Algorithms for Learning Sparse Additive Models with Interactions in High Dimensions

A function $f: \mathbb{R}^d \rightarrow \mathbb{R}$ is a Sparse Additive Model (SPAM), if it is of the form $f(\mathbf{x}) = \sum_{l \in \mathcal{S}}\phi_{l}(x_l)$ where $\mathcal{S} \subset [d]$, $|\mathcal{S}| \ll d$. Assuming $\phi$'s, $\mathcal{S}$ to be unknown, there exists extensive work for estimating $f$ from its samples. In this work, we consider a generalized version of SPAMs, that also allows for the presence of a sparse number of second order interaction terms. For some $\mathcal{S}_1 \subset [d], \mathcal{S}_2 \subset {[d] \choose 2}$, with $|\mathcal{S}_1| \ll d, |\mathcal{S}_2| \ll d^2$, the function $f$ is now assumed to be of the form: $\sum_{p \in \mathcal{S}_1}\phi_{p} (x_p) + \sum_{(l,l^{\prime}) \in \mathcal{S}_2}\phi_{(l,l^{\prime})} (x_l,x_{l^{\prime}})$. Assuming we have the freedom to query $f$ anywhere in its domain, we derive efficient algorithms that provably recover $\mathcal{S}_1,\mathcal{S}_2$ with finite sample bounds. Our analysis covers the noiseless setting where exact samples of $f$ are obtained, and also extends to the noisy setting where the queries are corrupted with noise. For the noisy setting in particular, we consider two noise models namely: i.i.d Gaussian noise and arbitrary but bounded noise. Our main methods for identification of $\mathcal{S}_2$ essentially rely on estimation of sparse Hessian matrices, for which we provide two novel compressed sensing based schemes. Once $\mathcal{S}_1, \mathcal{S}_2$ are known, we show how the individual components $\phi_p$, $\phi_{(l,l^{\prime})}$ can be estimated via additional queries of $f$, with uniform error bounds. Lastly, we provide simulation results on synthetic data that validate our theoretical findings.Comment: To appear in Information and Inference: A Journal of the IMA. Made following changes after review process: (a) Corrected typos throughout the text. (b) Corrected choice of sampling distribution in Section 5, see eqs. (5.2), (5.3). (c) More detailed comparison with existing work in Section 8. (d) Added Section B in appendix on roots of cubic equatio

Fast design optimization of UWB antenna with WLAN Band-Notch

In this paper, a methodology for rapid design optimization of an ultra-wideband ( UWB) monopole antenna with a lower WLAN band-notch is presented. The band-notch is realized using an open loop resonator implemented in the radiation patch of the antenna. Design optimization is a two stage process, with the first stage focused on the design of the antenna itself, and the second stage aiming at identification of the appropriate dimensions of the resonator with the purpose of allocating the band-notch in the desired frequency range. Both optimization stages are realized using surrogate-based optimization involving variable-fidelity electromagnetic ( EM) simulation models as well as an additive response correction ( first stage), and sequential approximate optimization ( second stage). The final antenna design is obtained at the CPU cost corresponding to only 23 high-fidelity EM antenna simulations