Jose Bento

Jose Bento, Ralph Furmaniak, Surjyendu Ray

The solution path of the 1D fused lasso for an $n$-dimensional input is piecewise linear with $\mathcal{O}(n)$ segments (Hoefling et al. 2010 and Tibshirani et al 2011). However, existing proofs of this bound do not hold for the weighted fused lasso. At the same time, results for the generalized lasso, of which the weighted fused lasso is a special case, allow $Ω(3^n)$ segments (Mairal et al. 2012). In this paper, we prove that the number of segments in the solution path of the weighted fused lasso is $\mathcal{O}(n^2)$, and that, for some instances, it is $Ω(n^2)$. We also give a new, very simple, proof of the $\mathcal{O}(n)$ bound for the fused lasso.

Jose Bento, Nate Derbinsky, Javier Alonso-Mora et al.

We describe a novel approach for computing collision-free \emph{global} trajectories for $p$ agents with specified initial and final configurations, based on an improved version of the alternating direction method of multipliers (ADMM). Compared with existing methods, our approach is naturally parallelizable and allows for incorporating different cost functionals with only minor adjustments. We apply our method to classical challenging instances and observe that its computational requirements scale well with $p$ for several cost functionals. We also show that a specialization of our algorithm can be used for {\em local} motion planning by solving the problem of joint optimization in velocity space.

Jose Bento, Morteza Ibrahimi

Consider the problem of learning the drift coefficient of a $p$-dimensional stochastic differential equation from a sample path of length $T$. We assume that the drift is parametrized by a high-dimensional vector, and study the support recovery problem when both $p$ and $T$ can tend to infinity. In particular, we prove a general lower bound on the sample-complexity $T$ by using a characterization of mutual information as a time integral of conditional variance, due to Kadota, Zakai, and Ziv. For linear stochastic differential equations, the drift coefficient is parametrized by a $p\times p$ matrix which describes which degrees of freedom interact under the dynamics. In this case, we analyze a $\ell_1$-regularized least squares estimator and prove an upper bound on $T$ that nearly matches the lower bound on specific classes of sparse matrices.