4,886 research outputs found
Regularized linear system identification using atomic, nuclear and kernel-based norms: the role of the stability constraint
Inspired by ideas taken from the machine learning literature, new
regularization techniques have been recently introduced in linear system
identification. In particular, all the adopted estimators solve a regularized
least squares problem, differing in the nature of the penalty term assigned to
the impulse response. Popular choices include atomic and nuclear norms (applied
to Hankel matrices) as well as norms induced by the so called stable spline
kernels. In this paper, a comparative study of estimators based on these
different types of regularizers is reported. Our findings reveal that stable
spline kernels outperform approaches based on atomic and nuclear norms since
they suitably embed information on impulse response stability and smoothness.
This point is illustrated using the Bayesian interpretation of regularization.
We also design a new class of regularizers defined by "integral" versions of
stable spline/TC kernels. Under quite realistic experimental conditions, the
new estimators outperform classical prediction error methods also when the
latter are equipped with an oracle for model order selection
Forecasting and Granger Modelling with Non-linear Dynamical Dependencies
Traditional linear methods for forecasting multivariate time series are not
able to satisfactorily model the non-linear dependencies that may exist in
non-Gaussian series. We build on the theory of learning vector-valued functions
in the reproducing kernel Hilbert space and develop a method for learning
prediction functions that accommodate such non-linearities. The method not only
learns the predictive function but also the matrix-valued kernel underlying the
function search space directly from the data. Our approach is based on learning
multiple matrix-valued kernels, each of those composed of a set of input
kernels and a set of output kernels learned in the cone of positive
semi-definite matrices. In addition to superior predictive performance in the
presence of strong non-linearities, our method also recovers the hidden dynamic
relationships between the series and thus is a new alternative to existing
graphical Granger techniques.Comment: Accepted for ECML-PKDD 201
Completing Low-Rank Matrices with Corrupted Samples from Few Coefficients in General Basis
Subspace recovery from corrupted and missing data is crucial for various
applications in signal processing and information theory. To complete missing
values and detect column corruptions, existing robust Matrix Completion (MC)
methods mostly concentrate on recovering a low-rank matrix from few corrupted
coefficients w.r.t. standard basis, which, however, does not apply to more
general basis, e.g., Fourier basis. In this paper, we prove that the range
space of an matrix with rank can be exactly recovered from few
coefficients w.r.t. general basis, though and the number of corrupted
samples are both as high as . Our model covers
previous ones as special cases, and robust MC can recover the intrinsic matrix
with a higher rank. Moreover, we suggest a universal choice of the
regularization parameter, which is . By our
filtering algorithm, which has theoretical guarantees, we can
further reduce the computational cost of our model. As an application, we also
find that the solutions to extended robust Low-Rank Representation and to our
extended robust MC are mutually expressible, so both our theory and algorithm
can be applied to the subspace clustering problem with missing values under
certain conditions. Experiments verify our theories.Comment: To appear in IEEE Transactions on Information Theor
Blind Multilinear Identification
We discuss a technique that allows blind recovery of signals or blind
identification of mixtures in instances where such recovery or identification
were previously thought to be impossible: (i) closely located or highly
correlated sources in antenna array processing, (ii) highly correlated
spreading codes in CDMA radio communication, (iii) nearly dependent spectra in
fluorescent spectroscopy. This has important implications --- in the case of
antenna array processing, it allows for joint localization and extraction of
multiple sources from the measurement of a noisy mixture recorded on multiple
sensors in an entirely deterministic manner. In the case of CDMA, it allows the
possibility of having a number of users larger than the spreading gain. In the
case of fluorescent spectroscopy, it allows for detection of nearly identical
chemical constituents. The proposed technique involves the solution of a
bounded coherence low-rank multilinear approximation problem. We show that
bounded coherence allows us to establish existence and uniqueness of the
recovered solution. We will provide some statistical motivation for the
approximation problem and discuss greedy approximation bounds. To provide the
theoretical underpinnings for this technique, we develop a corresponding theory
of sparse separable decompositions of functions, including notions of rank and
nuclear norm that specialize to the usual ones for matrices and operators but
apply to also hypermatrices and tensors.Comment: 20 pages, to appear in IEEE Transactions on Information Theor
- …