  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
Johnson-Lindenstrauss (J-L) 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 sub-Gaussian 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 J-L, RIP, and sketching, to
illustrate the generality of the results.
