Sparse Matrix-based Random Projection for Classification

As a typical dimensionality reduction technique, random projection can be simply implemented with linear projection, while maintaining the pairwise distances of high-dimensional data with high probability. Considering this technique is mainly exploited for the task of classification, this paper is developed to study the construction of random matrix from the viewpoint of feature selection, rather than of traditional distance preservation. This yields a somewhat surprising theoretical result, that is, the sparse random matrix with exactly one nonzero element per column, can present better feature selection performance than other more dense matrices, if the projection dimension is sufficiently large (namely, not much smaller than the number of feature elements); otherwise, it will perform comparably to others. For random projection, this theoretical result implies considerable improvement on both complexity and performance, which is widely confirmed with the classification experiments on both synthetic data and real data

On the Relationship of Optimal State Feedback and Disturbance Response Controllers

This paper studies the relationship between state feedback policies and disturbance response policies for the standard Linear Quadratic Regulator (LQR). For open-loop stable plants, we establish a simple relationship between the optimal state feedback controller $u_t=K_\star x_t$ and the optimal disturbance response controller $u_t=L^{(H)}_{\star;1}w_{t-1}+\cdots+L^{(H)}_{\star;H}w_{t-H}$ with $H$-order. Here $x_t, w_t, u_t$ stands for the state, disturbance, control action of the system, respectively. Our result shows that $L_{\star,1}^{(H)}$ is a good approximation of $K_\star$ and the approximation error $\|K_\star - L_{\star,1}^{(H)}\|$ decays exponentially with $H$. We further extend this result to LQR for open-loop unstable systems, when a pre-stabilizing controller $K_0$ is available