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## Section: New Results

### Axis 2: Learning general sparse additive models from point queries in high dimensions

Participant: Hemant Tyagi.

We consider the problem of learning a $d$-variate function $f$ defined on the cube ${\left[-1,1\right]}^{d}\subset {R}^{d}$, where the algorithm is assumed to have black box access to samples of $f$ within this domain. Denote ${S}_{r};r=1,\cdots ,{r}_{0}$ to be sets consisting of unknown $r$-wise interactions amongst the coordinate variables. We then focus on the setting where $f$ has an additive structure, i.e., it can be represented as

$f=\sum _{j\in {§}_{1}}{\phi }_{j}+\sum _{j\in {S}_{2}}{\phi }_{j}+\cdots +\sum _{j\in {S}_{{r}_{0}}}{\phi }_{j},$

where each ${\phi }_{j}$; $j\in {S}_{r}$ is at most $r$-variate for $1\le r\le {r}_{0}$. We derive randomized algorithms that query $f$ at carefully constructed set of points, and exactly recover each ${S}_{r}$ with high probability. In contrary to the previous work, our analysis does not rely on numerical approximation of derivatives by finite order differences.

This is joint work with Jan Vybiral (Czech Technical University, Prague) and was mostly done while Hemant Tyagi was affiliated to the Alan Turing Institute. It has now been published in an international journal [30].