Projected principal component analysis in factor models

This paper introduces a Projected Principal Component Analysis (Projected-PCA), which employs principal component analysis to the projected (smoothed) data matrix onto a given linear space spanned by covariates. When it applies to high-dimensional factor analysis, the projection removes noise components. We show that the unobserved latent factors can be more accurately estimated than the conventional PCA if the projection is genuine, or more precisely, when the factor loading matrices are related to the projected linear space. When the dimensionality is large, the factors can be estimated accurately even when the sample size is finite. We propose a flexible semiparametric factor model, which decomposes the factor loading matrix into the component that can be explained by subject-specific covariates and the orthogonal residual component. The covariates' effects on the factor loadings are further modeled by the additive model via sieve approximations. By using the newly proposed Projected-PCA, the rates of convergence of the smooth factor loading matrices are obtained, which are much faster than those of the conventional factor analysis. The convergence is achieved even when the sample size is finite and is particularly appealing in the high-dimension-low-sample-size situation. This leads us to developing nonparametric tests on whether observed covariates have explaining powers on the loadings and whether they fully explain the loadings. The proposed method is illustrated by both simulated data and the returns of the components of the S&P 500 index.Comment: Published at http://dx.doi.org/10.1214/15-AOS1364 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Estimation of functionals of sparse covariance matrices

High-dimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other Lr norms. Motivated by the computation of critical values of such tests, we investigate the difficulty of estimation the functionals of sparse correlation matrices. Specifically, we show that simple plug-in procedures based on thresholded estimators of correlation matrices are sparsity-adaptive and minimax optimal over a large class of correlation matrices. Akin to previous results on functional estimation, the minimax rates exhibit an elbow phenomenon. Our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics.National Science Foundation (U.S.) (Grant DMS-13-17308)National Science Foundation (U.S.) (Grant CAREER-DMS-1053987

Hierarchy Flow For High-Fidelity Image-to-Image Translation

Image-to-image (I2I) translation comprises a wide spectrum of tasks. Here we divide this problem into three levels: strong-fidelity translation, normal-fidelity translation, and weak-fidelity translation, indicating the extent to which the content of the original image is preserved. Although existing methods achieve good performance in weak-fidelity translation, they fail to fully preserve the content in both strong- and normal-fidelity tasks, e.g. sim2real, style transfer and low-level vision. In this work, we propose Hierarchy Flow, a novel flow-based model to achieve better content preservation during translation. Specifically, 1) we first unveil the drawbacks of standard flow-based models when applied to I2I translation. 2) Next, we propose a new design, namely hierarchical coupling for reversible feature transformation and multi-scale modeling, to constitute Hierarchy Flow. 3) Finally, we present a dedicated aligned-style loss for a better trade-off between content preservation and stylization during translation. Extensive experiments on a wide range of I2I translation benchmarks demonstrate that our approach achieves state-of-the-art performance, with convincing advantages in both strong- and normal-fidelity tasks. Code and models will be at https://github.com/WeichenFan/HierarchyFlow.Comment: arXiv admin note: text overlap with arXiv:2207.0190

Spectral Ranking Inferences based on General Multiway Comparisons

This paper studies the performance of the spectral method in the estimation and uncertainty quantification of the unobserved preference scores of compared entities in a very general and more realistic setup in which the comparison graph consists of hyper-edges of possible heterogeneous sizes and the number of comparisons can be as low as one for a given hyper-edge. Such a setting is pervasive in real applications, circumventing the need to specify the graph randomness and the restrictive homogeneous sampling assumption imposed in the commonly-used Bradley-Terry-Luce (BTL) or Plackett-Luce (PL) models. Furthermore, in the scenarios when the BTL or PL models are appropriate, we unravel the relationship between the spectral estimator and the Maximum Likelihood Estimator (MLE). We discover that a two-step spectral method, where we apply the optimal weighting estimated from the equal weighting vanilla spectral method, can achieve the same asymptotic efficiency as the MLE. Given the asymptotic distributions of the estimated preference scores, we also introduce a comprehensive framework to carry out both one-sample and two-sample ranking inferences, applicable to both fixed and random graph settings. It is noteworthy that it is the first time effective two-sample rank testing methods are proposed. Finally, we substantiate our findings via comprehensive numerical simulations and subsequently apply our developed methodologies to perform statistical inferences on statistics journals and movie rankings