Improving predictive models using non-spherical Gaussian priors

Anwar O. Nunez-Elizalde∗ (aone@berkeley.edu)

Alexander G. Huth∗ (alex.huth@berkeley.edu)

Jack L. Gallant (gallant@berkeley.edu)Helen Wills Neuroscience Institute, UC Berkeley

Berkeley, CA 94720 USA

AbstractPredictive models for neural or fMRI data areoften fit using regression methods that employpriors on the model parameters. One widelyused method is ridge regression, which em-ploys a Gaussian prior that has equal and in-dependent variance for all parameters (i.e. aspherical multivariate Gaussian). However, aspherical prior is not always appropriate: thereare many cases where expert knowledge or hy-potheses about the structure of the model pa-rameters could be used to construct a betterprior. Here we show that Tikhonov regres-sion with non-spherical Gaussian priors can im-prove several predictive models for fMRI data.This is particularly important when combiningfeature spaces into a single predictive model.Finally, we demonstrate a computationally ef-ficient method for incorporating non-sphericalpriors into regression models.

Keywords: encoding models; predictive models

IntroductionNon-spherical Gaussian priors can be employed us-ing a generalized form of ridge known as Tikhonovregression. Yet Tikhonov regression is not (explic-itly) used very often in neuroscience. Here we showthat Tikhonov regression with non-spherical Gaus-sian priors can improve several predictive models forfMRI data. We also show that many earlier stud-ies have implicitly used Tikhonov regression by lin-early transforming the regressors before performingridge regression. Our first result is based on anfMRI experiment in which subjects listened to sev-eral hours of natural stories (Huth, de Heer, Griffiths,Theunissen, & Gallant, 2016). A predictive model in

which the features were indicator variables for eachunique word provided poor predictions of held-outdata. However, performance was improved drasti-cally by employing a non-spherical Gaussian priorin which words that have similar meanings are likelyto be assigned similar weights. Our second resultcomes form an fMRI experiment in which subjectswatched hours of natural movies (Nishimoto et al.,2011). A predictive model in which low-level struc-tural and high-level categorical features were com-bined predicted held-out responses to novel stimulipoorly when using a spherical prior. We show thata non-spherical prior where the two feature spaceshave unequal variance performs better.

MethodsThe key insight underlying our results is thatTikhonov regression problems can be converted toa standard form (Hansen, 1998) that is more inter-pretable and yields more computationally efficientsolutions than the traditional formulation. The tra-ditional solution for Tikhonov regression is given by:β =

(X>X +λC>C

)−1 X>Y, where C is the penaltymatrix, and the prior distribution on β is multivari-ate normal with zero mean and covariance equalto

(λC>C

)−1. This traditional form is expensive

to compute. However, by a change of variables,A = XC−1, the problem is converted to standardform: β = C−1

(A>A+λI

)−1 A>Y. Thus, we seethat Tikhonov regression in standard form is iden-tical to ridge regression after projecting the regres-sors onto the inverse of the penalty matrix. The ma-trices are simultaneously diagonalizable, so we canefficiently test different values of without fully recom-puting any matrix inverse. Furthermore, this showsthat any model that uses ridge regression after ap-

plying a linear transformation to the regressors is im-plicitly doing Tikhonov regression. This suggests al-ternative interpretations of the models used in someprevious studies For example, the model used inHuth, et al., 2012, supplemented manually labeledvisual categories (e.g. dog) with higher-level cat-egories inferred using WordNet (e.g. canine, car-nivore, etc.). Because this operation is linear, themodel can be re-interpreted as Tikhonov regressionwhere the prior covariance between two parameters(e.g. dog and cat) is proportional to the number ofhigher-level categories that they have in common.

ResultsLanguage Experiment. We recorded BOLD fMRIdata from two subjects while they listened to hoursof natural narrative stories (Huth et al., 2016). Thestories were annotated with the exact time and iden-tity of each spoken word. We first fit a simple modelthat predicts BOLD timecourses in each voxel asa weighted sum across word timecourses, using aspherical prior on the weights. To extend this modelwe incorporated a non-spherical semantic prior, as-suming that words having similar meanings shouldhave similar weights. Semantic similarity was de-termined using a word embedding space based onword co-occurrence statistics across a large cor-pus of text (Wehbe et al., 2014). Models basedon the spherical and semantic priors were fit to oneset of data and then tested on a separate, held-outdataset. We found that the non-spherical seman-tic prior outperformed the spherical prior by a largemargin. This shows that theoretically motivated non-spherical priors can improve model performance.

Movie Experiment. We recorded BOLD fMRI datafrom three subjects while they watched naturalmovies. Motion-energy features were extracted fromthese movies using a spatio-temporal Gabor pyra-mid (Nishimoto et al., 2011), and visual object andaction categories present in each one second seg-ment of the movies were labeled by hand (Huth, etal., 2012). We fit a single model that predicts BOLDresponses as a linear combination of both motion-energy and category features. Applying a sphericalprior to this combined model yielded poor predic-tions. To address this issue we fit the same com-

bined model using a non-spherical prior in which allparameters are independent but the variance is dif-ferent for the two feature spaces. This imposes onespherical prior on the motion-energy features anda different spherical prior on the category features.Models based on the spherical and non-sphericalpriors were tested on a held-out dataset. The non-spherical prior provided far better predictions thanthe spherical prior (Fig. 1). This suggests that non-spherical priors are vital for optimally combining dif-ferent feature spaces into a single predictive model.

Figure 1: Movie experiment. Comparison of predic-tion accuracy for the same model fit with differentpriors. Left inset : Points that lie above the diago-nal line correspond to voxels that were more accu-rately predicted with a non-spherical prior. Flatmap:The prediction accuracy of the same model fit witha spherical (red) and a non-spherical (blue) prior isshown on a flattened view of the cortical sheet. Themodel fit with a non-spherical prior provides betterpredictions in most of visual cortex.

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Nishimoto, S., Vu, A. T., Naselaris, T., Benjamini, Y.,Yu, B., & Gallant, J. L. (2011). Reconstructingvisual experiences from brain activity evoked bynatural movies. Current Biology .

Wehbe, L., Murphy, B., Talukdar, P., Fyshe, A., Ram-das, A., & Mitchell, T. (2014). Simultaneously un-covering the patterns of brain regions involved indifferent story reading subprocesses. PloS one.