A subgaussian embedding theorem

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

1 Strong invertibility of subgaussian matrices In the last lecture, we derived an estimate for the smallest singular value of a subgaussian random matrix; Theorem 1. Let A be a n × n subgaussian matrix. Then, for any > 0, P(s n (A) < ε √ n) ≤ cε + Cn −1 2 (1) In particular, this implies s n (A) ∼ 1 √ n with high probability. However, (1) cannot show P(s n (A) < ε √ n) → 0 as → 0 because of the ...

Subgaussian random variables, Hoeffding’s inequality, and Cramér’s large deviation theorem

Embedding normed linear spaces into $C(X)$

‎It is well known that every (real or complex) normed linear space $L$ is isometrically embeddable into $C(X)$ for some compact Hausdorff space $X$‎. ‎Here $X$ is the closed unit ball of $L^*$ (the set of all continuous scalar-valued linear mappings on $L$) endowed with the weak$^*$ topology‎, ‎which is compact by the Banach--Alaoglu theorem‎. ‎We prove that the compact Hausdorff space $X$ can ...

Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem

An important theorem of Banaszczyk (Random Structures & Algorithms ‘98) states that for any sequence of vectors of `2 norm at most 1/5 and any convex body K of Gaussian measure 1/2 in R, there exists a signed combination of these vectors which lands inside K. A major open problem is to devise a constructive version of Banaszczyk’s vector balancing theorem, i.e. to find an efficient algorithm wh...

Dimensionality reduction with subgaussian matrices: a unified theory

We present a theory for Euclidean dimensionality reduction with subgaussian matrices which unifies several restricted isometry property and Johnson-Lindenstrauss type results obtained earlier for specific data sets. In particular, we recover and, in several cases, improve results for sets of sparse and structured sparse vectors, low-rank matrices and tensors, and smooth manifolds. In addition, ...