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We introduce a framework that enables efficient sampling from learned probability distributions for MRI reconstruction. Different from conventional deep learning-based MRI reconstruction techniques, samples are drawn from the posterior distribution given the measured k-space using the Markov chain Monte Carlo (MCMC) method. In addition to the maximum a posteriori (MAP) estimate for the image, which can be obtained with conventional methods, the minimum mean square error (MMSE) estimate and uncertainty maps can also be computed. The data-driven Markov chains are constructed from the generative model learned from a given image database and are independent of the forward operator that is used to model the k-space measurement. This provides flexibility because the method can be applied to k-space acquired with different sampling schemes or receive coils using the same pre-trained models. Furthermore, we use a framework based on a reverse diffusion process to be able to utilize advanced generative models. The performance of the method is evaluated on an open dataset using 10-fold accelerated acquisition.

Several studies have shown the ability of natural gradient descent to minimize the objective function more efficiently than ordinary gradient descent based methods. However, the bottleneck of this approach for training deep neural networks lies in the prohibitive cost of solving a large dense linear system corresponding to the Fisher Information Matrix (FIM) at each iteration. This has motivated various approximations of either the exact FIM or the empirical one. The most sophisticated of these is KFAC, which involves a Kronecker-factored block diagonal approximation of the FIM. With only a slight additional cost, a few improvements of KFAC from the standpoint of accuracy are proposed. The common feature of the four novel methods is that they rely on a direct minimization problem, the solution of which can be computed via the Kronecker product singular value decomposition technique. Experimental results on the three standard deep auto-encoder benchmarks showed that they provide more accurate approximations to the FIM. Furthermore, they outperform KFAC and state-of-the-art first-order methods in terms of optimization speed.

Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting where the two factors are generated from known component-wise independent prior distributions, and the statistician observes a (possibly noisy) component-wise function of their matrix product. In the limit where the dimensions of the matrices tend to infinity, but their ratios remain fixed, we expect to be able to derive closed form expressions for the optimal mean squared error on the estimation of the two factors. However, this remains a very involved mathematical and algorithmic problem. A related, but simpler, problem is extensive-rank matrix denoising, where one aims to reconstruct a matrix with extensive but usually small rank from noisy measurements. In this paper, we approach both these problems using high-temperature expansions at fixed order parameters. This allows to clarify how previous attempts at solving these problems failed at finding an asymptotically exact solution. We provide a systematic way to derive the corrections to these existing approximations, taking into account the structure of correlations particular to the problem. Finally, we illustrate our approach in detail on the case of extensive-rank matrix denoising. We compare our results with known optimal rotationally-invariant estimators, and show how exact asymptotic calculations of the minimal error can be performed using extensive-rank matrix integrals.

In a realistic wireless environment, the multi-antenna channel usually exhibits spatially correlation fading. This is more emphasized when a large number of antennas is densely deployed, known as holographic massive MIMO (multiple-input multiple-output). In the first part of this letter, we develop a channel model for holographic massive MIMO by considering both non-isotropic scattering and directive antennas. With a large number of antennas, it is difficult to obtain full knowledge of the spatial correlation matrix. In this case, channel estimation is conventionally done using the least-squares (LS) estimator that requires no prior information of the channel statistics or array geometry. In the second part of this letter, we propose a novel channel estimation scheme that exploits the array geometry to identify a subspace of reduced rank that covers the eigenspace of any spatial correlation matrix. The proposed estimator outperforms the LS estimator, without using any user-specific channel statistics.

The recent development of scintillation crystals combined with $\gamma$-rays sources opens the way to an imaging concept based on Compton scattering, namely Compton scattering tomography (CST). The associated inverse problem rises many challenges: non-linearity, multiple order-scattering and high level of noise. Already studied in the literature, these challenges lead unavoidably to uncertainty of the forward model. This work proposes to study exact and approximated forward models and develops two data-driven reconstruction algorithms able to tackle the inexactness of the forward model. The first one is based on the projective method called regularized sequential subspace optimization (RESESOP). We consider here a finite dimensional restriction of the semi-discrete forward model and show its well-posedness and regularisation properties. The second one considers the unsupervised learning method, deep image prior (DIP), inspired by the construction of the model uncertainty in RESESOP. The methods are validated on Monte-Carlo data.

This paper investigates why it is beneficial, when solving a problem, to search in the neighbourhood of a current solution. The paper identifies properties of problems and neighbourhoods that support two novel proofs that neighbourhood search is beneficial over blind search. These are: firstly a proof that search within the neighbourhood is more likely to find an improving solution in a single search step than blind search; and secondly a proof that a local improvement, using a sequence of neighbourhood search steps, is likely to achieve a greater improvement than a sequence of blind search steps. To explore the practical impact of these properties, a range of problem sets and neighbourhoods are generated, where these properties are satisfied to different degrees. Experiments reveal that the benefits of neighbourhood search vary dramatically in consequence. Random problems of a classical combinatorial optimisation problem are analysed, in order to demonstrate that the underlying theory is reflected in practice.

An l0-regularized linear regression for a sparse signal reconstruction is implemented based on the quadratic unconstrained binary optimization (QUBO) formulation. In this method, the signal values are quantized and expressed as bit sequences. By transforming l0-norm to a quadratic form of these bits, the fully quadratic objective function is provided and optimized by the solver specialized for QUBO, such as the quantum annealer. Numerical experiments with a commercial quantum annealer show that the proposed method performs slightly better than conventional methods based on orthogonal matching pursuit (OMP) and the least absolute shrinkage and selection operator (LASSO) under several limited conditions.

Collaborative filtering (CF), as a fundamental approach for recommender systems, is usually built on the latent factor model with learnable parameters to predict users' preferences towards items. However, designing a proper CF model for a given data is not easy, since the properties of datasets are highly diverse. In this paper, motivated by the recent advances in automated machine learning (AutoML), we propose to design a data-specific CF model by AutoML techniques. The key here is a new framework that unifies state-of-the-art (SOTA) CF methods and splits them into disjoint stages of input encoding, embedding function, interaction function, and prediction function. We further develop an easy-to-use, robust, and efficient search strategy, which utilizes random search and a performance predictor for efficient searching within the above framework. In this way, we can combinatorially generalize data-specific CF models, which have not been visited in the literature, from SOTA ones. Extensive experiments on five real-world datasets demonstrate that our method can consistently outperform SOTA ones for various CF tasks. Further experiments verify the rationality of the proposed framework and the efficiency of the search strategy. The searched CF models can also provide insights for exploring more effective methods in the future

Various 3D reconstruction methods have enabled civil engineers to detect damage on a road surface. To achieve the millimetre accuracy required for road condition assessment, a disparity map with subpixel resolution needs to be used. However, none of the existing stereo matching algorithms are specially suitable for the reconstruction of the road surface. Hence in this paper, we propose a novel dense subpixel disparity estimation algorithm with high computational efficiency and robustness. This is achieved by first transforming the perspective view of the target frame into the reference view, which not only increases the accuracy of the block matching for the road surface but also improves the processing speed. The disparities are then estimated iteratively using our previously published algorithm where the search range is propagated from three estimated neighbouring disparities. Since the search range is obtained from the previous iteration, errors may occur when the propagated search range is not sufficient. Therefore, a correlation maxima verification is performed to rectify this issue, and the subpixel resolution is achieved by conducting a parabola interpolation enhancement. Furthermore, a novel disparity global refinement approach developed from the Markov Random Fields and Fast Bilateral Stereo is introduced to further improve the accuracy of the estimated disparity map, where disparities are updated iteratively by minimising the energy function that is related to their interpolated correlation polynomials. The algorithm is implemented in C language with a near real-time performance. The experimental results illustrate that the absolute error of the reconstruction varies from 0.1 mm to 3 mm.

In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.