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Generalized linear models (GLMs) arguably represent the standard approach for statistical regression beyond the Gaussian likelihood scenario. When Bayesian formulations are employed, the general absence of a tractable posterior distribution has motivated the development of deterministic approximations, which are generally more scalable than sampling techniques. Among them, expectation propagation (EP) showed extreme accuracy, usually higher than many variational Bayes solutions. However, the higher computational cost of EP posed concerns about its practical feasibility, especially in high-dimensional settings. We address these concerns by deriving a novel efficient formulation of EP for GLMs, whose cost scales linearly in the number of covariates p. This reduces the state-of-the-art O(p^2 n) per-iteration computational cost of the EP routine for GLMs to O(p n min{p,n}), with n being the sample size. We also show that, for binary models and log-linear GLMs approximate predictive means can be obtained at no additional cost. To preserve efficient moment matching for count data, we propose employing a combination of log-normal Laplace transform approximations, avoiding numerical integration. These novel results open the possibility of employing EP in settings that were believed to be practically impossible. Improvements over state-of-the-art approaches are illustrated both for simulated and real data. The efficient EP implementation is available at //github.com/niccoloanceschi/EPglm.

We explore theoretical aspects of boundary conditions for lattice Boltzmann methods, focusing on a toy two-velocities scheme. By mapping lattice Boltzmann schemes to Finite Difference schemes, we facilitate rigorous consistency and stability analyses. We develop kinetic boundary conditions for inflows and outflows, highlighting the trade-off between accuracy and stability, which we successfully overcome. Stability is assessed using GKS (Gustafsson, Kreiss, and Sundstr{\"o}m) analysis and -- when this approach fails on coarse meshes -- spectral and pseudo-spectral analyses of the scheme's matrix that explain effects germane to low resolutions.

We make the interprecision transfers explicit in an algorithmic description of iterative refinement and obtain new insights into the algorithm. One example is the classic variant of iterative refinement where the matrix and the factorization are stored in a working precision and the residual is evaluated in a higher precision. In that case we make the observation that this algorithm will solve a promoted form of the original problem and thereby characterize the limiting behavior in a novel way and obtain a different version of the classic convergence analysis. We also discuss two approaches for interprecision transfer in the triangular solves.

The Lippmann--Schwinger--Lanczos (LSL) algorithm has recently been shown to provide an efficient tool for imaging and direct inversion of synthetic aperture radar data in multi-scattering environments \cite{DrMoZa3}, where the data set is limited to the monostatic, a.k.a. single input/single output (SISO) measurements. The approach is based on constructing data-driven estimates of internal fields via a reduced-order model (ROM) framework and then plugging them into the Lippmann-Schwinger integral equation. However, the approximations of the internal solutions may have more error due to missing the off diagonal elements of the multiple input/multiple output (MIMO) matrix valued transfer function. This, in turn, may result in multiple echoes in the image. Here we present a ROM-based data completion algorithm to mitigate this problem. First, we apply the LSL algorithm to the SISO data as in \cite{DrMoZa3} to obtain approximate reconstructions as well as the estimate of internal field. Next, we use these estimates to calculate a forward Lippmann-Schwinger integral to populate the missing off-diagonal data (the lifting step). Finally, to update the reconstructions, we solve the Lippmann-Schwinger equation using the original SISO data, where the internal fields are constructed from the lifted MIMO data. The steps of obtaining the approximate reconstructions and internal fields and populating the missing MIMO data entries can be repeated for complex models to improve the images even further. Efficiency of the proposed approach is demonstrated on 2D and 2.5D numerical examples, where we see reconstructions are improved substantially.

We derive the Alternating-Direction Implicit (ADI) method based on a commuting operator split and apply the results to the continuous time algebraic Lyapunov equation with low-rank constant term and approximate solution. Previously, it has been mandatory to start the low-rank ADI (LR-ADI) with an all-zero initial value. Our approach retains the known efficient iteration schemes of low-rank increments and residual to arbitrary low-rank initial values for the LR-ADI method. We further generalize some of the known properties of the LR-ADI for Lyapunov equations to larger classes of algorithms or problems. We investigate the performance of arbitrary initial values using two outer iterations in which LR-ADI is typically called. First, we solve an algebraic Riccati equation with the Newton method. Second, we solve a differential Riccati equation with a first-order Rosenbrock method. Numerical experiments confirm that the proposed new initial value of the alternating-directions implicit (ADI) can lead to a significant reduction in the total number of ADI steps, while also showing a 17% and 8x speed-up over the zero initial value for the two equation types, respectively.

Compared to mean regression and quantile regression, the literature on modal regression is very sparse. A unifying framework for Bayesian modal regression is proposed, based on a family of unimodal distributions indexed by the mode, along with other parameters that allow for flexible shapes and tail behaviors. Sufficient conditions for posterior propriety under an improper prior on the mode parameter are derived. Following prior elicitation, regression analysis of simulated data and datasets from several real-life applications are conducted. Besides drawing inference for covariate effects that are easy to interpret, prediction and model selection under the proposed Bayesian modal regression framework are also considered. Evidence from these analyses suggest that the proposed inference procedures are very robust to outliers, enabling one to discover interesting covariate effects missed by mean or median regression, and to construct much tighter prediction intervals than those from mean or median regression. Computer programs for implementing the proposed Bayesian modal regression are available at //github.com/rh8liuqy/Bayesian_modal_regression.

This contribution introduces a model order reduction approach for an advection-reaction problem with a parametrized reaction function. The underlying discretization uses an ultraweak formulation with an $L^2$-like trial space and an 'optimal' test space as introduced by Demkowicz et al. This ensures the stability of the discretization and in addition allows for a symmetric reformulation of the problem in terms of a dual solution which can also be interpreted as the normal equations of an adjoint least-squares problem. Classic model order reduction techniques can then be applied to the space of dual solutions which also immediately gives a reduced primal space. We show that the necessary computations do not require the reconstruction of any primal solutions and can instead be performed entirely on the space of dual solutions. We prove exponential convergence of the Kolmogorov $N$-width and show that a greedy algorithm produces quasi-optimal approximation spaces for both the primal and the dual solution space. Numerical experiments based on the benchmark problem of a catalytic filter confirm the applicability of the proposed method.

In the context of high-dimensional Gaussian linear regression for ordered variables, we study the variable selection procedure via the minimization of the penalized least-squares criterion. We focus on model selection where the penalty function depends on an unknown multiplicative constant commonly calibrated for prediction. We propose a new proper calibration of this hyperparameter to simultaneously control predictive risk and false discovery rate. We obtain non-asymptotic bounds on the False Discovery Rate with respect to the hyperparameter and we provide an algorithm to calibrate it. This algorithm is based on quantities that can typically be observed in real data applications. The algorithm is validated in an extensive simulation study and is compared with several existing variable selection procedures. Finally, we study an extension of our approach to the case in which an ordering of the variables is not available.

This paper considers the problem of minimizing a differentiable function with locally Lipschitz continuous gradient on the algebraic variety of real matrices of upper-bounded rank. This problem is known to enable the formulation of several machine learning and signal processing tasks such as collaborative filtering and signal recovery. Several definitions of stationarity exist for this nonconvex problem. Among them, Bouligand stationarity is the strongest first-order necessary condition for local optimality. This paper proposes a first-order algorithm that combines the well-known projected-projected gradient descent map with a rank reduction mechanism and generates a sequence in the variety whose accumulation points are Bouligand stationary. This algorithm compares favorably with the three other algorithms known in the literature to enjoy this stationarity property, regarding both the typical computational cost per iteration and empirically observed numerical performance. A framework to design hybrid algorithms enjoying the same property is proposed and illustrated through an example.

Conventional entity typing approaches are based on independent classification paradigms, which make them difficult to recognize inter-dependent, long-tailed and fine-grained entity types. In this paper, we argue that the implicitly entailed extrinsic and intrinsic dependencies between labels can provide critical knowledge to tackle the above challenges. To this end, we propose \emph{Label Reasoning Network(LRN)}, which sequentially reasons fine-grained entity labels by discovering and exploiting label dependencies knowledge entailed in the data. Specifically, LRN utilizes an auto-regressive network to conduct deductive reasoning and a bipartite attribute graph to conduct inductive reasoning between labels, which can effectively model, learn and reason complex label dependencies in a sequence-to-set, end-to-end manner. Experiments show that LRN achieves the state-of-the-art performance on standard ultra fine-grained entity typing benchmarks, and can also resolve the long tail label problem effectively.