Random feature (RF) mapping is an attractive and powerful technique for solving large-scale nonparametric regression. Yet, the existing theoretical analysis crucially relies on the i.i.d. assumption that individuals in the data are independent and identically distributed. It is still unclear whether learning accuracy would be compromised when the i.i.d. assumption is violated. This paper aims to provide theoretical understanding of the kernel ridge regression (KRR) with RFs for large-scale dependent data. Specifically, we consider two types of data dependence structure, namely, the $\tau$-mixing process with exponential decay coefficient, and that with polynomial decay coefficient. Theoretically, we prove that the kernel ridge estimator with RFs achieves the minimax optimality under the exponential decay scenario, but yields a sub-optimal result under the polynomial decay case. Our analysis further reveals how the decay rate of the $\tau$-mixing coefficient impacts the learning accuracy of the kernel ridge estimator with RFs. Extensive numerical experiments on both synthetic and real examples further validate our theoretical findings and support the effectiveness of the KRR with RFs in dealing with dependent data.

The problem of finding a unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low rank matrix$\boldsymbol Y\in \mathbb R ^{p\times n}$that admits a sparse representation. Specifically, we consider$\boldsymbol Y= \boldsymbol A \boldsymbol X $where the matrix$\boldsymbol A\in \mathbb R^{p\times r}$has full column rank, with$r < \min \{n,p\}$, and the matrix$\boldsymbol X\in \mathbb R^{r\times n}$is element-wise sparse. We prove that this low rank, sparse decomposition of$\boldsymbol Y$can be uniquely identified, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for its nonconvex optimization landscape shows that anystrictlocal solution is close to the ground truth, and can be recovered by a simple data-driven initialization followed with any second order descent algorithm. Our theoretical findings are corroborated by numerical experiments.1

The problem of finding the unique low dimensional decomposition of a given matrix has been a fundamental and recurrent problem in many areas. In this paper, we study the problem of seeking a unique decomposition of a low-rank matrix $Y\in \mathbb{R}^{p\times n}$ that admits a sparse representation. Specifically, we consider $ Y = AX\in \mathbb{R}^{p\times n}$ where the matrix $A\in \mathbb{R}^{p\times r}$ has full column rank, with $r < \min\{n,p\}$, and the matrix $X\in \mathbb{R}^{r\times n}$ is element-wise sparse. We prove that this sparse decomposition of $Y$ can be uniquely identified by recovering ground-truth $A$ column by column, up to some intrinsic signed permutation. Our approach relies on solving a nonconvex optimization problem constrained over the unit sphere. Our geometric analysis for the nonconvex optimization landscape shows that any {\em strict} local solution is close to the ground truth solution, and can be recovered by a simple data-driven initialization followed with any second-order descent algorithm. At last, we corroborate these theoretical results with numerical experiments

This work is devoted to the finite sample prediction risk analysis of a class of linear predictors of a response $Y\in \mathbb{R}$ from a high-dimensional random vector $X\in \mathbb{R}^p$ when $(X,Y)$ follows a latent factor regression model generated by a unobservable latent vector $Z$ of dimension less than $p$. Our primary contribution is in establishing finite sample risk bounds for prediction with the ubiquitous Principal Component Regression (PCR) method, under the factor regression model, with the number of principal components adaptively selected from the data---a form of theoretical guarantee that is surprisingly lacking from the PCR literature. To accomplish this, we prove a master theorem that establishes a risk bound for a large class of predictors, including the PCR predictor as a special case. This approach has the benefit of providing a unified framework for the analysis of a wide range of linear prediction methods, under the factor regression setting. In particular, we use our main theorem to recover known risk bounds for the minimum-norm interpolating predictor, which has received renewed attention in the past two years, and a prediction method tailored to a subclass of factor regression models with identifiable parameters. This model-tailored method can be interpreted as prediction via clusters with latent centers. To address the problem of selecting among a set of candidate predictors, we analyze a simple model selection procedure based on data-splitting, providing an oracle inequality under the factor model to prove that the performance of the selected predictor is close to the optimal candidate. We conclude with a detailed simulation study to support and complement our theoretical results.

Topic models have become popular tools for dimension reduction and exploratory analysis of text data which consists in observed frequencies of a vocabulary of $p$ words in $n$ documents, stored in a $p\times n$ matrix. The main premise is that the mean of this data matrix can be factorized into a product of two non-negative matrices: a $p\times K$ word-topic matrix $A$ and a $K\times n$ topic-document matrix $W$. This paper studies the estimation of $A$ that is possibly element-wise sparse, and the number of topics $K$ is unknown. In this under-explored context, we derive a new minimax lower bound for the estimation of such $A$ and propose a new computationally efficient algorithm for its recovery. We derive a finite sample upper bound for our estimator, and show that it matches the minimax lower bound in many scenarios. Our estimate adapts to the unknown sparsity of $A$ and our analysis is valid for any finite $n$, $p$, $K$ and document lengths. Empirical results on both synthetic data and semi-synthetic data show that our proposed estimator is a strong competitor of the existing state-of-the-art algorithms for both non-sparse $A$ and sparse $A$, and has superior performance is many scenarios of interest.