We present a generalization of the well-known problem of learning k-juntas in R^n, and a novel tensor algorithm for unraveling the structure of high-dimensional distributions. Our algorithm can be viewed as a higher-order extension of Principal Component Analysis (PCA). Our motivating problem is learning a labeling function in R^n, which is determined by an unknown k-dimensional subspace. This problem of learning a k-subspace junta is a common generalization of learning a k-junta (a function of k coordinates in R^n) and learning intersections of k halfspaces. In this context, we introduce an irrelevant noisy attributes model where the distribution over the "relevant" k-dimensional subspace is independent of the distribution over the (n-k)-dimensional "irrelevant" subspace orthogonal to it. We give a spectral tensor algorithm which identifies the relevant subspace, and thereby learns k-subspace juntas under some additional assumptions. We do this by exploiting the structure of local optima of higher moment tensors over the unit sphere; PCA finds the global optima of the second moment tensor (covariance matrix). Our main result is that when the distribution in the irrelevant (n-k)-dimensional subspace is any Gaussian, the complexity of our algorithm is T(k, ) + (n), where T is the complexity of learning the concept in k dimensions, and the polynomial is a function of the k-dimensional concept class being learned. This substantially generalizes existing results on learning low-dimensional concepts.
 Recent work on eigenvalues and eigenvectors for tensors of order @math has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form @math subject to @math , which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs. This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or @math -way array. Decompositions of higher-order tensors (i.e., @math -way arrays with @math ) have applications in psycho-metrics, chemometrics, signal processing, numerical linear algebra, computer vision, numerical analysis, data mining, neuroscience, graph analysis, and elsewhere. Two particular tensor decompositions can be considered to be higher-order extensions of the matrix singular value decomposition: CANDECOMP PARAFAC (CP) decomposes a tensor as a sum of rank-one tensors, and the Tucker decomposition is a higher-order form of principal component analysis. There are many other tensor decompositions, including INDSCAL, PARAFAC2, CANDELINC, DEDICOM, and PARATUCK2 as well as nonnegative variants of all of the above. The N-way Toolbox, Tensor Toolbox, and Multilinear Engine are examples of software packages for working with tensors.