The ellipsoid method is an algorithm that solves the (weak) feasibility and linear optimization problems for convex sets by making oracle calls to their (weak) separation problem. We observe that the previously known method for showing that this reduction can be done in fixed-point logic with counting (FPC) for linear and semidefinite programs applies to any family of explicitly bounded convex sets. We use this observation to show that the exact feasibility problem for semidefinite programs is expressible in the infinitary version of FPC. As a corollary we get that, for the isomorphism problem, the Lasserre/Sums-of-Squares semidefinite programming hierarchy of relaxations collapses to the Sherali-Adams linear programming hierarchy, up to a small loss in the degree.
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We describe two algorithms for multiplying n x n matrices using time and energy n^2 polylog(n) under basic models of classical physics. The first algorithm is for multiplying integer-valued matrices, and the second, quite different algorithm, is for Boolean matrix multiplication. We hope this work inspires a deeper consideration of physically plausible/realizable models of computing that might allow for algorithms which improve upon the runtimes and energy usages suggested by the parallel RAM model in which each operation requires one unit of time and one unit of energy.
Cosmological simulations play a crucial role in elucidating the effect of physical parameters on the statistics of fields and on constraining parameters given information on density fields. We leverage diffusion generative models to address two tasks of importance to cosmology -- as an emulator for cold dark matter density fields conditional on input cosmological parameters $\Omega_m$ and $\sigma_8$, and as a parameter inference model that can return constraints on the cosmological parameters of an input field. We show that the model is able to generate fields with power spectra that are consistent with those of the simulated target distribution, and capture the subtle effect of each parameter on modulations in the power spectrum. We additionally explore their utility as parameter inference models and find that we can obtain tight constraints on cosmological parameters.
We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, demonstrates enhanced generalization, sample efficiency, and interpretability, with less trainable parameters and computational costs.
Morphing quadrotors with four external actuators can adapt to different restricted scenarios by changing their geometric structure. However, previous works mainly focus on the improvements in structures and controllers, and existing planning algorithms don't consider the morphological modifications, which leads to safety and dynamic feasibility issues. In this paper, we propose a unified planning and control framework for morphing quadrotors to deform autonomously and efficiently. The framework consists of a milliseconds-level spatial-temporal trajectory optimizer that takes into account the morphological modifications of quadrotors. The optimizer can generate full-body safety trajectories including position and attitude. Additionally, it incorporates a nonlinear attitude controller that accounts for aerodynamic drag and dynamically adjusts dynamic parameters such as the inertia tensor and Center of Gravity. The controller can also online compute the thrust coefficient during morphing. Benchmark experiments compared with existing methods validate the robustness of the proposed controller. Extensive simulations and real-world experiments are performed to demonstrate the effectiveness of the proposed framework.
The purpose of this work is to present an effective tool for computing different QR-decompositions of a complex nonsingular square matrix. The concept of the discrete signal-induced heap transform (DsiHT, Grigoryan 2006) is used. This transform is fast, has a unique algorithm for any length of the input vector/signal and can be used with different complex basic 2x2 transforms. The DsiHT zeroes all components of the input signal while moving or heaping the energy of the signal into one component, such as the first. We describe three different types of QR-decompositions that use the basic transforms with the T, G, and M-type complex matrices we introduce, and also without matrices, but using analytical formulas. We also present the mixed QR-decomposition, when different type DsiHTs are used at different stages of the algorithm. The number of such decompositions is greater than 3^((N-1)), for an NxN complex matrix. Examples of the QR-decomposition are described in detail for the 4x4 and 6x6 complex matrices and compared with the known method of Householder transforms. The precision of the QR-decompositions of NxN matrices, when N are 6, 13, 17, 19, 21, 40, 64, 100, 128, 201, 256, and 400 is also compared. The MATLAB-based scripts of the codes for QR-decompositions by the described DsiHTs are given.
Nonparametric estimates of frequency response functions (FRFs) are often suitable for describing the dynamics of a mechanical system. If treating these estimates as measurement inputs, they can be used for parametric identification of, e.g., a gray-box model. Classical methods for nonparametric FRF estimation of MIMO systems require at least as many experiments as the system has inputs. Local parametric FRF estimation methods have been developed for avoiding multiple experiments. In this paper, these local methods are adapted and applied for estimating the FRFs of a 6-axes robotic manipulator, which is a nonlinear MIMO system operating in closed loop. The aim is to reduce the experiment time and amount of data needed for identification. The resulting FRFs are analyzed in an experimental study and compared to estimates obtained by classical MIMO techniques. It is furthermore shown that an accurate parametric model identification is possible based on local parametric FRF estimates and that the total experiment time can be significantly reduced.
To assess the quality of a probabilistic prediction for stochastic dynamical systems (SDSs), scoring rules assign a numerical score based on the predictive distribution and the measured state. In this paper, we propose an $\epsilon$-logarithm score that generalizes the celebrated logarithm score by considering a neighborhood with radius $\epsilon$. To begin with, we prove that the $\epsilon$-logarithm score is proper (the expected score is optimized when the predictive distribution meets the ground truth) based on discrete approximations. Then, we characterize the probabilistic predictability of an SDS by the optimal expected score and approximate it with an error of scale $\mathcal{O}(\epsilon)$. The approximation quantitatively shows how the system predictability is jointly determined by the neighborhood radius, the differential entropies of process noises, and the system dimension. In addition to the expected score, we also analyze the asymptotic behaviors of the score on individual trajectories. Specifically, we prove that the score on a trajectory will converge to the probabilistic predictability when the process noises are independent and identically distributed. Moreover, the convergence speed against the trajectory length $T$ is of scale $\mathcal{O}(T^{-\frac{1}{2}})$ in the sense of probability. Finally, we apply the predictability analysis to design unpredictable SDSs. Numerical examples are given to elaborate the results.
The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.
The existence of representative datasets is a prerequisite of many successful artificial intelligence and machine learning models. However, the subsequent application of these models often involves scenarios that are inadequately represented in the data used for training. The reasons for this are manifold and range from time and cost constraints to ethical considerations. As a consequence, the reliable use of these models, especially in safety-critical applications, is a huge challenge. Leveraging additional, already existing sources of knowledge is key to overcome the limitations of purely data-driven approaches, and eventually to increase the generalization capability of these models. Furthermore, predictions that conform with knowledge are crucial for making trustworthy and safe decisions even in underrepresented scenarios. This work provides an overview of existing techniques and methods in the literature that combine data-based models with existing knowledge. The identified approaches are structured according to the categories integration, extraction and conformity. Special attention is given to applications in the field of autonomous driving.
We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.
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