We consider the problem of sufficient dimension reduction (SDR) for multi-index models. The estimators of the central mean subspace in prior works either have slow (non-parametric) convergence rates, or rely on stringent distributional conditions (e.g., the covariate distribution $P_{\mathbf{X}}$ being elliptical symmetric). In this paper, we show that a fast parametric convergence rate of form $C_d \cdot n^{-1/2}$ is achievable via estimating the \emph{expected smoothed gradient outer product}, for a general class of distribution $P_{\mathbf{X}}$ admitting Gaussian or heavier distributions. When the link function is a polynomial with a degree of at most $r$ and $P_{\mathbf{X}}$ is the standard Gaussian, we show that the prefactor depends on the ambient dimension $d$ as $C_d \propto d^r$.

We introduce a multifidelity estimator of covariance matrices formulated as the solution to a regression problem on the manifold of symmetric positive definite matrices. The estimator is positive definite by construction, and the Mahalanobis distance minimized to obtain it possesses properties which enable practical computation. We show that our manifold regression multifidelity (MRMF) covariance estimator is a maximum likelihood estimator under a certain error model on manifold tangent space. More broadly, we show that our Riemannian regression framework encompasses existing multifidelity covariance estimators constructed from control variates. We demonstrate via numerical examples that our estimator can provide significant decreases, up to one order of magnitude, in squared estimation error relative to both single-fidelity and other multifidelity covariance estimators. Furthermore, preservation of positive definiteness ensures that our estimator is compatible with downstream tasks, such as data assimilation and metric learning, in which this property is essential.

We study a deep linear network endowed with a structure. It takes the form of a matrix $X$ obtained by multiplying $K$ matrices (called factors and corresponding to the action of the layers). The action of each layer (i.e. a factor) is obtained by applying a fixed linear operator to a vector of parameters satisfying a constraint. The number of layers is not limited. Assuming that $X$ is given and factors have been estimated, the error between the product of the estimated factors and $X$ (i.e. the reconstruction error) is either the statistical or the empirical risk. In this paper, we provide necessary and sufficient conditions on the network topology under which a stability property holds. The stability property requires that the error on the parameters defining the factors (i.e. the stability of the recovered parameters) scales linearly with the reconstruction error (i.e. the risk). Therefore, under these conditions on the network topology, any successful learning task leads to stably defined features and therefore interpretable layers/network.In order to do so, we first evaluate how the Segre embedding and its inverse distort distances. Then, we show that any deep structured linear network can be cast as a generic multilinear problem (that uses the Segre embedding). This is the {\em tensorial lifting}. Using the tensorial lifting, we provide necessary and sufficient conditions for the identifiability of the factors (up to a scale rearrangement). We finally provide the necessary and sufficient condition called \NSPlong~(because of the analogy with the usual Null Space Property in the compressed sensing framework) which guarantees that the stability property holds. We illustrate the theory with a practical example where the deep structured linear network is a convolutional linear network. As expected, the conditions are rather strong but not empty. A simple test on the network topology can be implemented to test if the condition holds.