We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

: This thesis applies neural network feature selection techniques to multivariate time series data to improve prediction of a target time series. Two approaches to feature selection are used. First, a subset enumeration method is used to determine which financial indicators are most useful for aiding in prediction of the S&P 500 futures daily price. The candidate indicators evaluated include RSI, Stochastics and several moving averages. Results indicate that the Stochastics and RSI indicators result in better prediction results than the moving averages. The second approach to feature selection is calculation of individual saliency metrics. A new decision boundary-based individual saliency metric, and a classifier independent saliency metric are developed and tested. Ruck's saliency metric, the decision boundary based saliency metric, and the classifier independent saliency metric are compared for a data set consisting of the RSI and Stochastics indicators as well as delayed closing price values. The decision based metric and the Ruck metric results are similar, but the classifier independent metric agrees with neither of the other metrics. The nine most salient features, determined by the decision boundary based metric, are used to train a neural network and the results are presented and compared to other published results. (AN)

/pdf/nonlinear-time-series-analysis-1j3a35n4y3.pdf

Nonlinear Time Series Analysis.

This paper develops a mathematical theory of super-resolution. Broadly speaking, superresolution is the problem of recovering the ne details of an object|the high end of its spectrum| from coarse scale information only|from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in [0; 1] and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-o fc. We show that one can super-resolve these point sources with innite precision|i.e. recover the exact locations and amplitudes|by solving a simple convex optimization problem, which can essentially be reformulated as a semidenite program. This holds provided that the distance between sources is at least 2=fc. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with innite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the super-resolution factor vary.

/pdf/towards-a-mathematical-theory-of-super-resolution-23m861muqo.pdf

Towards a Mathematical Theory of Super-resolution

This paper investigates the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0, 1]. An atomic norm minimization approach is proposed to exactly recover the unobserved samples and identify the unknown frequencies, which is then reformulated as an exact semidefinite program. Even with this continuous dictionary, it is shown that O(slog s log n) random samples are sufficient to guarantee exact frequency localization with high probability, provided the frequencies are well separated. Extensive numerical experiments are performed to illustrate the effectiveness of the proposed method.

Compressed Sensing Off the Grid

We consider the problem of estimating the frequency components of a mixture of s complex sinusoids from a random subset of n regularly spaced samples. Unlike previous work in compressed sensing, the frequencies are not assumed to lie on a grid, but can assume any values in the normalized frequency domain [0,1]. We propose an atomic norm minimization approach to exactly recover the unobserved samples. We reformulate this atomic norm minimization as an exact semidefinite program. Even with this continuous dictionary, we show that most sampling sets of size O(s log s log n) are sufficient to guarantee the exact frequency estimation with high probability, provided the frequencies are well separated. Numerical experiments are performed to illustrate the effectiveness of the proposed method.

Compressed Sensing off the Grid

Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.

Atomic Norm Denoising With Applications to Line Spectral Estimation

This paper establishes a nearly optimal algorithm for denoising a mixture of sinusoids from noisy equispaced samples. We derive our algorithm by viewing line spectral estimation as a sparse recovery problem with a continuous, infinite dictionary. We show how to compute the estimator via semidefinite programming and provide guarantees on its mean-squared error rate. We derive a complementary minimax lower bound on this estimation rate, demonstrating that our approach nearly achieves the best possible estimation error. Furthermore, we establish bounds on how well our estimator localizes the frequencies in the signal, showing that the localization error tends to zero as the number of samples grows. We verify our theoretical results in an array of numerical experiments, demonstrating that the semidefinite programming approach outperforms three classical spectral estimation techniques.

Near Minimax Line Spectral Estimation

The sub-Nyquist estimation of line spectra is a classical problem in signal processing, but currently popular subspace-based techniques have few guarantees in the presence of noise and rely on a priori knowledge about system model order. Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectrum estimation that provides theoretical guarantees for the mean-square-error performance in the presence of noise and without advance knowledge of the model order. We propose an abstract theory of denoising with atomic norms which is specialized to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials with guaranteed bounds on the mean-squared-error. In general, our proposed optimization problem has no known polynomial time solution, but we provide an efficient algorithm, called DAST, based on the Fast Fourier Transform that achieves nearly the same error rate. We compare DAST with Cadzow's canonical alternating projection algorithm, which performs marginally better under high signal-to-noise ratios when the model order is known exactly, and demonstrate experimentally that DAST outperforms other denoising techniques, including Cadzow's, over a wide range of signal-to-noise ratios.

http://www.optimization-online.org/DB_FILE//2012/04/3420.pdf

Badri Narayan Bhaskar

Papers

Compressed Sensing Off the Grid

Compressed Sensing off the Grid

Atomic Norm Denoising With Applications to Line Spectral Estimation

Near Minimax Line Spectral Estimation

Atomic norm denoising with applications to line spectral estimation