相关论文: Selecting Regularization Parameters for nuclear no…
We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with…
This paper studies the data-driven reconstruction of firing rate dynamics of brain activity described by linear-threshold network models. Identifying the system parameters directly leads to a large number of variables and a highly…
In this paper, the solution to the empirical risk minimization problem with $f$-divergence regularization (ERM-$f$DR) is presented and conditions under which the solution also serves as the solution to the minimization of the expected…
We consider the problem of recovering elements of a low-dimensional model from under-determined linear measurements. To perform recovery, we consider the minimization of a convex regularizer subject to a data fit constraint. Given a model,…
Fast Field-Cycling Nuclear Magnetic Resonance relaxometry is a non-destructive technique to investigate molecular dynamics and structure of systems having a wide range of applications such as environment, biology, and food. Besides a…
We consider the problem of finding the minimizer of a convex function $F: \mathbb R^d \rightarrow \mathbb R$ of the form $F(w) := \sum_{i=1}^n f_i(w) + R(w)$ where a low-rank factorization of $\nabla^2 f_i(w)$ is readily available. We…
In this paper, we present a rigorous framework for rational minimax approximation of matrix-valued functions that generalizes classical scalar approximation theory. Given sampled data $\{(x_\ell, {F}(x_\ell))\}_{\ell=1}^m$ where…
Recently theoretical guarantees have been obtained for matrix completion in the non-uniform sampling regime. In particular, if the sampling distribution aligns with the underlying matrix's leverage scores, then with high probability nuclear…
We develop a flexible framework for low-rank matrix estimation that allows us to transform noise models into regularization schemes via a simple bootstrap algorithm. Effectively, our procedure seeks an autoencoding basis for the observed…
Consider the following class of learning schemes: $$\hat{\boldsymbol{\beta}} := \arg\min_{\boldsymbol{\beta}}\;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbol{\beta}; y_j) + \lambda R(\boldsymbol{\beta}),\qquad\qquad (1) $$ where…
Recovering a low-rank signal matrix from its noisy observation, commonly known as matrix denoising, is a fundamental inverse problem in statistical signal processing. Matrix denoising methods are generally based on shrinkage or thresholding…
This paper presents a regularization technique incorporating a non-convex and non-smooth term, $\ell_{1}^{2}-\eta\ell_{2}^{2}$, with parameters $0<\eta\leq 1$ designed to address ill-posed linear problems that yield sparse solutions. We…
Model instability and poor prediction of long-term behavior are common problems when modeling dynamical systems using nonlinear "black-box" techniques. Direct optimization of the long-term predictions, often called simulation error…
In this work, we investigate data fitting problems with random noises. A randomized progressive iterative regularization method is proposed. It works well for large-scale matrix computations and converges in expectation to the least-squares…
A central challenge in machine learning is to understand how noise or measurement errors affect low-rank approximations, particularly in the spectral norm. This question is especially important in differentially private low-rank…
This paper considers a nuclear norm penalized estimator for panel data models with interactive effects. The low-rank interactive effects can be an approximate model and the rank of the best approximation unknown and grow with sample size.…
Although recovering an Euclidean distance matrix from noisy observations is a common problem in practice, how well this could be done remains largely unknown. To fill in this void, we study a simple distance matrix estimate based upon the…
Recently, the $\l_{p}$-norm regularization minimization problem $(P_{p}^{\lambda})$ has attracted great attention in compressed sensing. However, the $\l_{p}$-norm $\|x\|_{p}^{p}$ in problem $(P_{p}^{\lambda})$ is nonconvex and…
In image denoising problems, one widely-adopted approach is to minimize a regularized data-fit objective function, where the data-fit term is derived from a physical image acquisition model. Typically the regularizer is selected with two…
Matrix rank minimization problem is in general NP-hard. The nuclear norm is used to substitute the rank function in many recent studies. Nevertheless, the nuclear norm approximation adds all singular values together and the approximation…