This survey provides an overview of higher-order tensor decompositions, their applications, and available software. A tensor is a multidimensional or $N$-way array. Decompositions of higher-order tensors (i.e., $N$-way arrays with $N \geq 3$) 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.

/pdf/tensor-decompositions-and-applications-46vl2ih5gn.pdf

Tensor Decompositions and Applications

Full system simulation seeks to strike a balance between accuracy and performance. Many of its possibilities have been obvious to practitioners in both academia and industry for quite some time, perhaps decades, but Simics supports more of these possibilities within a single framework than other tools do. Simics is a platform for full system simulation that can run actual firmware and completely unmodified kernel and driver code. It is sufficiently abstract to achieve tolerable performance levels, and it provides both functional accuracy for running commercial workloads and sufficient timing accuracy to interface to detailed hardware models. Simics can also run a heterogeneous network of systems from different vendors within the same framework. Exceptionally fast, Simics can easily add new components and leverage older ones within a practical abstraction level. It offers a platform with a rich API and a powerful scripting environment for use in a broad range of applications.

/pdf/simics-a-full-system-simulation-platform-39nt38s605.pdf

Simics: A full system simulation platform

A symmetric tensor is a higher order generalization of a symmetric matrix. In this paper, we study various properties of symmetric tensors in relation to a decomposition into a symmetric sum of outer product of vectors. A rank-1 order-$k$ tensor is the outer product of $k$ nonzero vectors. Any symmetric tensor can be decomposed into a linear combination of rank-1 tensors, each of which is symmetric or not. The rank of a symmetric tensor is the minimal number of rank-1 tensors that is necessary to reconstruct it. The symmetric rank is obtained when the constituting rank-1 tensors are imposed to be themselves symmetric. It is shown that rank and symmetric rank are equal in a number of cases and that they always exist in an algebraically closed field. We will discuss the notion of the generic symmetric rank, which, due to the work of Alexander and Hirschowitz [J. Algebraic Geom., 4 (1995), pp. 201-222], is now known for any values of dimension and order. We will also show that the set of symmetric tensors of symmetric rank at most $r$ is not closed unless $r=1$.

/pdf/symmetric-tensors-and-symmetric-tensor-rank-1sfgx6fmsr.pdf

Symmetric Tensors and Symmetric Tensor Rank

A mu -(n*n,k) array code C over a field F is a k-dimensional linear space of n*n matrices over F such that every nonzero matrix in C has rank >or= mu . It is first shown that the dimension of such array codes must satisfy the Singleton-like bound k >

Maximum-rank array codes and their application to crisscross error correction

The realization of a digital recording system using partial-response class-IV signaling with maximum-likelihood sequence detection (MLSD) is described. To perform MLSD at the high data rates encountered in recording systems, a simple implementation of the Viterbi detector based on a difference-metric algorithm is developed. Decision-directed schemes for gain control and timing recovery, for tracking variations of the gain and timing phase during data readback, and for fast initial adjustment from a known preamble are presented. The dynamic behavior of the control algorithms was studied by computer simulations. Coding was used to facilitate timing recovery and gain control, to limit the path memory length of the Viterbi detector, and to allow fast and reliable startup of the receiver. The design and properties of rate 8/9 constrained codes are examined. The problem of equalization is addressed, and analog and combined analog/digital filter implementations are developed. A simple adaptive equalizer capable of compensating variations of the recording channel characteristics with track radius and/or head-to-medium distance is described. >

A PRML system for digital magnetic recording

Nonlinear distortion causes significant degradation at high linear recording densities. Nonlinear distortion cannot generally be corrected by equalization and is thus particularly important to correctly identify and quantify. A technique is described which is easily applied, clearly separates linear from nonlinear effects, can be related to the underlying physical processes, and provides a good basis for modeling the channel. The technique involves the recording of a binary pseudorandom sequence and the capture and processing of the playback waveform. The technique is used to identify four distinct nonlinear effects and to examine their behavior as a function of flying height and record current.

Identification of nonlinear write effects using pseudorandom sequences

The authors measured the ontrack and offtrack error-rate performance of several combinations of experimental heads and disks using a partial-response channel at approximately 1 Gb/in/sup 2/. The system components include a magnetoresistive head, a thin-film disk, and the partial response readback channel. Precompensation and equalization were used to tune the channel for best performance. Results indicate that these components can write and read data at 1 Gb/in/sup 2/ with good error rates and offtrack margins. >

Error rate performance of experimental gigabit per square inch recording components

A maximum-likelihood sequence decoder is used with a partial-response signaling system for processing sequences of sampled values from a communication channel or recording device. The partial-response signals can be duobinary, dicode, or partial-response class-IV. The maximum-likelihood decoding for each sequence can be based upon a two-state trellis. Instead of two survivor metrics for two states, only a difference metric is necessary.

Decoding the output signal of a partial-response class-IV communication or recording device channel

Method and apparatus for decoding the output signal of a partial-response communication or recording-device channel

Global properties of tensor rank

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