Stochastic Subsampling for Factorizing Huge Matrices

We present a matrix-factorization algorithm that scales to input matrices with both huge number of rows and columns. Learned factors may be sparse or dense and/or non-negative, which makes our algorithm suitable for dictionary learning, sparse component analysis, and non-negative matrix factorization. Our algorithm streams matrix columns while subsampling them to iteratively learn the matrix factors. At each iteration, the row dimension of a new sample is reduced by subsampling, resulting in lower time complexity compared to a simple streaming algorithm. Our method comes with convergence guarantees to reach a stationary point of the matrix-factorization problem. We demonstrate its efficiency on massive functional Magnetic Resonance Imaging data (2 TB), and on patches extracted from hyperspectral images (103 GB). For both problems, which involve different penalties on rows and columns, we obtain significant speed-ups compared to state-of-the-art algorithms.

Subsampled online matrix factorization with convergence guarantees

We present a matrix factorization algorithm that scales to input matrices that are large in both dimensions (i.e., that contains morethan 1TB of data). The algorithm streams the matrix columns while subsampling them, resulting in low complexity per iteration andreasonable memory footprint. In contrast to previous online matrix factorization methods, our approach relies on low-dimensional statistics from past iterates to control the extra variance introduced by subsampling. We present a convergence analysis that guarantees us to reach a stationary point of the problem. Large speed-ups can be obtained compared to previous online algorithms that do not perform subsampling, thanks to the feature redundancy that often exists in high-dimensional settings.

Deriving reproducible biomarkers from multi-site resting-state data: An Autism-based example

Resting-state functional Magnetic Resonance Imaging (R-fMRI) holds the promise to reveal functional biomarkers of neuropsychiatric disorders. However, extracting such biomarkers is challenging for complex multi-faceted neuropatholo-gies, such as autism spectrum disorders. Large multi-site datasets increase sample sizes to compensate for this complexity, at the cost of uncontrolled heterogeneity. This heterogeneity raises new challenges, akin to those face in realistic diagnostic applications. Here, we demonstrate the feasibility of inter-site classification of neuropsychiatric status, with an application to the Autism Brain Imaging Data Exchange (ABIDE) database, a large (N=871) multi-site autism dataset. For this purpose, we investigate pipelines that extract the most predictive biomarkers from the data. These R-fMRI pipelines build participant-specific connectomes from functionally-defined brain areas. Connectomes are then compared across participants to learn patterns of connectivity that differentiate typical controls from individuals with autism. We predict this neuropsychiatric status for participants from the same acquisition sites or different, unseen, ones. Good choices of methods for the various steps of the pipeline lead to 67% prediction accuracy on the full ABIDE data, which is significantly better than previously reported results. We perform extensive validation on multiple subsets of the data defined by different inclusion criteria. These enables detailed analysis of the factors contributing to successful connectome-based prediction. First, prediction accuracy improves as we include more subjects, up to the maximum amount of subjects available. Second, the definition of functional brain areas is of paramount importance for biomarker discovery: brain areas extracted from large R-fMRI datasets outperform reference atlases in the classification tasks.

Assessing and tuning brain decoders: cross-validation, caveats, and guidelines

Decoding, ie prediction from brain images or signals, calls for empirical evaluation of its predictive power. Such evaluation is achieved via cross-validation, a method also used to tune decoders' hyper-parameters. This paper is a review on cross-validation procedures for decoding in neuroimaging. It includes a didactic overview of the relevant theoretical considerations. Practical aspects are highlighted with an extensive empirical study of the common decoders in within-and across-subject predictions, on multiple datasets --anatomical and functional MRI and MEG-- and simulations. Theory and experiments outline that the popular " leave-one-out " strategy leads to unstable and biased estimates, and a repeated random splits method should be preferred. Experiments outline the large error bars of cross-validation in neuroimaging settings: typical confidence intervals of 10%. Nested cross-validation can tune decoders' parameters while avoiding circularity bias. However we find that it can be more favorable to use sane defaults, in particular for non-sparse decoders.

Social-sparsity brain decoders: faster spatial sparsity

Spatially-sparse predictors are good models for brain decoding: they give accurate predictions and their weight maps are interpretable as they focus on a small number of regions. However, the state of the art, based on total variation or graph-net, is computationally costly. Here we introduce sparsity in the local neighborhood of each voxel with social-sparsity, a structured shrinkage operator. We find that, on brain imaging classification problems, social-sparsity performs almost as well as total-variation models and better than graph-net, for a fraction of the computational cost. It also very clearly outlines predictive regions. We give details of the model and the algorithm.

Dictionary Learning for Massive Matrix Factorization

Sparse matrix factorization is a popular tool to obtain interpretable data decompositions, which are also effective to perform data completion or denoising. Its applicability to large datasets has been addressed with online and randomized methods, that reduce the complexity in one of the matrix dimension, but not in both of them. In this paper, we tackle very large matrices in both dimensions. We propose a new factoriza-tion method that scales gracefully to terabyte-scale datasets, that could not be processed by previous algorithms in a reasonable amount of time. We demonstrate the efficiency of our approach on massive functional Magnetic Resonance Imaging (fMRI) data, and on matrix completion problems for recommender systems, where we obtain significant speed-ups compared to state-of-the art coordinate descent methods.

Compressed Online Dictionary Learning for Fast fMRI Decomposition

We present a method for fast resting-state fMRI spatial decomposi-tions of very large datasets, based on the reduction of the temporal dimension before applying dictionary learning on concatenated individual records from groups of subjects. Introducing a measure of correspondence between spatial decompositions of rest fMRI, we demonstrates that time-reduced dictionary learning produces result as reliable as non-reduced decompositions. We also show that this reduction significantly improves computational scalability.

Fast clustering for scalable statistical analysis on structured images

The use of brain images as markers for diseases or behavioral differences is challenged by the small effects size and the ensuing lack of power, an issue that has incited researchers to rely more systematically on large cohorts. Coupled with resolution increases, this leads to very large datasets. A striking example in the case of brain imaging is that of the Human Connectome Project: 20 Terabytes of data and growing. The resulting data deluge poses severe challenges regarding the tractability of some processing steps (discriminant analysis, multivariate models) due to the memory demands posed by these data. In this work, we revisit dimension reduction approaches, such as random projections, with the aim of replacing costly function evaluations by cheaper ones while decreasing the memory requirements. Specifically, we investigate the use of alternate schemes, based on fast clustering, that are well suited for signals exhibiting a strong spatial structure, such as anatomical and functional brain images. Our contribution is twofold: i) we propose a linear-time clustering scheme that bypasses the percolation issues inherent in these algorithms and thus provides compressions nearly as good as traditional quadratic-complexity variance-minimizing clustering schemes, ii) we show that cluster-based compression can have the virtuous effect of removing high-frequency noise, actually improving subsequent estimations steps. As a consequence, the proposed approach yields very accurate models on several large-scale problems yet with impressive gains in computational efficiency, making it possible to analyze large datasets.

Region segmentation for sparse decompositions: better brain parcellations from rest fMRI

Functional Magnetic Resonance Images acquired during resting-state provide information about the functional organization of the brain through measuring correlations between brain areas. Independent components analysis is the reference approach to estimate spatial components from weakly structured data such as brain signal time courses; each of these components may be referred to as a brain network and the whole set of components can be conceptualized as a brain functional atlas. Recently, new methods using a sparsity prior have emerged to deal with low signal-to-noise ratio data. However, even when using sophisticated priors, the results may not be very sparse and most often do not separate the spatial components into brain regions. This work presents post-processing techniques that automatically sparsify brain maps and separate regions properly using geometric operations, and compares these techniques according to faithfulness to data and stability metrics. In particular, among threshold-based approaches, hysteresis thresholding and random walker segmentation, the latter improves significantly the stability of both dense and sparse models.

Machine Learning for Neuroimaging with Scikit-Learn

Statistical machine learning methods are increasingly used for neuroimaging data analysis. Their main virtue is their ability to model high-dimensional datasets, e.g. multivariate analysis of activation images or resting-state time series. Supervised learning is typically used in decoding or encoding settings to relate brain images to behavioral or clinical observations, while unsupervised learning can uncover hidden structures in sets of images (e.g. resting state functional MRI) or find sub-populations in large cohorts. By considering different functional neuroimaging applications, we illustrate how scikit-learn, a Python machine learning library, can be used to perform some key analysis steps. Scikit-learn contains a very large set of statistical learning algorithms, both supervised and unsupervised, and its application to neuroimaging data provides a versatile tool to study the brain.