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We tackle the problem of nonparametric variable selection with a focus ondiscovering interactions between variables. With $p$ variables there are$O(p^s)$ possible order-$s$ interactions making exhaustive search infeasible.It is nonetheless possible to identify the variables involved in interactionswith only linear computation cost, $O(p)$. The trick is to maximize a class ofparametrized nonparametric dependence measures which we call metric learningobjectives; the landscape of these nonconvex objective functions is sensitiveto interactions but the objectives themselves do not explicitly modelinteractions. Three properties make metric learning objectives highlyattractive:  (a) The stationary points of the objective are automatically sparse (i.e.performs selection) -- no explicit $\ell_1$ penalization is needed.  (b) All stationary points of the objective exclude noise variables with highprobability.  (c) Guaranteed recovery of all signal variables without needing to reach theobjective's global maxima or special stationary points.  The second and third properties mean that all our theoretical results applyin the practical case where one uses gradient ascent to maximize the metriclearning objective. While not all metric learning objectives enjoy goodstatistical power, we design an objective based on $\ell_1$ kernels that doesexhibit favorable power: it recovers (i) main effects with $n \sim \log p$samples, (ii) hierarchical interactions with $n \sim \log p$ samples and (iii)order-$s$ pure interactions with $n \sim p^{2(s-1)}\log p$ samples.

A Self-Penalizing Objective Function for Scalable Interaction Detection