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Variational regularization is commonly used to solve linear inverse problems, and involves augmenting a data fidelity by a regularizer. The regularizer is used to promote a priori information and is weighted by a regularization parameter. Selection of an appropriate regularization parameter is critical, with various choices leading to very different reconstructions. Classical strategies used to determine a suitable parameter value include the discrepancy principle and the L-curve criterion, and in recent years a supervised machine learning approach called bilevel learning has been employed. Bilevel learning is a powerful framework to determine optimal parameters and involves solving a nested optimization problem. While previous strategies enjoy various theoretical results, the well-posedness of bilevel learning in this setting is still an open question. In particular, a necessary property is positivity of the determined regularization parameter. In this work, we provide a new condition that better characterizes positivity of optimal regularization parameters than the existing theory. Numerical results verify and explore this new condition for both small and high-dimensional problems.

Ensemble transform Kalman filtering (ETKF) data assimilation is often used to combine available observations with numerical simulations to obtain statistically accurate and reliable state representations in dynamical systems. However, it is well known that the commonly used Gaussian distribution assumption introduces biases for state variables that admit discontinuous profiles, which are prevalent in nonlinear partial differential equations. This investigation designs a new structurally informed non-Gaussian prior that exploits statistical information from the simulated state variables. In particular, we construct a new weighting matrix based on the second moment of the gradient information of the state variable to replace the prior covariance matrix used for model/data compromise in the ETKF data assimilation framework. We further adapt our weighting matrix to include information in discontinuity regions via a clustering technique. Our numerical experiments demonstrate that this new approach yields more accurate estimates than those obtained using ETKF on shallow water equations, even when ETKF is enhanced with inflation and localization techniques.

Cinematic audio source separation is a relatively new subtask of audio source separation, with the aim of extracting the dialogue stem, the music stem, and the effects stem from their mixture. In this work, we developed a model generalizing the Bandsplit RNN for any complete or overcomplete partitions of the frequency axis. Psycho-acoustically motivated frequency scales were used to inform the band definitions which are now defined with redundancy for more reliable feature extraction. A loss function motivated by the signal-to-noise ratio and the sparsity-promoting property of the 1-norm was proposed. We additionally exploit the information-sharing property of a common-encoder setup to reduce computational complexity during both training and inference, improve separation performance for hard-to-generalize classes of sounds, and allow flexibility during inference time with easily detachable decoders. Our best model sets the state of the art on the Divide and Remaster dataset with performance above the ideal ratio mask for the dialogue stem.

Composed image retrieval, a task involving the search for a target image using a reference image and a complementary text as the query, has witnessed significant advancements owing to the progress made in cross-modal modeling. Unlike the general image-text retrieval problem with only one alignment relation, i.e., image-text, we argue for the existence of two types of relations in composed image retrieval. The explicit relation pertains to the reference image & complementary text-target image, which is commonly exploited by existing methods. Besides this intuitive relation, the observations during our practice have uncovered another implicit yet crucial relation, i.e., reference image & target image-complementary text, since we found that the complementary text can be inferred by studying the relation between the target image and the reference image. Regrettably, existing methods largely focus on leveraging the explicit relation to learn their networks, while overlooking the implicit relation. In response to this weakness, We propose a new framework for composed image retrieval, termed dual relation alignment, which integrates both explicit and implicit relations to fully exploit the correlations among the triplets. Specifically, we design a vision compositor to fuse reference image and target image at first, then the resulted representation will serve two roles: (1) counterpart for semantic alignment with the complementary text and (2) compensation for the complementary text to boost the explicit relation modeling, thereby implant the implicit relation into the alignment learning. Our method is evaluated on two popular datasets, CIRR and FashionIQ, through extensive experiments. The results confirm the effectiveness of our dual-relation learning in substantially enhancing composed image retrieval performance.

We derive optimality conditions for the optimum sample allocation problem in stratified sampling, formulated as the determination of the fixed strata sample sizes that minimize the total cost of the survey, under the assumed level of variance of the stratified $\pi$ estimator of the population total (or mean) and one-sided upper bounds imposed on sample sizes in strata. In this context, we presume that the variance function is of some generic form that, in particular, covers the case of the simple random sampling without replacement design in strata. The optimality conditions mentioned above will be derived from the Karush-Kuhn-Tucker conditions. Based on the established optimality conditions, we provide a formal proof of the optimality of the existing procedure, termed here as LRNA, which solves the allocation problem considered. We formulate the LRNA in such a way that it also provides the solution to the classical optimum allocation problem (i.e. minimization of the estimator's variance under a fixed total cost) under one-sided lower bounds imposed on sample sizes in strata. In this context, the LRNA can be considered as a counterparty to the popular recursive Neyman allocation procedure that is used to solve the classical problem of an optimum sample allocation with added one-sided upper bounds. Ready-to-use R-implementation of the LRNA is available through our stratallo package, which is published on the Comprehensive R Archive Network (CRAN) package repository.

There have been recent advances in the analysis and visualization of 3D symmetric tensor fields, with a focus on the robust extraction of tensor field topology. However, topological features such as degenerate curves and neutral surfaces do not live in isolation. Instead, they intriguingly interact with each other. In this paper, we introduce the notion of {\em topological graph} for 3D symmetric tensor fields to facilitate global topological analysis of such fields. The nodes of the graph include degenerate curves and regions bounded by neutral surfaces in the domain. The edges in the graph denote the adjacency information between the regions and degenerate curves. In addition, we observe that a degenerate curve can be a loop and even a knot and that two degenerate curves (whether in the same region or not) can form a link. We provide a definition and theoretical analysis of individual degenerate curves in order to help understand why knots and links may occur. Moreover, we differentiate between wedges and trisectors, thus making the analysis more detailed about degenerate curves. We incorporate this information into the topological graph. Such a graph can not only reveal the global structure in a 3D symmetric tensor field but also allow two symmetric tensor fields to be compared. We demonstrate our approach by applying it to solid mechanics and material science data sets.

We consider the problem of inferring the underlying graph topology from smooth graph signals in a novel but practical scenario where data are located in distributed clients and are privacy-sensitive. The main difficulty of this task lies in how to utilize the potentially heterogeneous data of all isolated clients under privacy constraints. Towards this end, we propose a framework where personalized graphs for local clients as well as a consensus graph are jointly learned. The personalized graphs match local data distributions, thereby mitigating data heterogeneity, while the consensus graph captures the global information. We next devise a tailored algorithm to solve the induced problem without violating privacy constraints, i.e., all private data are processed locally. To further enhance privacy protection, we introduce differential privacy (DP) into the proposed algorithm to resist privacy attacks when transmitting model updates. Theoretically, we establish provable convergence analyses for the proposed algorithms, including that with DP. Finally, extensive experiments on both synthetic and real-world data are carried out to validate the proposed framework. Experimental results illustrate that our approach can learn graphs effectively in the target scenario.

Discrete state spaces represent a major computational challenge to statistical inference, since the computation of normalisation constants requires summation over large or possibly infinite sets, which can be impractical. This paper addresses this computational challenge through the development of a novel generalised Bayesian inference procedure suitable for discrete intractable likelihood. Inspired by recent methodological advances for continuous data, the main idea is to update beliefs about model parameters using a discrete Fisher divergence, in lieu of the problematic intractable likelihood. The result is a generalised posterior that can be sampled from using standard computational tools, such as Markov chain Monte Carlo, circumventing the intractable normalising constant. The statistical properties of the generalised posterior are analysed, with sufficient conditions for posterior consistency and asymptotic normality established. In addition, a novel and general approach to calibration of generalised posteriors is proposed. Applications are presented on lattice models for discrete spatial data and on multivariate models for count data, where in each case the methodology facilitates generalised Bayesian inference at low computational cost.

Learning latent representations of nodes in graphs is an important and ubiquitous task with widespread applications such as link prediction, node classification, and graph visualization. Previous methods on graph representation learning mainly focus on static graphs, however, many real-world graphs are dynamic and evolve over time. In this paper, we present Dynamic Self-Attention Network (DySAT), a novel neural architecture that operates on dynamic graphs and learns node representations that capture both structural properties and temporal evolutionary patterns. Specifically, DySAT computes node representations by jointly employing self-attention layers along two dimensions: structural neighborhood and temporal dynamics. We conduct link prediction experiments on two classes of graphs: communication networks and bipartite rating networks. Our experimental results show that DySAT has a significant performance gain over several different state-of-the-art graph embedding baselines.