6 research outputs found
Complex-Valued Random Vectors and Channels: Entropy, Divergence, and Capacity
Recent research has demonstrated significant achievable performance gains by
exploiting circularity/non-circularity or propeness/improperness of
complex-valued signals. In this paper, we investigate the influence of these
properties on important information theoretic quantities such as entropy,
divergence, and capacity. We prove two maximum entropy theorems that strengthen
previously known results. The proof of the former theorem is based on the
so-called circular analog of a given complex-valued random vector. Its
introduction is supported by a characterization theorem that employs a minimum
Kullback-Leibler divergence criterion. In the proof of latter theorem, on the
other hand, results about the second-order structure of complex-valued random
vectors are exploited. Furthermore, we address the capacity of multiple-input
multiple-output (MIMO) channels. Regardless of the specific distribution of the
channel parameters (noise vector and channel matrix, if modeled as random), we
show that the capacity-achieving input vector is circular for a broad range of
MIMO channels (including coherent and noncoherent scenarios). Finally, we
investigate the situation of an improper and Gaussian distributed noise vector.
We compute both capacity and capacity-achieving input vector and show that
improperness increases capacity, provided that the complementary covariance
matrix is exploited. Otherwise, a capacity loss occurs, for which we derive an
explicit expression.Comment: 33 pages, 1 figure, slightly modified version of first paper revision
submitted to IEEE Trans. Inf. Theory on October 31, 201
Wiener filters in canonical coordinates for transform coding, filtering, and quantizing
Includes bibliographical references.Canonical correlations are used to decompose the Wiener filter into a whitening transform coder, a canonical filter, and a coloring transform decoder. The outputs of the whitening transform coder are called canonical coordinates; these are the coordinates that are reduced in rank and quantized in our finite-precision version of the Gauss-Markov theorem. Canonical correlations are, in fact, cosines of the canonical angles between a source vector and a measurement vector. They produce new formulas for error covariance, spectral flatness, and entropy.This work supported by the National Science Foundation under Contract MIP-9529050 and by the Office of Naval Research under Contract N00014-89-J-1070
Informative Data Fusion: Beyond Canonical Correlation Analysis
Multi-modal data fusion is a challenging but common problem arising in fields such as economics, statistical signal processing, medical imaging, and machine learning. In such applications, we have access to multiple datasets that use different data modalities to describe some system feature. Canonical correlation analysis (CCA) is a multidimensional joint dimensionality reduction algorithm for exactly two datasets. CCA finds a linear transformation for each feature vector set such that the correlation between the two transformed feature sets is maximized. These linear transformations are easily found by solving the SVD of a matrix that only involves the covariance and cross-covariance matrices of the feature vector sets. When these covariance matrices
are unknown, an empirical version of CCA substitutes sample covariance estimates formed from training data. However, when the number of training samples is less than the combined dimension of the datasets, CCA fails to reliably detect correlation between the datasets.
This thesis explores the the problem of detecting correlations from data modeled by the ubiquitous signal-plus noise data model. We present a modification to CCA, which we call informative CCA (ICCA) that first projects each dataset onto a low-dimensional informative signal subspace. We verify the superior performance of ICCA on real-world datasets and argue the optimality of trim-then-fuse over fuse-then-trim correlation analysis strategies. We provide a significance test for the correlations returned by ICCA and derive improved estimates of the population canonical vectors using insights from random matrix theory. We then extend the analysis of CCA to regularized CCA (RCCA) and demonstrate that setting the regularization parameter to infinity results in the best performance and has the same solution as taking the SVD of the cross-covariance matrix of the two datasets. Finally, we apply the ideas learned from ICCA to multiset CCA (MCCA), which analyzes correlations for more than two datasets. There are multiple formulations of multiset CCA (MCCA), each using a different combination of objective function and constraint function to describe a notion of multiset correlation. We consider MAXVAR, provide an informative version of the algorithm, which we call informative MCCA (IMCCA), and demonstrate its superiority on a real-world dataset.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113419/1/asendorf_1.pd