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Multi-modal Magnetic Resonance Imaging (MRI) offers complementary diagnostic information, but some modalities are limited by the long scanning time. To accelerate the whole acquisition process, MRI reconstruction of one modality from highly undersampled k-space data with another fully-sampled reference modality is an efficient solution. However, the misalignment between modalities, which is common in clinic practice, can negatively affect reconstruction quality. Existing deep learning-based methods that account for inter-modality misalignment perform better, but still share two main common limitations: (1) The spatial alignment task is not adaptively integrated with the reconstruction process, resulting in insufficient complementarity between the two tasks; (2) the entire framework has weak interpretability. In this paper, we construct a novel Deep Unfolding Network with Spatial Alignment, termed DUN-SA, to appropriately embed the spatial alignment task into the reconstruction process. Concretely, we derive a novel joint alignment-reconstruction model with a specially designed cross-modal spatial alignment term. By relaxing the model into cross-modal spatial alignment and multi-modal reconstruction tasks, we propose an effective algorithm to solve this model alternatively. Then, we unfold the iterative steps of the proposed algorithm and design corresponding network modules to build DUN-SA with interpretability. Through end-to-end training, we effectively compensate for spatial misalignment using only reconstruction loss, and utilize the progressively aligned reference modality to provide inter-modality prior to improve the reconstruction of the target modality. Comprehensive experiments on three real datasets demonstrate that our method exhibits superior reconstruction performance compared to state-of-the-art methods.

In recent years, the Adaptive Antoulas-Anderson AAA algorithm has established itself as the method of choice for solving rational approximation problems. Data-driven Model Order Reduction (MOR) of large-scale Linear Time-Invariant (LTI) systems represents one of the many applications in which this algorithm has proven to be successful since it typically generates reduced-order models (ROMs) efficiently and in an automated way. Despite its effectiveness and numerical reliability, the classical AAA algorithm is not guaranteed to return a ROM that retains the same structural features of the underlying dynamical system, such as the stability of the dynamics. In this paper, we propose a novel algebraic characterization for the stability of ROMs with transfer function obeying the AAA barycentric structure. We use this characterization to formulate a set of convex constraints on the free coefficients of the AAA model that, whenever verified, guarantee by construction the asymptotic stability of the resulting ROM. We suggest how to embed such constraints within the AAA optimization routine, and we validate experimentally the effectiveness of the resulting algorithm, named stabAAA, over a set of relevant MOR applications.

We study operator - or noncommutative - variants of constraint satisfaction problems (CSPs). These higher-dimensional variants are a core topic of investigation in quantum information, where they arise as nonlocal games and entangled multiprover interactive proof systems (MIP*). The idea of higher-dimensional relaxations of CSPs is also important in the classical literature. For example since the celebrated work of Goemans and Williamson on Max-Cut, higher dimensional vector relaxations have been central in the design of approximation algorithms for classical CSPs. We introduce a framework for designing approximation algorithms for noncommutative CSPs. Prior to this work Max-$2$-Lin$(k)$ was the only family of noncommutative CSPs known to be efficiently solvable. This work is the first to establish approximation ratios for a broader class of noncommutative CSPs. In the study of classical CSPs, $k$-ary decision variables are often represented by $k$-th roots of unity, which generalise to the noncommutative setting as order-$k$ unitary operators. In our framework, using representation theory, we develop a way of constructing unitary solutions from SDP relaxations, extending the pioneering work of Tsirelson on XOR games. Then, we introduce a novel rounding scheme to transform these solutions to order-$k$ unitaries. Our main technical innovation here is a theorem guaranteeing that, for any set of unitary operators, there exists a set of order-$k$ unitaries that closely mimics it. As an integral part of the rounding scheme, we prove a random matrix theory result that characterises the distribution of the relative angles between eigenvalues of random unitaries using tools from free probability.

Permutation tests are widely recognized as robust alternatives to tests based on normal theory. Random permutation tests have been frequently employed to assess the significance of variables in linear models. Despite their widespread use, existing random permutation tests lack finite-sample and assumption-free guarantees for controlling type I error in partial correlation tests. To address this ongoing challenge, we have developed a conformal test through permutation-augmented regressions, which we refer to as PALMRT. PALMRT not only achieves power competitive with conventional methods but also provides reliable control of type I errors at no more than $2\alpha$, given any targeted level $\alpha$, for arbitrary fixed designs and error distributions. We have confirmed this through extensive simulations. Compared to the cyclic permutation test (CPT) and residual permutation test (RPT), which also offer theoretical guarantees, PALMRT does not compromise as much on power or set stringent requirements on the sample size, making it suitable for diverse biomedical applications. We further illustrate the differences in a long-Covid study where PALMRT validated key findings previously identified using the t-test after multiple corrections, while both CPT and RPT suffered from a drastic loss of power and failed to identify any discoveries. We endorse PALMRT as a robust and practical hypothesis test in scientific research for its superior error control, power preservation, and simplicity. An R package for PALMRT is available at \url{//github.com/LeyingGuan/PairedRegression}.

Resonance based numerical schemes are those in which cancellations in the oscillatory components of the equation are taken advantage of in order to reduce the regularity required of the initial data to achieve a particular order of error and convergence. We investigate the potential for the derivation of resonance based schemes in the context of nonlinear stochastic PDEs. By comparing the regularity conditions required for error analysis to traditional exponential schemes we demonstrate that at orders less than $ \mathcal{O}(t^2) $, the techniques are successful and provide a significant gain on the regularity of the initial data, while at orders greater than $ \mathcal{O}(t^2) $, that the resonance based techniques does not achieve any gain. This is due to limitations in the explicit path-wise analysis of stochastic integrals. As examples of applications of the method, we present schemes for the Sch\"odinger equation and Manakov system accompanied by local error and stability analysis as well as proof of global convergence in both the strong and path-wise sense.

We consider the problem of synchronizing a multi-agent system (MAS) composed of several identical linear systems connected through a directed graph.To design a suitable controller, we construct conditions based on Bilinear Matrix Inequalities (BMIs) that ensure state synchronization.Since these conditions are non-convex, we propose an iterative algorithm based on a suitable relaxation that allows us to formulate Linear Matrix Inequality (LMI) conditions.As a result, the algorithm yields a common static state-feedback matrix for the controller that satisfies general linear performance constraints.Our results are achieved under the mild assumption that the graph is time-invariant and connected.

In this paper we deal with a second order evolution inclusion involving a multivalued term generated by a Clarke subdifferential of a locally Lipschitz potential. For this problem we construct a double step time-semidiscrete approximation, known as the Rothe scheme. We study a sequence of solutions of the semidiscrete approximate problems and provide its weak convergence to a limit element that is a solution of the original problem.

Phase field models are gradient flows with their energy naturally dissipating in time. In order to preserve this property, many numerical schemes have been well-studied. In this paper we consider a well-known method, namely the exponential integrator method (EI). In the literature a few works studied several EI schemes for various phase field models and proved the energy dissipation by either requiring a strong Lipschitz condition on the nonlinear source term or certain $L^\infty$ bounds on the numerical solutions (maximum principle). However for phase field models such as the (non-local) Cahn-Hilliard equation, the maximum principle no longer exists. As a result, solving such models via EI schemes remains open for a long time. In this paper we aim to give a systematic approach on applying EI-type schemes to such models by solving the Cahn-Hilliard equation with a first order EI scheme and showing the energy dissipation. In fact second order EI schemes can be handled similarly and we leave the discussion in a subsequent paper. To our best knowledge, this is the first work to handle phase field models without assuming any strong Lipschitz condition or $L^\infty$ boundedness. Furthermore, we will analyze the $L^2$ error and present some numerical simulations to demonstrate the dynamics.

We propose a simple multivariate normality test based on Kac-Bernstein's characterization, which can be conducted by utilising existing statistical independence tests for sums and differences of data samples. We also perform its empirical investigation, which reveals that for high-dimensional data, the proposed approach may be more efficient than the alternative ones. The accompanying code repository is provided at \url{//shorturl.at/rtuy5}.

Many complex tasks and environments can be decomposed into simpler, independent parts. Discovering such underlying compositional structure has the potential to expedite adaptation and enable compositional generalization. Despite progress, our most powerful systems struggle to compose flexibly. While most of these systems are monolithic, modularity promises to allow capturing the compositional nature of many tasks. However, it is unclear under which circumstances modular systems discover this hidden compositional structure. To shed light on this question, we study a teacher-student setting with a modular teacher where we have full control over the composition of ground truth modules. This allows us to relate the problem of compositional generalization to that of identification of the underlying modules. We show theoretically that identification up to linear transformation purely from demonstrations is possible in hypernetworks without having to learn an exponential number of module combinations. While our theory assumes the infinite data limit, in an extensive empirical study we demonstrate how meta-learning from finite data can discover modular solutions that generalize compositionally in modular but not monolithic architectures. We further show that our insights translate outside the teacher-student setting and demonstrate that in tasks with compositional preferences and tasks with compositional goals hypernetworks can discover modular policies that compositionally generalize.