergent properties arise from this dynamic system, including locomotion, navigation, transportation, and competitive or cooperative interactions. Through the control of a few link parameters, link-bots show rich usefulness by performing a variety of divergent tasks, including traversing or obstructing narrow spaces, passing by or enclosing objects, and propelling loads in both forward and backward directions. The reconfigurable nature of the link-bot suggests that our approach may significantly contribute to the development of programmable soft robotic systems with minimal information and materials at any scale.

Humans excel at discovering regular structures from limited samples and applying inferred rules to novel settings. We investigate whether modern generative models can similarly learn underlying rules from finite samples and perform reasoning through conditional sampling. Inspired by Raven's Progressive Matrices task, we designed GenRAVEN dataset, where each sample consists of three rows, and one of 40 relational rules governing the object position, number, or attributes applies to all rows. We trained generative models to learn the data distribution, where samples are encoded as integer arrays to focus on rule learning. We compared two generative model families: diffusion (EDM, DiT, SiT) and autoregressive models (GPT2, Mamba). We evaluated their ability to generate structurally consistent samples and perform panel completion via unconditional and conditional sampling. We found diffusion models excel at unconditional generation, producing more novel and consistent samples from scratch and memorizing less, but performing less well in panel completion, even with advanced conditional sampling methods. Conversely, autoregressive models excel at completing missing panels in a rule-consistent manner but generate less consistent samples unconditionally. We observe diverse data scaling behaviors: for both model families, rule learning emerges at a certain dataset size - around 1000s examples per rule. With more training data, diffusion models improve both their unconditional and conditional generation capabilities. However, for autoregressive models, while panel completion improves with more training data, unconditional generation consistency declines. Our findings highlight complementary capabilities and limitations of diffusion and autoregressive models in rule learning and reasoning tasks, suggesting avenues for further research into their mechanisms and potential for human-like reasoning.