Minimax frequency domain performance and robustness optimization of linear feedback systems

It is shown that feedback system design objectives, such as disturbance attenuation and rejection, power and bandwidth limitation, and robustness, may be expressed in terms of required bounds of the sensitivity function and its complement on the imaginary axis. This leads to a minimax frequency domain optimization problem, whose solution is reduced to the solution of a polynomial equation

An Oracle Inequality for Quasi-Bayesian Non-Negative Matrix Factorization

The aim of this paper is to provide some theoretical understanding of quasi-Bayesian aggregation methods non-negative matrix factorization. We derive an oracle inequality for an aggregated estimator. This result holds for a very general class of prior distributions and shows how the prior affects the rate of convergence.Comment: This is the corrected version of the published paper P. Alquier, B. Guedj, An Oracle Inequality for Quasi-Bayesian Non-negative Matrix Factorization, Mathematical Methods of Statistics, 2017, vol. 26, no. 1, pp. 55-67. Since then Arnak Dalalyan (ENSAE) found a mistake in the proofs. We fixed the mistake at the price of a slightly different logarithmic term in the boun

A new SVD approach to optimal topic estimation

In the probabilistic topic models, the quantity of interest---a low-rank matrix consisting of topic vectors---is hidden in the text corpus matrix, masked by noise, and Singular Value Decomposition (SVD) is a potentially useful tool for learning such a matrix. However, different rows and columns of the matrix are usually in very different scales and the connection between this matrix and the singular vectors of the text corpus matrix are usually complicated and hard to spell out, so how to use SVD for learning topic models faces challenges. We overcome the challenges by introducing a proper Pre-SVD normalization of the text corpus matrix and a proper column-wise scaling for the matrix of interest, and by revealing a surprising Post-SVD low-dimensional {\it simplex} structure. The simplex structure, together with the Pre-SVD normalization and column-wise scaling, allows us to conveniently reconstruct the matrix of interest, and motivates a new SVD-based approach to learning topic models. We show that under the popular probabilistic topic model \citep{hofmann1999}, our method has a faster rate of convergence than existing methods in a wide variety of cases. In particular, for cases where documents are long or $n$ is much larger than $p$, our method achieves the optimal rate. At the heart of the proofs is a tight element-wise bound on singular vectors of a multinomially distributed data matrix, which do not exist in literature and we have to derive by ourself. We have applied our method to two data sets, Associated Process (AP) and Statistics Literature Abstract (SLA), with encouraging results. In particular, there is a clear simplex structure associated with the SVD of the data matrices, which largely validates our discovery.Comment: 73 pages, 8 figures, 6 tables; considered two different VH algorithm, OVH and GVH, and provided theoretical analysis for each algorithm; re-organized upper bound theory part; added the subsection of comparing error rate with other existing methods; provided another improved version of error analysis through Bernstein inequality for martingale