arXiv subjects

Jose Bento

Publications and source records attributed to Jose Bento.

5 recordsLinked to original sources

arXiv2018

On the Complexity of the Weighted Fused Lasso

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 $\Omega(3^n)$ 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 $\Omega(n^2)$. We also give a new, very simple, proof of the $\mathcal{O}(n)$ bound for the fused lasso.

cs.LG↗

arXiv2018

A Family of Tractable Graph Distances

Important data mining problems such as nearest-neighbor search and clustering admit theoretical guarantees when restricted to objects embedded in a metric space. Graphs are ubiquitous, and clustering and classification over graphs arise in diverse areas, including, e.g., image processing and social networks. Unfortunately, popular distance scores used in these applications, that scale over large graphs, are not metrics and thus come with no guarantees. Classic graph distances such as, e.g., the chemical and the CKS distance are arguably natural and intuitive, and are indeed also metrics, but they are intractable: as such, their computation does not scale to large graphs. We define a broad family of graph distances, that includes both the chemical and the CKS distance, and prove that these are all metrics. Crucially, we show that our family includes metrics that are tractable. Moreover, we extend these distances by incorporating auxiliary node attributes, which is important in practice, while maintaining both the metric property and tractability.

math.CO↗

arXiv2013

A message-passing algorithm for multi-agent trajectory planning

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.

cs.AI↗

arXiv2013

Support Recovery for the Drift Coefficient of High-Dimensional Diffusions

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.

cs.IT↗

arXiv2009

Which graphical models are difficult to learn?

We consider the problem of learning the structure of Ising models (pairwise binary Markov random fields) from i.i.d. samples. While several methods have been proposed to accomplish this task, their relative merits and limitations remain somewhat obscure. By analyzing a number of concrete examples, we show that low-complexity algorithms systematically fail when the Markov random field develops long-range correlations. More precisely, this phenomenon appears to be related to the Ising model phase transition (although it does not coincide with it).

stat.ML↗