Related papers: Regularization of Discrete Ill-Conditioned Problem…
The recently developed data-driven eigenmatrix method shows very promising reconstruction accuracy in sparse recovery for a wide range of kernel functions and random sample locations. However, its current implementation can lead to…
Algorithmic stability is a classical framework for analyzing the generalization error of learning algorithms. It predicts that an algorithm has small generalization error if it is insensitive to small perturbations in the training set such…
We consider the problem of recovering a lowrank matrix M from a small number of random linear measurements. A popular and useful example of this problem is matrix completion, in which the measurements reveal the values of a subset of the…
We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the…
In this paper we study general $l_p$ regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of first- and second-order stationary points, and hence also of local minimizers of the $l_p$…
Inverse problems are fundamental in fields like medical imaging, geophysics, and computerized tomography, aiming to recover unknown quantities from observed data. However, these problems often lack stability due to noise and…
Multi-task learning has been observed by many researchers, which supposes that different tasks can share a low-rank common yet latent subspace. It means learning multiple tasks jointly is better than learning them independently. In this…
For an ill-posed inverse problem, particularly with incomplete and limited measurement data, regularization is an essential tool for stabilizing the inverse problem. Among various forms of regularization, the lp penalty term provides a…
The instability of embedding spaces across model retraining cycles presents significant challenges to downstream applications using user or item embeddings derived from recommendation systems as input features. This paper introduces a novel…
Regularization-based approaches for injecting constraints in Machine Learning (ML) were introduced to improve a predictive model via expert knowledge. We tackle the issue of finding the right balance between the loss (the accuracy of the…
Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space-dimensions. The matrix…
In this work, we propose a method to efficiently find the regularization parameter for low-rank MMSE filters based on a Kronecker-product representation. We show that the regularization parameter is surprisingly linked to the problem of…
In this paper, we propose new randomization based algorithms for large scale linear discrete ill-posed problems with general-form regularization: ${\min} \|Lx\|$ subject to ${\min} \|Ax - b\|$, where $L$ is a regularization matrix. Our…
Regularization plays a pivotal role when facing the challenge of solving ill-posed inverse problems, where the number of observations is smaller than the ambient dimension of the object to be estimated. A line of recent work has studied…
State-of-the-art neural networks can be trained to become remarkable solutions to many problems. But while these architectures can express symbolic, perfect solutions, trained models often arrive at approximations instead. We show that the…
This paper considers a large class of linear operator equations, including linear boundary value problems for partial differential equations, and treats them as linear recovery problems for objects from their data. Well-posedness of the…
In recent years, new regularization methods based on (deep) neural networks have shown very promising empirical performance for the numerical solution of ill-posed problems, e.g., in medical imaging and imaging science. Due to the…
Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for…
In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method…
In this paper, we study the inverse problem for a class of abstract ultraparabolic equations which is well-known to be ill-posed. We employ some elementary results of semi-group theory to present the formula of solution, then show the…