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We explore a notion of bent sequence attached to the data consisting of an Hadamard matrix of order $n$ defined over the complex $q^{th}$ roots of unity, an eigenvalue of that matrix, and a Galois automorphism from the cyclotomic field of order $q.$ In particular we construct self-dual bent sequences for various $q\le 60$ and lengths $n\le 21.$ Computational construction methods comprise the resolution of polynomial systems by Groebner bases and eigenspace computations. Infinite families can be constructed from regular Hadamard matrices, Bush-type Hadamard matrices, and generalized Boolean bent functions.As an application, we estimate the covering radius of the code attached to that matrix over $\Z_q.$ We derive a lower bound on that quantity for the Chinese Euclidean metric when bent sequences exist. We give the Euclidean distance spectrum, and bound above the covering radius of an attached spherical code, depending on its strength as a spherical design.

We establish a general convergence theory of the Rayleigh--Ritz method and the refined Rayleigh--Ritz method for computing some simple eigenpair ($\lambda_{*},x_{*}$) of a given analytic nonlinear eigenvalue problem (NEP). In terms of the deviation $\varepsilon$ of $x_{*}$ from a given subspace $\mathcal{W}$, we establish a priori convergence results on the Ritz value, the Ritz vector and the refined Ritz vector, and present sufficient convergence conditions for them. The results show that, as $\varepsilon\rightarrow 0$, there is a Ritz value that unconditionally converges to $\lambda_*$ and the corresponding refined Ritz vector does so too but the Ritz vector may fail to converge and even may not be unique. We also present an error bound for the approximate eigenvector in terms of the computable residual norm of a given approximate eigenpair, and give lower and upper bounds for the error of the refined Ritz vector and the Ritz vector as well as for that of the corresponding residual norms. These results nontrivially extend some convergence results on these two methods for the linear eigenvalue problem to the NEP. Examples are constructed to illustrate some of the results.

A convincing feature of least-squares finite element methods is the built-in a posteriori error estimator for any conforming discretization. In order to generalize this property to discontinuous finite element ansatz functions, this paper introduces a least-squares principle on piecewise Sobolev functions for the solution of the Poisson model problem in 2D with mixed boundary conditions. It allows for fairly general discretizations including standard piecewise polynomial ansatz spaces on triangular and polygonal meshes. The presented scheme enforces the interelement continuity of the piecewise polynomials by additional least-squares residuals. A side condition on the normal jumps of the flux variable requires a vanishing integral mean and enables a natural weighting of the jump in the least-squares functional in terms of the mesh size. This avoids over-penalization with additional regularity assumptions on the exact solution as usually present in the literature on discontinuous LSFEM. The proof of the built-in a posteriori error estimation for the over-penalized scheme is presented as well. All results in this paper are robust with respect to the size of the domain guaranteed by a suitable weighting of the residuals in the least-squares functional. Numerical experiments exhibit optimal convergence rates of the adaptive mesh-refining algorithm for various polynomial degrees.

We give a simple and computationally efficient algorithm that, for any constant $\varepsilon>0$, obtains $\varepsilon T$-swap regret within only $T = \mathsf{polylog}(n)$ rounds; this is an exponential improvement compared to the super-linear number of rounds required by the state-of-the-art algorithm, and resolves the main open problem of [Blum and Mansour 2007]. Our algorithm has an exponential dependence on $\varepsilon$, but we prove a new, matching lower bound. Our algorithm for swap regret implies faster convergence to $\varepsilon$-Correlated Equilibrium ($\varepsilon$-CE) in several regimes: For normal form two-player games with $n$ actions, it implies the first uncoupled dynamics that converges to the set of $\varepsilon$-CE in polylogarithmic rounds; a $\mathsf{polylog}(n)$-bit communication protocol for $\varepsilon$-CE in two-player games (resolving an open problem mentioned by [Babichenko-Rubinstein'2017, Goos-Rubinstein'2018, Ganor-CS'2018]; and an $\tilde{O}(n)$-query algorithm for $\varepsilon$-CE (resolving an open problem of [Babichenko'2020] and obtaining the first separation between $\varepsilon$-CE and $\varepsilon$-Nash equilibrium in the query complexity model). For extensive-form games, our algorithm implies a PTAS for $\mathit{normal}$ $\mathit{form}$ $\mathit{correlated}$ $\mathit{equilibria}$, a solution concept often conjectured to be computationally intractable (e.g. [Stengel-Forges'08, Fujii'23]).

We construct a graph with $n$ vertices where the smoothed runtime of the 3-FLIP algorithm for the 3-Opt Local Max-Cut problem can be as large as $2^{\Omega(\sqrt{n})}$. This provides the first example where a local search algorithm for the Max-Cut problem can fail to be efficient in the framework of smoothed analysis. We also give a new construction of graphs where the runtime of the FLIP algorithm for the Local Max-Cut problem is $2^{\Omega(n)}$ for any pivot rule. This graph is much smaller and has a simpler structure than previous constructions.

The Bayesian statistical framework provides a systematic approach to enhance the regularization model by incorporating prior information about the desired solution. For the Bayesian linear inverse problems with Gaussian noise and Gaussian prior, we propose a new iterative regularization algorithm that belongs to subspace projection regularization (SPR) methods. By treating the forward model matrix as a linear operator between the two underlying finite dimensional Hilbert spaces with new introduced inner products, we first introduce an iterative process that can generate a series of valid solution subspaces. The SPR method then projects the original problem onto these solution subspaces to get a series of low dimensional linear least squares problems, where an efficient procedure is developed to update the solutions of them to approximate the desired solution of the original problem. With the new designed early stopping rules, this iterative algorithm can obtain a regularized solution with a satisfied accuracy. Several theoretical results about the algorithm are established to reveal the regularization properties of it. We use both small-scale and large-scale inverse problems to test the proposed algorithm and demonstrate its robustness and efficiency. The most computationally intensive operations in the proposed algorithm only involve matrix-vector products, making it highly efficient for large-scale problems.

Causal representation learning algorithms discover lower-dimensional representations of data that admit a decipherable interpretation of cause and effect; as achieving such interpretable representations is challenging, many causal learning algorithms utilize elements indicating prior information, such as (linear) structural causal models, interventional data, or weak supervision. Unfortunately, in exploratory causal representation learning, such elements and prior information may not be available or warranted. Alternatively, scientific datasets often have multiple modalities or physics-based constraints, and the use of such scientific, multimodal data has been shown to improve disentanglement in fully unsupervised settings. Consequently, we introduce a causal representation learning algorithm (causalPIMA) that can use multimodal data and known physics to discover important features with causal relationships. Our innovative algorithm utilizes a new differentiable parametrization to learn a directed acyclic graph (DAG) together with a latent space of a variational autoencoder in an end-to-end differentiable framework via a single, tractable evidence lower bound loss function. We place a Gaussian mixture prior on the latent space and identify each of the mixtures with an outcome of the DAG nodes; this novel identification enables feature discovery with causal relationships. Tested against a synthetic and a scientific dataset, our results demonstrate the capability of learning an interpretable causal structure while simultaneously discovering key features in a fully unsupervised setting.

We introduce a novel structure-preserving method in order to approximate the compressible ideal Magnetohydrodynamics (MHD) equations. This technique addresses the MHD equations using a non-divergence formulation, where the contributions of the magnetic field to the momentum and total mechanical energy are treated as source terms. Our approach uses the Marchuk-Strang splitting technique and involves three distinct components: a compressible Euler solver, a source-system solver, and an update procedure for the total mechanical energy. The scheme allows for significant freedom on the choice of Euler's equation solver, while the magnetic field is discretized using a curl-conforming finite element space, yielding exact preservation of the involution constraints. We prove that the method preserves invariant domain properties, including positivity of density, positivity of internal energy, and the minimum principle of the specific entropy. If the scheme used to solve Euler's equation conserves total energy, then the resulting MHD scheme can be proven to preserve total energy. Similarly, if the scheme used to solve Euler's equation is entropy-stable, then the resulting MHD scheme is entropy stable as well. In our approach, the CFL condition does not depend on magnetosonic wave-speeds, but only on the usual maximum wave speed from Euler's system. To validate the effectiveness of our method, we solve a variety of ideal MHD problems, showing that the method is capable of delivering high-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.

Conformal inference is a fundamental and versatile tool that provides distribution-free guarantees for many machine learning tasks. We consider the transductive setting, where decisions are made on a test sample of $m$ new points, giving rise to $m$ conformal $p$-values. {While classical results only concern their marginal distribution, we show that their joint distribution follows a P\'olya urn model, and establish a concentration inequality for their empirical distribution function.} The results hold for arbitrary exchangeable scores, including {\it adaptive} ones that can use the covariates of the test+calibration samples at training stage for increased accuracy. We demonstrate the usefulness of these theoretical results through uniform, in-probability guarantees for two machine learning tasks of current interest: interval prediction for transductive transfer learning and novelty detection based on two-class classification.

We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unified framework. We build on the Bayesian approximation error (BAE) approach, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.