Communications inspired linear discriminant analysis

We study the problem of supervised linear dimensionality reduction, taking an information-theoretic viewpoint. The linear projection matrix is designed by maximizing the mutual information between the projected signal and the class label. By harnessing a recent theoretical result on the gradient of mutual information, the above optimization problem can be solved directly using gradient descent, without requiring simplification of the objective function. Theoretical analysis and empirical comparison are made between the proposed method and two closely related methods, and comparisons are also made with a method in which Rényi entropy is used to define the mutual information (in this case the gradient may be computed simply, under a special parameter setting). Relative to these alternative approaches, the proposed method achieves promising results on real datasets. Copyright 2012 by the author(s)/owner(s)

Discrimination on the Grassmann Manifold: Fundamental Limits of Subspace Classifiers

We present fundamental limits on the reliable classification of linear and affine subspaces from noisy, linear features. Drawing an analogy between discrimination among subspaces and communication over vector wireless channels, we propose two Shannon-inspired measures to characterize asymptotic classifier performance. First, we define the classification capacity, which characterizes necessary and sufficient conditions for the misclassification probability to vanish as the signal dimension, the number of features, and the number of subspaces to be discerned all approach infinity. Second, we define the diversity-discrimination tradeoff which, by analogy with the diversity-multiplexing tradeoff of fading vector channels, characterizes relationships between the number of discernible subspaces and the misclassification probability as the noise power approaches zero. We derive upper and lower bounds on these measures which are tight in many regimes. Numerical results, including a face recognition application, validate the results in practice.Comment: 19 pages, 4 figures. Revised submission to IEEE Transactions on Information Theor

Neural-inspired sensors enable sparse, efficient classification of spatiotemporal data

Sparse sensor placement is a central challenge in the efficient characterization of complex systems when the cost of acquiring and processing data is high. Leading sparse sensing methods typically exploit either spatial or temporal correlations, but rarely both. This work introduces a new sparse sensor optimization that is designed to leverage the rich spatiotemporal coherence exhibited by many systems. Our approach is inspired by the remarkable performance of flying insects, which use a few embedded strain-sensitive neurons to achieve rapid and robust flight control despite large gust disturbances. Specifically, we draw on nature to identify targeted neural-inspired sensors on a flapping wing to detect body rotation. This task is particularly challenging as the rotational twisting mode is three orders-of-magnitude smaller than the flapping modes. We show that nonlinear filtering in time, built to mimic strain-sensitive neurons, is essential to detect rotation, whereas instantaneous measurements fail. Optimized sparse sensor placement results in efficient classification with approximately ten sensors, achieving the same accuracy and noise robustness as full measurements consisting of hundreds of sensors. Sparse sensing with neural inspired encoding establishes a new paradigm in hyper-efficient, embodied sensing of spatiotemporal data and sheds light on principles of biological sensing for agile flight control.Comment: 21 pages, 19 figure

Compressive Classification

This paper derives fundamental limits associated with compressive classification of Gaussian mixture source models. In particular, we offer an asymptotic characterization of the behavior of the (upper bound to the) misclassification probability associated with the optimal Maximum-A-Posteriori (MAP) classifier that depends on quantities that are dual to the concepts of diversity gain and coding gain in multi-antenna communications. The diversity, which is shown to determine the rate at which the probability of misclassification decays in the low noise regime, is shown to depend on the geometry of the source, the geometry of the measurement system and their interplay. The measurement gain, which represents the counterpart of the coding gain, is also shown to depend on geometrical quantities. It is argued that the diversity order and the measurement gain also offer an optimization criterion to perform dictionary learning for compressive classification applications.Comment: 5 pages, 3 figures, submitted to the 2013 IEEE International Symposium on Information Theory (ISIT 2013

Generalizing Lieb's Concavity Theorem via Operator Interpolation

We introduce the notion of $k$-trace and use interpolation of operators to prove the joint concavity of the function $(A,B)\mapsto\text{Tr}_k\big[(B^\frac{qs}{2}K^*A^{ps}KB^\frac{qs}{2})^{\frac{1}{s}}\big]^\frac{1}{k}$, which generalizes Lieb's concavity theorem from trace to a class of homogeneous functions $\text{Tr}_k[\cdot]^\frac{1}{k}$. Here $\text{Tr}_k[A]$ denotes the $k_{\text{th}}$ elementary symmetric polynomial of the eigenvalues of $A$. This result gives an alternative proof for the concavity of $A\mapsto\text{Tr}_k\big[\exp(H+\log A)\big]^\frac{1}{k}$ that was obtained and used in a recent work to derive expectation estimates and tail bounds on partial spectral sums of random matrices