This paper addresses local path re-planning for $n$-dimensional systems by introducing an informed sampling scheme and cost function to achieve collision avoidance with minimum deviation from an (optimal) nominal path. The proposed informed subset consists of the union of ellipsoids along the specified nominal path, such that the subset efficiently encapsulates all points along the nominal path. The cost function penalizes large deviations from the nominal path, thereby ensuring current safety in the face of potential collisions while retaining most of the overall efficiency of the nominal path. The proposed method is demonstrated on scenarios related to the navigation of autonomous marine crafts.
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Gradient-based first-order convex optimization algorithms find widespread applicability in a variety of domains, including machine learning tasks. Motivated by the recent advances in fixed-time stability theory of continuous-time dynamical systems, we introduce a generalized framework for designing accelerated optimization algorithms with strongest convergence guarantees that further extend to a subclass of non-convex functions. In particular, we introduce the \emph{GenFlow} algorithm and its momentum variant that provably converge to the optimal solution of objective functions satisfying the Polyak-{\L}ojasiewicz (PL) inequality, in a fixed-time. Moreover for functions that admit non-degenerate saddle-points, we show that for the proposed GenFlow algorithm, the time required to evade these saddle-points is bounded uniformly for all initial conditions. Finally, for strongly convex-strongly concave minimax problems whose optimal solution is a saddle point, a similar scheme is shown to arrive at the optimal solution again in a fixed-time. The superior convergence properties of our algorithm are validated experimentally on a variety of benchmark datasets.
We develop the theoretical foundations of a generalized Gromov-Hausdorff distance between functions on networks that has recently been applied to various subfields of topological data analysis and optimal transport. These functional representations of networks, or networks for short, specialize in the finite setting to (possibly asymmetric) adjacency matrices and derived representations such as distance or kernel matrices. Existing literature utilizing these constructions cannot, however, benefit from continuous formulations because the continuum limits of finite networks under this distance are not well-understood. For example, while there are currently numerous persistent homology methods on finite networks, it is unclear if these methods produce well-defined persistence diagrams in the infinite setting. We resolve this situation by introducing the collection of compact networks that arises by taking continuum limits of finite networks and developing sampling results showing that this collection admits well-defined persistence diagrams. Compared to metric spaces, the isomorphism class of the generalized Gromov-Hausdorff distance over networks is rather complex, and contains representatives having different cardinalities and different topologies. We provide an exact characterization of a suitable notion of isomorphism for compact networks as well as alternative, stronger characterizations under additional topological regularity assumptions. Toward data applications, we describe a unified framework for developing quantitatively stable network invariants, provide basic examples, and cast existing results on the stability of persistent homology methods in this extended framework. To illustrate our theoretical results, we introduce a model of directed circles with finite reversibility and characterize their Dowker persistence diagrams.
Derived datasets can be defined implicitly or explicitly. An implicit definition (of dataset O in terms of datasets I) is a logical specification involving the source data I and the interface data O. It is a valid definition of O in terms of I, if any two models of the specification agreeing on I agree on O. In contrast, an explicit definition is a query that produces O from I. Variants of Beth's theorem state that one can convert implicit definitions to explicit ones. Further, this conversion can be done effectively given a proof witnessing implicit definability in a suitable proof system. We prove the analogous implicit-to-explicit result for nested relations: implicit definitions, given in the natural logic for nested relations, can be converted to explicit definitions in the nested relational calculus (NRC) We first provide a model-theoretic argument for this result, which makes some additional connections that may be of independent interest. between NRC queries, interpretations, a standard mechanisms for defining structure-to-structure translation in logic, and between interpretations and implicit to definability ``up to unique isomorphism''. The latter connection makes use of a variation of a result of Gaifman. We also provide a proof-theoretic result that provides an effective argument: from a proof witnessing implicit definability, we can efficiently produce an NRC definition. This will involve introducing the appropriate proof system for reasoning with nested sets, along with some auxiliary Beth-type results for this system. As a consequence, we can effectively extract rewritings of NRC queries in terms of NRC views, given a proof witnessing that the query is determined by the views.
In this paper, we consider the problem where a drone has to collect semantic information to classify multiple moving targets. In particular, we address the challenge of computing control inputs that move the drone to informative viewpoints, position and orientation, when the information is extracted using a ``black-box'' classifier, e.g., a deep learning neural network. These algorithms typically lack of analytical relationships between the viewpoints and their associated outputs, preventing their use in information-gathering schemes. To fill this gap, we propose a novel attention-based architecture, trained via Reinforcement Learning (RL), that outputs the next viewpoint for the drone favoring the acquisition of evidence from as many unclassified targets as possible while reasoning about their movement, orientation, and occlusions. Then, we use a low-level MPC controller to move the drone to the desired viewpoint taking into account its actual dynamics. We show that our approach not only outperforms a variety of baselines but also generalizes to scenarios unseen during training. Additionally, we show that the network scales to large numbers of targets and generalizes well to different movement dynamics of the targets.
Exploratory factor analysis (EFA) has been widely used to learn the latent structure underlying multivariate data. Rotation and regularised estimation are two classes of methods in EFA that are widely used to find interpretable loading matrices. This paper proposes a new family of oblique rotations based on component-wise $L^p$ loss functions $(0 < p\leq 1)$ that is closely related to an $L^p$ regularised estimator. Model selection and post-selection inference procedures are developed based on the proposed rotation method. When the true loading matrix is sparse, the proposed method tends to outperform traditional rotation and regularised estimation methods in terms of statistical accuracy and computational cost. Since the proposed loss functions are non-smooth, an iteratively reweighted gradient projection algorithm is developed to solve the optimisation problem. Theoretical results are developed that establish the statistical consistency of the estimation, model selection, and post-selection inference. The proposed method is evaluated and compared with regularised estimation and traditional rotation methods via simulation studies. It is further illustrated by an application to the big-five personality assessment.
This paper considers the problem of kernel regression and classification with possibly unobservable response variables in the data, where the mechanism that causes the absence of information is unknown and can depend on both predictors and the response variables. Our proposed approach involves two steps: In the first step, we construct a family of models (possibly infinite dimensional) indexed by the unknown parameter of the missing probability mechanism. In the second step, a search is carried out to find the empirically optimal member of an appropriate cover (or subclass) of the underlying family in the sense of minimizing the mean squared prediction error. The main focus of the paper is to look into the theoretical properties of these estimators. The issue of identifiability is also addressed. Our methods use a data-splitting approach which is quite easy to implement. We also derive exponential bounds on the performance of the resulting estimators in terms of their deviations from the true regression curve in general Lp norms, where we also allow the size of the cover or subclass to diverge as the sample size n increases. These bounds immediately yield various strong convergence results for the proposed estimators. As an application of our findings, we consider the problem of statistical classification based on the proposed regression estimators and also look into their rates of convergence under different settings. Although this work is mainly stated for kernel-type estimators, they can also be extended to other popular local-averaging methods such as nearest-neighbor estimators, and histogram estimators.
We consider the idealized setting of gradient flow on the population risk for infinitely wide two-layer ReLU neural networks (without bias), and study the effect of symmetries on the learned parameters and predictors. We first describe a general class of symmetries which, when satisfied by the target function $f^*$ and the input distribution, are preserved by the dynamics. We then study more specific cases. When $f^*$ is odd, we show that the dynamics of the predictor reduces to that of a (non-linearly parameterized) linear predictor, and its exponential convergence can be guaranteed. When $f^*$ has a low-dimensional structure, we prove that the gradient flow PDE reduces to a lower-dimensional PDE. Furthermore, we present informal and numerical arguments that suggest that the input neurons align with the lower-dimensional structure of the problem.
Non-orthogonal multiple access (NOMA) assisted semi-grant-free (SGF) transmission has recently received significant research attention due to its outstanding ability of serving grant-free (GF) users with grant-based (GB) users' spectrum, which greatly improves the spectrum efficiency and effectively relieves the massive access problem of 5G and beyond networks. In this paper, we first study the outage performance of the greedy best user scheduling SGF scheme (BU-SGF) by considering the impacts of Rayleigh fading, path loss, and random user locations. In order to tackle the admission fairness problem of the BU-SGF scheme, we propose a fair SGF scheme by applying cumulative distribution function (CDF)-based scheduling (CS-SGF), in which the GF user with the best channel relative to its own statistics will be admitted. Moreover, by employing the theories of order statistics and stochastic geometry, the outage performances of both BU-SGF and CS-SGF schemes are analyzed. Theoretical results show that both schemes can achieve full diversity orders only when the served users' data rate is capped, which severely limits the rate performance of SGF schemes. To further address this issue, we propose a distributed power control strategy to relax such data rate constraint, and derive analytical expressions of the two schemes' outage performances under this strategy. Finally, simulation results validate the fairness performance of the proposed CS-SGF scheme, the effectiveness of the power control strategy, and the accuracy of the theoretical analyses.
We propose a novel method for automatic reasoning on knowledge graphs based on debate dynamics. The main idea is to frame the task of triple classification as a debate game between two reinforcement learning agents which extract arguments -- paths in the knowledge graph -- with the goal to promote the fact being true (thesis) or the fact being false (antithesis), respectively. Based on these arguments, a binary classifier, called the judge, decides whether the fact is true or false. The two agents can be considered as sparse, adversarial feature generators that present interpretable evidence for either the thesis or the antithesis. In contrast to other black-box methods, the arguments allow users to get an understanding of the decision of the judge. Since the focus of this work is to create an explainable method that maintains a competitive predictive accuracy, we benchmark our method on the triple classification and link prediction task. Thereby, we find that our method outperforms several baselines on the benchmark datasets FB15k-237, WN18RR, and Hetionet. We also conduct a survey and find that the extracted arguments are informative for users.
In recent years, mobile devices have gained increasingly development with stronger computation capability and larger storage. Some of the computation-intensive machine learning and deep learning tasks can now be run on mobile devices. To take advantage of the resources available on mobile devices and preserve users' privacy, the idea of mobile distributed machine learning is proposed. It uses local hardware resources and local data to solve machine learning sub-problems on mobile devices, and only uploads computation results instead of original data to contribute to the optimization of the global model. This architecture can not only relieve computation and storage burden on servers, but also protect the users' sensitive information. Another benefit is the bandwidth reduction, as various kinds of local data can now participate in the training process without being uploaded to the server. In this paper, we provide a comprehensive survey on recent studies of mobile distributed machine learning. We survey a number of widely-used mobile distributed machine learning methods. We also present an in-depth discussion on the challenges and future directions in this area. We believe that this survey can demonstrate a clear overview of mobile distributed machine learning and provide guidelines on applying mobile distributed machine learning to real applications.
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