Accelerating permutation testing in voxel-wise analysis through subspace tracking: A new plugin for SnPM.

Permutation testing is a non-parametric method for obtaining the max null distribution used to compute corrected p-values that provide strong control of false positives. In neuroimaging, however, the computational burden of running such an algorithm can be significant. We find that by viewing the permutation testing procedure as the construction of a very large permutation testing matrix, T, one can exploit structural properties derived from the data and the test statistics to reduce the runtime under certain conditions. In particular, we see that T is low-rank plus a low-variance residual. This makes T a good candidate for low-rank matrix completion, where only a very small number of entries of T (∼0.35% of all entries in our experiments) have to be computed to obtain a good estimate. Based on this observation, we present RapidPT, an algorithm that efficiently recovers the max null distribution commonly obtained through regular permutation testing in voxel-wise analysis. We present an extensive validation on a synthetic dataset and four varying sized datasets against two baselines: Statistical NonParametric Mapping (SnPM13) and a standard permutation testing implementation (referred as NaivePT). We find that RapidPT achieves its best runtime performance on medium sized datasets (50≤n≤200), with speedups of 1.5× - 38× (vs. SnPM13) and 20x-1000× (vs. NaivePT). For larger datasets (n≥200) RapidPT outperforms NaivePT (6× - 200×) on all datasets, and provides large speedups over SnPM13 when more than 10000 permutations (2× - 15×) are needed. The implementation is a standalone toolbox and also integrated within SnPM13, able to leverage multi-core architectures when available.

The Incremental Multiresolution Matrix Factorization Algorithm.

Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric matrices - an important aspect in the success of many vision problems. Our new algorithm, the incremental multiresolution matrix factorization, uncovers such structure one feature at a time, and hence scales well to large matrices. We describe how this multiscale analysis goes much farther than what a direct "global" factorization of the data can identify. We evaluate the efficacy of the resulting factorizations for relative leveraging within regression tasks using medical imaging data. We also use the factorization on representations learned by popular deep networks, providing evidence of their ability to infer semantic relationships even when they are not explicitly trained to do so. We show that this algorithm can be used as an exploratory tool to improve the network architecture, and within numerous other settings in vision.

When can Multi-Site Datasets be Pooled for Regression? Hypothesis Tests, ℓ 2-consistency and Neuroscience Applications.

Many studies in biomedical and health sciences involve small sample sizes due to logistic or financial constraints. Often, identifying weak (but scientifically interesting) associations between a set of predictors and a response necessitates pooling datasets from multiple diverse labs or groups. While there is a rich literature in statistical machine learning to address distributional shifts and inference in multi-site datasets, it is less clear when such pooling is guaranteed to help (and when it does not) - independent of the inference algorithms we use. In this paper, we present a hypothesis test to answer this question, both for classical and high dimensional linear regression. We precisely identify regimes where pooling datasets across multiple sites is sensible, and how such policy decisions can be made via simple checks executable on each site before any data transfer ever happens. With a focus on Alzheimer's disease studies, we present empirical results showing that in regimes suggested by our analysis, pooling a local dataset with data from an international study improves power.

Experimental Design on a Budget for Sparse Linear Models and Applications.

Budget constrained optimal design of experiments is a well studied problem. Although the literature is very mature, not many strategies are available when these design problems appear in the context of sparse linear models commonly encountered in high dimensional machine learning. In this work, we study this budget constrained design where the underlying regression model involves a ℓ1-regularized linear function. We propose two novel strategies: the first is motivated geometrically whereas the second is algebraic in nature. We obtain tractable algorithms for this problem which also hold for a more general class of sparse linear models. We perform a detailed set of experiments, on benchmarks and a large neuroimaging study, showing that the proposed models are effective in practice. The latter experiment suggests that these ideas may play a small role in informing enrollment strategies for similar scientific studies in the future.

Hypothesis Testing in Unsupervised Domain Adaptation with Applications in Alzheimer's Disease.

Consider samples from two different data sources [Formula: see text] and [Formula: see text]. We only observe their transformed versions [Formula: see text] and [Formula: see text], for some known function class h(·) and g(·). Our goal is to perform a statistical test checking if Psource = Ptarget while removing the distortions induced by the transformations. This problem is closely related to domain adaptation, and in our case, is motivated by the need to combine clinical and imaging based biomarkers from multiple sites and/or batches - a fairly common impediment in conducting analyses with much larger sample sizes. We address this problem using ideas from hypothesis testing on the transformed measurements, wherein the distortions need to be estimated in tandem with the testing. We derive a simple algorithm and study its convergence and consistency properties in detail, and provide lower-bound strategies based on recent work in continuous optimization. On a dataset of individuals at risk for Alzheimer's disease, our framework is competitive with alternative procedures that are twice as expensive and in some cases operationally infeasible to implement.

Imaging-based enrichment criteria using deep learning algorithms for efficient clinical trials in mild cognitive impairment.

The mild cognitive impairment (MCI) stage of Alzheimer's disease (AD) may be optimal for clinical trials to test potential treatments for preventing or delaying decline to dementia. However, MCI is heterogeneous in that not all cases progress to dementia within the time frame of a trial and some may not have underlying AD pathology. Identifying those MCIs who are most likely to decline during a trial and thus most likely to benefit from treatment will improve trial efficiency and power to detect treatment effects. To this end, using multimodal, imaging-derived, inclusion criteria may be especially beneficial. Here, we present a novel multimodal imaging marker that predicts future cognitive and neural decline from [F-18]fluorodeoxyglucose positron emission tomography (PET), amyloid florbetapir PET, and structural magnetic resonance imaging, based on a new deep learning algorithm (randomized denoising autoencoder marker, rDAm). Using ADNI2 MCI data, we show that using rDAm as a trial enrichment criterion reduces the required sample estimates by at least five times compared with the no-enrichment regime and leads to smaller trials with high statistical power, compared with existing methods.

A Projection free method for Generalized Eigenvalue Problem with a nonsmooth Regularizer.

Eigenvalue problems are ubiquitous in computer vision, covering a very broad spectrum of applications ranging from estimation problems in multi-view geometry to image segmentation. Few other linear algebra problems have a more mature set of numerical routines available and many computer vision libraries leverage such tools extensively. However, the ability to call the underlying solver only as a "black box" can often become restrictive. Many 'human in the loop' settings in vision frequently exploit supervision from an expert, to the extent that the user can be considered a subroutine in the overall system. In other cases, there is additional domain knowledge, side or even partial information that one may want to incorporate within the formulation. In general, regularizing a (generalized) eigenvalue problem with such side information remains difficult. Motivated by these needs, this paper presents an optimization scheme to solve generalized eigenvalue problems (GEP) involving a (nonsmooth) regularizer. We start from an alternative formulation of GEP where the feasibility set of the model involves the Stiefel manifold. The core of this paper presents an end to end stochastic optimization scheme for the resultant problem. We show how this general algorithm enables improved statistical analysis of brain imaging data where the regularizer is derived from other 'views' of the disease pathology, involving clinical measurements and other image-derived representations.

Speeding up Permutation Testing in Neuroimaging.

Multiple hypothesis testing is a significant problem in nearly all neuroimaging studies. In order to correct for this phenomena, we require a reliable estimate of the Family-Wise Error Rate (FWER). The well known Bonferroni correction method, while simple to implement, is quite conservative, and can substantially under-power a study because it ignores dependencies between test statistics. Permutation testing, on the other hand, is an exact, non-parametric method of estimating the FWER for a given α-threshold, but for acceptably low thresholds the computational burden can be prohibitive. In this paper, we show that permutation testing in fact amounts to populating the columns of a very large matrix P. By analyzing the spectrum of this matrix, under certain conditions, we see that P has a low-rank plus a low-variance residual decomposition which makes it suitable for highly sub-sampled - on the order of 0.5% - matrix completion methods. Based on this observation, we propose a novel permutation testing methodology which offers a large speedup, without sacrificing the fidelity of the estimated FWER. Our evaluations on four different neuroimaging datasets show that a computational speedup factor of roughly 50× can be achieved while recovering the FWER distribution up to very high accuracy. Further, we show that the estimated α-threshold is also recovered faithfully, and is stable.