We study the fundamental problem of PAC learning positive linear combinations of $k$
arbitrary ReLU activations on $\mathbb{R}^d$ with respect to the Gaussian distribution, in $L_2$
loss. We give the first fully-polynomial time algorithm for this learning task.
Specifically, our algorithm draws $\mathrm{poly}(d, k, 1/\epsilon)$ samples,
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In an earlier note (arXiv:2606.11090) I showed that the border determinantal complexity of the power sum x_1^d + ... + x_N^d is ?(Nd). The proof counted the simple solutions of an explicit polynomial system and compared that number with what a small determinant can produce. This left two questions. Does ... more >>>

The rational identity testing problem, RIT, asks whether an input \emph{rational circuit}, i.e.\ a noncommutative arithmetic circuit that has inversion gates, computes the zero element of the free skew field. For rational \emph{formulas} the problem is known to be in deterministic polynomial time \cite{GGOW20}, \cite{IQS18}. For rational circuits of ... more >>>