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In this paper we consider an orthonormal basis, generated by a tensor product of Fourier basis functions, half period cosine basis functions, and the Chebyshev basis functions. We deal with the approximation problem in high dimensions related to this basis and design a fast algorithm to multiply with the underlying matrix, consisting of rows of the non-equidistant Fourier matrix, the non-equidistant cosine matrix and the non-equidistant Chebyshev matrix, and its transposed. This leads us to an ANOVA (analysis of variance) decomposition for functions with partially periodic boundary conditions through using the Fourier basis in some dimensions and the half period cosine basis or the Chebyshev basis in others. We consider sensitivity analysis in this setting, in order to find an adapted basis for the underlying approximation problem. More precisely, we find the underlying index set of the multidimensional series expansion. Additionally, we test this ANOVA approximation with mixed basis at numerical experiments, and refer to the advantage of interpretable results.

We consider linear random coefficient regression models, where the regressors are allowed to have a finite support. First, we investigate identifiability, and show that the means and the variances and covariances of the random coefficients are identified from the first two conditional moments of the response given the covariates if the support of the covariates, excluding the intercept, contains a Cartesian product with at least three points in each coordinate. We also discuss ientification of higher-order mixed moments, as well as partial identification in the presence of a binary regressor. Next we show the variable selection consistency of the adaptive LASSO for the variances and covariances of the random coefficients in finite and moderately high dimensions. This implies that the estimated covariance matrix will actually be positive semidefinite and hence a valid covariance matrix, in contrast to the estimate arising from a simple least squares fit. We illustrate the proposed method in a simulation study.

In this paper, we propose and analyze the least squares finite element methods for the linear elasticity interface problem in the stress-displacement system on unfitted meshes. We consider the cases that the interface is $C^2$ or polygonal, and the exact solution $(\sigma,u)$ belongs to $H^s(div; \Omega_0 \cup \Omega_1) \times $H^{1+s}(\Omega_0 \cup \Omega_1)$ with $s > 1/2$. Two types of least squares functionals are defined to seek the numerical solution. The first is defined by simply applying the $L^2$ norm least squares principle, and requires the condition $s \geq 1$. The second is defined with a discrete minus norm, which is related to the inner product in $H^{-1/2}(\Gamma)$. The use of this discrete minus norm results in a method of optimal convergence rates and allows the exact solution has the regularity of any $s > 1/2$. The stability near the interface for both methods is guaranteed by the ghost penalty bilinear forms and we can derive the robust condition number estimates. The convergence rates under $L^2$ norm and the energy norm are derived for both methods. We illustrate the accuracy and the robustness of the proposed methods by a series of numerical experiments for test problems in two and three dimensions.

The ability to reliably predict the future quality of a wireless channel, as seen by the media access control layer, is a key enabler to improve performance of future industrial networks that do not rely on wires. Knowing in advance how much channel behavior may change can speed up procedures for adaptively selecting the best channel, making the network more deterministic, reliable, and less energy-hungry, possibly improving device roaming capabilities at the same time. To this aim, popular approaches based on moving averages and regression were compared, using multiple key performance indicators, on data captured from a real Wi-Fi setup. Moreover, a simple technique based on a linear combination of outcomes from different techniques was presented and analyzed, to further reduce the prediction error, and some considerations about lower bounds on achievable errors have been reported. We found that the best model is the exponential moving average, which managed to predict the frame delivery ratio with a 2.10\% average error and, at the same time, has lower computational complexity and memory consumption than the other models we analyzed.

The vehicle routing problem with time windows (VRPTW) is a classic optimization problem that arises in many different areas, such as logistics and transportation. The goal of the VRPTW is to find the shortest possible route for a fleet of vehicles to visit a set of destinations. In recent years, there has been growing interest in using variational quantum algorithms (VQAs), to find approximate solutions to problems that can be formulated as quadratic unconstrained binary optimization (QUBO) problems. In this work, we formulate the VRPTW as a QUBO and apply a quantum variational approach to the VRPTW using our earlier suggested encoding scheme described in [1] to reduce drastically the number of qubits required. We evaluate our approach on a set of VRPTW instances ranging from 11 to 3964 routes constructed with data provided by researchers from ExxonMobil. We compare the solutions obtained with standard full encoding approaches for which the max problems size possible in NISQ era are of the order of 20-30 routes. We run our algorithms in simulators as well as cloud quantum hardware provided by IBMQ, AWS (Rigetti) and IonQ and benchmark our results against each other as well as on the simulators. We show that our approach can find approximate solutions to the VRPTW that are comparable to the solutions found by quantum algorithms using the full encoding. Our results suggest that our unique encoding approach, provides a promising approach to drastically reducing the number of qubits required to find decent approximate solutions for industry-based optimization problems.

Simultaneous localization and mapping (SLAM) stands as one of the critical challenges in robot navigation. Recent advancements suggest that methods based on supervised learning deliver impressive performance in front-end odometry, while traditional optimization-based methods still play a vital role in the back-end for minimizing estimation drift. In this paper, we found that such decoupled paradigm can lead to only sub-optimal performance, consequently curtailing system capabilities and generalization potential. To solve this problem, we proposed a novel self-supervised learning framework, imperative SLAM (iSLAM), which fosters reciprocal correction between the front-end and back-end, thus enhancing performance without necessitating any external supervision. Specifically, we formulate a SLAM system as a bi-level optimization problem so that the two components are bidirectionally connected. As a result, the front-end model is able to learn global geometric knowledge obtained through pose graph optimization by back-propagating the residuals from the back-end. This significantly improves the generalization ability of the entire system and thus achieves the accuracy improvement up to 45%. To the best of our knowledge, iSLAM is the first SLAM system showing that the front-end and back-end can learn jointly and mutually contribute to each other in a self-supervised manner.

Understanding quantum channels and the strange behavior of their capacities is a key objective of quantum information theory. Here we study a remarkably simple, low-dimensional, single-parameter family of quantum channels with exotic quantum information-theoretic features. As the simplest example from this family, we focus on a qutrit-to-qutrit channel that is intuitively obtained by hybridizing together a simple degradable channel and a completely useless qubit channel. Such hybridizing makes this channel's capacities behave in a variety of interesting ways. For instance, the private and classical capacity of this channel coincide and can be explicitly calculated, even though the channel does not belong to any class for which the underlying information quantities are known to be additive. Moreover, the quantum capacity of the channel can be computed explicitly, given a clear and compelling conjecture is true. This "spin alignment conjecture," which may be of independent interest, is proved in certain special cases and additional numerical evidence for its validity is provided. Finally, we generalize the qutrit channel in two ways, and the resulting channels and their capacities display similarly rich behavior. In the companion paper [Phys. Rev. Lett. 130, 200801 (2023); arXiv:2202.08377], we further show that the qutrit channel demonstrates superadditivity when transmitting quantum information jointly with a variety of assisting channels, in a manner unknown before.

The shortest path problem is a typical problem in graph theory with wide potential applications. The state-of-the-art single-source shortest paths algorithm on the weight graph is the $\Delta$-stepping algorithm, which can efficiently process weighted graphs in parallel. DAWN is an algorithm that addresses the shortest path problem on unweighted graphs, and we propose a weighted version that can handle graphs with weights edges, while maintaining the high scalability and parallelism features as DAWN. The novel version requires $O(\mu m)$ and $O(\mu \cdot E_{wcc})$ times on the connected and unconnected graphs for SSSP problems, respectively. $E_{wcc}$ denote the number of edges included in the largest weakly connected component, and $\mu$ is a constant denoting the average number of path transformations in the tasks. We tested the weighted version on the real graphs from Stanford Network Analysis Platform and SuiteSparse Matrix Collection, which outperformed the solution of $\Delta$-stepping algorithm from Gunrock, achieving a speedup of 43.163$\times$.

Big data programming frameworks have become increasingly important for the development of applications for which performance and scalability are critical. In those complex frameworks, optimizing code by hand is hard and time-consuming, making automated optimization particularly necessary. In order to automate optimization, a prerequisite is to find suitable abstractions to represent programs; for instance, algebras based on monads or monoids to represent distributed data collections. Currently, however, such algebras do not represent recursive programs in a way which allows for analyzing or rewriting them. In this paper, we extend a monoid algebra with a fixpoint operator for representing recursion as a first class citizen and show how it enables new optimizations. Experiments with the Spark platform illustrate performance gains brought by these systematic optimizations.

A core capability of intelligent systems is the ability to quickly learn new tasks by drawing on prior experience. Gradient (or optimization) based meta-learning has recently emerged as an effective approach for few-shot learning. In this formulation, meta-parameters are learned in the outer loop, while task-specific models are learned in the inner-loop, by using only a small amount of data from the current task. A key challenge in scaling these approaches is the need to differentiate through the inner loop learning process, which can impose considerable computational and memory burdens. By drawing upon implicit differentiation, we develop the implicit MAML algorithm, which depends only on the solution to the inner level optimization and not the path taken by the inner loop optimizer. This effectively decouples the meta-gradient computation from the choice of inner loop optimizer. As a result, our approach is agnostic to the choice of inner loop optimizer and can gracefully handle many gradient steps without vanishing gradients or memory constraints. Theoretically, we prove that implicit MAML can compute accurate meta-gradients with a memory footprint that is, up to small constant factors, no more than that which is required to compute a single inner loop gradient and at no overall increase in the total computational cost. Experimentally, we show that these benefits of implicit MAML translate into empirical gains on few-shot image recognition benchmarks.