Related papers: Phase transition and higher order analysis of $L_q…
We study the problem of estimating $\beta \in \mathbb{R}^p$ from its noisy linear observations $y= X\beta+ w$, where $w \sim N(0, \sigma_w^2 I_{n\times n})$, under the following high-dimensional asymptotic regime: given a fixed number…
We consider the problem of recovering a $k$-sparse signal ${\mbox{$\beta$}}_0\in\mathbb{R}^p$ from noisy observations $\bf y={\bf X}\mbox{$\beta$}_0+{\bf w}\in\mathbb{R}^n$. One of the most popular approaches is the $l_1$-regularized least…
The class of Lq-regularized least squares (LQLS) are considered for estimating a p-dimensional vector \b{eta} from its n noisy linear observations y = X\b{eta}+w. The performance of these schemes are studied under the high-dimensional…
We consider a sparse linear regression model Y=X\beta^{*}+W where X has a Gaussian entries, W is the noise vector with mean zero Gaussian entries, and \beta^{*} is a binary vector with support size (sparsity) k. Using a novel conditional…
We study the estimation of $\beta$ for the nonlinear model $y = f(X\sp{\top}\beta) + \epsilon$ when $f$ is a nonlinear transformation that is known, $\beta$ has sparse nonzero coordinates, and the number of observations can be much smaller…
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…
In this paper, we aim at recovering an unknown signal x0 from noisy L1measurements y=Phi*x0+w, where Phi is an ill-conditioned or singular linear operator and w accounts for some noise. To regularize such an ill-posed inverse problem, we…
Consider the noisy underdetermined system of linear equations: y=Ax0 + z0, with n x N measurement matrix A, n < N, and Gaussian white noise z0 ~ N(0,\sigma^2 I). Both y and A are known, both x0 and z0 are unknown, and we seek an…
We discuss stability for a class of learning algorithms with respect to noisy labels. The algorithms we consider are for regression, and they involve the minimization of regularized risk functionals, such as L(f) := 1/N sum_i…
$l^q$-regularization has been demonstrated to be an attractive technique in machine learning and statistical modeling. It attempts to improve the generalization (prediction) capability of a machine (model) through appropriately shrinking…
Within the context of recursive least squares (RLS) parameter estimation, the goal of the present paper is to study the effect of regularization-induced bias on the transient and asymptotic accuracy of the parameter estimates. We consider…
We study the statistics of the condition number $\kappa=\lambda_{\mathrm{max}}/\lambda_{\mathrm{min}}$ (the ratio between largest and smallest squared singular values) of $N\times M$ Gaussian random matrices. Using a Coulomb fluid…
We focus on the high-dimensional linear regression problem, where the algorithmic goal is to efficiently infer an unknown feature vector $\beta^*\in\mathbb{R}^p$ from its linear measurements, using a small number $n$ of samples. Unlike most…
Consider the following class of learning schemes: \begin{equation} \label{eq:main-problem1} \hat{\boldsymbol{\beta}} := \underset{\boldsymbol{\beta} \in \mathcal{C}}{\arg\min} \;\sum_{j=1}^n \ell(\boldsymbol{x}_j^\top\boldsymbol{\beta};…
We consider a stochastic process with long-range dependence perturbed by multiplicative noise. The marginal distributions of both the original process and the noise have regularly-varying tails, with tail indices $\alpha,\alpha'>0$,…
We investigate the local quantum uncertainty (LQU) between a block of L qubits and one single qubit in a composite system of n qubits driven through a quantum phase transition (QPT). A first-order QPT is analytically considered through a…
In this paper, we study extended linear regression approaches for quantum state tomography based on regularization techniques. For unknown quantum states represented by density matrices, performing measurements under certain basis yields…
We consider the problem of recovering a vector $\beta_o \in \mathbb{R}^p$ from $n$ random and noisy linear observations $y= X\beta_o + w$, where $X$ is the measurement matrix and $w$ is noise. The LASSO estimate is given by the solution to…
We consider a high dimensional linear regression problem where the goal is to efficiently recover an unknown vector $\beta^*$ from $n$ noisy linear observations $Y=X\beta^*+W \in \mathbb{R}^n$, for known $X \in \mathbb{R}^{n \times p}$ and…
An observed $K$-dimensional series $\left\{ y_{n}\right\} _{n=1}^{N}$ is expressed in terms of a lower $p$-dimensional latent series called factors $f_{n}$ and random noise $\varepsilon_{n}$. The equation, $y_{n}=Qf_{n}+\varepsilon_{n}$ is…