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Optimal feedback control (OFC) is a theory from the motor control literature that explains how humans move their body to achieve a certain goal, e.g., pointing with the finger. OFC is based on the assumption that humans aim to control their body optimally, within the constraints imposed by body, environment, and task. In this paper, we explain how this theory can be applied to understanding Human-Computer Interaction (HCI) in the case of pointing. We propose that the human body and computer dynamics can be interpreted as a single dynamical system. The system state is controlled by the user via muscle control signals, and estimated from observations. Between-trial variability arises from signal-dependent control noise and observation noise. We compare four different models from optimal control theory and evaluate to what degree these models can replicate movements in the case of mouse pointing. We introduce a procedure to identify parameters that best explain observed user behavior. To support HCI researchers in simulating, analyzing, and optimizing interaction movements, we provide the Python toolbox OFC4HCI. We conclude that OFC presents a powerful framework for HCI to understand and simulate motion of the human body and of the interface on a moment by moment basis.

Safety is critical in autonomous robotic systems. A safe control law ensures forward invariance of a safe set (a subset in the state space). It has been extensively studied regarding how to derive a safe control law with a control-affine analytical dynamic model. However, in complex environments and tasks, it is challenging and time-consuming to obtain a principled analytical model of the system. In these situations, data-driven learning is extensively used and the learned models are encoded in neural networks. How to formally derive a safe control law with Neural Network Dynamic Models (NNDM) remains unclear due to the lack of computationally tractable methods to deal with these black-box functions. In fact, even finding the control that minimizes an objective for NNDM without any safety constraint is still challenging. In this work, we propose MIND-SIS (Mixed Integer for Neural network Dynamic model with Safety Index Synthesis), the first method to derive safe control laws for NNDM. The method includes two parts: 1) SIS: an algorithm for the offline synthesis of the safety index (also called as barrier function), which uses evolutionary methods and 2) MIND: an algorithm for online computation of the optimal and safe control signal, which solves a constrained optimization using a computationally efficient encoding of neural networks. It has been theoretically proved that MIND-SIS guarantees forward invariance and finite convergence. And it has been numerically validated that MIND-SIS achieves safe and optimal control of NNDM. From our experiments, the optimality gap is less than $10^{-8}$, and the safety constraint violation is $0$.

Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution has conservation properties, i.e.: (1) no conservation, (2) subdomain conservation, (3) grid-level conservation, and (4) exact conservation. This treatment results in sharper and fully computable estimates when mass is conserved either at the grid level or exactly, with a comparable structure to those obtained from grid-based a posteriori techniques. We demonstrate the practical effectiveness of our theoretical results through numerical experiments using four different discretization methods for synthetic problems and applications based on benchmarks of flow in fractured porous media.

Covariance estimation for matrix-valued data has received an increasing interest in applications. Unlike previous works that rely heavily on matrix normal distribution assumption and the requirement of fixed matrix size, we propose a class of distribution-free regularized covariance estimation methods for high-dimensional matrix data under a separability condition and a bandable covariance structure. Under these conditions, the original covariance matrix is decomposed into a Kronecker product of two bandable small covariance matrices representing the variability over row and column directions. We formulate a unified framework for estimating bandable covariance, and introduce an efficient algorithm based on rank one unconstrained Kronecker product approximation. The convergence rates of the proposed estimators are established, and the derived minimax lower bound shows our proposed estimator is rate-optimal under certain divergence regimes of matrix size. We further introduce a class of robust covariance estimators and provide theoretical guarantees to deal with heavy-tailed data. We demonstrate the superior finite-sample performance of our methods using simulations and real applications from a gridded temperature anomalies dataset and a S&P 500 stock data analysis.

In this paper, we introduce reduced-bias estimators for the estimation of the tail index of a Pareto-type distribution. This is achieved through the use of a regularised weighted least squares with an exponential regression model for log-spacings of top order statistics. The asymptotic properties of the proposed estimators are investigated analytically and found to be asymptotically unbiased, consistent and normally distributed. Also, the finite sample behaviour of the estimators are studied through a simulations theory. The proposed estimators were found to yield low bias and MSE. In addition, the proposed estimators are illustrated through the estimation of the tail index of the underlying distribution of claims from the insurance industry.

While the theoretical analysis of evolutionary algorithms (EAs) has made significant progress for pseudo-Boolean optimization problems in the last 25 years, only sporadic theoretical results exist on how EAs solve permutation-based problems. To overcome the lack of permutation-based benchmark problems, we propose a general way to transfer the classic pseudo-Boolean benchmarks into benchmarks defined on sets of permutations. We then conduct a rigorous runtime analysis of the permutation-based $(1+1)$ EA proposed by Scharnow, Tinnefeld, and Wegener (2004) on the analogues of the \textsc{LeadingOnes} and \textsc{Jump} benchmarks. The latter shows that, different from bit-strings, it is not only the Hamming distance that determines how difficult it is to mutate a permutation $\sigma$ into another one $\tau$, but also the precise cycle structure of $\sigma \tau^{-1}$. For this reason, we also regard the more symmetric scramble mutation operator. We observe that it not only leads to simpler proofs, but also reduces the runtime on jump functions with odd jump size by a factor of $\Theta(n)$. Finally, we show that a heavy-tailed version of the scramble operator, as in the bit-string case, leads to a speed-up of order $m^{\Theta(m)}$ on jump functions with jump size~$m$.%

Binding operation is fundamental to many cognitive processes, such as cognitive map formation, relational reasoning, and language comprehension. In these processes, two different modalities, such as location and objects, events and their contextual cues, and words and their roles, need to be bound together, but little is known about the underlying neural mechanisms. Previous works introduced a binding model based on quadratic functions of bound pairs, followed by vector summation of multiple pairs. Based on this framework, we address following questions: Which classes of quadratic matrices are optimal for decoding relational structures? And what is the resultant accuracy? We introduce a new class of binding matrices based on a matrix representation of octonion algebra, an eight-dimensional extension of complex numbers. We show that these matrices enable a more accurate unbinding than previously known methods when a small number of pairs are present. Moreover, numerical optimization of a binding operator converges to this octonion binding. We also show that when there are a large number of bound pairs, however, a random quadratic binding performs as well as the octonion and previously-proposed binding methods. This study thus provides new insight into potential neural mechanisms of binding operations in the brain.

We recall some of the history of the information-theoretic approach to deriving core results in probability theory and indicate parts of the recent resurgence of interest in this area with current progress along several interesting directions. Then we give a new information-theoretic proof of a finite version of de Finetti's classical representation theorem for finite-valued random variables. We derive an upper bound on the relative entropy between the distribution of the first $k$ in a sequence of $n$ exchangeable random variables, and an appropriate mixture over product distributions. The mixing measure is characterised as the law of the empirical measure of the original sequence, and de Finetti's result is recovered as a corollary. The proof is nicely motivated by the Gibbs conditioning principle in connection with statistical mechanics, and it follows along an appealing sequence of steps. The technical estimates required for these steps are obtained via the use of a collection of combinatorial tools known within information theory as `the method of types.'

Classic machine learning methods are built on the $i.i.d.$ assumption that training and testing data are independent and identically distributed. However, in real scenarios, the $i.i.d.$ assumption can hardly be satisfied, rendering the sharp drop of classic machine learning algorithms' performances under distributional shifts, which indicates the significance of investigating the Out-of-Distribution generalization problem. Out-of-Distribution (OOD) generalization problem addresses the challenging setting where the testing distribution is unknown and different from the training. This paper serves as the first effort to systematically and comprehensively discuss the OOD generalization problem, from the definition, methodology, evaluation to the implications and future directions. Firstly, we provide the formal definition of the OOD generalization problem. Secondly, existing methods are categorized into three parts based on their positions in the whole learning pipeline, namely unsupervised representation learning, supervised model learning and optimization, and typical methods for each category are discussed in detail. We then demonstrate the theoretical connections of different categories, and introduce the commonly used datasets and evaluation metrics. Finally, we summarize the whole literature and raise some future directions for OOD generalization problem. The summary of OOD generalization methods reviewed in this survey can be found at //out-of-distribution-generalization.com.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.