We study interpolation inequalities between H\"older Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $\mu$ and $\mu^\star$ have $\beta$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and $\mathcal{M}^\star$ respectively, then the H\"older IPMs $d_{\mathcal{H}^\gamma_1}$ of smoothness $\gamma\geq 1$ and $d_{\mathcal{H}^\eta_1}$ of smoothness $\eta>\gamma$, satisfy $d_{ \mathcal{H}_1^{\gamma}}(\mu,\mu^\star)\lesssim d_{ \mathcal{H}_1^{\eta}}(\mu,\mu^\star)^\frac{\beta+\gamma}{\beta+\eta}$, up to logarithmic factors. We provide an application of this result to high-dimensional inference. These functional inequalities turn out to be a key tool for density estimation on unknown submanifold. In particular, it allows to build the first estimator attaining optimal rates of estimation for all the distances $d_{\mathcal{H}_1^\gamma}$, $\gamma \in [1,\infty)$ simultaneously.
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Automating psychological hypothesis generation with AI: when large language models meet causal graph
Leveraging the synergy between causal knowledge graphs and a large language model (LLM), our study introduces a groundbreaking approach for computational hypothesis generation in psychology. We analyzed 43,312 psychology articles using a LLM to extract causal relation pairs. This analysis produced a specialized causal graph for psychology. Applying link prediction algorithms, we generated 130 potential psychological hypotheses focusing on `well-being', then compared them against research ideas conceived by doctoral scholars and those produced solely by the LLM. Interestingly, our combined approach of a LLM and causal graphs mirrored the expert-level insights in terms of novelty, clearly surpassing the LLM-only hypotheses (t(59) = 3.34, p=0.007 and t(59) = 4.32, p<0.001, respectively). This alignment was further corroborated using deep semantic analysis. Our results show that combining LLM with machine learning techniques such as causal knowledge graphs can revolutionize automated discovery in psychology, extracting novel insights from the extensive literature. This work stands at the crossroads of psychology and artificial intelligence, championing a new enriched paradigm for data-driven hypothesis generation in psychological research.
As Neural Radiance Field (NeRF) implementations become faster, more efficient and accurate, their applicability to real world mapping tasks becomes more accessible. Traditionally, 3D mapping, or scene reconstruction, has relied on expensive LiDAR sensing. Photogrammetry can perform image-based 3D reconstruction but is computationally expensive and requires extremely dense image representation to recover complex geometry and photorealism. NeRFs perform 3D scene reconstruction by training a neural network on sparse image and pose data, achieving superior results to photogrammetry with less input data. This paper presents an evaluation of two NeRF scene reconstructions for the purpose of estimating the diameter of a vertical PVC cylinder. One of these are trained on commodity iPhone data and the other is trained on robot-sourced imagery and poses. This neural-geometry is compared to state-of-the-art lidar-inertial SLAM in terms of scene noise and metric-accuracy.
This study proposes a unified theory and statistical learning approach for traffic conflict detection, addressing the long-existing call for a consistent and comprehensive methodology to evaluate the collision risk emerged in road user interactions. The proposed theory assumes a context-dependent probabilistic collision risk and frames conflict detection as estimating the risk by statistical learning from observed proximities and contextual variables. Three primary tasks are integrated: representing interaction context from selected observables, inferring proximity distributions in different contexts, and applying extreme value theory to relate conflict intensity with conflict probability. As a result, this methodology is adaptable to various road users and interaction scenarios, enhancing its applicability without the need for pre-labelled conflict data. Demonstration experiments are executed using real-world trajectory data, with the unified metric trained on lane-changing interactions on German highways and applied to near-crash events from the 100-Car Naturalistic Driving Study in the U.S. The experiments demonstrate the methodology's ability to provide effective collision warnings, generalise across different datasets and traffic environments, cover a broad range of conflicts, and deliver a long-tailed distribution of conflict intensity. This study contributes to traffic safety by offering a consistent and explainable methodology for conflict detection applicable across various scenarios. Its societal implications include enhanced safety evaluations of traffic infrastructures, more effective collision warning systems for autonomous and driving assistance systems, and a deeper understanding of road user behaviour in different traffic conditions, contributing to a potential reduction in accident rates and improving overall traffic safety.
We introduce a new observational setting for Positive Unlabeled (PU) data where the observations at prediction time are also labeled. This occurs commonly in practice -- we argue that the additional information is important for prediction, and call this task "augmented PU prediction". We allow for labeling to be feature dependent. In such scenario, Bayes classifier and its risk is established and compared with a risk of a classifier which for unlabeled data is based only on predictors. We introduce several variants of the empirical Bayes rule in such scenario and investigate their performance. We emphasise dangers (and ease) of applying classical classification rule in the augmented PU scenario -- due to no preexisting studies, an unaware researcher is prone to skewing the obtained predictions. We conclude that the variant based on recently proposed variational autoencoder designed for PU scenario works on par or better than other considered variants and yields advantage over feature-only based methods in terms of accuracy for unlabeled samples.
Generalised quantifiers, which include Henkin's branching quantifiers, have been introduced by Mostowski and Lindstr\"om and developed as a substantial topic application of logic, especially model theory, to linguistics with work by Barwise, Cooper, Keenan. In this paper, we mainly study the proof theory of some non-standard quantifiers as second order formulae . Our first example is the usual pair of first order quantifiers (for all / there exists) when individuals are viewed as individual concepts handled by second order deductive rules. Our second example is the study of a second order translation of the simplest branching quantifier: ``A member of each team and a member of each board of directors know each other", for which we propose a second order treatment.
This article presents a concise proof of the famous Benford's law when the distribution has a Riemann integrable probability density function and provides a criterion to judge whether a distribution obeys the law. The proof is intuitive and elegant, accessible to anyone with basic knowledge of calculus, revealing that the law originates from the basic property of the human number system. The criterion can bring great convenience to the field of fraud detection.
The C-Orientation problem asks whether it is possible to orient an undirected graph to a directed phylogenetic network of a desired class C, and to find such an orientation if one exists. The problem can arise when visualising evolutionary data, for example, because popular phylogenetic network reconstruction methods such as Neighbor-Net are distance-based and thus inevitably produce undirected graphs. The complexity of C-Orientation remains open for many classes C, including binary tree-child networks, and practical methods are still lacking. In this paper, we propose an exponential but practically efficient FPT algorithm for C-Orientation, which is parameterised by the reticulation number and the maximum size of minimal basic cycles used in the computation. We also present a very fast heuristic for Tree-Child Orientation. To evaluate the empirical performance of the proposed methods, we compared their accuracy and execution time for Tree-Child Orientation with those of an exponential time C-orientation algorithm from the literature. Our experiments show that the proposed exact algorithm is significantly faster than the state-of-the-art exponential time algorithm. The proposed heuristic runs even faster but the accuracy decreases as the reticulation number increases.
It is a notorious open question whether integer programs (IPs), with an integer coefficient matrix $M$ whose subdeterminants are all bounded by a constant $\Delta$ in absolute value, can be solved in polynomial time. We answer this question in the affirmative if we further require that, by removing a constant number of rows and columns from $M$, one obtains a submatrix $A$ that is the transpose of a network matrix. Our approach focuses on the case where $A$ arises from $M$ after removing $k$ rows only, where $k$ is a constant. We achieve our result in two main steps, the first related to the theory of IPs and the second related to graph minor theory. First, we derive a strong proximity result for the case where $A$ is a general totally unimodular matrix: Given an optimal solution of the linear programming relaxation, an optimal solution to the IP can be obtained by finding a constant number of augmentations by circuits of $[A\; I]$. Second, for the case where $A$ is transpose of a network matrix, we reformulate the problem as a maximum constrained integer potential problem on a graph $G$. We observe that if $G$ is $2$-connected, then it has no rooted $K_{2,t}$-minor for $t = \Omega(k \Delta)$. We leverage this to obtain a tree-decomposition of $G$ into highly structured graphs for which we can solve the problem locally. This allows us to solve the global problem via dynamic programming.
In this work we study the numerical approximation of a class of ergodic Backward Stochastic Differential Equations. These equations are formulated in an infinite horizon framework and provide a probabilistic representation for elliptic Partial Differential Equations of ergodic type. In order to build our numerical scheme, we put forward a new representation of the PDE solution by using a classical probabilistic representation of the gradient. Then, based on this representation, we propose a fully implementable numerical scheme using a Picard iteration procedure, a grid space discretization and a Monte-Carlo approximation. Up to a limiting technical condition that guarantee the contraction of the Picard procedure, we obtain an upper bound for the numerical error. We also provide some numerical experiments that show the efficiency of this approach for small dimensions.
Motivated by the application of saddlepoint approximations to resampling-based statistical tests, we prove that a Lugananni-Rice style approximation for conditional tail probabilities of averages of conditionally independent random variables has vanishing relative error. We also provide a general condition on the existence and uniqueness of the solution to the corresponding saddlepoint equation. The results are valid under a broad class of distributions involving no restrictions on the smoothness of the distribution function. The derived saddlepoint approximation formula can be directly applied to resampling-based hypothesis tests, including bootstrap, sign-flipping and conditional randomization tests. Our results extend and connect several classical saddlepoint approximation results. On the way to proving our main results, we prove a new conditional Berry-Esseen inequality for the sum of conditionally independent random variables, which may be of independent interest.
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