We propose a new method for learning videos by aggregating multiple models by sequentially extracting video clips from untrimmed video. The proposed method reduces the correlation between clips by feeding clips to multiple models in turn and synchronizes these models through federated learning. Experimental results show that the proposed method improves the performance compared to the no synchronization.
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We propose a new concept of lifts of reversible diffusion processes and show that various well-known non-reversible Markov processes arising in applications are lifts in this sense of simple reversible diffusions. Furthermore, we introduce a concept of non-asymptotic relaxation times and show that these can at most be reduced by a square root through lifting, generalising a related result in discrete time. Finally, we demonstrate how the recently developed approach to quantitative hypocoercivity based on space-time Poincar\'e inequalities can be rephrased and simplified in the language of lifts and how it can be applied to find optimal lifts.
Standard techniques such as leave-one-out cross-validation (LOOCV) might not be suitable for evaluating the predictive performance of models incorporating structured random effects. In such cases, the correlation between the training and test sets could have a notable impact on the model's prediction error. To overcome this issue, an automatic group construction procedure for leave-group-out cross validation (LGOCV) has recently emerged as a valuable tool for enhancing predictive performance measurement in structured models. The purpose of this paper is (i) to compare LOOCV and LGOCV within structured models, emphasizing model selection and predictive performance, and (ii) to provide real data applications in spatial statistics using complex structured models fitted with INLA, showcasing the utility of the automatic LGOCV method. First, we briefly review the key aspects of the recently proposed LGOCV method for automatic group construction in latent Gaussian models. We also demonstrate the effectiveness of this method for selecting the model with the highest predictive performance by simulating extrapolation tasks in both temporal and spatial data analyses. Finally, we provide insights into the effectiveness of the LGOCV method in modelling complex structured data, encompassing spatio-temporal multivariate count data, spatial compositional data, and spatio-temporal geospatial data.
Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to 1). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, that history-dependent (Markovian) policies strictly outperform stationary policies for average optimality in s-rectangular RMDPs. We also study Blackwell optimality for sa-rectangular RMDPs, where we show that {\em approximate} Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games.
One of the main challenges for interpreting black-box models is the ability to uniquely decompose square-integrable functions of non-independent random inputs into a sum of functions of every possible subset of variables. However, dealing with dependencies among inputs can be complicated. We propose a novel framework to study this problem, linking three domains of mathematics: probability theory, functional analysis, and combinatorics. We show that, under two reasonable assumptions on the inputs (non-perfect functional dependence and non-degenerate stochastic dependence), it is always possible to decompose such a function uniquely. This generalizes the well-known Hoeffding decomposition. The elements of this decomposition can be expressed using oblique projections and allow for novel interpretability indices for evaluation and variance decomposition purposes. The properties of these novel indices are studied and discussed. This generalization offers a path towards a more precise uncertainty quantification, which can benefit sensitivity analysis and interpretability studies whenever the inputs are dependent. This decomposition is illustrated analytically, and the challenges for adopting these results in practice are discussed.
We present a new methodology for decomposing flows with multiple transports that further extends the shifted proper orthogonal decomposition (sPOD). The sPOD tries to approximate transport-dominated flows by a sum of co-moving data fields. The proposed methods stem from sPOD but optimize the co-moving fields directly and penalize their nuclear norm to promote low rank of the individual data in the decomposition. Furthermore, we add a robustness term to the decomposition that can deal with interpolation error and data noises. Leveraging tools from convex optimization, we derive three proximal algorithms to solve the decomposition problem. We report a numerical comparison with existing methods against synthetic data benchmarks and then show the separation ability of our methods on 1D and 2D incompressible and reactive flows. The resulting methodology is the basis of a new analysis paradigm that results in the same interpretability as the POD for the individual co-moving fields.
Several mixed-effects models for longitudinal data have been proposed to accommodate the non-linearity of late-life cognitive trajectories and assess the putative influence of covariates on it. No prior research provides a side-by-side examination of these models to offer guidance on their proper application and interpretation. In this work, we examined five statistical approaches previously used to answer research questions related to non-linear changes in cognitive aging: the linear mixed model (LMM) with a quadratic term, LMM with splines, the functional mixed model, the piecewise linear mixed model, and the sigmoidal mixed model. We first theoretically describe the models. Next, using data from two prospective cohorts with annual cognitive testing, we compared the interpretation of the models by investigating associations of education on cognitive change before death. Lastly, we performed a simulation study to empirically evaluate the models and provide practical recommendations. Except for the LMM-quadratic, the fit of all models was generally adequate to capture non-linearity of cognitive change and models were relatively robust. Although spline-based models have no interpretable nonlinearity parameters, their convergence was easier to achieve, and they allow graphical interpretation. In contrast, piecewise and sigmoidal models, with interpretable non-linear parameters, may require more data to achieve convergence.
This paper introduces a new method for the efficient computation of oscillatory multidimensional lattice sums in geometries with boundaries. Such sums are ubiquitous in both pure and applied mathematics, and have immediate applications in condensed matter physics and topological quantum physics. The challenge in their evaluation results from the combination of singular long-range interactions with the loss of translational invariance caused by the boundaries, rendering standard tools ineffective. Our work shows that these lattice sums can be generated from a generalization of the Riemann zeta function to multidimensional non-periodic lattice sums. We put forth a new representation of this zeta function together with a numerical algorithm that ensures exponential convergence across an extensive range of geometries. Notably, our method's runtime is influenced only by the complexity of the considered geometries and not by the number of particles, providing the foundation for efficient simulations of macroscopic condensed matter systems. We showcase the practical utility of our method by computing interaction energies in a three-dimensional crystal structure with $3\times 10^{23}$ particles. Our method's accuracy is demonstrated through extensive numerical experiments. A reference implementation is provided online along with this article.
We present a new method for causal discovery in linear structural vector autoregressive models. We adapt an idea designed for independent observations to the case of time series while retaining its favorable properties, i.e., explicit error control for false causal discovery, at least asymptotically. We apply our method to several real-world bivariate time series datasets and discuss its findings which mostly agree with common understanding. The arrow of time in a model can be interpreted as background knowledge on possible causal mechanisms. Hence, our ideas could be extended to incorporating different background knowledge, even for independent observations.
Lensless illumination single-pixel imaging with a multicore fiber (MCF) is a computational imaging technique that enables potential endoscopic observations of biological samples at cellular scale. In this work, we show that this technique is tantamount to collecting multiple symmetric rank-one projections (SROP) of an interferometric matrix--a matrix encoding the spectral content of the sample image. In this model, each SROP is induced by the complex sketching vector shaping the incident light wavefront with a spatial light modulator (SLM), while the projected interferometric matrix collects up to $O(Q^2)$ image frequencies for a $Q$-core MCF. While this scheme subsumes previous sensing modalities, such as raster scanning (RS) imaging with beamformed illumination, we demonstrate that collecting the measurements of $M$ random SLM configurations--and thus acquiring $M$ SROPs--allows us to estimate an image of interest if $M$ and $Q$ scale log-linearly with the image sparsity level This demonstration is achieved both theoretically, with a specific restricted isometry analysis of the sensing scheme, and with extensive Monte Carlo experiments. On a practical side, we perform a single calibration of the sensing system robust to certain deviations to the theoretical model and independent of the sketching vectors used during the imaging phase. Experimental results made on an actual MCF system demonstrate the effectiveness of this imaging procedure on a benchmark image.
This study addresses a class of linear mixed-integer programming (MILP) problems that involve uncertainty in the objective function parameters. The parameters are assumed to form a random vector, whose probability distribution can only be observed through a finite training data set. Unlike most of the related studies in the literature, we also consider uncertainty in the underlying data set. The data uncertainty is described by a set of linear constraints for each random sample, and the uncertainty in the distribution (for a fixed realization of data) is defined using a type-1 Wasserstein ball centered at the empirical distribution of the data. The overall problem is formulated as a three-level distributionally robust optimization (DRO) problem. First, we prove that the three-level problem admits a single-level MILP reformulation, if the class of loss functions is restricted to biaffine functions. Secondly, it turns out that for several particular forms of data uncertainty, the outlined problem can be solved reasonably fast by leveraging the nominal MILP problem. Finally, we conduct a computational study, where the out-of-sample performance of our model and computational complexity of the proposed MILP reformulation are explored numerically for several application domains.
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