{"title": "Unified representation of tractography and diffusion-weighted MRI data using sparse multidimensional arrays", "book": "Advances in Neural Information Processing Systems", "page_first": 4340, "page_last": 4351, "abstract": "Recently, linear formulations and convex optimization methods have been proposed to predict diffusion-weighted Magnetic Resonance Imaging (dMRI) data given estimates of brain connections generated using tractography algorithms. The size of the linear models comprising such methods grows with both dMRI data and connectome resolution, and can become very large when applied to modern data. In this paper, we introduce a method to encode dMRI signals and large connectomes, i.e., those that range from hundreds of thousands to millions of fascicles (bundles of neuronal axons), by using a sparse tensor decomposition. We show that this tensor decomposition accurately approximates the Linear Fascicle Evaluation (LiFE) model, one of the recently developed linear models. We provide a theoretical analysis of the accuracy of the sparse decomposed model, LiFESD, and demonstrate that it can reduce the size of the model significantly. Also, we develop algorithms to implement the optimisation solver using the tensor representation in an efficient way.", "full_text": "Uni\ufb01ed representation of tractography and\ndi\ufb00usion-weighted MRI data using sparse\n\nmultidimensional arrays\n\nCesar F. Caiafa\u2217\n\nDepartment of Psychological and Brain Sciences\nIndiana University (47405) Bloomington, IN, USA\n\nIAR - CCT La Plata, CONICET / CIC-PBA\n\n(1894) V. Elisa, ARGENTINA\n\nccaiafa@gmail.com\n\nOlaf Sporns\n\nDepartment of Psychological and Brain Sciences\nIndiana University (47405) Bloomington, IN, USA\n\nosporns@indiana.edu\n\nAndrew J. Saykin\n\nDepartment of Radiology - Indiana University\n\nSchool of Medicine. (46202) Indianapolis, IN, USA\n\nasaykin@iupui.edu\n\nFranco Pestilli\u2020\n\nDepartment of Psychological and Brain Sciences\nIndiana University (47405) Bloomington, IN, USA\n\nfranpest@indiana.edu\n\nAbstract\n\nRecently, linear formulations and convex optimization methods have been\nproposed to predict di\ufb00usion-weighted Magnetic Resonance Imaging (dMRI)\ndata given estimates of brain connections generated using tractography\nalgorithms. The size of the linear models comprising such methods grows\nwith both dMRI data and connectome resolution, and can become very\nlarge when applied to modern data. In this paper, we introduce a method\nto encode dMRI signals and large connectomes, i.e., those that range from\nhundreds of thousands to millions of fascicles (bundles of neuronal axons), by\nusing a sparse tensor decomposition. We show that this tensor decomposition\naccurately approximates the Linear Fascicle Evaluation (LiFE) model, one\nof the recently developed linear models. We provide a theoretical analysis of\nthe accuracy of the sparse decomposed model, LiFESD, and demonstrate that\nit can reduce the size of the model signi\ufb01cantly. Also, we develop algorithms\nto implement the optimization solver using the tensor representation in an\ne\ufb03cient way.\n\n\u2217http://web.fi.uba.ar/~ccaiafa/Cesar.html\n\u2020http://www.brain-life.org/plab/\n\n31st Conference on Neural Information Processing Systems (NIPS 2017), Long Beach, CA, USA.\n\n\f1 Introduction\n\nMultidimensional arrays, hereafter referred to as tensors, are useful mathematical objects to\nmodel a variety of problems in machine learning [2, 47] and neuroscience [27, 8, 50, 48, 3, 26,\n13]. Tensor decomposition algorithms have a long history of applications in signal processing,\nhowever, only recently their relation to sparse representations has started to be explored\n[35, 11]. In this work, we present a sparse tensor decomposition model and its associated\nalgorithm applied to di\ufb00usion-weighted Magnetic Resonance Imaging (dMRI).\nDi\ufb00usion-weighted MRI allows us to estimate structural brain connections in-vivo by mea-\nsuring the di\ufb00usion of water molecules at di\ufb00erent spatial directions. Brain connections\nare comprised of a set of fascicles describing the putative position and orientation of the\nneuronal axons bundles wrapped by myelin sheaths traveling within the living human brain\n[25]. The process by which fascicles (the connectome) are identi\ufb01ed from dMRI measurements\nis called tractography. Tractography and dMRI are the primary methods for mapping struc-\ntural brain networks and white matter tissue properties in living human brains [6, 46, 34].\nDespite current limits and criticisms, through these methods we have learned much about\nthe macrostructural organization of the human brain, such that network neuroscience has\nbecome one of the fastest-growing scienti\ufb01c \ufb01elds [38, 43, 44].\nIn recent years, a large variety of tractography algorithms have been proposed and tested\non modern datasets such as the Human Connectome Project (HCP) [45]. However, it\nhas been established that the estimated anatomical properties of the fascicles depend on\ndata type, tractography algorithm and parameters settings [32, 39, 7]. Such variability in\nestimates makes it di\ufb03cult to trust a single algorithm for all applications, and calls for\nroutine statistical evaluation methods of brain connectomes [32]. For this reason, linear\nmethods based on convex optimization have been proposed for connectome evaluation [32, 39]\nand simultaneous connectome and white matter microstructure estimation [15]. However,\nthese methods can require substantial computational resources (memory and computation\nload) making it prohibitive to apply them to the highest resolution datasets.\nIn this article, we propose a method to encode brain connectomes in multidimensional arrays\nand perform statistical evaluation e\ufb03ciently on high-resolution datasets. The article is\norganized as follows: in section 2, the connectome encoding method is introduced; in section\n2.1, a linear formulation of the connectome evaluation problem is described; in section 3, the\napproximated tensor decomposed model is introduced; in section 3.3, we derive a theoretical\nbound of the approximation error and compute the theoretical compression factor obtained\nwith the tensor decomposition; in section 4 we develop algorithms to make the operations\nneeded for solving the connectome evaluation optimization problem; in section 5 we present\nexperimental results using high resolution in vivo datasets; \ufb01nally, in section 6, the main\nconclusions of our work are outlined.\n\n2 Encoding brain connectomes into multidimensional array\n\nstructures.\n\nWe propose a framework to encode brain connectome data (both dMRI and white matter\nfascicles) into tensors [12, 11, 23] to allow fast and e\ufb03cient mathematical operations on the\nstructure of the connectome. Here, we introduce the tensor encoding framework and show\nhow it can be used to implement recent methods for statistical evaluation of tractography\n[32]. More speci\ufb01cally, we demonstrate that the framework can be used to approximate the\nLinear Fascicle Evaluation model [32] with high accuracy while reducing the size of the model\nsubstantially (with measured compression factors up to 40x). Hereafter, we refer to the new\ntensor encoding method as ENCODE [10]. ENCODE maps fascicles from their natural brain\nspace (Fig. 1(a)) into a three dimensional sparse tensor \u03a6 (Fig. 1(b)). The \ufb01rst dimension\nof \u03a6 (1st mode) encodes each individual white matter fascicle\u2019s orientation at each position\nalong their path through the brain. Individual segments (nodes) in a fascicle are coded as\nnon-zero entries in the sparse array (dark-blue cubes in Fig. 1(b)). The second dimension\nof \u03a6 (2nd mode) encodes each fascicle\u2019s spatial position within dMRI data volume (voxels).\nSlices in this second dimension represent single voxels (cyan lateral slice in Fig. 1(b)). The\n\n2\n\n\fthird dimension (3rd mode) encodes the indices of each fascicle within the connectome. Full\nfascicles are encoded as \u03a6 frontal slices (c.f., yellow and blue in Fig. 1(b)).\n\nFigure 1: The ENCODE method: mapping structural connectomes from natural brain space to\ntensor space. (a) Two example white matter fascicles (f1 and f2) passing through three voxels (v1,\nv2 and v3). (b) Encoding of the two fascicles in a three dimensional tensor. The non-zero entries in\n\u03a6 indicate fascicle\u2019s orientation (1st mode), position (voxel, 2nd mode) and identity (3rd mode).\n\nBelow we demonstrate how to use ENCODE to integrate connectome each fascicle\u2019s structure\nand measured dMRI signal into a single tensor decomposition model. We then show how to\nuse this decompositon model to implement very e\ufb03ciently a recent model for tractography\nevaluation, the linear fascicle evaluation method, also referred to as LiFE [32]. Before\nintroducing the tensor decomposition method, we brie\ufb02y describe the LiFE model, as this is\nneeded to explain the model decomposition using the ENCODE method. We then calculate\nthe theoretical bounds to accuracy and compression factor that can be achieved using\nENCODE and tensor decomposition. Finally, we report the results of experiments on real\ndata and validate the theoretical calculations.\n\n2.1 Statistical evaluation for brain connectomes by convex optimization.\nThe Linear Fascicle Evaluation (LiFE) method was introduced to compute the statistical\nerror of the fascicles comprising a structural brain connectome in predicting the measured\ndi\ufb00usion signal [32]. The fundamental idea behind LiFE is that a connectome should contain\nfascicles whose trajectories represent the measured di\ufb00usion signal well. LiFE implements\na method for connectome evaluation that can be used, among other things, to eliminate\ntracked fascicles that do not predict well the di\ufb00usion signal. LiFE takes as input the set of\nfascicles generated by using tractography methods (the candidate connectome) and returns\nas output the subset of fascicles that best predict the measured dMRI signal (the optimized\nconnectome). Fascicles are scored with respect to how well their trajectories represent the\nmeasured di\ufb00usion signal in the voxels along the their path. To do so, weights are assigned\nto each fascicle using convex optimization. Fascicles assigned a weight of zero are removed\nfrom the connectome, as their contribution to predicting the di\ufb00usion signal is null. The\nfollowing linear system describes the equation of LiFE (see Fig. 2(a)):\n\n(2.1)\nwhere y \u2208 RN\u03b8 Nv is a vector containing the demeaned signal yi = \u00afS(\u03b8ni , vi) measured\nat all white-matter voxels vi \u2208 V = {1, 2, . . . , Nv} and across all di\ufb00usion directions \u03b8n \u2208\n\u0398 = {\u03b81, \u03b82, . . . , \u03b8N\u03b8} \u2282 R3, and w \u2208 RNf contains the weights for each fascicle in the\nconnectome.\nMatrix M \u2208 RN\u03b8 Nv\u00d7Nf contains, at column f, the predicted demeaned signal contributed\nby fascicle f at all voxels V and across all directions \u0398:\n\ny \u2248 Mw,\n\n(2.2)\nS0(v) is de\ufb01ned as the non di\ufb00usion-weighted signal and Of(\u03b8, vf) is the orientation distri-\nbution function [32] of fascicle f at di\ufb00usion direction \u03b8, i.e.\n\nM(i, f) = S0(vi)Of(\u03b8ni , vf).\n\nOf(\u03b8, vf) = e\u2212b(\u03b8T vf )2 \u2212 1\n\ne\u2212b(\u03b8T\n\nn vf )2\n\n,\n\n(2.3)\n\nX\n\nN\u03b8\n\n\u03b8n\u2208\u0398\n\n3\n\nOrientationVoxelsFasciclesnon-zero entryfasciclefasciclevoxel(b) (a)\f(a) The predicted signal y \u2208 RN\u03b8 Nv in\nFigure 2: The Linear Fascicle Evaluation (LiFE) model.\nall voxels and gradient directions is obtained by multiplying matrix M \u2208 RN\u03b8 Nv\u00d7Nf by the vector\nof weights w \u2208 RNf (see equation 2.1).\n(c) The\npredicted di\ufb00usion signal yv \u2208 RN\u03b8 at voxel v is approximated by a nonnegative weighted linear\ncombination of the predicted signals for the fascicles in the voxel.\n\n(b) A voxel containing two fascicles, f1 and f2.\n\nwhere the simple \u201cstick\u201d di\ufb00usion tensor model [31] was used and vector vf \u2208 R3 is de\ufb01ned\nas the spatial orientation of the fascicle in that voxel.\nWhereas vector y and matrix M in equation (2.1) are fully determined by the dMRI\nmeasurements and the output of a tractography algorithm, respectively, the vector of weights\nw needs to be estimated by solving a Non-Negative Least squares (NNLS) optimization\nproblem, which is de\ufb01ned as follows:\n\n(cid:18)1\n2ky \u2212 Mwk2\n\n(cid:19)\n\nmin\nw\n\nsubject to wf \u2265 0,\u2200f.\n\n(2.4)\n\nAs a result, a sparse non-negative vector of weights w is obtained. Whereas nonzero weights\ncorrespond to fascicles that contribute to predict the measured dMRI signal, fascicles with\nzero weight make no contribution to predicting the measurements and can be eliminated.\nIn this way, LiFE identi\ufb01es the fascicles supported by the data in a candidate connectome\nproviding a principled approach to evaluate connectomes in terms of prediction error as well\nas the number of non-zero weighted fascicles.\nA noticeable property of the LiFE method is that the size of matrix M in equation (2.1)\ncan require tens of gigabytes for full-brain connectomes, even when using optimized sparse\nmatrix formats [19]. Below we show how to use ENCODE to implement a sparse tensor\ndecomposition [9, 11] of matrix M. This decomposition allows accurate approximation of\nthe original LiFE model with dramatic reduction in memory requirements.\n\n3 Theoretical results: Tensor decomposition and approximation\n\nof the linear model for tractography evaluation.\n\nWe describe the theoretical approach to factorizing the LiFE model, eq. (2.1). We note\nthat matrix M \u2208 RN\u03b8Nv\u00d7Nf (Fig. 2(a)) can be rewritten as a tensor (3D-array) M \u2208\nRN\u03b8\u00d7Nv\u00d7Nf by decoupling the gradient direction and voxel indices into separate indices, i.e.\nM(ni, vi, f) = M(i, f), where ni = {1, 2, . . . , N\u03b8}, vi = {1, 2, . . . , Nv} and f = {1, 2, . . . , Nf}.\nThus, equation (2.1) can be rewritten in tensor form as follows:\n\n(3.1)\nwhere Y \u2208 RN\u03b8\u00d7Nv is obtained by converting vector y \u2208 RN\u03b8 Nv into a matrix (matricization)\nand \u201c\u00d7n\u201d is the tensor-by-matrix product in mode-n [23], more speci\ufb01cally, the mode-3\n\nY \u2248 M \u00d73 wT ,\n\n4\n\nVoxelvoxelEmpty entries (zero values)(b) (a)(c) \fproduct in the above equation is de\ufb01ned as follows: Y(n, v) =PNf\n\nf=1 M(n, v, f)wf. Below,\nwe show how to approximate the tensor model in equation (3.1) using a sparse Tucker\ndecomposition [9] by \ufb01rst focusing on the dMRI signal in individual voxels and then across\nvoxels.\n\nFigure 3: The LiFESD model:\n(a) Each block Mv of matrix M (a lateral slice in tensor M) is\nfactorized by using a dictionary of di\ufb00usion signal predictions D and a sparse matrix of coe\ufb03cients\n\u03a6v.\n(b) LiFESD model is written as a Tucker decomposition model with a sparse core tensor \u03a6 and\nfactors D (mode-1) and wT (mode-3).\n(c). The maximum distance between a fascicle orientation\nvector v and its approximation va is determined by the discretization of azimuth (\u2206\u03b1) and elevation\n(\u2206\u03b2) spherical coordinates. More speci\ufb01cally, for \u2206\u03b1 = \u2206\u03b2 = \u03c0/L, the maximum discretization\nerror is k\u2206vk \u2264 \u03c0\u221a\n2L\n\n.\n\n3.1 Approximation of the linear model within individual brain voxels.\nWe focus on writing the linear formulation of the di\ufb00usion prediction model (Fig. 2(b)-(c))\nby restricting equation (3.1) to individual voxels, v:\nyv \u2248 Mvw,\n\n(3.2)\nwhere vector yv = Y(:, v) \u2208 RN\u03b8 and matrix Mv = M(:, v, :) \u2208 RN\u03b8\u00d7Nf , correspond to a\ncolumn in Y and a lateral slice in tensor M, respectively. We propose to factorize matrix\nMv as follows\n(3.3)\nwhere matrix D \u2208 RN\u03b8\u00d7Na is a dictionary of di\ufb00usion predictions whose columns (atoms)\ncorrespond to precomputed fascicle orientations, and \u03a6v \u2208 RNa\u00d7Nf is a sparse matrix\nwhose non-zero entries, \u03a6v(a, f), indicate the orientation of fascicle f in voxel v, which\nis approximated by atom a (see Fig. 3(a) for an example of a voxel v as shown in Fig.\n2(b)-(c)). For computing the di\ufb00usion predictions, we use a discrete grid in the sphere\nby uniformly sampling the spherical coordinates using L points in azimuth and elevation\ncoordinates (Fig. 2(c)).\n\nMv \u2248 \u02c6Mv = D\u03a6v,\n\n3.2 Approximation of the linear model across multiple brain voxels.\nBy applying the approximation introduced in equation (3.3) to every slice in tensor M in\nequation 3.1, we obtain the following tensor Sparse Decomposed LiFE model, hereafter\nreferred to as LiFESD (Fig. 3(b)):\n\n(3.4)\nwhere D is a common factor in mode-1, i.e., it multiplies all lateral slices. It is noted that, the\nformula in the above equation (3.4), is a particular case of the Tucker decomposition [42, 16]\nwhere the core tensor \u03a6 is sparse [9, 11], and only factors in mode-1 (D) and mode-3 (wT )\n\nY \u2248 \u03a6 \u00d71 D \u00d73 wT ,\n\n5\n\nEmpty entries (zero values)(a) Max. discretization error, (c) (b) \fare present. By comparing equations (3.4) and (3.1) we de\ufb01ne the LiFESD approximated\ntensor model as\n(3.5)\n\n\u02c6M = \u03a6 \u00d71 D\n\n3.3 Theoretical bound for model decomposition accuracy and data\n\ncompression.\n\nIn this section, we derive a theoretical bound on the accuracy of LiFESD compared to the\noriginal LiFE model (Proposition 3.1) and we theoretically analyze the compression factor\nassociated to the factorized tensor approximation (Proposition 3.2). Hereafter, we assume\nthat, in a given connectome having Nf fascicles, each fascicle has a \ufb01xed number of nodes\n(Nn), and the di\ufb00usion weighted measurements were taken on N\u03b8 gradient directions with\na gradient strength b. The proofs of the propositions can be found in the Supplementary\nmaterial.\nProposition 3.1 (accuracy). For a given connectome, and dictionary D obtained by\nuniformly sampling the azimuth-elevation (\u03b1, \u03b2) space using \u2206\u03b1 = \u2206\u03b2 = \u03c0/L (see Fig.\n3(c)), the following upper bound on the Frobenius norm based model error is veri\ufb01ed:\n\nkM \u2212 \u02c6MkF \u2264 2b\u03c0p6Nf NnN\u03b8\n\nL\n\n.\n\n(3.6)\n\nThe importance of this theoretical result is that the error is inversely proportional to the\ndiscretization parameter L, which allows one to design the decomposed model so that a\nprescribed accuracy is met.\nProposition 3.2 (size reduction). For a given connectome, and a dictionary D \u2208 RN\u03b8\u00d7Na\ncontaining Na atoms (columns of matrix D), the achieved compression factor is\n\n(cid:18) 4\n\n(cid:19)\u22121\n\nCF =\n\n\u2212 Na\n3NnNf\n\n3N\u03b8\n\n,\n\n(3.7)\n\nwhere CF = C(M)/C( \u02c6M), with C(M) and C( \u02c6M) being the storage costs of LiFE and\nLiFESD models, respectively.\nIt is noted that, usually 3NnNf (cid:29) Na, which implies that the compression factor can be\napproximated by CF \u2248 3N\u03b84 , i.e., it is proportional to the number of gradient directions N\u03b8.\n4 Model optimization using tensor encoding.\nOnce the LiFESD model has been built, the \ufb01nal step to validate a connectome requires\n\ufb01nding the non-negative weights that least-squares \ufb01t the measured di\ufb00usion data. This is\na convex optimization problem that can be solved using a variety of NNLS optimization\nalgorithms. We used a NNLS algorithm based on \ufb01rst-order methods specially designed for\nlarge scale problems [22]. Next, we show how to exploit the decomposed LiFESD model in\nthe optimization.\nThe gradient of the original objective function for the LiFE model can be written as follows:\n\n\u2207w\n\n(4.1)\nwhere M \u2208 RN\u03b8Nv\u00d7Nf is the original LiFE model, w \u2208 RNf the fascicle weights and\ny \u2208 RN\u03b8Nv the demeaned di\ufb00usion signal. Because the decomposed version does not\nexplicitly store M, below we describe how to perform two basic operations (y = Mw and\nw = MT y) using the sparse decomposition.\n\n= MT Mw \u2212 2MT y,\n\n(cid:18)1\n2ky \u2212 Mwk2\n\n(cid:19)\n\n4.1 Computing y = Mw\nUsing equation (3.1) we can see that the product Mw can be computed using equation\n(3.4) and vectorizing the result, i.e. y = vec(Y), where vec() stands for the vectorization\n\n6\n\n\foperation, i.e., to convert a matrix to a vector by stacking its columns in a long vector. In\nAlgorithm 1, we present the steps for computing y = Mw in an e\ufb03cient way.\n\nAlgorithm 1 : y = M_times_w(\u03a6,D,w)\nRequire: Decomposition components (\u03a6, D and vector w \u2208 RNf ).\nEnsure: y = Mw\n1: Y = \u03a6 \u00d73 wT ; the result is a large but very sparse matrix (Na \u00d7 Nv)\n2: Y = DY; the result is a relatively small matrix (N\u03b8 \u00d7 Nv)\n3: y = vec(Y)\n4: return y;\n\n4.2 Computing w = MT y\nThe product w = MT y can be computed using LiFESD in the following way:\n\nw = MT y = M(3)y = \u03a6(3)(I \u2297 DT )y,\n\n(4.2)\nwhere M(3) \u2208 RNf\u00d7N\u03b8 Nv and \u03a6(3) \u2208 RNf\u00d7NaNv are the unfolding matrices [23] of tensors\nM \u2208 RN\u03b8\u00d7Nv\u00d7Nf and \u03a6 \u2208 RNa\u00d7Nv\u00d7Nf , respectively; \u2297 is the Kronecker product and I is\nthe (Nv \u00d7 Nv) identity matrix. Equation (4.2) can be written also as follows [9]:\n\nw = \u03a6(3)vec(DT Y).\n\n(4.3)\n\nBecause matrix \u03a6(3) is very sparse, we avoid computing the large and dense matrix DT Y\nand compute instead only its blocks that are being multiplied by the non-zero entries in\n\u03a6(3). This allows maintaining e\ufb03cient memory usage and limits the number of CPU cycles\nneeded. In Algorithm 2, we present the steps for computing w = MT y in an e\ufb03cient way.\n\nAlgorithm 2 : w = Mtransp_times_y(\u03a6,D,y)\nRequire: Decomposition components (\u03a6, D) and vector y \u2208 RN\u03b8 Nv.\nEnsure: w = MT y\n1: Y \u2208 RN\u03b8\u00d7Nv \u2190 y \u2208 RN\u03b8 Nv; reshape vector y into a matrix Y\n2: [a, v, f , c] = get_nonzero_entries(\u03a6); a(n), v(n), f(n), c(n) indicate the atom, the voxel, the\n3: w = 0 \u2208 RNf ; Initialize weights with zeros\n4: for n = 1 to Nn do\nw(f(n)) = w(f(n)) + DT (:, a(n))Y(:, v(n))c(n);\n5:\n6: end for\n7: return w;\n\nfascicle and the entry in tensor \u03a6 associated to node n, respectively, with n = 1, 2, . . . , Nn;\n\n5 Experimental results: Validation of the theoretical bounds for\n\nmodel decomposition accuracy and data compression.\n\nHere, we validate our theoretical \ufb01ndings by using dMRI data from subjects in a public\nsource (the Stanford dataset [32]). The data were collected using N\u03b8 = 96 (STN96, \ufb01ve\nsubjects) and N\u03b8 = 150 (STN150, one subject) directions with b-value b = 2, 000s/mm2. We\nperformed tractography using these data and both, probabilistic and deterministic methods,\nin combination with Constrained Spherical Deconvolution (CSD) and the di\ufb00usion tensor\nmodel (DTI) [41, 17, 5]. We generated candidate connectomes with Nf = 500, 000 fascicles\nper brain brain. See for [10, 32, 39] for additional details on data preprocessing.\nWe \ufb01rst analyzed the accuracy of the approximated model (LiFESD) as a function of the\nparameter, L, which describes the number of fascicles orientations encoded in the dictionary D.\nIn theory, the larger the number of atoms in D the higher the accuracy of the approximation.\nWe show that model error (de\ufb01ned as eM = kM\u2212 \u02c6MkF\n) decreases as a function of the\nkMkF\nparameter L for all subjects in the dataset Fig. 4(a). This result validates the theoretical\nupper bound in Proposition 3.1. We also solved the convex optimization problem of equation\n\n7\n\n\f(2.4) for both, LiFE and LiFESD, and estimated the error in the weights assigned to each\nfascicle by the two models (we computed the error in weights as follows ew = kw\u2212 \u02c6wk\nkwk ). Fig.\n4(b) shows the error ew as a function of the parameter L. It is noted that for L > 180 the\nerror is lower than 0.1% in all subjects.\n\nFigure 4: Experimental results:\n(a) The model error eM in approximating the matrix M with\nLiFESD is inversely proportional to the parameter L as predicted by our Proposition 3.1 (eM \u2248 C/L\nwas \ufb01tted to the data with C = 27.78 and a \ufb01tting error equal to 2.94%).\n(b) Error in the weights\nobtained by LiFESD compared with original LiFE\u2019s weights, ew, as a function of parameter L.\n(c)-(d) Model size (GB) scales linearly with the number of directions N\u03b8 and the number of fascicles\nNf, however it increases much faster in the LiFE model compared to the LiFESD model. LiFESD\nwas computed using L = 360. (e)-(f) Probabilistic and deterministic connectomes validated with\nLiFESD for a HCP subject.\n(g) Comparison of the Root-mean-squared-error (r.m.s, as de\ufb01ned in\n[32]) obtained in all voxels for probabilistic and deterministic connectomes. The averaged r.m.s.e\nare 361.12 and 423.06 for the probabilistic and deterministic cases, respectively.\n\nHaving experimentally demonstrated that model approximation error decreases as function\nof L, we move on to demonstrate the magnitude of model compression achieved by the\ntensor decomposition approach. To do so, we \ufb01xed L = 360 and computed the model size for\nboth, LiFE and LiFESD, as a function of the number of gradient directions N\u03b8 (Fig. 4(c))\nand fascicles Nf (Fig. 4(d)). Results show that, as predicted by our theoretical results in\nProposition 3.2, model size scales linearly with the number of directions for both, LiFE and\nLiFESD, but that the di\ufb00erence in slope is profound. Experimentally measured compression\nratios raise up to approximately 40 as it is the case for the subjects in the STN150 dataset\n(Nf = 500, 000 and N\u03b8 = 150).\n\n8\n\nProbabilistic(121,050 fascicles)Deterministic(64,134 fascicles)100050010r.m.s.e (det)r.m.s.e (prob)3x100Probability00.511.5Subject 1Subject 2Subject 3Subject 4Subject 5STN150STN96Matrix based LiFELiFESD (L=360)Matrix based LiFELiFESD (L=360)010203040Model size (GB)(c) 0501001500102030401,000,000100,00010,0001,00023459018036072023459018036072000.511.5(%)Model error(%)Weights error (d) 0.1%0.1%(g)(e)(f)105001000(a)(b)-3Model size (GB)\fFinally, we show an example comparison between two connectomes obtained by applying\nprobabilistic [17] and deterministic [4] tracking algorithms to one brain dataset (a single\nsubject) from the Human Connectome Project dataset [45], with N\u03b8 = 90, Nv = 267, 306\nand Nf = 500, 000. Figs. 4e-f show the detected 20 major tracts in a human brain using\nonly the fascicles with nonzero weigths. In this case, the probabilistic connectome has more\nfascicles (121, 050) than the deterministic one (64, 134). Moreover, we replicate previous\nresults demonstrating that probabilistic connectomes have lower error than the deterministic\none in a majority of the voxels (see Fig. 4(g)).\n\n6 Conclusions\n\nWe introduced a method to encode brain connectomes in multidimensional arrays and\ndecomposition approach that can accurately approximate the linear model for connectome\nevaluation used in the LiFE method [32]. We demonstrate that the decomposition approach\ndramatically reduces the memory requirements of the LiFE model, approximately from 40GB\nto 1GB, with a small model approximation error of less than 1%. The compactness of the\ndecomposed LIFE model has important implications for other computational problems. For\nexample, model optimization can be implemented by using operations involving tensorial\noperations avoiding the use of large matrices such as M and using instead the sparse tensor\nand prediction dictionary (\u03a6 and D respectively).\nMultidimensional tensors and decomposition methods have been used to help investigators\nmake sense of large multimodal datasets [27, 11]. Yet to date these methods have found\nonly a few applications in neuroscience, such as performing multi-subject, clustering and\nelectroencephalography analyses [49, 48, 3, 28, 26, 13, 8]. Generally, decomposition methods\nhave been used to \ufb01nd compact representations of complex data by estimating the combination\nof a limited number of common meaningful factors that best \ufb01t the data [24, 27, 23]. We\npropose a new application that, instead of using the decomposition to estimate latent factors,\nit encodes the structure of the problem explicitly.\nThe new application of tensor decomposition proposed here has the potential to improve\nfuture generations of models of connectomics, tractography evaluation and microstructure\n[32, 15, 36, 39]. Improving these models will allow going beyond the current limitations of\nthe state of the art methods [14]. Finally, tensorial representations for brain imaging data\nhave the potential to contribute advancing the application of machine learning algorithms to\nmapping the human connectome [18, 37, 21, 20, 30, 1, 51, 29, 40, 33].\n\nAcknowledgments\nThis research was supported by (NSF IIS-1636893; BCS-1734853; NIH ULTTR001108) to F.P.\nData provided by Stanford University (NSF BCS 1228397). F.P. were partially supported by\nthe Indiana University Areas of Emergent Research initiative Learning: Brains, Machines,\nChildren.\n\nReferences\n[1] Daniel C Alexander, Darko Zikic, Aurobrata Ghosh, Ryutaro Tanno, Viktor Wottschel, Jiaying\nZhang, Enrico Kaden, Tim B Dyrby, Stamatios N Sotiropoulos, Hui Zhang, and Antonio\nCriminisi. Image quality transfer and applications in di\ufb00usion MRI. Human Brain Mapping\nJournal, pages 1\u201365, March 2017.\n\n[2] Animashree Anandkumar, Rong Ge 0001, Daniel J Hsu, and Sham M Kakade. A tensor\napproach to learning mixed membership community models. Journal of Machine Learning\nResearch (JMLR), 15:2239\u20132312, 2014.\n\n[3] Michael Barnathan, Vasileios Megalooikonomou, Christos Faloutsos, Scott Faro, and Feroze B\nMohamed. TWave: High-order analysis of functional MRI. Human Brain Mapping Journal,\n58(2):537\u2013548, September 2011.\n\n[4] P J Basser, S Pajevic, C Pierpaoli, J Duda, and A Aldroubi. In vivo \ufb01ber tractography using\n\nDT-MRI data. Magnetic Resonance in Medicine, 44(4):625\u2013632, October 2000.\n\n9\n\n\f[5] PJ Basser, J Mattiello, and D Lebihan. Estimation of the e\ufb00ective self-di\ufb00usion tensor from\nthe NMR spin echo. Journal of Magnetic Resonance, Series B, 103(3):247\u2013254, January 1994.\n[6] Danielle S Bassett and Olaf Sporns. Network neuroscience. Nature Neuroscience, 20(3):353\u2013364,\n\nFebruary 2017.\n\n[7] Matteo Bastiani, Nadim Jon Shah, Rainer Goebel, and Alard Roebroeck. Human cortical\nconnectome reconstruction from di\ufb00usion weighted MRI: the e\ufb00ect of tractography algorithm.\nHuman Brain Mapping Journal, 62(3):1732\u20131749, 2012.\n\n[8] C F Beckmann and S M Smith. Tensorial extensions of independent component analysis for\n\nmultisubject FMRI analysis. NeuroImage, 25(1):294\u2013311, March 2005.\n\n[9] Cesar F Caiafa and A Cichocki. Computing Sparse representations of multidimensional signals\n\nusing Kronecker bases. Neural Computation, pages 186\u2013220, December 2012.\n\n[10] Cesar F Caiafa and Franco Pestilli. Multidimensional encoding of brain connectomes. Scienti\ufb01c\n\nReports, 7(1):11491, September 2017.\n\n[11] Andrzej Cichocki, Danilo Mandic, Lieven De Lathauwer, Guoxu Zhou, Qibin Zhao, Cesar\nCaiafa, and Anh Huy Phan. Tensor decompositions for signal processing applications: from\ntwo-way to multiway component analysis. IEEE Signal Processing Magazine, 32:145\u2013163, March\n2015.\n\n[12] Pierre Comon. Tensors : A brief introduction. IEEE Signal Processing Magazine, 31(3):44\u201353,\n\nApril 2014.\n\n[13] Fengyu Cong, Qiu-Hua Lin, Li-Dan Kuang, Xiao-Feng Gong, Piia Astikainen, and Tapani\nRistaniemi. Tensor decomposition of EEG signals: a brief review. Journal of neuroscience\nmethods, 248:59\u201369, 2015.\n\n[14] Alessandro Daducci, Alessandro Dal Palu, Maxime Descoteaux, and Jean-Philippe Thiran.\nMicrostructure Informed Tractography: Pitfalls and Open Challenges. Frontiers in Neuroscience,\n10(8):1374\u201313, June 2016.\n\n[15] Alessandro Daducci, Alessandro Dal Pal\u00f9, Alia Lemkaddem, and Jean-Philippe Thiran. COM-\nMIT: Convex optimization modeling for microstructure informed tractography. Medical Imaging,\nIEEE Transactions on, 34(1):246\u2013257, January 2015.\n\n[16] Lieven De Lathauwer, Bart De Moor, and Joos Vandewalle. A multilinear singular value\n\ndecomposition. SIAM J. Matrix Anal. Appl, 21(4):1253\u20131278, 2000.\n\n[17] M Descoteaux, R Deriche, T R Knosche, and A Anwander. Deterministic and Probabilistic\nTractography Based on Complex Fibre Orientation Distributions. Medical Imaging, IEEE\nTransactions on, 28(2):269\u2013286, January 2009.\n\n[18] Andrew T Drysdale, Logan Grosenick, Jonathan Downar, Katharine Dunlop, Farrokh Mansouri,\nYue Meng, Robert N Fetcho, Benjamin Zebley, Desmond J Oathes, Amit Etkin, Alan F\nSchatzberg, Keith Sudheimer, Jennifer Keller, Helen S Mayberg, Faith M Gunning, George S\nAlexopoulos, Michael D Fox, Alvaro Pascual-Leone, Henning U Voss, B J Casey, Marc J Dubin,\nand Conor Liston. Resting-state connectivity biomarkers de\ufb01ne neurophysiological subtypes of\ndepression. Nature Medicine, pages 1\u201316, December 2016.\n\n[19] John R Gilbert, Cleve Moler, and Robert Schreiber. Sparse matrices in matlab: design and\nimplementation. SIAM Journal on Matrix Analysis and Applications, 13(1):333\u2013356, January\n1992.\n\n[20] Matthew F Glasser, Timothy S Coalson, Emma C Robinson, Carl D Hacker, John Harwell, Essa\nYacoub, Kamil Ugurbil, Jesper Andersson, Christian F Beckmann, Mark Jenkinson, Stephen M\nSmith, and David C Van Essen. A multi-modal parcellation of human cerebral cortex. Nature\nPublishing Group, 536(7615):171\u2013178, August 2016.\n\n[21] Heather Cody Hazlett, Hongbin Gu, Brent C Munsell, Sun Hyung Kim, Martin Styner, Jason J\nWol\ufb00, Jed T Elison, Meghan R Swanson, Hongtu Zhu, Kelly N Botteron, D Louis Collins,\nJohn N Constantino, Stephen R Dager, Annette M Estes, Alan C Evans, Vladimir S Fonov,\nGuido Gerig, Penelope Kostopoulos, Robert C McKinstry, Juhi Pandey, Sarah Paterson, John R\nPruett, Robert T Schultz, Dennis W Shaw, Lonnie Zwaigenbaum, and Joseph Piven. Early\nbrain development in infants at high risk for autism spectrum disorder. Nature Publishing\nGroup, 542(7641):348\u2013351, February 2017.\n\n10\n\n\f[22] Dongmin Kim, Suvrit Sra, and Inderjit S Dhillon. A non-monotonic method for large-scale\nnon-negative least squares. Optimization Methods and Software, 28(5):1012\u20131039, October\n2013.\n\n[23] TG Kolda and BW Bader. Tensor decompositions and applications. SIAM Review, 51(3):455\u2013\n\n500, 2009.\n\n[24] Pieter M Kroonenberg. Applied Multiway Data Analysis. John Wiley & Sons, February 2008.\n[25] Junning Li, Yonggang Shi, and Arthur W Toga. Mapping Brain Anatomical Connectivity Using\nDi\ufb00usion Magnetic Resonance Imaging: Structural connectivity of the human brain. IEEE\nSignal Processing Magazine, 33(3):36\u201351, April 2016.\n\n[26] F Miwakeichi, E Mart\u00ednez-Montes, PA Vald\u00e9s-Sosa, N Nishiyama, H Mizuhara, and Y Yam-\naguchi. Decomposing EEG Data into Space\u2013time\u2013frequency Components using Parallel Factor\nAnalysis. NeuroImage, 22(3):1035\u20131045, July 2004.\n\n[27] M M\u00f8rup. Applications of tensor (multiway array) factorizations and decompositions in data\nmining. Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery, 1(1):24\u201340,\nJanuary 2011.\n\n[28] Morten M\u00f8rup, Lars Kai Hansen, Christoph S Herrmann, Josef Parnas, and Sidse M. Arnfred.\nParallel Factor Analysis as an exploratory tool for wavelet transformed event-related EEG.\nHuman Brain Mapping Journal, 29(3):938\u2013947, 2006.\n\n[29] Gemma L Nedjati-Gilani, Torben Schneider, Matt G Hall, Niamh Cawley, Ioana Hill, Olga\nCiccarelli, Ivana Drobnjak, Claudia A M Gandini Wheeler-Kingshott, and Daniel C Alexander.\nMachine learning based compartment models with permeability for white matter microstructure\nimaging. Human Brain Mapping Journal, 150:119\u2013135, April 2017.\n\n[30] Peter Florian Neher, Marc-Alexandre Cote, Jean-Christophe Houde, Maxime Descoteaux, and\nKlaus H Maier-Hein. Fiber tractography using machine learning. bioRxiv, pages 1\u201320, January\n2017.\n\n[31] Eleftheria Panagiotaki, Torben Schneider, Bernard Siow, Matt G Hall, Mark F Lythgoe, and\nDaniel C Alexander. Compartment models of the di\ufb00usion MR signal in brain white matter: A\ntaxonomy and comparison. Human Brain Mapping Journal, 59(3):2241\u20132254, February 2012.\n[32] Franco Pestilli, Jason D Yeatman, Ariel Rokem, Kendrick N Kay, and Brian A Wandell.\nEvaluation and statistical inference for human connectomes. Nature Methods, 11(10):1058\u20131063,\nSeptember 2014.\n\n[33] Ariel Rokem, Hiromasa Takemura, Andrew S Bock, K Suzanne Scherf, Marlene Behrmann,\nBrian A Wandell, Ione Fine, Holly Bridge, and Franco Pestilli. The visual white matter: The\napplication of di\ufb00usion MRI and \ufb01ber tractography to vision science. Journal of Vision, 17(2):4,\nFebruary 2017.\n\n[34] Ariel Rokem, Jason D Yeatman, Franco Pestilli, Kendrick N Kay, Aviv Mezer, Stefan van der\nWalt, and Brian A Wandell. Evaluating the accuracy of di\ufb00usion MRI models in white matter.\nPLoS ONE, 10(4):e0123272, April 2015.\n\n[35] Parikshit Shah, Nikhil S Rao, and Gongguo Tang. Sparse and Low-Rank Tensor Decomposition.\n\nNIPS, 2015.\n\n[36] Robert E Smith, Jacques-Donald Tournier, Fernando Calamante, and Alan Connelly. SIFT2:\nEnabling dense quantitative assessment of brain white matter connectivity using streamlines\ntractography. Human Brain Mapping Journal, 119(C):338\u2013351, October 2015.\n\n[37] Stephen M Smith, Thomas E Nichols, Diego Vidaurre, Anderson M Winkler, Timothy E J\nBehrens, Matthew F Glasser, Kamil Ugurbil, Deanna M Barch, David C Van Essen, and\nKarla L Miller. A positive-negative mode of population covariation links brain connectivity,\ndemographics and behavior. Nature Publishing Group, 18(11):1565\u20131567, September 2015.\n\n[38] Olaf Sporns. Making sense of brain network data. Nature Methods, 10(6):491\u2013493, May 2013.\n[39] Hiromasa Takemura, Cesar F Caiafa, Brian A Wandell, and Franco Pestilli. Ensemble Tractog-\n\nraphy. PLoS Computational Biology, 12(2):e1004692\u2013, February 2016.\n\n11\n\n\f[40] Chantal M W Tax, Tom Dela Haije, Andrea Fuster, Carl-Fredrik Westin, Max A Viergever, Luc\nFlorack, and Alexander Leemans. Sheet Probability Index (SPI): Characterizing the geometrical\norganization of the white matter with di\ufb00usion MRI. Human Brain Mapping Journal, pages\n1\u201353, July 2016.\n\n[41] J-Donald Tournier, Fernando Calamante, and Alan Connelly. MRtrix: Di\ufb00usion tractography\nin crossing \ufb01ber regions. International Journal of Imaging Systems and Technology, 22(1):53\u201366,\nFebruary 2012.\n\n[42] L R Tucker. Some mathematical notes on three-mode factor analysis. Psychometrika, 31(3):279\u2013\n\n311, September 1966.\n\n[43] M P Van den Heuvel and O Sporns. Rich-Club Organization of the Human Connectome.\n\nJournal of Neuroscience, 31(44):15775\u201315786, November 2011.\n\n[44] Martijn P Van den Heuvel, Edward T Bullmore, and Olaf Sporns. Comparative Connectomics.\n\nTrends in Cognitive Sciences, 20(5):345\u2013361, 2016.\n\n[45] David C Van Essen, Stephen M Smith, Deanna M Barch, Timothy E J Behrens, Essa Yacoub,\nKamil Ugurbil, and for the WU-Minn HCP Consortium. The WU-Minn Human Connectome\nProject: An overview. NeuroImage, 80(C):62\u201379, October 2013.\n\n[46] Brian A Wandell. Clarifying Human White Matter. Annual Review of Neuroscience, 39(1):103\u2013\n\n128, July 2016.\n\n[47] Kishan Wimalawarne, Masashi Sugiyama, and Ryota Tomioka. Multitask learning meets tensor\n\nfactorization - task imputation via convex optimization. NIPS, 2014.\n\n[48] Yeyang Yu, Jin Jin, Feng Liu, and Stuart Crozier. Multidimensional Compressed Sensing MRI\nUsing Tensor Decomposition-Based Sparsifying Transform. PLoS ONE, 9(6):e98441, June 2014.\n[49] Qibin Zhao, C F Caiafa, D P. Mandic, Z C Chao, Y Nagasaka, N Fujii, Liqing Zhang, and\nA Cichocki. Higher Order Partial Least Squares (HOPLS): A Generalized Multilinear Regression\nMethod. IEEE Transactions on Pattern Analysis and Machine Intelligence, 35(7):1660\u20131673,\nMay 2013.\n\n[50] Qibin Zhao, Cesar F Caiafa, Danilo P Mandic, Liqing Zhang, Tonio Ball, Andreas Schulze-\nbonhage, and Andrzej S Cichocki. Multilinear Subspace Regression: An Orthogonal Tensor\nDecomposition Approach. In J Shawe-Taylor, R S Zemel, P L Bartlett, F Pereira, and K Q\nWeinberger, editors, Advances in Neural Information Processing Systems 24, pages 1269\u20131277.\nCurran Associates, Inc., 2011.\n\n[51] D Zhu, N Jahanshad, B C Riedel, and L Zhan. Population learning of structural connectivity\nby white matter encoding and decoding. In 2016 IEEE 13th International Symposium on\nBiomedical Imaging (ISBI), pages 554\u2013558. IEEE, 2016.\n\n12\n\n\f", "award": [], "sourceid": 2264, "authors": [{"given_name": "Cesar", "family_name": "Caiafa", "institution": "Indiana University"}, {"given_name": "Olaf", "family_name": "Sporns", "institution": "Department of Psychological and Brain Sciences - Indiana University"}, {"given_name": "Andrew", "family_name": "Saykin", "institution": "IUPUI"}, {"given_name": "Franco", "family_name": "Pestilli", "institution": "Indiana University"}]}