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We advance the state of the art in Mixed-Integer Linear Programming (MILP) formulations for Guillotine 2D Cutting Problems by (i) adapting a previously known reduction to our preprocessing phase and by (ii) enhancing a previous formulation by cutting down its size and symmetries. Our focus is the Guillotine 2D Knapsack Problem with orthogonal and unrestricted cuts, constrained demand, unlimited stages, and no rotation -- however, the formulation may be adapted to many related problems. The code is available. Concerning the set of 59 instances used to benchmark the original formulation, and summing the statistics for all models generated, the enhanced formulation has only a small fraction of the variables and constraints of the original model (respectively, 3.07% and 8.35%). The enhanced formulation also takes about 4 hours to solve all instances while the original formulation takes 12 hours to solve 53 of them (the other six runs hit a three-hour time limit each). We integrate, to both formulations, a pricing framework proposed for the original formulation; the enhanced formulation keeps a significant advantage in this situation. Finally, in a recently proposed set of 80 harder instances, the enhanced formulation (with and without the pricing framework) found: 22 optimal solutions for the unrestricted problem (5 already known, 17 new); 22 optimal solutions for the restricted problem (all are new and they are not the same 22 of the optimal unrestricted solutions); better lower bounds for 25 instances; better upper bounds for 58 instances.

Stochastic approximation (SA) and stochastic gradient descent (SGD) algorithms are work-horses for modern machine learning algorithms. Their constant stepsize variants are preferred in practice due to fast convergence behavior. However, constant step stochastic iterative algorithms do not converge asymptotically to the optimal solution, but instead have a stationary distribution, which in general cannot be analytically characterized. In this work, we study the asymptotic behavior of the appropriately scaled stationary distribution, in the limit when the constant stepsize goes to zero. Specifically, we consider the following three settings: (1) SGD algorithms with smooth and strongly convex objective, (2) linear SA algorithms involving a Hurwitz matrix, and (3) nonlinear SA algorithms involving a contractive operator. When the iterate is scaled by $1/\sqrt{\alpha}$, where $\alpha$ is the constant stepsize, we show that the limiting scaled stationary distribution is a solution of an integral equation. Under a uniqueness assumption (which can be removed in certain settings) on this equation, we further characterize the limiting distribution as a Gaussian distribution whose covariance matrix is the unique solution of a suitable Lyapunov equation. For SA algorithms beyond these cases, our numerical experiments suggest that unlike central limit theorem type results: (1) the scaling factor need not be $1/\sqrt{\alpha}$, and (2) the limiting distribution need not be Gaussian. Based on the numerical study, we come up with a formula to determine the right scaling factor, and make insightful connection to the Euler-Maruyama discretization scheme for approximating stochastic differential equations.

In this note, we investigate how well we can reconstruct the best rank-$r$ approximation of a large matrix from a small number of its entries. We show that even if a data matrix is of full rank and cannot be approximated well by a low-rank matrix, its best low-rank approximations may still be reliably computed or estimated from a small number of its entries. This is especially relevant from a statistical viewpoint: the best low-rank approximations to a data matrix are often of more interest than itself because they capture the more stable and oftentimes more reproducible properties of an otherwise complicated data-generating model. In particular, we investigate two agnostic approaches: the first is based on spectral truncation; and the second is a projected gradient descent based optimization procedure. We argue that, while the first approach is intuitive and reasonably effective, the latter has far superior performance in general. We show that the error depends on how close the matrix is to being of low rank. Both theoretical and numerical evidence is presented to demonstrate the effectiveness of the proposed approaches.

The arrival of Immersive Virtual and Augmented Reality hardware to the consumer market suggests seamless multi-modal communication between human participants and autonomous interactive characters is an achievable goal in the near future. This possibility is further reinforced by the rapid improvements in the automated analysis of speech, facial expressions and body language, as well as improvements in character animation and speech synthesis techniques. However, we do not have a formal theory that allows us to compare, on one side, interactive social scenarios among human users and autonomous virtual characters and, on the other side, pragmatic inference mechanisms as they occur in non-mediated communication. Grices' and Sperbers' model of inferential communication does explain the nature of everyday communication through cognitive mechanisms that support spontaneous inferences performed in pragmatic communication. However, such a theory is not based on a mathematical framework with a precision comparable to classical information theory. To address this gap, in this article we introduce a Mathematical Theory of Inferential Communication (MaTIC). MaTIC formalises some assumptions of inferential communication, it explores its theoretical consequences and outlines the practical steps needed to use it in different application scenarios.

Source identification problems have multiple applications in engineering such as the identification of fissures in materials, determination of sources in electromagnetic fields or geophysical applications, detection of contaminant sources, among others. In this work we are concerned with the determination of a time-dependent source in a transport equation from noisy data measured at a fixed position. By means of Fourier techniques can be shown that the problem is ill-posed in the sense that the solution exists but it does not vary continuously with the data. A number of different techniques were developed by other authors to approximate the solution. In this work, we consider a family of parametric regularization operators to deal with the ill-posedness of the problem. We proposed a manner to select the regularization parameter as a function of noise level in data in order to obtain a regularized solution that approximate the unknown source. We find a H\"older type bound for the error of the approximated source when the unknown function is considered to be bounded in a given norm. Numerical examples illustrate the convergence and stability of the method.

Regulating artificial intelligence (AI) has become necessary in light of its deployment in high-risk scenarios. This paper explores the proposal to extend legal personhood to AI and robots, which had not yet been examined through the lens of the general public. We present two studies (N = 3,559) to obtain people's views of electronic legal personhood vis-\`a-vis existing liability models. Our study reveals people's desire to punish automated agents even though these entities are not recognized any mental state. Furthermore, people did not believe automated agents' punishment would fulfill deterrence nor retribution and were unwilling to grant them legal punishment preconditions, namely physical independence and assets. Collectively, these findings suggest a conflict between the desire to punish automated agents and its perceived impracticability. We conclude by discussing how future design and legal decisions may influence how the public reacts to automated agents' wrongdoings.

The automatic road roller, as a popular type of construction robot, has attracted much interest from both the industry and the research community in recent years. However, when it comes to tunnels where the degeneration issues are prone to happen, it is still a challenging problem to provide an accurate positioning result for the robot. In this paper, we aim to deal with this problem by fusing LiDAR and UWB measurements based on optimization. In the proposed localization method, the directions of non-degeneration will be constrained and the covariance of UWB reconstruction will be introduced to improve the accuracy of localization. Apart from these, a method that can extract the feature of the inner wall of tunnels to assist positioning is also presented in this paper. To evaluate the effectiveness of the proposed method, three experiments with real road roller were carried out and the results show that our method can achieve better performance than the existing methods and can be applied to automatic road roller working inside tunnels. Finally, we discuss the feasibility of deploying the system in real applications and make several recommendations.

Attention-based encoder-decoder architectures such as Listen, Attend, and Spell (LAS), subsume the acoustic, pronunciation and language model components of a traditional automatic speech recognition (ASR) system into a single neural network. In our previous work, we have shown that such architectures are comparable to state-of-the-art ASR systems on dictation tasks, but it was not clear if such architectures would be practical for more challenging tasks such as voice search. In this work, we explore a variety of structural and optimization improvements to our LAS model which significantly improve performance. On the structural side, we show that word piece models can be used instead of graphemes. We introduce a multi-head attention architecture, which offers improvements over the commonly-used single-head attention. On the optimization side, we explore techniques such as synchronous training, scheduled sampling, label smoothing, and minimum word error rate optimization, which are all shown to improve accuracy. We present results with a unidirectional LSTM encoder for streaming recognition. On a 12,500 hour voice search task, we find that the proposed changes improve the WER of the LAS system from 9.2% to 5.6%, while the best conventional system achieve 6.7% WER. We also test both models on a dictation dataset, and our model provide 4.1% WER while the conventional system provides 5% WER.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.