55,711 research outputs found
Paved with Good Intentions: Analysis of a Randomized Block Kaczmarz Method
The block Kaczmarz method is an iterative scheme for solving overdetermined
least-squares problems. At each step, the algorithm projects the current
iterate onto the solution space of a subset of the constraints. This paper
describes a block Kaczmarz algorithm that uses a randomized control scheme to
choose the subset at each step. This algorithm is the first block Kaczmarz
method with an (expected) linear rate of convergence that can be expressed in
terms of the geometric properties of the matrix and its submatrices. The
analysis reveals that the algorithm is most effective when it is given a good
row paving of the matrix, a partition of the rows into well-conditioned blocks.
The operator theory literature provides detailed information about the
existence and construction of good row pavings. Together, these results yield
an efficient block Kaczmarz scheme that applies to many overdetermined
least-squares problem
Block-diagonal covariance selection for high-dimensional Gaussian graphical models
Gaussian graphical models are widely utilized to infer and visualize networks
of dependencies between continuous variables. However, inferring the graph is
difficult when the sample size is small compared to the number of variables. To
reduce the number of parameters to estimate in the model, we propose a
non-asymptotic model selection procedure supported by strong theoretical
guarantees based on an oracle inequality and a minimax lower bound. The
covariance matrix of the model is approximated by a block-diagonal matrix. The
structure of this matrix is detected by thresholding the sample covariance
matrix, where the threshold is selected using the slope heuristic. Based on the
block-diagonal structure of the covariance matrix, the estimation problem is
divided into several independent problems: subsequently, the network of
dependencies between variables is inferred using the graphical lasso algorithm
in each block. The performance of the procedure is illustrated on simulated
data. An application to a real gene expression dataset with a limited sample
size is also presented: the dimension reduction allows attention to be
objectively focused on interactions among smaller subsets of genes, leading to
a more parsimonious and interpretable modular network.Comment: Accepted in JAS
Attention and Anticipation in Fast Visual-Inertial Navigation
We study a Visual-Inertial Navigation (VIN) problem in which a robot needs to
estimate its state using an on-board camera and an inertial sensor, without any
prior knowledge of the external environment. We consider the case in which the
robot can allocate limited resources to VIN, due to tight computational
constraints. Therefore, we answer the following question: under limited
resources, what are the most relevant visual cues to maximize the performance
of visual-inertial navigation? Our approach has four key ingredients. First, it
is task-driven, in that the selection of the visual cues is guided by a metric
quantifying the VIN performance. Second, it exploits the notion of
anticipation, since it uses a simplified model for forward-simulation of robot
dynamics, predicting the utility of a set of visual cues over a future time
horizon. Third, it is efficient and easy to implement, since it leads to a
greedy algorithm for the selection of the most relevant visual cues. Fourth, it
provides formal performance guarantees: we leverage submodularity to prove that
the greedy selection cannot be far from the optimal (combinatorial) selection.
Simulations and real experiments on agile drones show that our approach ensures
state-of-the-art VIN performance while maintaining a lean processing time. In
the easy scenarios, our approach outperforms appearance-based feature selection
in terms of localization errors. In the most challenging scenarios, it enables
accurate visual-inertial navigation while appearance-based feature selection
fails to track robot's motion during aggressive maneuvers.Comment: 20 pages, 7 figures, 2 table
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