Optimality of Graphlet Screening in High Dimensional Variable Selection

Consider a linear regression model where the design matrix X has n rows and p columns. We assume (a) p is much large than n, (b) the coefficient vector beta is sparse in the sense that only a small fraction of its coordinates is nonzero, and (c) the Gram matrix G = X'X is sparse in the sense that each row has relatively few large coordinates (diagonals of G are normalized to 1). The sparsity in G naturally induces the sparsity of the so-called graph of strong dependence (GOSD). We find an interesting interplay between the signal sparsity and the graph sparsity, which ensures that in a broad context, the set of true signals decompose into many different small-size components of GOSD, where different components are disconnected. We propose Graphlet Screening (GS) as a new approach to variable selection, which is a two-stage Screen and Clean method. The key methodological innovation of GS is to use GOSD to guide both the screening and cleaning. Compared to m-variate brute-forth screening that has a computational cost of p^m, the GS only has a computational cost of p (up to some multi-log(p) factors) in screening. We measure the performance of any variable selection procedure by the minimax Hamming distance. We show that in a very broad class of situations, GS achieves the optimal rate of convergence in terms of the Hamming distance. Somewhat surprisingly, the well-known procedures subset selection and the lasso are rate non-optimal, even in very simple settings and even when their tuning parameters are ideally set

On the Approximation and Complexity of Deep Neural Networks to Invariant Functions

Recent years have witnessed a hot wave of deep neural networks in various domains; however, it is not yet well understood theoretically. A theoretical characterization of deep neural networks should point out their approximation ability and complexity, i.e., showing which architecture and size are sufficient to handle the concerned tasks. This work takes one step on this direction by theoretically studying the approximation and complexity of deep neural networks to invariant functions. We first prove that the invariant functions can be universally approximated by deep neural networks. Then we show that a broad range of invariant functions can be asymptotically approximated by various types of neural network models that includes the complex-valued neural networks, convolutional neural networks, and Bayesian neural networks using a polynomial number of parameters or optimization iterations. We also provide a feasible application that connects the parameter estimation and forecasting of high-resolution signals with our theoretical conclusions. The empirical results obtained on simulation experiments demonstrate the effectiveness of our method

Triple condensate halo from water droplets impacting on cold surfaces

Understanding the dynamics in the deposition of water droplets onto solid surfaces is of importance from both fundamental and practical viewpoints. While the deposition of a water droplet onto a heated surface is extensively studied, the characteristics of depositing a droplet onto a cold surface and the phenomena leading to such behavior remain elusive. Here we report the formation of a triple condensate halo observed during the deposition of a water droplet onto a cold surface, due to the interplay between droplet impact dynamics and vapor diffusion. Two subsequent condensation stages occur during the droplet spreading and cooling processes, engendering this unique condensate halo with three distinctive bands. We further proposed a scaling model to interpret the size of each band, and the model is validated by the experiments of droplets with different impact velocity and varying substrate temperature. Our experimental and theoretical investigation of the droplet impact dynamics and the associated condensation unravels the mass and heat transfer among droplet, vapor and substrate, offer a new sight for designing of heat exchange devices

Orbital angular momentum mode-demultiplexing scheme with partial angular receiving aperture

For long distance orbital angular momentum (OAM) based transmission, the conventional whole beam receiving scheme encounters the difficulty of large aperture due to the divergence of OAM beams. We propose a novel partial receiving scheme, using a restricted angular aperture to receive and demultiplex multi-OAM-mode beams. The scheme is theoretically analyzed to show that a regularly spaced OAM mode set remain orthogonal and therefore can be de-multiplexed. Experiments have been carried out to verify the feasibility. This partial receiving scheme can serve as an effective method with both space and cost savings for the OAM communications. It is applicable to both free space OAM optical communications and radio frequency (RF) OAM communications