General linear model concerns the statistical problem of estimating a vector x from the vector of measurements y=Ax+e, where A is a given design matrix whose rows correspond to individual measurements and e represents errors in measurements. Popular iterative algorithms, e.g. message passing, used in this context requires a "warm start", meaning they must be initialized better than a random guess. In practice, it is often the case that a spectral estimator, i.e. the principal component of a certain matrix built from Y, serves as such an initialization. In this talk, we discuss the theoretical aspect of the spectral estimator and present a theorem on its performance guarantee. Our result gives a threshold for the sample complexity, that is, how many measurements are needed for a warm start to be obtainable, as well as a concrete estimator. If time permits, we will also discuss our method based on (vector-valued) Approximate Message Passing.
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