6 research outputs found
Widely Linear Kernels for Complex-Valued Kernel Activation Functions
Complex-valued neural networks (CVNNs) have been shown to be powerful
nonlinear approximators when the input data can be properly modeled in the
complex domain. One of the major challenges in scaling up CVNNs in practice is
the design of complex activation functions. Recently, we proposed a novel
framework for learning these activation functions neuron-wise in a
data-dependent fashion, based on a cheap one-dimensional kernel expansion and
the idea of kernel activation functions (KAFs). In this paper we argue that,
despite its flexibility, this framework is still limited in the class of
functions that can be modeled in the complex domain. We leverage the idea of
widely linear complex kernels to extend the formulation, allowing for a richer
expressiveness without an increase in the number of adaptable parameters. We
test the resulting model on a set of complex-valued image classification
benchmarks. Experimental results show that the resulting CVNNs can achieve
higher accuracy while at the same time converging faster.Comment: Accepted at ICASSP 201
Learning ground states of gapped quantum Hamiltonians with Kernel Methods
Neural network approaches to approximate the ground state of quantum
hamiltonians require the numerical solution of a highly nonlinear optimization
problem. We introduce a statistical learning approach that makes the
optimization trivial by using kernel methods. Our scheme is an approximate
realization of the power method, where supervised learning is used to learn the
next step of the power iteration. We show that the ground state properties of
arbitrary gapped quantum hamiltonians can be reached with polynomial resources
under the assumption that the supervised learning is efficient. Using kernel
ridge regression, we provide numerical evidence that the learning assumption is
verified by applying our scheme to find the ground states of several
prototypical interacting many-body quantum systems, both in one and two
dimensions, showing the flexibility of our approach
Widely-Linear MMSE Estimation of Complex-Valued Graph Signals
In this paper, we consider the problem of recovering random graph signals
with complex values. For general Bayesian estimation of complex-valued vectors,
it is known that the widely-linear minimum mean-squared-error (WLMMSE)
estimator can achieve a lower mean-squared-error (MSE) than that of the linear
minimum MSE (LMMSE) estimator. Inspired by the WLMMSE estimator, in this paper
we develop the graph signal processing (GSP)-WLMMSE estimator, which minimizes
the MSE among estimators that are represented as a two-channel output of a
graph filter, i.e. widely-linear GSP estimators. We discuss the properties of
the proposed GSP-WLMMSE estimator. In particular, we show that the MSE of the
GSP-WLMMSE estimator is always equal to or lower than the MSE of the GSP-LMMSE
estimator. The GSP-WLMMSE estimator is based on diagonal covariance matrices in
the graph frequency domain, and thus has reduced complexity compared with the
WLMMSE estimator. This property is especially important when using the
sample-mean versions of these estimators that are based on a training dataset.
We then state conditions under which the low-complexity GSP-WLMMSE estimator
coincides with the WLMMSE estimator. In the simulations, we investigate two
synthetic estimation problems (with linear and nonlinear models) and the
problem of state estimation in power systems. For these problems, it is shown
that the GSP-WLMMSE estimator outperforms the GSP-LMMSE estimator and achieves
similar performance to that of the WLMMSE estimator.Comment: This work has been submitted to the IEEE for possible publication.
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The Generalized Complex Kernel Least-Mean-Square Algorithm
We propose a novel adaptive kernel based regression method for complex-valued
signals: the generalized complex-valued kernel least-mean-square (gCKLMS). We
borrow from the new results on widely linear reproducing kernel Hilbert space
(WL-RKHS) for nonlinear regression and complex-valued signals, recently
proposed by the authors. This paper shows that in the adaptive version of the
kernel regression for complex-valued signals we need to include another kernel
term, the so-called pseudo-kernel. This new solution is endowed with better
representation capabilities in complex-valued fields, since it can efficiently
decouple the learning of the real and the imaginary part. Also, we review
previous realizations of the complex KLMS algorithm and its augmented version
to prove that they can be rewritten as particular cases of the gCKLMS.
Furthermore, important conclusions on the kernels design are drawn that help to
greatly improve the convergence of the algorithms. In the experiments, we
revisit the nonlinear channel equalization problem to highlight the better
convergence of the gCKLMS compared to previous solutions. Also, the flexibility
of the proposed generalized approach is tested in a second experiment with
non-independent real and imaginary parts. The results illustrate the
significant performance improvements of the gCKLMS approach when the
complex-valued signals have different properties for the real and imaginary
parts.Comment: Submitted to IEEE Transactions on Signal Processin