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Register a new userHigh-dimensional $p$-norms
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Added by
Gerard Biau
Publication date
2013
fields
Mathematical Statistics
and research's language is
English
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Let $bX=(X_1, hdots, X_d)$ be a $mathbb R^d$-valued random vector with i.i.d. components, and let $VertbXVert_p= (sum_{j=1}^d|X_j|^p)^{1/p}$ be its $p$-norm, for $p>0$. The impact of letting $d$ go to infinity on $VertbXVert_p$ has surprising consequences, which may dramatically affect high-dimensional data processing. This effect is usually referred to as the {it distance concentration phenomenon} in the computational learning literature. Despite a growing interest in this important question, previous work has essentially characterized the problem in terms of numerical experiments and incomplete mathematical statements. In the present paper, we solidify some of the arguments which previously appeared in the literature and offer new insights into the phenomenon.
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Testing for white noise is a classical yet important problem in statistics, especially for diagnostic checks in time series modeling and linear regression. For high-dimensional time series in the sense that the dimension $p$ is large in relation to the sample size $T$, the popular omnibus tests including the multivariate Hosking and Li-McLeod tests are extremely conservative, leading to substantial power loss. To develop more relevant tests for high-dimensional cases, we propose a portmanteau-type test statistic which is the sum of squared singular values of the first $q$ lagged sample autocovariance matrices. It, therefore, encapsulates all the serial correlations (upto the time lag $q$) within and across all component series. Using the tools from random matrix theory and assuming both $p$ and $T$ diverge to infinity, we derive the asymptotic normality of the test statistic under both the null and a specific VMA(1) alternative hypothesis. As the actual implementation of the test requires the knowledge of three characteristic constants of the population cross-sectional covariance matrix and the value of the fourth moment of the standardized innovations, non trivial estimations are proposed for these parameters and their integration leads to a practically usable test. Extensive simulation confirms the excellent finite-sample performance of the new test with accurate size and satisfactory power for a large range of finite $(p,T)$ combinations, therefore ensuring wide applicability in practice. In particular, the new tests are consistently superior to the traditional Hosking and Li-McLeod tests.
Multidimensional Scaling (MDS) is a classical technique for embedding data in low dimensions, still in widespread use today. Originally introduced in the 1950s, MDS was not designed with high-dimensional data in mind; while it remains popular with data analysis practitioners, no doubt it should be adapted to the high-dimensional data regime. In this paper we study MDS under modern setting, and specifically, high dimensions and ambient measurement noise. We show that, as the ambient noise level increase, MDS suffers a sharp breakdown that depends on the data dimension and noise level, and derive an explicit formula for this breakdown point in the case of white noise. We then introduce MDS+, an extremely simple variant of MDS, which applies a carefully derived shrinkage nonlinearity to the eigenvalues of the MDS similarity matrix. Under a loss function measuring the embedding quality, MDS+ is the unique asymptotically optimal shrinkage function. We prove that MDS+ offers improved embedding, sometimes significantly so, compared with classical MDS. Furthermore, MDS+ does not require external estimates of the embedding dimension (a famous difficulty in classical MDS), as it calculates the optimal dimension into which the data should be embedded.
In this paper we discuss the estimation of a nonparametric component $f_1$ of a nonparametric additive model $Y=f_1(X_1) + ...+ f_q(X_q) + epsilon$. We allow the number $q$ of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating $f_1$ in the oracle model $Z= f_1(X_1) + epsilon$, for which the additive components $f_2,dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing estimator of $f_1$ and state finite-sample bounds for the difference between our estimator and some smoothing estimators $hat f_1^{text{(oracle)}}$ in the oracle model. In an asymptotic setting these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimators; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of $f_2,dots,f_q$ has no effect on estimation accuracy. Our first step is to estimate $f_1$ with an undersmoothed estimator based on near-orthogonal projections with a group Lasso bias correction. We then construct pseudo responses $hat Y$ by evaluating a debiased modification of our undersmoothed estimator of $f_1$ at the design points. In the second step the smoothing method of the oracle estimator $hat f_1^{text{(oracle)}}$ is applied to a nonparametric regression problem with responses $hat Y$ and covariates $X_1$. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We present simulation results demonstrating close-to-oracle performance of our estimator in practical applications.
Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, this paper establishes a new central limit theorem (CLT) for a linear spectral statistic (LSS) of high dimensional sample correlation matrices for the case where the dimension p and the sample size $n$ are comparable. This result is of independent interest in large dimensional random matrix theory. Meanwhile, we apply the linear spectral statistic to an independence test for $p$ random variables, and then an equivalence test for p factor loadings and $n$ factors in a factor model. The finite sample performance of the proposed test shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from different cities in China is also conducted.
We focus on the high dimensional linear regression $Ysimmathcal{N}(Xbeta^{*},sigma^{2}I_{n})$, where $beta^{*}inmathds{R}^{p}$ is the parameter of interest. In this setting, several estimators such as the LASSO and the Dantzig Selector are known to satisfy interesting properties whenever the vector $beta^{*}$ is sparse. Interestingly both of the LASSO and the Dantzig Selector can be seen as orthogonal projections of 0 into $mathcal{DC}(s)={betainmathds{R}^{p},|X(Y-Xbeta)|_{infty}leq s}$ - using an $ell_{1}$ distance for the Dantzig Selector and $ell_{2}$ for the LASSO. For a well chosen $s>0$, this set is actually a confidence region for $beta^{*}$. In this paper, we investigate the properties of estimators defined as projections on $mathcal{DC}(s)$ using general distances. We prove that the obtained estimators satisfy oracle properties close to the one of the LASSO and Dantzig Selector. On top of that, it turns out that these estimators can be tuned to exploit a different sparsity or/and slightly different estimation objectives.
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