Robust Hypothesis Testing with a Relative Entropy Tolerance

This paper considers the design of a minimax test for two hypotheses where the actual probability densities of the observations are located in neighborhoods obtained by placing a bound on the relative entropy between actual and nominal densities. The minimax problem admits a saddle point which is characterized. The robust test applies a nonlinear transformation which flattens the nominal likelihood ratio in the vicinity of one. Results are illustrated by considering the transmission of binary data in the presence of additive noise.Comment: 14 pages, 5 figures, submitted to the IEEE Transactions on Information Theory, July 2007, revised April 200

A Contraction Analysis of the Convergence of Risk-Sensitive Filters

A contraction analysis of risk-sensitive Riccati equations is proposed. When the state-space model is reachable and observable, a block-update implementation of the risk-sensitive filter is used to show that the N-fold composition of the Riccati map is strictly contractive with respect to the Riemannian metric of positive definite matrices, when N is larger than the number of states. The range of values of the risk-sensitivity parameter for which the map remains contractive can be estimated a priori. It is also found that a second condition must be imposed on the risk-sensitivity parameter and on the initial error variance to ensure that the solution of the risk-sensitive Riccati equation remains positive definite at all times. The two conditions obtained can be viewed as extending to the multivariable case an earlier analysis of Whittle for the scalar case.Comment: 22 pages, 6 figure

Robust Kalman Filtering: Asymptotic Analysis of the Least Favorable Model

We consider a robust filtering problem where the robust filter is designed according to the least favorable model belonging to a ball about the nominal model. In this approach, the ball radius specifies the modeling error tolerance and the least favorable model is computed by performing a Riccati-like backward recursion. We show that this recursion converges provided that the tolerance is sufficiently small

Robust State Space Filtering under Incremental Model Perturbations Subject to a Relative Entropy Tolerance

This paper considers robust filtering for a nominal Gaussian state-space model, when a relative entropy tolerance is applied to each time increment of a dynamical model. The problem is formulated as a dynamic minimax game where the maximizer adopts a myopic strategy. This game is shown to admit a saddle point whose structure is characterized by applying and extending results presented earlier in [1] for static least-squares estimation. The resulting minimax filter takes the form of a risk-sensitive filter with a time varying risk sensitivity parameter, which depends on the tolerance bound applied to the model dynamics and observations at the corresponding time index. The least-favorable model is constructed and used to evaluate the performance of alternative filters. Simulations comparing the proposed risk-sensitive filter to a standard Kalman filter show a significant performance advantage when applied to the least-favorable model, and only a small performance loss for the nominal model

Spectral action on noncommutative torus

The spectral action on noncommutative torus is obtained, using a Chamseddine--Connes formula via computations of zeta functions. The importance of a Diophantine condition is outlined. Several results on holomorphic continuation of series of holomorphic functions are obtained in this context.Comment: 57 page