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.
This work focuses on the space-time reduced-order modeling (ROM) method for solving large-scale uncertainty quantification (UQ) problems with multiple random coefficients. In contrast with the traditional space ROM approach, which performs dimension reduction in the spatial dimension, the space-time ROM approach performs dimension reduction on both the spatial and temporal domains, and thus enables accurate approximate solutions at a low cost. We incorporate the space-time ROM strategy with various classical stochastic UQ propagation methods such as stochastic Galerkin and Monte Carlo. Numerical results demonstrate that our methodology has significant computational advantages compared to state-of-the-art ROM approaches. By testing the approximation errors, we show that there is no obvious loss of simulation accuracy for space-time ROM given its high computational efficiency.
We study the best low-rank Tucker decomposition of symmetric tensors, advocating a straightforward projected gradient descent (PGD) method for its computation. The main application of interest is in decomposing higher-order multivariate moments, which are symmetric tensors. We develop scalable adaptations of the basic PGD method and higher-order eigenvalue decomposition (HOEVD) to decompose sample moment tensors. With the help of implicit and streaming techniques, we evade the overhead cost of building and storing the moment tensor. Such reductions make computing the Tucker decomposition realizable for large data instances in high dimensions. Numerical experiments demonstrate the efficiency of the algorithms and the applicability of moment tensor decompositions to real-world datasets. Last, we study the convergence on the Grassmannian manifold, and prove that the update sequence derived by the PGD solver achieves first and second-order criticality.
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