Cross-validation is usually employed to evaluate the performance of a given statistical methodology. When such a methodology depends on a number of tuning parameters, cross-validation proves to be helpful to select the parameters that optimize the estimated performance. In this paper, however, a very different and nonstandard use of cross-validation is investigated. Instead of focusing on the cross-validated parameters, the main interest is switched to the estimated value of the error criterion at optimal performance. It is shown that this approach is able to provide consistent and efficient estimates of some density functionals, with the noteworthy feature that these estimates do not rely on the choice of any further tuning parameter, so that, in that sense, they can be considered to be purely empirical. Here, a base case of application of this new paradigm is developed in full detail, while many other possible extensions are hinted as well.
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Robotic exploration has long captivated researchers aiming to map complex environments efficiently. Techniques such as potential fields and frontier exploration have traditionally been employed in this pursuit, primarily focusing on solitary agents. Recent advancements have shifted towards optimizing exploration efficiency through multiagent systems. However, many existing approaches overlook critical real-world factors, such as broadcast range limitations, communication costs, and coverage overlap. This paper addresses these gaps by proposing a distributed maze exploration strategy (CU-LVP) that assumes constrained broadcast ranges and utilizes Voronoi diagrams for better area partitioning. By adapting traditional multiagent methods to distributed environments with limited broadcast ranges, this study evaluates their performance across diverse maze topologies, demonstrating the efficacy and practical applicability of the proposed method. The code and experimental results supporting this study are available in the following repository: //github.com/manouslinard/multiagent-exploration/.
Differential abundance analysis is a key component of microbiome studies. While dozens of methods for it exist, currently, there is no consensus on the preferred methods. Correctness of results in differential abundance analysis is an ambiguous concept that cannot be evaluated without employing simulated data, but we argue that consistency of results across datasets should be considered as an essential quality of a well-performing method. We compared the performance of 14 differential abundance analysis methods employing datasets from 54 taxonomic profiling studies based on 16S rRNA gene or shotgun sequencing. For each method, we examined how the results replicated between random partitions of each dataset and between datasets from independent studies. While certain methods showed good consistency, some widely used methods were observed to produce a substantial number of conflicting findings. Overall, the highest consistency without unnecessary reduction in sensitivity was attained by analyzing relative abundances with a non-parametric method (Wilcoxon test or ordinal regression model) or linear regression (MaAsLin2). Comparable performance was also attained by analyzing presence/absence of taxa with logistic regression.
The problems of optimal recovering univariate functions and their derivatives are studied. To solve these problems, two variants of the truncation method are constructed, which are order-optimal both in the sense of accuracy and in terms of the amount of involved Galerkin information. For numerical summation, it has been established how the parameters characterizing the problem being solved affect its stability.
Identifiability of statistical models is a key notion in unsupervised representation learning. Recent work of nonlinear independent component analysis (ICA) employs auxiliary data and has established identifiable conditions. This paper proposes a statistical model of two latent vectors with single auxiliary data generalizing nonlinear ICA, and establishes various identifiability conditions. Unlike previous work, the two latent vectors in the proposed model can have arbitrary dimensions, and this property enables us to reveal an insightful dimensionality relation among two latent vectors and auxiliary data in identifiability conditions. Furthermore, surprisingly, we prove that the indeterminacies of the proposed model has the same as \emph{linear} ICA under certain conditions: The elements in the latent vector can be recovered up to their permutation and scales. Next, we apply the identifiability theory to a statistical model for graph data. As a result, one of the identifiability conditions includes an appealing implication: Identifiability of the statistical model could depend on the maximum value of link weights in graph data. Then, we propose a practical method for identifiable graph embedding. Finally, we numerically demonstrate that the proposed method well-recovers the latent vectors and model identifiability clearly depends on the maximum value of link weights, which supports the implication of our theoretical results
Predictive posterior densities (PPDs) are of interest in approximate Bayesian inference. Typically, these are estimated by simple Monte Carlo (MC) averages using samples from the approximate posterior. We observe that the signal-to-noise ratio (SNR) of such estimators can be extremely low. An analysis for exact inference reveals SNR decays exponentially as there is an increase in (a) the mismatch between training and test data, (b) the dimensionality of the latent space, or (c) the size of the test data relative to the training data. Further analysis extends these results to approximate inference. To remedy the low SNR problem, we propose replacing simple MC sampling with importance sampling using a proposal distribution optimized at test time on a variational proxy for the SNR and demonstrate that this yields greatly improved estimates.
We present a class of high-order Eulerian-Lagrangian Runge-Kutta finite volume methods that can numerically solve Burgers' equation with shock formations, which could be extended to general scalar conservation laws. Eulerian-Lagrangian (EL) and semi-Lagrangian (SL) methods have recently seen increased development and have become a staple for allowing large time-stepping sizes. Yet, maintaining relatively large time-stepping sizes post shock formation remains quite challenging. Our proposed scheme integrates the partial differential equation on a space-time region partitioned by linear approximations to the characteristics determined by the Rankine-Hugoniot jump condition. We trace the characteristics forward in time and present a merging procedure for the mesh cells to handle intersecting characteristics due to shocks. Following this partitioning, we write the equation in a time-differential form and evolve with Runge-Kutta methods in a method-of-lines fashion. High-resolution methods such as ENO and WENO-AO schemes are used for spatial reconstruction. Extension to higher dimensions is done via dimensional splitting. Numerical experiments demonstrate our scheme's high-order accuracy and ability to sharply capture post-shock solutions with large time-stepping sizes.
The problem of binary hypothesis testing between two probability measures is considered. New sharp bounds are derived for the best achievable error probability of such tests based on independent and identically distributed observations. Specifically, the asymmetric version of the problem is examined, where different requirements are placed on the two error probabilities. Accurate nonasymptotic expansions with explicit constants are obtained for the error probability, using tools from large deviations and Gaussian approximation. Examples are shown indicating that, in the asymmetric regime, the approximations suggested by the new bounds are significantly more accurate than the approximations provided by either of the two main earlier approaches -- normal approximation and error exponents.
In many statistical modeling problems, such as classification and regression, it is common to encounter sparse and blocky coefficients. Sparse fused Lasso is specifically designed to recover these sparse and blocky structured features, especially in cases where the design matrix has ultrahigh dimensions, meaning that the number of features significantly surpasses the number of samples. Quantile loss is a well-known robust loss function that is widely used in statistical modeling. In this paper, we propose a new sparse fused lasso classification model, and develop a unified multi-block linearized alternating direction method of multipliers algorithm that effectively selects sparse and blocky features for regression and classification. Our algorithm has been proven to converge with a derived linear convergence rate. Additionally, our algorithm has a significant advantage over existing methods for solving ultrahigh dimensional sparse fused Lasso regression and classification models due to its lower time complexity. Note that the algorithm can be easily extended to solve various existing fused Lasso models. Finally, we present numerical results for several synthetic and real-world examples, which demonstrate the robustness, scalability, and accuracy of the proposed classification model and algorithm
In the analysis of spatially resolved transcriptomics data, detecting spatially variable genes (SVGs) is crucial. Numerous computational methods exist, but varying SVG definitions and methodologies lead to incomparable results. We review 31 state-of-the-art methods, categorizing SVGs into three types: overall, cell-type-specific, and spatial-domain-marker SVGs. Our review explains the intuitions underlying these methods, summarizes their applications, and categorizes the hypothesis tests they use in the trade-off between generality and specificity for SVG detection. We discuss challenges in SVG detection and propose future directions for improvement. Our review offers insights for method developers and users, advocating for category-specific benchmarking.
Optogenetics is widely used to study the effects of neural circuit manipulation on behavior. However, the paucity of causal inference methodological work on this topic has resulted in analysis conventions that discard information, and constrain the scientific questions that can be posed. To fill this gap, we introduce a nonparametric causal inference framework for analyzing "closed-loop" designs, which use dynamic policies that assign treatment based on covariates. In this setting, standard methods can introduce bias and occlude causal effects. Building on the sequentially randomized experiments literature in causal inference, our approach extends history-restricted marginal structural models for dynamic regimes. In practice, our framework can identify a wide range of causal effects of optogenetics on trial-by-trial behavior, such as, fast/slow-acting, dose-response, additive/antagonistic, and floor/ceiling. Importantly, it does so without requiring negative controls, and can estimate how causal effect magnitudes evolve across time points. From another view, our work extends "excursion effect" methods--popular in the mobile health literature--to enable estimation of causal contrasts for treatment sequences greater than length one, in the presence of positivity violations. We derive rigorous statistical guarantees, enabling hypothesis testing of these causal effects. We demonstrate our approach on data from a recent study of dopaminergic activity on learning, and show how our method reveals relevant effects obscured in standard analyses.
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