In this paper, we give the first subexponential (and in fact quasi-polynomial time) reconstruction algorithm for depth 3 circuits of top fan-in 3 ($\Sigma\Pi\Sigma(3)$ circuits) over the fields $\mathbb{R}$ and $\mathbb C$. Concretely, we show that given blackbox access to an $n$-variate polynomial $f$ computed by a $\Sigma\Pi\Sigma(3)$ circuit of size $s$, there is a randomized algorithm that runs in time quasi-poly($n,s$) and outputs a generalized $\Sigma\Pi\Sigma(3)$ circuit computing $f$. The size $s$ includes the bit complexity of coefficients appearing in the circuit.

Depth 3 circuits of constant fan-in ($\Sigma\Pi\Sigma(k)$ circuits) and closely related models have been extensively studied in the context of polynomial identity testing (PIT). The study of PIT for these models led to an understanding of the structure of identically zero $\Sigma\Pi\Sigma(3)$ circuits and $\Sigma\Pi\Sigma(k)$ circuits using some very elegant connections to discrete geometry, specifically the Sylvester-Gallai Theorem, and colorful and high dimensional variants of them. Despite a lot of progress on PIT for $\Sigma\Pi\Sigma(k)$ circuits and more recently on PIT for depth 4 circuits of bounded top and bottom fan-in, reconstruction algorithms for $\Sigma\Pi\Sigma(k)$ circuits has proven to be extremely challenging.

In this paper, we build upon the structural results for identically zero $\Sigma\Pi\Sigma(3)$ circuits that bound their rank, and prove stronger structural properties of $\Sigma\Pi\Sigma(3)$ circuits (again using connections to discrete geometry). One such result is a bound on the number of codimension 3 subspaces on which a polynomial computed by an $\Sigma\Pi\Sigma(3)$ circuit can vanish on. Armed with the new structural results, we provide the first reconstruction algorithms for $\Sigma\Pi\Sigma(3)$ circuits over $\mathbb R$ and $\mathbb C$.

Our work extends the work of [Sinha, CCC 2016] who provided a reconstruction algorithm for $\Sigma\Pi\Sigma(2)$ circuits over $\mathbb R$ and $\mathbb C$ as well as the works of [Shpilka, STOC 2007] who provided a reconstruction algorithms for $\Sigma\Pi\Sigma(2)$ circuits in the setting of small finite fields, and [Karnin-Shpilka, CCC 2009] who provided reconstruction algorithms for $\Sigma\Pi\Sigma(k)$ circuits in the setting of small finite fields.