Optimally Weighted PCA for High-Dimensional Heteroscedastic Data

Modern applications increasingly involve high-dimensional and heterogeneous data, e.g., datasets formed by combining numerous measurements from myriad sources. Principal Component Analysis (PCA) is a classical method for reducing dimensionality by projecting such data onto a low-dimensional subspace capturing most of their variation, but PCA does not robustly recover underlying subspaces in the presence of heteroscedastic noise. Specifically, PCA suffers from treating all data samples as if they are equally informative. This paper analyzes a weighted variant of PCA that accounts for heteroscedasticity by giving samples with larger noise variance less influence. The analysis provides expressions for the asymptotic recovery of underlying low-dimensional components from samples with heteroscedastic noise in the high-dimensional regime, i.e., for sample dimension on the order of the number of samples. Surprisingly, it turns out that whitening the noise by using inverse noise variance weights is suboptimal. We derive optimal weights, characterize the performance of weighted PCA, and consider the problem of optimally collecting samples under budget constraints.Comment: 52 pages, 13 figure

A Bayesian Heteroscedastic GLM with Application to fMRI Data with Motion Spikes

We propose a voxel-wise general linear model with autoregressive noise and heteroscedastic noise innovations (GLMH) for analyzing functional magnetic resonance imaging (fMRI) data. The model is analyzed from a Bayesian perspective and has the benefit of automatically down-weighting time points close to motion spikes in a data-driven manner. We develop a highly efficient Markov Chain Monte Carlo (MCMC) algorithm that allows for Bayesian variable selection among the regressors to model both the mean (i.e., the design matrix) and variance. This makes it possible to include a broad range of explanatory variables in both the mean and variance (e.g., time trends, activation stimuli, head motion parameters and their temporal derivatives), and to compute the posterior probability of inclusion from the MCMC output. Variable selection is also applied to the lags in the autoregressive noise process, making it possible to infer the lag order from the data simultaneously with all other model parameters. We use both simulated data and real fMRI data from OpenfMRI to illustrate the importance of proper modeling of heteroscedasticity in fMRI data analysis. Our results show that the GLMH tends to detect more brain activity, compared to its homoscedastic counterpart, by allowing the variance to change over time depending on the degree of head motion

Uncertainty in multitask learning: joint representations for probabilistic MR-only radiotherapy planning

Multi-task neural network architectures provide a mechanism that jointly integrates information from distinct sources. It is ideal in the context of MR-only radiotherapy planning as it can jointly regress a synthetic CT (synCT) scan and segment organs-at-risk (OAR) from MRI. We propose a probabilistic multi-task network that estimates: 1) intrinsic uncertainty through a heteroscedastic noise model for spatially-adaptive task loss weighting and 2) parameter uncertainty through approximate Bayesian inference. This allows sampling of multiple segmentations and synCTs that share their network representation. We test our model on prostate cancer scans and show that it produces more accurate and consistent synCTs with a better estimation in the variance of the errors, state of the art results in OAR segmentation and a methodology for quality assurance in radiotherapy treatment planning.Comment: Early-accept at MICCAI 2018, 8 pages, 4 figure

Large-scale Heteroscedastic Regression via Gaussian Process

Heteroscedastic regression considering the varying noises among observations has many applications in the fields like machine learning and statistics. Here we focus on the heteroscedastic Gaussian process (HGP) regression which integrates the latent function and the noise function together in a unified non-parametric Bayesian framework. Though showing remarkable performance, HGP suffers from the cubic time complexity, which strictly limits its application to big data. To improve the scalability, we first develop a variational sparse inference algorithm, named VSHGP, to handle large-scale datasets. Furthermore, two variants are developed to improve the scalability and capability of VSHGP. The first is stochastic VSHGP (SVSHGP) which derives a factorized evidence lower bound, thus enhancing efficient stochastic variational inference. The second is distributed VSHGP (DVSHGP) which (i) follows the Bayesian committee machine formalism to distribute computations over multiple local VSHGP experts with many inducing points; and (ii) adopts hybrid parameters for experts to guard against over-fitting and capture local variety. The superiority of DVSHGP and SVSHGP as compared to existing scalable heteroscedastic/homoscedastic GPs is then extensively verified on various datasets.Comment: 14 pages, 15 figure

HeMPPCAT: Mixtures of Probabilistic Principal Component Analysers for Data with Heteroscedastic Noise

Mixtures of probabilistic principal component analysis (MPPCA) is a well-known mixture model extension of principal component analysis (PCA). Similar to PCA, MPPCA assumes the data samples in each mixture contain homoscedastic noise. However, datasets with heterogeneous noise across samples are becoming increasingly common, as larger datasets are generated by collecting samples from several sources with varying noise profiles. The performance of MPPCA is suboptimal for data with heteroscedastic noise across samples. This paper proposes a heteroscedastic mixtures of probabilistic PCA technique (HeMPPCAT) that uses a generalized expectation-maximization (GEM) algorithm to jointly estimate the unknown underlying factors, means, and noise variances under a heteroscedastic noise setting. Simulation results illustrate the improved factor estimates and clustering accuracies of HeMPPCAT compared to MPPCA