It is well known that the minimum ℓ2-norm solution of the convex LASSO model, sayx⋆, is a continuous piecewise linear function of the regularization parameter λ, and its signed sparsity pattern is constant within each linear piece (Osborne 2000, Efron et al. 2004). The current study is an extension of this classic result, proving that the aforementioned properties extend to the min-norm solution mapx⋆(y, λ), whereyis the observed signal, for a generalization of LASSO termed the scaled generalized minimax concave (sGMC) model. The sGMC model adopts a nonconvex debiased variant of the ℓ1-norm as sparse regularizer, but its objective function is overall-convex. Based on the geometric properties ofx⋆(y, λ), we propose an extension of the least angle regression (LARS) algorithm, which iteratively computes the closed-form expression ofx⋆(y, λ) in each linear zone. Under suitable conditions, the proposed algorithm provably obtains the whole solution mapx⋆(y, λ) within finite iterations. Numerical experiments demonstrate the efficiency and reduced estimation error of the proposed algorithm compared to the conventional LARS. Notably, our proof techniques for establishing continuity and piecewise linearity ofx⋆(y, λ) are novel, and they lead to two side contributions: (a) our proofs establish continuity of the sGMC solution set as a set-valued mapping of (y, λ); (b) to prove piecewise linearity and piecewise constant sparsity pattern ofx⋆(y, λ), we do not require any assumption that previous work relies on (whereas to prove some additional properties ofx⋆(y, λ), we use a different set of assumptions from previous work).

The generalized minimax concave (GMC) penalty is a nonconvex sparse regularizer which can preserve the overall-convexity of the sparse least squares problem. In this paper, we study the solution path of a special but important instance of the GMC model termed the scaled GMC (sGMC) model. We show that despite the nonconvexity of the regularizer, there exists a solution path of the sGMC model which is piecewise linear as a function of the regularization parameter, and we propose an efficient algorithm for computing a solution path of this type. Our algorithm is an extension of the well-known least angle regression (LARS) algorithm for LASSO, hence we term the proposed algorithm LARS-sGMC. The proposed algorithm is provably correct and finitely terminating under suitable assumptions. Numerical experiments verify the correctness of LARS-sGMC, and demonstrate the usefulness of LARS-sGMC (with proper model selection criterion) for finding the optimal regularization parameter of the sGMC model.

In this paper, we propose a novel sparse adaptive filtering algorithm termed compensated shrinkage affine projection algorithm (CS-APA). Our cost function is the sum of a time-varying data fidelity term and a difference-of-convex (DC) type nonconvex sparse regularizer. The regularizer includes the well known MC and SCAD penalty as special instances, thus leading to sparse estimation with small bias. Leveraging the DC structure of the regularizer, the nonconvex forward-backward splitting algorithm can be applied to the cost function, whereby the proposed CS-APA is derived. We present several favourable properties of CS-APA, including its mean stability analysis. Numerical examples demonstrate the superiority of CS-APA with comparisons to existing methods.

Contrast enhanced ultrasound is a radiation-free imaging modality which uses encapsulated gas microbubbles for improved visualization of the vascular bed deep within the tissue. It has recently been used to enable imaging with unprecedented subwavelength spatial resolution by relying on super-resolution techniques. A typical preprocessing step in super-resolution ultrasound is to separate the microbubble signal from the cluttering tissue signal. This step has a crucial impact on the final image quality. Here, we propose a new approach to clutter removal based on robust principle component analysis (PCA) and deep learning. We begin by modeling the acquired contrast enhanced ultrasound signal as a combination of low rank and sparse components. This model is used in robust PCA and was previously suggested in the context of ultrasound Doppler processing and dynamic magnetic resonance imaging. We then illustrate that an iterative algorithm based on this model exhibits improved separation of microbubble signal from the tissue signal over commonly practiced methods. Next, we apply the concept of deep unfolding to suggest a deep network architecture tailored to our clutter filtering problem which exhibits improved convergence speed and accuracy with respect to its iterative counterpart. We compare the performance of the suggested deep network on both simulations and in-vivo rat brain scans, with a commonly practiced deep-network architecture and with the fast iterative shrinkage algorithm. We show that our architecture exhibits better image quality and contrast.

Sparse and low-rank decomposition, also known as robust principle component analysis, has been applied successfully in numerous applications. Typically, this approach leads to a minimization problem which is solved using iterative algorithms. Drawing inspiration from recurrent networks, in recent years deep-learning strategies have been extended to mimic the behavior of iterative algorithms, with reduced complexity. In this work, we propose an extension of these deep architectures to robust principle component analysis in which fully-connected layers are replaced with convolutional ones. This strategy offers spatial invariance and significant reduction in the number of learned parameters. We then apply the proposed method to contrast-enhanced ultrasound, in which low-rank tissue signal needs to be removed in order to visualize blood vessels. We demonstrate the effectiveness of our approach on simulations and in-vivo rat brain scans. The resulting images exhibit improved visual quality and contrast compared with images obtained by commonly practiced methods.