38 research outputs found
Provable Sparse Tensor Decomposition
We propose a novel sparse tensor decomposition method, namely Tensor
Truncated Power (TTP) method, that incorporates variable selection into the
estimation of decomposition components. The sparsity is achieved via an
efficient truncation step embedded in the tensor power iteration. Our method
applies to a broad family of high dimensional latent variable models, including
high dimensional Gaussian mixture and mixtures of sparse regressions. A
thorough theoretical investigation is further conducted. In particular, we show
that the final decomposition estimator is guaranteed to achieve a local
statistical rate, and further strengthen it to the global statistical rate by
introducing a proper initialization procedure. In high dimensional regimes, the
obtained statistical rate significantly improves those shown in the existing
non-sparse decomposition methods. The empirical advantages of TTP are confirmed
in extensive simulated results and two real applications of click-through rate
prediction and high-dimensional gene clustering.Comment: To Appear in JRSS-
Parameter Estimation in Gaussian Mixture Models with Malicious Noise, without Balanced Mixing Coefficients
We consider the problem of estimating means of two Gaussians in a 2-Gaussian
mixture, which is not balanced and is corrupted by noise of an arbitrary
distribution. We present a robust algorithm to estimate the parameters,
together with upper bounds on the numbers of samples required for the estimate
to be correct, where the bounds are parametrised by the dimension, ratio of the
mixing coefficients, a measure of the separation of the two Gaussians, related
to Mahalanobis distance, and a condition number of the covariance matrix. In
theory, this is the first sample-complexity result for imbalanced mixtures
corrupted by adversarial noise. In practice, our algorithm outperforms the
vanilla Expectation-Maximisation (EM) algorithm in terms of estimation error