Random Tessellations, Restricted Isometric Embeddings, and One Bit Sensing

We obtain mproved bounds for one bit sensing. For instance, let $ K_s$ denote the set of $ s$-sparse unit vectors in the sphere $ \mathbb S ^{n}$ in dimension $ n+1$ with sparsity parameter $ 0 < s < n+1$ and assume that $ 0 < \delta < 1$. We show that for $ m \gtrsim \delta ^{-2} s \log \frac ns$, the one-bit map $$ x \mapsto \bigl[ {sgn} \langle x,g_j \rangle \bigr] _{j=1} ^{m}, $$ where $ g_j$ are iid gaussian vectors on $ \mathbb R ^{n+1}$, with high probability has $ \delta $-RIP from $ K_s$ into the $ m$-dimensional Hamming cube. These bounds match the bounds for the {linear} $ \delta $-RIP given by $ x \mapsto \frac 1m[\langle x,g_j \rangle ] _{j=1} ^{m} $, from the sparse vectors in $ \mathbb R ^{n}$ into $ \ell ^{1}$. In other words, the one bit and linear RIPs are equally effective. There are corresponding improvements for other one-bit properties, such as the sign-product RIP property.