84 research outputs found
Structured Random Matrices
Random matrix theory is a well-developed area of probability theory that has
numerous connections with other areas of mathematics and its applications. Much
of the literature in this area is concerned with matrices that possess many
exact or approximate symmetries, such as matrices with i.i.d. entries, for
which precise analytic results and limit theorems are available. Much less well
understood are matrices that are endowed with an arbitrary structure, such as
sparse Wigner matrices or matrices whose entries possess a given variance
pattern. The challenge in investigating such structured random matrices is to
understand how the given structure of the matrix is reflected in its spectral
properties. This chapter reviews a number of recent results, methods, and open
problems in this direction, with a particular emphasis on sharp spectral norm
inequalities for Gaussian random matrices.Comment: 46 pages; to appear in IMA Volume "Discrete Structures: Analysis and
Applications" (Springer
Smooth analysis of the condition number and the least singular value
Let \a be a complex random variable with mean zero and bounded variance.
Let be the random matrix of size whose entries are iid copies of
\a and be a fixed matrix of the same size. The goal of this paper is to
give a general estimate for the condition number and least singular value of
the matrix , generalizing an earlier result of Spielman and Teng for
the case when \a is gaussian.
Our investigation reveals an interesting fact that the "core" matrix does
play a role on tail bounds for the least singular value of . This
does not occur in Spielman-Teng studies when \a is gaussian.
Consequently, our general estimate involves the norm .
In the special case when is relatively small, this estimate is nearly
optimal and extends or refines existing results.Comment: 20 pages. An erratum to the published version has been adde
Estimation in high dimensions: a geometric perspective
This tutorial provides an exposition of a flexible geometric framework for
high dimensional estimation problems with constraints. The tutorial develops
geometric intuition about high dimensional sets, justifies it with some results
of asymptotic convex geometry, and demonstrates connections between geometric
results and estimation problems. The theory is illustrated with applications to
sparse recovery, matrix completion, quantization, linear and logistic
regression and generalized linear models.Comment: 56 pages, 9 figures. Multiple minor change
User-friendly tail bounds for sums of random matrices
This paper presents new probability inequalities for sums of independent,
random, self-adjoint matrices. These results place simple and easily verifiable
hypotheses on the summands, and they deliver strong conclusions about the
large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for
the norm of a sum of random rectangular matrices follow as an immediate
corollary. The proof techniques also yield some information about matrix-valued
martingales.
In other words, this paper provides noncommutative generalizations of the
classical bounds associated with the names Azuma, Bennett, Bernstein, Chernoff,
Hoeffding, and McDiarmid. The matrix inequalities promise the same diversity of
application, ease of use, and strength of conclusion that have made the scalar
inequalities so valuable.Comment: Current paper is the version of record. The material on Freedman's
inequality has been moved to a separate note; other martingale bounds are
described in Caltech ACM Report 2011-0
Mental Health and School Functioning for Girls in the Child Welfare System : the Mediating Role of Future Orientation and School Engagement
This study investigated the association between mental health problems and academic and behavioral school functioning for adolescent girls in the child welfare system and determined whether school engagement and future orientation meditated the relationship. Participants were 231 girls aged between 12 and 19 who had been involved with the child welfare system. Results indicated that 39% of girls reported depressive symptoms in the clinical range and 54% reported posttraumatic symptoms in the clinical range. The most common school functioning problems reported were failing a class (41%) and physical fights with other students (35%). Participants reported a mean number of 1.7 school functioning problems. Higher levels of depression and PTSD were significantly associated with more school functioning problems. School engagement fully mediated the relationship between depression and school functioning and between PTSD and school functioning, both models controlling for age, race, and placement stability. Future orientation was not significantly associated with school functioning problems at the bivariate level. Findings suggest that school engagement is a potentially modifiable target for interventions aiming to ameliorate the negative influence of mental health problems on school functioning for adolescent girls with histories of abuse or neglect
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