International audienceThe characteristics of the model dynamics are critical in the performance of (ensemble) Kalmanfilters. In particular, as emphasized in the seminal work of Anna Trevisan and coauthors, theerror covariance matrix is asymptotically supported by the unstable-neutral subspace only, i.e., itis spanned by the backward Lyapunov vectors with nonnegative exponents. This behavior is at thecore of algorithms known as assimilation in the unstable subspace, although a formal proof was stillmissing. This paper provides the analytical proof of the convergence of the Kalman filter covariancematrix onto the unstable-neutral subspace when the dynamics and the observation operator are linearand when the dynamical model is error free, for any, possibly rank-deficient, initial error covariancematrix. The rate of convergence is provided as well. The derivation is based on an expression thatexplicitly relates the error covariances at an arbitrary time to the initial ones. It is also shown thatif the unstable and neutral directions of the model are sufficiently observed and if the column spaceof the initial covariance matrix has a nonzero projection onto all of the forward Lyapunov vectorsassociated with the unstable and neutral directions of the dynamics, the covariance matrix of theKalman filter collapses onto an asymptotic sequence which is independent of the initial covariances.Numerical results are also shown to illustrate and support the theoretical findings

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Abstract. We prove that for linear, discrete, time-varying, deterministic system (perfect model) with noisy outputs, the Riccati transformation in the Kalman filter asymptotically bounds the rank of the forecast and the analysis error covariance matrices to be less than or equal to the number of non-negative Lyapunov exponents of the system. Further, the support of these error covariance matrices is shown to be confined to the space spanned by the unstable-neutral backward Lyapunov vectors, providing the theoretical justification for the methodology of the algorithms that perform assimilation only in the unstable-neutral subspace. The equivalent property of the autonomous system is investigated as a special case.

We present a new method for compact representation of large image datasets. Our method is based on treating small patches from a 2-D image as matrices as opposed to the conventional vectorial representation, and encoding these patches as sparse projections onto a set of exemplar orthonormal bases, which are learned a priori from a training set. The end result is a low-error, highly compact image/patch representation that has significant theoretical merits and compares favorably with existing techniques (including JPEG) on experiments involving the compression of ORL and Yale face databases, as well as a database of miscellaneous natural images. In the context of learning multiple orthonormal bases, we show the easy tunability of our method to efficiently represent patches of different complexities. Furthermore, we show that our method is extensible in a theoretically sound manner to higher-order matrices ("tensors"). We demonstrate applications of this theory to compression of well-known color image datasets such as the GaTech and CMU-PIE face databases and show performance competitive with JPEG. Lastly, we also analyze the effect of image noise on the performance of our compression schemes.