Robots are notoriously difficult to design because of complex interdependencies between their physical structure, sensory and motor layouts, and behavior. Despite this, almost every detail of every robot built to date has been manually determined by a human designer after several months or years of iterative ideation, prototyping, and testing. Inspired by evolutionary design in nature, the automated design of robots using evolutionary algorithms has been attempted for two decades, but it too remains inefficient: days of supercomputing are required to design robots in simulation that, when manufactured, exhibit desired behavior. Here we show for the first time de-novo optimization of a robot's structure to exhibit a desired behavior, within seconds on a single consumer-grade computer, and the manufactured robot's retention of that behavior. Unlike other gradient-based robot design methods, this algorithm does not presuppose any particular anatomical form; starting instead from a randomly-generated apodous body plan, it consistently discovers legged locomotion, the most efficient known form of terrestrial movement. If combined with automated fabrication and scaled up to more challenging tasks, this advance promises near instantaneous design, manufacture, and deployment of unique and useful machines for medical, environmental, vehicular, and space-based tasks.
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As the development of formal proofs is a time-consuming task, it is important to devise ways of sharing the already written proofs to prevent wasting time redoing them. One of the challenges in this domain is to translate proofs written in proof assistants based on impredicative logics to proof assistants based on predicative logics, whenever impredicativity is not used in an essential way. In this paper we present a transformation for sharing proofs with a core predicative system supporting prenex universe polymorphism (like in Agda). It consists in trying to elaborate a potentially impredicative term into a predicative universe polymorphic term as general as possible. The use of universe polymorphism is justified by the fact that mapping each universe to a fixed one in the target theory is not sufficient in most cases. During the algorithm, we need to solve unification problems in the equational theory of universe levels. In order to do this, we give a complete characterization of when a single equation admits a most general unifier. This characterization is then employed in an algorithm which uses a constraint-postponement strategy to solve unification problems. The proposed translation is of course partial, but in practice allows one to translate many proofs that do not use impredicativity in an essential way. Indeed, it was implemented in the tool Predicativize and then used to translate semi-automatically many non-trivial developments from Matita's arithmetic library to Agda, including proofs of Bertrand's Postulate and Fermat's Little Theorem, which (as far as we know) were not available in Agda yet.
Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal $a$-$b$ separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P = NP.
Statistical techniques are needed to analyse data structures with complex dependencies such that clinically useful information can be extracted. Individual-specific networks, which capture dependencies in complex biological systems, are often summarized by graph-theoretical features. These features, which lend themselves to outcome modelling, can be subject to high variability due to arbitrary decisions in network inference and noise. Correlation-based adjacency matrices often need to be sparsified before meaningful graph-theoretical features can be extracted, requiring the data analysts to determine an optimal threshold.. To address this issue, we propose to incorporate a flexible weighting function over the full range of possible thresholds to capture the variability of graph-theoretical features over the threshold domain. The potential of this approach, which extends concepts from functional data analysis to a graph-theoretical setting, is explored in a plasmode simulation study using real functional magnetic resonance imaging (fMRI) data from the Autism Brain Imaging Data Exchange (ABIDE) Preprocessed initiative. The simulations show that our modelling approach yields accurate estimates of the functional form of the weight function, improves inference efficiency, and achieves a comparable or reduced root mean square prediction error compared to competitor modelling approaches. This assertion holds true in settings where both complex functional forms underlie the outcome-generating process and a universal threshold value is employed. We demonstrate the practical utility of our approach by using resting-state fMRI data to predict biological age in children. Our study establishes the flexible modelling approach as a statistically principled, serious competitor to ad-hoc methods with superior performance.
One of the central problems in neuroscience is understanding how brain structure relates to function. Naively one can relate the direct connections of white matter fiber tracts between brain regions of interest (ROIs) to the increased co-activation in the same pair of ROIs, but the link between structural and functional connectomes (SCs and FCs) has proven to be much more complex. To learn a realistic generative model characterizing population variation in SCs, FCs, and the SC-FC coupling, we develop a graph auto-encoder that we refer to as Staf-GATE. We trained Staf-GATE with data from the Human Connectome Project (HCP) and show state-of-the-art performance in predicting FC and joint generation of SC and FC. In addition, as a crucial component of the proposed approach, we provide a masking-based algorithm to extract interpretable inferences about SC-FC coupling. Our interpretation methods identified important SC subnetworks for FC coupling and relating SC and FC with sex.
This article investigates synthetic model-predictive control (MPC) problems to demonstrate that an increased precision of the internal prediction model (PM) automatially entails an improvement of the controller as a whole. In contrast to reinforcement learning (RL), MPC uses the PM to predict subsequent states of the controlled system (CS), instead of directly recommending suitable actions. To assess how the precision of the PM translates into the quality of the model-predictive controller, we compare a DNN-based PM to the optimal baseline PM for three well-known control problems of varying complexity. The baseline PM achieves perfect accuracy by accessing the simulation of the CS itself. Based on the obtained results, we argue that an improvement of the PM will always improve the controller as a whole, without considering the impact of other components such as action selection (which, in this article, relies on evolutionary optimization).
Conjugate gradient is an efficient algorithm for solving large sparse linear systems. It has been utilized to accelerate the computation in Bayesian analysis for many large-scale problems. This article discusses the applications of conjugate gradient in Bayesian computation, with a focus on sparse regression and spatial analysis. A self-contained introduction of conjugate gradient is provided to facilitate potential applications in a broader range of problems.
We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.
Advances in survival analysis have facilitated unprecedented flexibility in data modeling, yet there remains a lack of tools for graphically illustrating the influence of continuous covariates on predicted survival outcomes. We propose the utilization of a colored contour plot to depict the predicted survival probabilities over time, and provide a Shiny app and R package as implementations of this tool. Our approach is capable of supporting conventional models, including the Cox and Fine-Gray models. However, its capability shines when coupled with cutting-edge machine learning models such as deep neural networks.
We study various aspects of the first-order transduction quasi-order, which provides a way of measuring the relative complexity of classes of structures based on whether one can encode the other using a formula of first-order (FO) logic. In contrast with the conjectured simplicity of the transduction quasi-order for monadic second-order logic, the FO-transduction quasi-order is very complex; in particular, we prove that the quotient partial order is not a lattice, although it is a bounded distributive join-semilattice, as is the subposet of additive classes. Many standard properties from structural graph theory and model theory naturally appear in this quasi-order. For example, we characterize transductions of paths, cubic graphs, and cubic trees in terms of bandwidth, bounded degree, and treewidth. We establish that the classes of all graphs with pathwidth at most~$k$, for $k\geq 1$, form a strict hierarchy in the FO-transduction quasi-order and leave open whether same is true for treewidth. This leads to considering whether properties admit maximum or minimum classes in this quasi-order. We prove that many properties do not admit a maximum class, and that star forests are the minimum class that is not a transduction of a class with bounded degree, which can be seen as an instance of transduction duality. We close with a notion of dense analogues of sparse classes, and discuss several related conjectures. As a ubiquitous tool in our results, we prove a normal form for FO-transductions that manifests the locality of FO logic. This is among several other technical results about FO-transductions which we anticipate being broadly useful.
Counterfactual prediction methods are required when a model will be deployed in a setting where treatment policies differ from the setting where the model was developed, or when the prediction question is explicitly counterfactual. However, estimating and evaluating counterfactual prediction models is challenging because one does not observe the full set of potential outcomes for all individuals. Here, we discuss how to tailor a model to a counterfactual estimand, how to assess the model's performance, and how to perform model and tuning parameter selection. We also provide identifiability results for measures of performance for a potentially misspecified counterfactual prediction model based on training and test data from the same (factual) source population. Last, we illustrate the methods using simulation and apply them to the task of developing a statin-na\"{i}ve risk prediction model for cardiovascular disease.
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