14 research outputs found
Independent Process Analysis without A Priori Dimensional Information
Recently, several algorithms have been proposed for independent subspace
analysis where hidden variables are i.i.d. processes. We show that these
methods can be extended to certain AR, MA, ARMA and ARIMA tasks. Central to our
paper is that we introduce a cascade of algorithms, which aims to solve these
tasks without previous knowledge about the number and the dimensions of the
hidden processes. Our claim is supported by numerical simulations. As a
particular application, we search for subspaces of facial components.Comment: 9 pages, 2 figure
Oversampling in shift-invariant spaces with a rational sampling period
8 pages, no figures.It is well known that, under appropriate hypotheses, a sampling formula allows us to recover any function in a principal shift-invariant space from its samples taken with sampling period one. Whenever the generator of the shift-invariant space satisfies the Strang-Fix conditions of order r, this formula also provides an approximation scheme of order r valid for smooth functions. In this paper we obtain sampling formulas sharing the same features by using a rational sampling period less than one. With the use of this oversampling technique, there is not one but an infinite number of sampling formulas. Whenever the generator has compact support, among these formulas it is possible to find one whose associated reconstruction functions have also compact support.This work has been supported by the Grant MTM2009-08345 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología
A matrix pencil approach to the existence of compactly supported reconstruction functions in average sampling
The aim of this work is to solve a question raised for average sampling in shift-invariant spaces by using the well-known matrix pencil theory. In many common situations in sampling theory, the available data are samples of some convolution operator acting on the function itself: this leads to the problem of average sampling, also known as generalized sampling. In this paper we deal with the existence of a sampling formula involving these samples and having reconstruction functions with compact support. Thus, low computational complexity is involved and truncation errors are avoided. In practice, it is accomplished by means of a FIR filter bank. An answer is given in the light of the generalized sampling theory by using the oversampling technique: more samples than strictly necessary are used. The original problem reduces to finding a polynomial left inverse of a polynomial matrix intimately related to the sampling problem which, for a suitable choice of the sampling period, becomes a matrix pencil. This matrix pencil approach allows us to obtain a practical method for computing the compactly supported reconstruction functions for the important case where the oversampling rate is minimum. Moreover, the optimality of the obtained solution is established
Complex independent process analysis
We present a general framework for the search of hidden independent processes in the complex domain. The task is to estimate the hidden independent multidimensional complex-valued components observing only the mixture of the processes driven by them. In our model (i) the hidden independent processes can be multidimensional, they may be subject to (ii) moving averaging, or may evolve in an autoregressive manner, or (iii) they can be non-stationary. These assumptions are covered by integrated autoregressive moving average processes and thus our task is to solve their complex extensions. We show how to reduce the undercomplete version of complex integrated autoregressive moving average processes to real independent subspace analysis that we can solve. Simulations illustrate the working of the algorithm
Undercomplete Blind Subspace Deconvolution via Linear Prediction
We present a novel solution technique for the blind subspace deconvolution
(BSSD) problem, where temporal convolution of multidimensional hidden
independent components is observed and the task is to uncover the hidden
components using the observation only. We carry out this task for the
undercomplete case (uBSSD): we reduce the original uBSSD task via linear
prediction to independent subspace analysis (ISA), which we can solve. As it
has been shown recently, applying temporal concatenation can also reduce uBSSD
to ISA, but the associated ISA problem can easily become `high dimensional'
[1]. The new reduction method circumvents this dimensionality problem. We
perform detailed studies on the efficiency of the proposed technique by means
of numerical simulations. We have found several advantages: our method can
achieve high quality estimations for smaller number of samples and it can cope
with deeper temporal convolutions.Comment: 12 page
Generic Invertibility of Multidimensional FIR Filter Banks and MIMO Systems
We study the invertibility of M-variate Laurent polynomial N × P matrices. Such matrices represent multidimensional systems in various settings such as filter banks, multiple-input multiple-output systems, and multirate systems. Given an N × P Laurent polynomial matrix H(z1,..., zM) of degree at most k, we want to find a P × N Laurent polynomial left inverse matrix G(z) of H(z) such that G(z)H(z) = I. We provide computable conditions to test the invertibility and propose algorithms to find a particular inverse. The main result of this paper is to prove that H(z) is generically invertible when N −P ≥ M; whereas when N −P < M, then H(z) is generically noninvertible. As a result, we propose an algorithm to find a particular inverse of a Laurent polynomial matrix that is faster than current algorithms known to us
Blind, MIMO system estimation based on PARAFAC decomposition of higher order output tensors
IEEE Transactions on Signal Processing, 54(11): pp. 4156-4168.We present a novel framework for the identification of
a multiple-input multiple-output (MIMO) system driven by white,
mutually independent unobservable inputs. Samples of the system
frequency response are obtained based on parallel factorization
(PARAFAC) of three- or four-way tensors constructed based on,
respectively, third- or fourth-order cross spectra of the system outputs.
The main difficulties in frequency-domain methods are frequency-
dependent permutation and filtering ambiguities.We show
that the information available in the higher order spectra allows
for the ambiguities to be resolved up to a constant scaling and permutation
ambiguities and a linear phase ambiguity. Important features
of the proposed approach are that it does not require channel
length information, needs no phase unwrapping, and unlike the
majority of existing methods, needs no prewhitening of the system
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