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Tiefeng Jiang

Bio: Tiefeng Jiang is an academic researcher from University of Minnesota. The author has contributed to research in topics: Random matrix & Random variable. The author has an hindex of 24, co-authored 65 publications receiving 2307 citations. Previous affiliations of Tiefeng Jiang include Jilin University & Hong Kong University of Science and Technology.


Papers
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Book
01 Jan 2004
TL;DR: In this paper, the authors studied the limiting spectral measure of large symmetric random matrices of linear algebraic structure and proved the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γ H, γ M and γ T of unbounded support.
Abstract: We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {X k } of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {X ij } j>i of zero mean and unit variance, scaling the eigenvalues by √n we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γ H , γ M and γ T of unbounded support. The moments of γ H and γ T are the sum of volumes of solids related to Eulerian numbers, whereas γ M has a bounded smooth density given by the free convolution of the semicircle and normal densities. For symmetric Markov matrices generated by i.i.d. random variables {X iy } y>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at -m. If m = 0, and the fourth moment is finite, we prove that the spectral norm of M n scaled by √2n log n converges almost surely to 1.

240 citations

Journal ArticleDOI
TL;DR: In this paper, the limiting laws of the coherence of an n × p random matrix in the high-dimensional setting where p can be much larger than n were derived and applied to the construction of compressed sensing matrices.
Abstract: Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an n × p random matrix in the high-dimensional setting where p can be much larger than n. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high-dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.

173 citations

Journal ArticleDOI
TL;DR: In this article, the asymptotic distribution of Ln is derived by using the Chen-Stein Poisson approximation method for the non-Gaussian case, and the test statistic is chosen as Ln=max i≠j|ρij|.
Abstract: Let Xn=(xij) be an n by p data matrix, where the n rows form a random sample of size n from a certain p-dimensional population distribution. Let Rn=(ρij) be the p×p sample correlation matrix of Xn; that is, the entry ρij is the usual Pearson”s correlation coefficient between the ith column of Xn and jth column of Xn. For contemporary data both n and p are large. When the population is a multivariate normal we study the test that H0: the p variates of the population are uncorrelated. A test statistic is chosen as Ln=max i≠j|ρij|. The asymptotic distribution of Ln is derived by using the Chen–Stein Poisson approximation method. Similar results for the non-Gaussian case are also derived.

145 citations

Posted Content
TL;DR: The limiting laws of the coherence of an n× p random matrix in the high-dimensional setting where p can be much larger than n are derived and the law of large numbers and the limiting distribution are derived.
Abstract: Testing covariance structure is of significant interest in many areas of statistical analysis and construction of compressed sensing matrices is an important problem in signal processing. Motivated by these applications, we study in this paper the limiting laws of the coherence of an $n\times p$ random matrix in the high-dimensional setting where $p$ can be much larger than $n$. Both the law of large numbers and the limiting distribution are derived. We then consider testing the bandedness of the covariance matrix of a high dimensional Gaussian distribution which includes testing for independence as a special case. The limiting laws of the coherence of the data matrix play a critical role in the construction of the test. We also apply the asymptotic results to the construction of compressed sensing matrices.

139 citations

Journal Article
TL;DR: In this article, the authors studied the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in Rp as the number of points n → ∞, while the dimension p is either fixed or growing with n.
Abstract: This paper studies the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in Rp as the number of points n → ∞, while the dimension p is either fixed or growing with n. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that "all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and machine learning and connections with some open problems in physics and mathematics are also discussed.

129 citations


Cited by
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Proceedings ArticleDOI
22 Jan 2006
TL;DR: Some of the major results in random graphs and some of the more challenging open problems are reviewed, including those related to the WWW.
Abstract: We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

7,116 citations

Journal ArticleDOI
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

5,689 citations

Book
21 Dec 2009
TL;DR: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial) as mentioned in this paper.
Abstract: The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

1,289 citations

01 Jan 1996

1,282 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied a random Groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process and showed that shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.
Abstract: We study a certain random groeth model in two dimensions closely related to the one-dimensional totally asymmetric exclusion process. The results show that the shape fluctuations, appropriately scaled, converges in distribution to the Tracy-Widom largest eigenvalue distribution for the Gaussian Unitary Ensemble.

1,031 citations