Non-Asymptotic Theory of Random Matrices Lecture 18: Strong invertibility of subgaussian matrices and Small ball probability via arithmetic progression

Non-Asymptotic Theory of Random Matrices Lecture 19: Small Ball Probability via Sum-Sets

Non-Asymptotic Theory of Random Matrices Lecture 16: Invertibility of Gaussian Matrices and Compressible/Incompressible Vectors

We begin this lecture by asking why should an arbitrary n × n Gaussian matrix A be invertible? That is, does there exist a lower bound on the smallest singular value s n (A) = inf x∈S n−1 Ax 2 ≥ c √ n where c > 0 is an absolute constant. There are two reasons (or cases) which we will pursue in this lecture. 1. In Lecture 15 we saw that the invertibility of rectangular (i.e., non-square) Gaussia...

Non-Asymptotic Theory of Random Matrices

1 Definition The topic in this lecture is Subgaussian random variables. We start with the definition, and discuss some properties they hold. Definition 1 (Subgaussian random variables). A random variable X is subgaussian if ∃c, C such that P(|x| > t) ≤ Ce −ct 2 ∀t ≥ 0. (1) As the name suggests, the notion of subgaussian random variables is a generalization of Gaussian random variables. Both the...

Notes on Free Probability Theory

Lecture 1. Free Independence and Free Harmonic Analysis. 2 1.1. Probability spaces. 2 1.2. Non-commutative probability spaces. 3 1.3. Classical independence. 4 1.4. Free products of non-commutative probability spaces. 5 1.5. Free Fock space. 6 1.6. Free Central Limit Theorem. 8 1.7. Free Harmonic Analysis. 10 1.8. Further topics. 16 Lecture 2. Random Matrices and Free Probability. 17 2.1. Rando...

Non-Asymptotic Theory of Random Matrices

Let A be an n × n subgaussian matrix (entries are i.i.d. subgaussian r.v’s with variance 1). There are two reasons for the invertibility of A, depending on the nature of the unit vector on which A is acting – either compressible or incompressible. We recall that compressible vectors are those whose distance is at most some constant ρ from the set of (δn)-sparse vectors, and incompressible vecto...