In this paper, we give the first subexponential (in fact, quasi-polynomial time) reconstruction algorithm for depth-3 circuits of any constant top fan-in ($\Sigma\Pi\Sigma(k)$ circuits) over $\mathbb R$, $\mathbb C$, or any large characteristic finite field $\mathbb F$. More explicitly, we show that for any constant $k$, given black-box access to an $n$-variate polynomial $f$ computed by a $\Sigma\Pi\Sigma(k)$ circuit of size $s$, there is a randomized algorithm that runs in time quasi-poly($n,s$) and outputs a generalized $\Sigma\Pi\Sigma(k)$ circuit computing $f$. The size $s$ includes the bit complexity of coefficients appearing in the circuit: this is the max bit complexity if the field is $\mathbb R$ or $\mathbb C$, and $\log |\mathbb F|$ if the field is finite.

Depth-3 circuits of constant fan-in ($\Sigma\Pi\Sigma(k)$ circuits) and closely related models have been very well studied in the context of polynomial identity testing (PIT). In this paper, we build upon the structural results for identically zero $\Sigma\Pi\Sigma(k)$ circuits that were studied in the context of PIT. Using connections to discrete geometry, we prove new structural properties of \emph{vanishing spaces} of polynomials computed by such circuits.

Prior to our work, the only subexponential reconstruction algorithm for $\Sigma\Pi\Sigma(k)$ circuits is by [Karnin--Shpilka, CCC 2009]. However, the run time is quasipolynomial in $|\mathbb F|$, and hence this is only efficient over small finite fields. Over general (potentially exponentially large size) finite fields, efficient reconstruction algorithms were only known for $k=2$ ([Sinha, ITCS 2022]); and over $\mathbb R$ and $\mathbb C$, they were only known for $k=2$ ([Sinha, CCC 2016]) and $k=3$ ([Saraf--Shringi, CCC 2025]).