Approximate eigenvalue decompositions of orthonormal and symmetric transformations with a few Householder reflectors Abstract   The ability to decompose a signal in an orthonormal basis (a set of orthogonal components, each normalized to have unit length) using a fast numerical procedure rests at the heart of many signal processing methods and applications. The classic examples are the Fourier and wavelet transforms that enjoy numerically efficient implementations (FFT and FWT, respectively). Unfortunately, orthonormal transformations are in general unstructured, and therefore they do not enjoy low computational complexity properties. In this paper, based on Householder reflectors, we introduce a class of orthonormal matrices that are numerically efficient to manipulate: we control the complexity of matrixvector multiplications with these matrices using a given parameter. We provide numerical algorithms that approximate any orthonormal or symmetric transform with a new orthonormal or symmetric structure made up of products of a given number of Householder reflectors. We show analyses and numerical evidence to highlight the accuracy of the proposed approximations and provide an application to the case of learning fast Mahalanobis distance metric transformations.

Spectrum Activity Estimation by Partitionblind Block Partitioned Tensor Decomposition We consider a scenario where signals that can be modeled as finite sums of linearly modulated signals that are nonorthogonal in both time and frequency are observed by a sensor. Under this model, when an antidiagonal slice of the trispectum of the sum of these signals is computed for multiple segments in time and stacked into a frontal symmetric block partitioned tensor (FSBPT) whose factor matrices characterize the received power spectrum and transmission activities of the signals. We propose a new alternating soft thresholding decomposition strategy that is tailored to FSBPTs and, unlike existing BPT decomposition algorithms, simultaneously estimates the factor partitions which is necessary since in our signal model the partitioning is not known a priori. We demonstrate the algorithmxe2x80x99s performance via Monte Carlo simulations. We find that our proposed nonblind algorithm is able to estimate the tensor partitions and moderately accurately estimate the factor matrices under a range of factor matrix column collinearity values.