A simple proof of Kashin’s decomposition theorem∗†

Compressive Sensing techniques are used in a short proof of Kashin’s decomposition theorem generalized to `p-spaces for p ≤ 1. The proof is based on the observation that the null-space of a properly-sized matrix with restricted isometry property is almost Euclidean when endowed with the `p-quasinorm. Kashin’s decomposition theorem states that, for any integer m ≥ 1, `2m 1 is the orthogonal sum of two almost Euclidean m-dimensional spaces. This was first established by Kashin in [4] before Szarek gave a short proof in [6]. This note presents another short proof inspired by the theory of Compressive Sensing [2, 3] — the argument is in fact very similar to the one of [5]. Although we settle for large m and give up explicit constants, we are able to extend the result to `p-spaces with p ≤ 1 using this method. Theorem 1. For any integer m sufficiently large, the space R2m contains two orthogonal subspaces E and E⊥ of dimension m such that, given any 0 < p ≤ 1, there exist constants αp, βp > 0 making the inequalities (1) αpm‖x‖2 ≤ ‖x‖p ≤ βpm‖x‖2 hold for all x ∈ E and all x ∈ E⊥. Proof. We only concentrate on the leftmost inequality in (1), since the rightmost inequality holds with βp := 21/p−1/2 regardless of the subspace E of R2m. Let G be an m × m matrix whose entries are independent Gaussian random variables with mean zero and variance 1/m. We define two full-rank m× (2m) matrices by A := [ I G ] , B := [ G> − I ] , and we consider the m-dimensional space E := kerA. We readily observe that E⊥ = kerB using dimension arguments and the fact that E⊥ = imA> ⊆ kerB because BA> = 0. We are going to show that, given any 0 < ε < 1 and any vector x ∈ R2m, the matrices M = A and M = B satisfy the concentration inequality (2) Pr (∣∣‖Mx‖22 − ‖x‖22∣∣ ≥ ε‖x‖22) ≤ 2 exp (− c(ε)m), where c(ε) > 0 is a constant depending only on ε. Fixing 0 < δ < 1, say δ := 3/5, it is known in Compressive Sensing that this implies the existence of constants c1, c2 > 0 such that both M = A and M = B satisfy the sth order restricted isometry property (3) (1− δ)‖x‖2 ≤ ‖Mx‖2 ≤ (1 + δ)‖x‖2 for all s-sparse x ∈ R ∗2000 Mathematics Subject Classification: 46B20, 46B09 †