What is the direct connection between LWE and GapSVP? Learning with Errors Problem (LWE): Given a polynomial number of random noisy linear equations $b_i$ in the form of pairs

$$
 (a_i, \quad b_i = \langle s, a_i \rangle + e_i)
$$
where $a_i \in \mathbb{Z}^n_q$ are uniformly random, $e_i$ is sampled from some noise distribution such that $e_i \ll q$, and  $\langle s, a_i \rangle$ denotes the inner product of $s$ and $a_i$, find the secret vector $s \in \mathbb{Z}^n_q$.

This problem is supposed to rely on the hardness of GapSVP. In Regev's paper, he showed a complicated reduction from GapSVP. Is there a way to see the immediate connection between the two problems? What are $a_i$, $b_i$ and $\langle s, a_i \rangle$ with regards to a lattice?

Thanks!
Taking $n$ LWE samples, the associated vector $\mathbf{b} = (b_1, \dots, b_n)$ can be written as $A\mathbf{s}+\mathbf{e}$ with $A^T = (\mathbf{a}_1 | \dots | \mathbf{a}_n)$ and $\mathbf{e} = (e_1, \dots, e_n)$ a small noise vector. The vector $A\mathbf{s}$ could be interpreted as a lattice point in the lattice $\mathcal{L}(\mathbf{a}_1, \dots, \mathbf{a}_n)$, with unknown coefficients defined by the secret vector $\mathbf{s}$, and $\mathbf{e}$ is small so $\mathbf{b}$ is pretty close to this lattice point. This corresponds to a CVP/BDD-style problem, where you want to find the closest lattice vector to $\mathbf{b}$ in this lattice. 

Finally, Kannan's embedding technique shows how to translate CVP instances to (Gap-)SVP instances. Quoting Hanrot, Pujol, and Stehle, at the end of Section 4:



I'm not sure if this is also how Regev reduces GapSVP to LWE, but it's probably the most intuitive way to see how CVP-style problems and SVP-style problems are usually related.