An Open Problem on Sparse Representations in Unions of Bases

We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of sparks are generally combinatorial. For unions of several orthonormal bases, two lower bounds on the spark via the mutual coherence were established in previous work. We constructively prove that both of them are tight. Our main results give positive answers to Gribonval and Nielsens open problem on sparse representations in unions of orthonormal bases. Constructive proofs rely on a family of mutually unbiased bases which first appears in quantum information theory.

CARlasso: An R package for the estimation of sparse microbial networks with predictors

Microbiome data analyses require statistical tools that can simultaneously decode microbes reactions to the environment and interactions among microbes. We introduce CARlasso, the first user-friendly open-source and publicly available R package to fit a chain graph model for the inference of sparse microbial networks that represent both interactions among nodes and effects of a set of predictors. Unlike in standard regression approaches, the edges represent the correct conditional structure among responses and predictors that allows the incorporation of prior knowledge from controlled experiments. In addition, CARlasso 1) enforces sparsity in the network via LASSO; 2) allows for an adaptive extension to include different shrinkage to different edges; 3) is computationally inexpensive through an efficient Gibbs sampling algorithm so it can equally handle small and big data; 4) allows for continuous, binary, counting and compositional responses via proper hierarchical structure, and 5) has a similar syntax to lm for ease of use. The package also supports Bayesian graphical LASSO and several of its hierarchical models as well as lower level one-step sampling functions of the CAR-LASSO model for users to extend.

The Effect of the Prior and the Experimental Design on the Inference of the Precision Matrix in Gaussian Chain Graph Models

Here, we investigate whether (and how) experimental design could aid in the estimation of the precision matrix in a Gaussian chain graph model, especially the interplay between the design, the effect of the experiment and prior knowledge about the effect. We approximate the marginal posterior precision of the precision matrix via Laplace approximation under different priors: a flat prior, the conjugate prior Normal-Wishart, the unconfounded prior Normal-Matrix Generalized Inverse Gaussian (MGIG) and a general independent prior. We show that the approximated posterior precision is not a function of the design matrix for the cases of the Normal-Wishart and flat prior, but it is for the cases of the Normal-MGIG and the general independent prior. However, for the Normal-MGIG and the general independent prior, we find a sharp upper bound on the approximated posterior precision that does not involve the design matrix which translates into a bound on the information that could be extracted from a given experiment. We confirm the theoretical findings via a simulation study comparing the Steins loss difference between random versus no experiment (design matrix equal to zero). Our findings provide practical advice for domain scientists conducting experiments to decode the relationships between a multidimensional response and a set of predictors.

Bayesian Conditional Auto-Regressive LASSO Models to Learn Sparse Microbial Networks with Predictors

Microbiome data analyses require statistical models that can simultaneously decode microbes reactions to the environment and interactions among microbes. While a multiresponse linear regression model seems like a straightforward solution, we argue that treating it as a graphical model is flawed given that the regression coefficient matrix does not encode the conditional dependence structure between response and predictor nodes because it does not represent the adjacency matrix. This observation is especially important in biological settings when we have prior knowledge on the edges from specific experimental interventions that can only be properly encoded under a conditional dependence model. Here, we propose a chain graph model with two sets of nodes (predictors and responses) whose solution yields a graph with edges that indeed represent conditional dependence and thus, agrees with the experimenters intuition on the average behavior of nodes under treatment. The solution to our model is sparse via Bayesian LASSO and is also guaranteed to be the sparse solution to a Conditional Auto-Regressive (CAR) model. In addition, we propose an adaptive extension so that different shrinkage can be applied to different edges to incorporate edge-specific prior knowledge. Our model is computationally inexpensive through an efficient Gibbs sampling algorithm and can account for binary, counting, and compositional responses via appropriate hierarchical structure. We apply our model to a human gut and a soil microbial compositional datasets and we highlight that CAR-LASSO can estimate biologically meaningful network structures in the data. The CAR-LASSO software is available as an R package at https://github.com/YunyiShen/CAR-LASSO.

Two-phase crystallisation in a carpet of inertial spinners

We study the dynamics of torque driven spherical spinners settled on a surface, and demonstrate that hydrodynamic interactions at finite Reynolds numbers can lead to a concentration dependent and non-uniform crystallisation. At semi-dilute concentrations, we observe a rapid formation of a uniform hexagonal structure in the spinner monolayer. We attribute this to repulsive hydrodynamic interactions created by the secondary flow of the spinning particles. Increasing the surface coverage leads to a state with two co-existing spinner densities. The uniform hexagonal structure deviates into a high density crystalline structure surrounded by a continuous lower density hexatically ordered state. We show that this phase separation occurs due to a non-monotonic hydrodynamic repulsion, arising from a concentration dependent spinning frequency.

Inertias of entanglement witnesses

Entanglement witnesses (EWs) are a fundamental tool for the detection of entanglement. We study the inertias of EWs, i.e., the triplet of the numbers of negative, zero, and positive eigenvalues respectively. We focus on the EWs constructed by the partial transposition of states with non-positive partial transposes. We provide a method to generate more inertias from a given inertia by the relevance between inertias. Based on that we exhaust all the inertias for EWs in each qubit-qudit system. We apply our results to propose a separability criterion in terms of the rank of the partial transpose of state. We also connect our results to tripartite genuinely entangled states and the classification of states with non-positive partial transposes. Additionally, the inertias of EWs constructed by X-states are clarified.

Quantifying socioeconomic activities and weather effects on the global spread of COVID-19 epidemic

The COVID-19 has caused more than three million infections and over two hundred thousand deaths by April 20201. Limiting socioeconomic activities (SA) is among the most adopted governmental mitigating efforts to combat the transmission of the virus, though the degree varies dramatically among different regimes2. This study aims to quantify the contribution from the SA and weather conditions to the transmission of COVID-19 at global scale. Ruling out the unobservable factors including medical facilities and other control policies (MOC) through region-by-time fixed effects3,4, we show that the limiting SA has a leading contribution to lower the reproductive number by 18.3%, while weather conditions, including ultraviolet, relative humidity, and wind explain a smaller amount of variation. Temperature might have a non-monotonic impact on the transmission. We further show that in developed countries5 and China, the SA effect is more pronounced whereas the weather effect is significantly downplayed possibly because people tend to stay indoors most of the time with a controlled climate. We finally estimate the reduced reproductive number and the population spared from infections due to restricting SA at 40,964, 180,336, 174,494, in China, United States, and Europe respectively. From late January to mid-April, all regions, except for China, Australia, and south Korea show a steep upward trend of spared infections due to restricting SA. US and Europe, in particular, show far steeper upward trends of spared infections in the analyzed timeframe, signaling a greater risk of reopening the economy too soon.

Modeling and Enhancing Low-quality Retinal Fundus Images

Retinal fundus images are widely used for the clinical screening and diagnosis of eye diseases. However, fundus images captured by operators with various levels of experience have a large variation in quality. Low-quality fundus images increase uncertainty in clinical observation and lead to the risk of misdiagnosis. However, due to the special optical beam of fundus imaging and structure of the retina, natural image enhancement methods cannot be utilized directly to address this. In this paper, we first analyze the ophthalmoscope imaging system and simulate a reliable degradation of major inferior-quality factors, including uneven illumination, image blurring, and artifacts. Then, based on the degradation model, a clinically oriented fundus enhancement network (cofe-Net) is proposed to suppress global degradation factors, while simultaneously preserving anatomical retinal structures and pathological characteristics for clinical observation and analysis. Experiments on both synthetic and real images demonstrate that our algorithm effectively corrects low-quality fundus images without losing retinal details. Moreover, we also show that the fundus correction method can benefit medical image analysis applications, e.g., retinal vessel segmentation and optic disc/cup detection.

Exploiting Semantics for Face Image Deblurring

In this paper, we propose an effective and efficient face deblurring algorithm by exploiting semantic cues via deep convolutional neural networks. As the human faces are highly structured and share unified facial components (e.g., eyes and mouths), such semantic information provides a strong prior for restoration. We incorporate face semantic labels as input priors and propose an adaptive structural loss to regularize facial local structures within an end-to-end deep convolutional neural network. Specifically, we first use a coarse deblurring network to reduce the motion blur on the input face image. We then adopt a parsing network to extract the semantic features from the coarse deblurred image. Finally, the fine deblurring network utilizes the semantic information to restore a clear face image. We train the network with perceptual and adversarial losses to generate photo-realistic results. The proposed method restores sharp images with more accurate facial features and details. Quantitative and qualitative evaluations demonstrate that the proposed face deblurring algorithm performs favorably against the state-of-the-art methods in terms of restoration quality, face recognition and execution speed.

Human-Aware Motion Deblurring

This paper proposes a human-aware deblurring model that disentangles the motion blur between foreground (FG) humans and background (BG). The proposed model is based on a triple-branch encoder-decoder architecture. The first two branches are learned for sharpening FG humans and BG details, respectively; while the third one produces global, harmonious results by comprehensively fusing multi-scale deblurring information from the two domains. The proposed model is further endowed with a supervised, human-aware attention mechanism in an end-to-end fashion. It learns a soft mask that encodes FG human information and explicitly drives the FG/BG decoder-branches to focus on their specific domains. To further benefit the research towards Human-aware Image Deblurring, we introduce a large-scale dataset, named HIDE, which consists of 8,422 blurry and sharp image pairs with 65,784 densely annotated FG human bounding boxes. HIDE is specifically built to span a broad range of scenes, human object sizes, motion patterns, and background complexities. Extensive experiments on public benchmarks and our dataset demonstrate that our model performs favorably against the state-of-the-art motion deblurring methods, especially in capturing semantic details.