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We consider a class of statistical estimation problems in which we are given a random data matrix ${\boldsymbol X}\in {\mathbb R}^{n\times d}$ (and possibly some labels ${\boldsymbol y}\in{\mathbb R}^n$) and would like to estimate a coefficient vector ${\boldsymbol \theta}\in{\mathbb R}^d$ (or possibly a constant number of such vectors). Special cases include low-rank matrix estimation and regularized estimation in generalized linear models (e.g., sparse regression). First order methods proceed by iteratively multiplying current estimates by ${\boldsymbol X}$ or its transpose. Examples include gradient descent or its accelerated variants. Celentano, Montanari, Wu proved that for any constant number of iterations (matrix vector multiplications), the optimal first order algorithm is a specific approximate message passing algorithm (known as `Bayes AMP'). The error of this estimator can be characterized in the high-dimensional asymptotics $n,d\to\infty$, $n/d\to\delta$, and provides a lower bound to the estimation error of any first order algorithm. Here we present a simpler proof of the same result, and generalize it to broader classes of data distributions and of first order algorithms, including algorithms with non-separable nonlinearities. Most importantly, the new proof technique does not require to construct an equivalent tree-structured estimation problem, and is therefore susceptible of a broader range of applications.

This study introduces an ability-based method for personalized keyboard generation, wherein an individual's own movement and human-computer interaction data are used to automatically compute a personalized virtual keyboard layout. Our approach integrates a multidirectional point-select task to characterize cursor control over time, distance, and direction. The characterization is automatically employed to develop a computationally efficient keyboard layout that prioritizes each user's movement abilities through capturing directional constraints and preferences. We evaluated our approach in a study involving 16 participants without motor impairments using inertial sensing and facial electromyography as an access method, resulting in significantly increased communication rates using the personalized keyboard (52.0 bits/min) when compared to a generically optimized keyboard (47.9 bits/min). Our results demonstrate the ability to effectively characterize an individual's movement abilities to design a personalized keyboard for improved communication. This work underscores the importance of integrating a user's motor abilities when designing virtual interfaces.

In backward error analysis, an approximate solution to an equation is compared to the exact solution to a nearby "modified" equation. In numerical ordinary differential equations, the two agree up to any power of the step size. If the differential equation has a geometric property then the modified equation may share it. In this way, known properties of differential equations can be applied to the approximation. But for partial differential equations, the known modified equations are of higher order, limiting applicability of the theory. Therefore, we study symmetric solutions of discretized partial differential equations that arise from a discrete variational principle. These symmetric solutions obey infinite-dimensional functional equations. We show that these equations admit second-order modified equations which are Hamiltonian and also possess first-order Lagrangians in modified coordinates. The modified equation and its associated structures are computed explicitly for the case of rotating travelling waves in the nonlinear wave equation.

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the asymptotic linear rate of convergence, it does not explain the non-linear behavior of these algorithms in the non-asymptotic regime. In this letter, we take into account the effect of the first-order approximation error and present a closed-form bound on the convergence in terms of the number of iterations required for the distance between the iterate and the limit point to reach an arbitrarily small fraction of the initial distance. Our bound includes two terms: one corresponds to the number of iterations required for the linearized version of the fixed-point operator and the other corresponds to the overhead associated with the approximation error. With a focus on the convergence in the scalar case, the tightness of the proposed bound is proven for positively quadratic first-order difference equations.

The numerical solution of an ordinary differential equation can be interpreted as the exact solution of a nearby modified equation. Investigating the behaviour of numerical solutions by analysing the modified equation is known as backward error analysis. If the original and modified equation share structural properties, then the exact and approximate solution share geometric features such as the existence of conserved quantities. Conjugate symplectic methods preserve a modified symplectic form and a modified Hamiltonian when applied to a Hamiltonian system. We show how a blended version of variational and symplectic techniques can be used to compute modified symplectic and Hamiltonian structures. In contrast to other approaches, our backward error analysis method does not rely on an ansatz but computes the structures systematically, provided that a variational formulation of the method is known. The technique is illustrated on the example of symmetric linear multistep methods with matrix coefficients.

This paper presents and analyzes a discontinuous Galerkin method for the incompressible three-phase flow problem in porous media. We use a first order time extrapolation which allows us to solve the equations implicitly and sequentially. We show that the discrete problem is well-posed, and obtain a priori error estimates. Our numerical results validate the theoretical results, i.e. the algorithm converges with first order.

This paper focuses on the expected difference in borrower's repayment when there is a change in the lender's credit decisions. Classical estimators overlook the confounding effects and hence the estimation error can be magnificent. As such, we propose another approach to construct the estimators such that the error can be greatly reduced. The proposed estimators are shown to be unbiased, consistent, and robust through a combination of theoretical analysis and numerical testing. Moreover, we compare the power of estimating the causal quantities between the classical estimators and the proposed estimators. The comparison is tested across a wide range of models, including linear regression models, tree-based models, and neural network-based models, under different simulated datasets that exhibit different levels of causality, different degrees of nonlinearity, and different distributional properties. Most importantly, we apply our approaches to a large observational dataset provided by a global technology firm that operates in both the e-commerce and the lending business. We find that the relative reduction of estimation error is strikingly substantial if the causal effects are accounted for correctly.

We present R-LINS, a lightweight robocentric lidar-inertial state estimator, which estimates robot ego-motion using a 6-axis IMU and a 3D lidar in a tightly-coupled scheme. To achieve robustness and computational efficiency even in challenging environments, an iterated error-state Kalman filter (ESKF) is designed, which recursively corrects the state via repeatedly generating new corresponding feature pairs. Moreover, a novel robocentric formulation is adopted in which we reformulate the state estimator concerning a moving local frame, rather than a fixed global frame as in the standard world-centric lidar-inertial odometry(LIO), in order to prevent filter divergence and lower computational cost. To validate generalizability and long-time practicability, extensive experiments are performed in indoor and outdoor scenarios. The results indicate that R-LINS outperforms lidar-only and loosely-coupled algorithms, and achieve competitive performance as the state-of-the-art LIO with close to an order-of-magnitude improvement in terms of speed.

Sentence splitting is a major simplification operator. Here we present a simple and efficient splitting algorithm based on an automatic semantic parser. After splitting, the text is amenable for further fine-tuned simplification operations. In particular, we show that neural Machine Translation can be effectively used in this situation. Previous application of Machine Translation for simplification suffers from a considerable disadvantage in that they are over-conservative, often failing to modify the source in any way. Splitting based on semantic parsing, as proposed here, alleviates this issue. Extensive automatic and human evaluation shows that the proposed method compares favorably to the state-of-the-art in combined lexical and structural simplification.

We consider the exploration-exploitation trade-off in reinforcement learning and we show that an agent imbued with a risk-seeking utility function is able to explore efficiently, as measured by regret. The parameter that controls how risk-seeking the agent is can be optimized exactly, or annealed according to a schedule. We call the resulting algorithm K-learning and show that the corresponding K-values are optimistic for the expected Q-values at each state-action pair. The K-values induce a natural Boltzmann exploration policy for which the `temperature' parameter is equal to the risk-seeking parameter. This policy achieves an expected regret bound of $\tilde O(L^{3/2} \sqrt{S A T})$, where $L$ is the time horizon, $S$ is the number of states, $A$ is the number of actions, and $T$ is the total number of elapsed time-steps. This bound is only a factor of $L$ larger than the established lower bound. K-learning can be interpreted as mirror descent in the policy space, and it is similar to other well-known methods in the literature, including Q-learning, soft-Q-learning, and maximum entropy policy gradient, and is closely related to optimism and count based exploration methods. K-learning is simple to implement, as it only requires adding a bonus to the reward at each state-action and then solving a Bellman equation. We conclude with a numerical example demonstrating that K-learning is competitive with other state-of-the-art algorithms in practice.