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Zhiyi Chi

  1. Effects of statistical dependence on multiple testing under a hidden Markov model.

    Thu, 03/10/2011 - 16:01 — SomNambul
    Authors: Zhiyi Chi
    Subjects: Statistics
    Abstract

    The performance of multiple hypothesis testing is known to be affected by the
    statistical dependence among random variables involved. The mechanisms
    responsible for this, however, are not well understood. We study the effects of
    the dependence structure of a finite state hidden Markov model (HMM) on the
    likelihood ratios critical for optimal multiple testing on the hidden states.
    Various convergence results are obtained for the likelihood ratios as the
    observations of the HMM form an increasing long chain.

  2. Stochastic Lipschitz continuity for high dimensional Lasso with multiple linear covariate structures or hidden linear covariates.

    Mon, 11/08/2010 - 16:01 — SomNambul
    Authors: Zhiyi Chi
    Subjects: Statistics
    Abstract

    Two extensions of generalized linear models are considered. In the first one,
    response variables depend on multiple linear combinations of covariates. In the
    second one, only response variables are observed while the linear covariates
    are missing. We derive stochastic Lipschitz continuity results for the loss
    functions involved in the regression problems and apply them to get bounds on
    estimation error for Lasso. Multivariate comparison results on Rademacher
    complexity are obtained as tools to establish the stochastic Lipschitz
    continuity results.

  3. A local stochastic Lipschitz condition with application to Lasso for high dimensional generalized linear models.

    Tue, 09/07/2010 - 15:02 — SomNambul
    Authors: Zhiyi Chi
    Subjects: Statistics
    Abstract

    For regularized estimation, the upper tail behavior of the random Lipschitz
    coefficient associated with empirical loss functions is known to play an
    important role in the error bound of Lasso for high dimensional generalized
    linear models. The upper tail behavior is known for linear models but much less
    so for nonlinear models. We establish exponential type inequalities for the
    upper tail of the coefficient and illustrate an application of the results to
    Lasso likelihood estimation for high dimensional generalized linear models.

  4. On $\ell_1$-regularized estimation for nonlinear models that have sparse underlying linear structures.

    Thu, 11/26/2009 - 16:01 — SomNambul
    Authors: Zhiyi Chi
    Subjects: Statistics
    Abstract

    In a recent work (arXiv:0910.2517), for nonlinear models with sparse
    underlying linear structures, we studied the error bounds of
    $\ell_0$-regularized estimation. In this note, we show that
    $\ell_1$-regularized estimation in some important cases can achieve the same
    order of error bounds as those in the aforementioned work.

  5. $L_0$ regularized estimation for nonlinear models that have sparse underlying linear structures.

    Fri, 10/16/2009 - 04:10 — SomNambul
    Authors: Zhiyi Chi
    Subjects: Statistics
    Abstract

    We study the estimation of $\beta$ for the nonlinear model $y =
    f(X\sp{\top}\beta) + \epsilon$ when $f$ is a nonlinear transformation that is
    known, $\beta$ has sparse nonzero coordinates, and the number of observations
    can be much smaller than that of parameters ($n\ll p$). We show that in order
    to bound the $L_2$ error of the $L_0$ regularized estimator $\hat\beta$, i.e.,
    $\|\hat\beta - \beta\|_2$, it is sufficient to establish two conditions.

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