The Kronecker product is an important matrix operation with a wide range of applications in signal processing, graph theory, quantum computing and deep learning. In this work, we introduce a generalization of the fast Johnson–Lindenstrauss projection for embedding vectors with Kronecker product structure, the Kronecker fast Johnson–Lindenstrauss transform (KFJLT). The KFJLT reduces the embedding cost by an exponential factor of the standard fast Johnson–Lindenstrauss transform’s cost when applied to vectors with Kronecker structure, by avoiding explicitly forming the full Kronecker products. We prove that this computational gain comes with only a small price in embedding power: consider a finite set of $p$ points in a tensor product of $d$ constituent Euclidean spaces $\bigotimes _{k=d}^{1}{\mathbb{R}}^{n_k}$, and let $N = \prod _{k=1}^{d}n_k$. With high probability, a random KFJLT matrix of dimension $m \times N$ embeds the set of points up to multiplicative distortion $(1\pm \varepsilon )$ provided $m \gtrsim \varepsilon ^{-2} \, \log ^{2d - 1} (p) \, \log N$. We conclude by describing a direct application of the KFJLT to the efficient solution of large-scale Kronecker-structured least squares problems for fitting the CP tensor decomposition.
Background: Triptonide (TN) was recently proved to have anti-tumor effects. The current study explored whether TN inhibited thyroid cancer and the possible underlying mechanism.Methods: MDA-T68 and BCPAP cells were treated by TN. Cell viability, migration and invasion rate were detected by MTT and Transwell. Protein expressions were determined by Western blot and mRNA expressions were detected by Real-time Quantitative PCR (qPCR).Results: TN at the concentration higher than 50 nmol/L inhibited cell viability, migration and invasion of MDA-T68 and BCPAP cells, and astrocyte elevated gene (AEG-1) expression, was decreased by TN at the concentration higher than 50 nmol/L. Furthermore, AEG-1 overexpression inhibited cell viability, migration and invasion capacity of MDA-T68 and BCPAP cells, while TN reduced AEG-1 expression, and weaken the effect of AEG-1 overexpression on cell viability, migration and invasion capacities. Moreover, TN depressed the increase of matrix metalloproteinase (MMP) 2, MMP9 and N-cadherin expressions caused by AEG-1 overexpression. Meanwhile, E-cadherin expression reduced by AEG-1 overexpression was increased by TN.Conclusions: TN could inhibit the metastasis potential of thyroid cancer cells through inhibiting the expression of AEG-1. Our findings reveal the mechanism of TN in the treatment of thyroid cancer, which should be further explored in the study of thyroid cancer.
In this work, we study a tensor-structured random sketching matrix to project a largescale convex optimization problem to a much lower-dimensional counterpart, which leads to huge memory and computation savings. We show that while maintaining the prediction error between a random estimator and the true solution with high probability, the dimension of the projected problem obtains optimal dependence in terms of the geometry of the constraint set. Moreover, the tensor structure and sparsity pattern of the structured random matrix yields extra computational advantage. Our analysis is based on probability chaining theory, which allows us to obtain an almost sharp estimate for the sketching dimension of convex optimization problems. Consequences of our main result are demonstrated in a few concrete examples, including unconstrained linear regressions and sparse recovery problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.