
On complex Gaussian random fields, Gaussian quadratic forms and sample distance multivariance
The paper contains results in three areas: First we present a general es...
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Improved bounds for the RIP of Subsampled Circulant matrices
In this paper, we study the restricted isometry property of partial rand...
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The Tensor Quadratic Forms
We consider the following data perturbation model, where the covariates ...
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SubGaussian Matrices on Sets: Optimal Tail Dependence and Applications
Random linear mappings are widely used in modern signal processing, comp...
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On Learnability under General Stochastic Processes
Statistical learning theory under independent and identically distribute...
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Isotropic vectors over global fields
We present a complete suite of algorithms for finding isotropic vectors ...
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Simple Analysis of JohnsonLindenstrauss Transform under Neuroscience Constraints
The paper reanalyzes a version of the celebrated JohnsonLindenstrauss ...
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Random Quadratic Forms with Dependence: Applications to Restricted Isometry and Beyond
Several important families of computational and statistical results in machine learning and randomized algorithms rely on uniform bounds on quadratic forms of random vectors or matrices. Such results include the JohnsonLindenstrauss (JL) Lemma, the Restricted Isometry Property (RIP), randomized sketching algorithms, and approximate linear algebra. The existing results critically depend on statistical independence, e.g., independent entries for random vectors, independent rows for random matrices, etc., which prevent their usage in dependent or adaptive modeling settings. In this paper, we show that such independence is in fact not needed for such results which continue to hold under fairly general dependence structures. In particular, we present uniform bounds on random quadratic forms of stochastic processes which are conditionally independent and subGaussian given another (latent) process. Our setup allows general dependencies of the stochastic process on the history of the latent process and the latent process to be influenced by realizations of the stochastic process. The results are thus applicable to adaptive modeling settings and also allows for sequential design of random vectors and matrices. We also discuss stochastic process based forms of JL, RIP, and sketching, to illustrate the generality of the results.
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