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讲座简介：
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Abstract: The singular value decomposition (SVD) and principal component analysis (PCA) provide a key tool for unsupervised dimension reduction. The rapid growth in data size and dimensionality has urged the need for developing efficient large-scale SVD and PCA algorithms. In recent years, there are emerging new techniques in numerical linear algebra, called randomized algorithms or random sketching, for high dimensional and large scale problems. Randomized SVD based on one-time sketching has been studied, and its potential has been demonstrated for computing a low-rank SVD. Instead of exploring different single random sketching techniques, we propose a Monte Carlo type integrated SVD algorithm based on multiple random sketches. The proposed integration algorithm takes multiple random sketches and then integrates the results obtained from the multiple sketches. The integrated SVD can achieve higher accuracy and lower stochastic variations. For supervised approach, the sufficient dimension reduction (SDR) has been continuing an active research field. When estimating the central subspace (CS), inverse regression based SDR methods involve solving a generalized eigenvalue problem, which can be problematic under the large-p-small-n situation. To overcome the large-p-small-n problem in SDR, we combine the idea of statistical inference with random sketching to propose a new SDR method, called ``integrated random-partition SDR (iRP-SDR)''. Asymptotic properties as well as numerical examples will be provided.
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