An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Let $K_{r+1}$ be a complete graph with $r+1$ vertices. The $r$-uniform hypergraph $K_{r+1}^+$ is obtained from $K_{r+1}$ by enlarging each edge of $K_{r+1}$ with $r-2$ new vertices disjoint from $V(K_{r+1})$ such that distinct edges of $K_{r+1}$ are enlarged by distinct vertices. Let $H$ be a $K_{r+1}^+$-free linear $r$-uniform hypergraph with $n$ vertices. In this paper, we prove that when $n$ is sufficiently large, the spectral radius $\rho (H)$ of the adjacency tensor of $H$ is no more than $\frac{n}{r}$, i.e., $\rho (H)\leq \frac{n}{r}$, with equality if and only if $r|n$ and $H$ is a transversal design, where the transversal design is the balanced $r$-partite $r$-uniform hypergraph such that each pair of vertices from distinct parts is contained in one hyperedge exactly. An immediate corollary of this result is that $ex_r^{lin}(n,K_{r+1}^+)= \frac{n^2}{r^2}$ for sufficiently large $n$ and $r|n$, where $ex_r^{lin}(n,K_{r+1}^+)$ is the maximum number of edges of an $n$-vertex $K_{r+1}^+$-free linear $r$-uniform hypergraph, i.e., the linear Turán number of $K_{r+1}^+$.

Decomposition is integral to most image processing algorithms and often required in texture analysis. We present a new approach using a recent 2-dimensional exponential analysis technique. Exponential analysis offers the advantage of sparsity in the model and continuity in the parameters. This results in a much more compact representation of textures when compared to traditional Fourier or wavelet transform techniques. Our experiments include synthetic as well as real texture images from standard benchmark datasets. The results outperform FFT in representing texture patterns with significantly fewer terms while retaining RMSE values after reconstruction. The underlying periodic complex exponential model works best for texture patterns that are homogeneous. We demonstrate the usefulness of the method in two common vision processing application examples, namely texture classification and defect detection.

MOTIVATION: Systematic identification of molecular targets among known drugs plays an essential role in drug repurposing and understanding of their unexpected side effects. Computational approaches for prediction of drug-target interactions (DTIs) are highly desired in comparison to traditional experimental assays. Furthermore, recent advances of multiomics technologies and systems biology approaches have generated large-scale heterogeneous, biological networks, which offer unexpected opportunities for network-based identification of new molecular targets among known drugs. RESULTS: In this study, we present a network-based computational framework, termed AOPEDF, an arbitrary-order proximity embedded deep forest approach, for prediction of DTIs. AOPEDF learns a low-dimensional vector representation of features that preserve arbitrary-order proximity from a highly integrated, heterogeneous biological network connecting drugs, targets (proteins) and diseases. In total, we construct a heterogeneous network by uniquely integrating 15 networks covering chemical, genomic, phenotypic and network profiles among drugs, proteins/targets and diseases. Then, we build a cascade deep forest classifier to infer new DTIs. Via systematic performance evaluation, AOPEDF achieves high accuracy in identifying molecular targets among known drugs on two external validation sets collected from DrugCentral [area under the receiver operating characteristic curve (AUROC) = 0.868] and ChEMBL (AUROC = 0.768) databases, outperforming several state-of-the-art methods. In a case study, we showcase that multiple molecular targets predicted by AOPEDF are associated with mechanism-of-action of substance abuse disorder for several marketed drugs (such as aripiprazole, risperidone and haloperidol). AVAILABILITY AND IMPLEMENTATION: Source code and data can be downloaded from https://github.com/ChengF-Lab/AOPEDF.