Abstract-There are two types of problems in the theory of least squares signal processing: parameter estimation and signal extraction. Parameter estimation is called ''inversion'' and signal extraction is called ''filtering.'' In this paper, we present a unified theory of rank shaping for solving overdetermined and underdetermined versions of these problems. We develop several data-dependent rank-shaping methods and evaluate their performance. Our key result is a data-adaptive Wiener filter that automatically adjusts its gains to accommodate realizations that are a priori unlikely. The adaptive filter dramatically outperforms the Wiener filter on atypical realizations and just slightly underperforms it on typical realizations. This is the most one can hope for in a data-adaptive filter.[1], Marquardt [2], and Stein [3]; however, our data adaptive shrinkage takes place mode-by-mode.Our philosophy in this paper is that with clairvoyant side information (which we do not have), we could improve on least squares for estimating signals and parameters. A natural inclination is then to try to steal this clairvoyant information from the data. We show that this is extremely risky, that naive methods cannot work, and that only sophisticated, conservative deviations from Wiener filtering can work. The result is a nonlinear filter that uses mode-dependent, nonlinear companders to estimate something akin to Wiener gain.where y is a noisy N x 1 observation of the signal x. The matrix H is the N x p model matrix, and fl is the p x 1 parameter vector. Geometrically, the signal x lies in the rankp subspace (H), illustrated in Fig. 1. The signal x can be thought of as a linear combination of columns of H A. The Linear Statistical ModelThe linear statistical model is a signal-plus-noise model: the observations consist of a model or signal component and an error or noise component. Moreover, the signal component satisfies a set of linear equations. This leads to the model Each hi might be a mode in a system. We wish to determine the weights ()i. Alternatively, the observation y could be a noisy version of some modulated information fl that we are trying to estimate. The linear model also arises in curve-fitting problems such as polynomial interpolation.We will make extensive use of the singular value decomposition of H, namely H = U~VT, where~is the diagonal matrix of singular values ai. In the overdetermined case, where

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We present a data-driven solution to the terminalhitting stochastic reachability problem for a Markov control process. We employ a nonparametric representation of the stochastic kernel as a conditional distribution embedding within a reproducing kernel Hilbert space (RKHS). This representation avoids intractable integrals in the dynamic recursion of the stochastic reachability problem since the expectations can be calculated as an inner product within the RKHS. We demonstrate this approach on a high-dimensional chain of integrators and on Clohessy-Wiltshire-Hill dynamics.

We present a data-driven algorithm for efficiently computing stochastic control policies for general joint chance constrained optimal control problems. Our approach leverages the theory of kernel distribution embeddings, which allows representing expectation operators as inner products in a reproducing kernel Hilbert space. This framework enables approximately reformulating the original problem using a dataset of observed trajectories from the system without imposing prior assumptions on the parameterization of the system dynamics or the structure of the uncertainty. By optimizing over a finite subset of stochastic open-loop control trajectories, we relax the original problem to a linear program over the control parameters that can be efficiently solved using standard convex optimization techniques. We demonstrate our proposed approach in simulation on a system with nonlinear non-Markovian dynamics navigating in a cluttered environment.

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