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To enable low-rank tensor completion and factorization, this paper puts forth a novel tensor rank regularization method based on the ℓ1,2-norm of the tensor's parallel factor analysis (PARAFAC) factors. Specifically, for an N-way tensor, upon collecting the magnitudes of its rank-1 components in a vector, the proposed regularizer controls the tensor's rank by inducing sparsity in the vector of magnitudes...
We give an overview of recent developments in numerical optimization-based computation of tensor decompositions that have led to the release of Tensorlab 3.0 in March 2016 (www.tensorlab.net). By careful exploitation of tensor product structure in methods such as quasi-Newton and nonlinear least squares, good convergence is combined with fast computation. A modular approach extends the computation...
State-of-the-art narrowband noise cancellation techniques utilise the generalised eigenvalue decomposition (GEVD) for multi-channel Wiener filtering, which can be applied to independent frequency bins in order to achieve broadband processing. Here we investigate the extension of the GEVD to broadband, polynomial matrices, akin to strategies that have already been developed by McWhirter et. al on the...
Large datasets usually contain redundant information and summarizing these datasets is important for better data interpretation. Higher-order data reduction is usually achieved through low-rank tensor approximation which assumes that the data lies near a linear subspace across each mode. However, non-linearities in the data cannot be captured well by linear methods. In this paper, we propose a multiscale...
This paper considers the problem of reconstructing a N × N low rank positive semidefinite Toeplitz matrix from a noisy compressed sketch of size O(√r) × O (√r) where r << N is the rank of the matrix. A novel algorithm is proposed which only exploits a positive semidefinite (PSD) constraint to denoise the compressed sketch using a simple least squares approach. A major advantage of our algorithm...
Several combined signal processing applications such as the joint processing of EEG and MEG data can benefit from coupled tensor decompositions, for instance, the coupled CP (Canonical Polyadic) decomposition. The coupled CP decomposition jointly decomposes tensors that have at least one factor matrix in common. The SECSI (Semi-Algebraic framework for approximate CP decomposition via SImultaneaous...
Directed networks are pervasive both in nature and engineered systems, often underlying the complex behavior observed in biological systems, microblogs and social interactions over the web, as well as global financial markets. Since their explicit structures are often unobservable, in order to facilitate network analytics, one generally resorts to approaches capitalizing on measurable nodal processes...
The recently developed super-resolution framework by Candes enables direction-of-arrival (DOA) estimation from a sparse spatial power spectrum in the continuous domain with infinite precision in the noise-free case. By means of atomic norm minimization (ANM), the discretization of the spatial domain is no longer required, which overcomes the basis mismatch problem in conventional sparse signal recovery...
In this paper, we consider the problem of estimating the principal subspace of data in decentralized sensing systems with resource constraints, where the sensors only transmit a single bit to the fusion center to minimize communication costs. In particular, the data covariance matrix is modeled as a low-rank Toeplitz positive semidefinite (PSD) matrix, which arises in applications such as array signal...
Sampling of bandlimited graph signals has well-documented merits for dimensionality reduction, affordable storage, and online processing of streaming network data. Most existing sampling methods are designed to minimize the error incurred when reconstructing the original signal from its samples. Oftentimes these parsimonious signals serve as inputs to computationally-intensive linear transformations...
Neural data analysis has increasingly incorporated causal information to study circuit connectivity. Dimensional reduction forms the basis of most analyses of large multivariate time series. Here, we present a new, multitaper-based decomposition for stochastic, multivariate time series that acts on the covariance of the time series at all lags, C (τ), as opposed to standard methods that decompose...
In recent years, several algorithms for the iterative calculation of a polynomial matrix eigenvalue decomposition (PEVD) have been introduced. The PEVD is a generalisation of the ordinary EVD and uses paraunitary operations to diagonalise a parahermitian matrix. This paper addresses potential computational savings that can be applied to existing cyclic-by-row approaches for the PEVD. These savings...
This paper considers decentralized consensus optimization problems where different summands of a global objective function are available at nodes of a network. The method of multipliers is well studied for centralized optimization; however, it is not applicable to decentralized optimization problems since the augmented Lagrangian is not decomposable. We propose ESOM as a decentralized primal-dual...
We describe a method for unmixing mixtures of ‘freely’ independent random variables in a manner analogous to the independent component analysis (ICA) based method for unmixing independent random variables from their additive mixture. Random matrices play the role of free random variables in this context so the method we develop, which we call Free component analysis (FCA), unmixes matrices from an...
In this paper, we study robust principal component analysis on tensors, in the setting where frame-wise outliers exist. We propose a convex formulation to decompose a tensor into a low rank component and a frame-wise sparse component. Theoretically, we guarantee that exact subspace recovery and outlier identification can be achieved under mild model assumptions. Compared with entry-wise outlier pursuit...
Real convolutional lattices over the ring of integers Z are considered in this paper. We study the stability of convolutional lattices under sphere decoding. A new stable family of time-alternating convolutional lattices is proposed. The structure, the parameters, and a performance example are shown for time-alternating convolutional lattices. These lattices can be used as constituent blocks in concatenated...
Given data that lies in a union of low-dimensional subspaces, the problem of subspace clustering aims to learn — in an unsupervised manner — the membership of the data to their respective subspaces. State-of-the-art subspace clustering methods typically adopt a two-step procedure. In the first step, an affinity measure among data points is constructed, usually by exploiting some form of data self-representation...
In this paper we propose an efficient hardware architecture for computation of matrix inversion of positive definite matrices. The algorithm chosen is LDL decomposition followed directly by equation system solving using back substitution. The architecture combines a high throughput with an efficient utilization of its hardware units. We also report FPGA implementation results that show that the architecture...
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