766 research outputs found
Pose-graph SLAM sparsification using factor descent
Since state of the art simultaneous localization and mapping (SLAM) algorithms are not constant time, it is often necessary to reduce the problem size while keeping as much of the original graph’s information content. In graph SLAM, the problem is reduced by removing nodes and rearranging factors. This is normally faced locally: after selecting a node to be removed, its Markov blanket sub-graph is isolated, the node is marginalized and its dense result is sparsified. The aim of sparsification is to compute an approximation of the dense and non-relinearizable result of node marginalization with a new set of factors. Sparsification consists on two processes: building the topology of new factors, and finding the optimal parameters that best approximate the original dense distribution. This best approximation can be obtained through minimization of the Kullback-Liebler divergence between the two distributions. Using simple topologies such as Chow-Liu trees, there is a closed form for the optimal solution. However, a tree is oftentimes too sparse and produces bad distribution approximations. On the contrary, more populated topologies require nonlinear iterative optimization. In the present paper, the particularities of pose-graph SLAM are exploited for designing new informative topologies and for applying the novel factor descent iterative optimization method for sparsification. Several experiments are provided comparing the proposed topology methods and factor descent optimization with state-of-the-art methods in synthetic and real datasets with regards to approximation accuracy and computational cost.Peer ReviewedPostprint (author's final draft
Analyzing sparse dictionaries for online learning with kernels
Many signal processing and machine learning methods share essentially the
same linear-in-the-parameter model, with as many parameters as available
samples as in kernel-based machines. Sparse approximation is essential in many
disciplines, with new challenges emerging in online learning with kernels. To
this end, several sparsity measures have been proposed in the literature to
quantify sparse dictionaries and constructing relevant ones, the most prolific
ones being the distance, the approximation, the coherence and the Babel
measures. In this paper, we analyze sparse dictionaries based on these
measures. By conducting an eigenvalue analysis, we show that these sparsity
measures share many properties, including the linear independence condition and
inducing a well-posed optimization problem. Furthermore, we prove that there
exists a quasi-isometry between the parameter (i.e., dual) space and the
dictionary's induced feature space.Comment: 10 page
Approximation errors of online sparsification criteria
Many machine learning frameworks, such as resource-allocating networks,
kernel-based methods, Gaussian processes, and radial-basis-function networks,
require a sparsification scheme in order to address the online learning
paradigm. For this purpose, several online sparsification criteria have been
proposed to restrict the model definition on a subset of samples. The most
known criterion is the (linear) approximation criterion, which discards any
sample that can be well represented by the already contributing samples, an
operation with excessive computational complexity. Several computationally
efficient sparsification criteria have been introduced in the literature, such
as the distance, the coherence and the Babel criteria. In this paper, we
provide a framework that connects these sparsification criteria to the issue of
approximating samples, by deriving theoretical bounds on the approximation
errors. Moreover, we investigate the error of approximating any feature, by
proposing upper-bounds on the approximation error for each of the
aforementioned sparsification criteria. Two classes of features are described
in detail, the empirical mean and the principal axes in the kernel principal
component analysis.Comment: 10 page
Sampling Random Spanning Trees Faster than Matrix Multiplication
We present an algorithm that, with high probability, generates a random
spanning tree from an edge-weighted undirected graph in
time (The notation hides
factors). The tree is sampled from a distribution
where the probability of each tree is proportional to the product of its edge
weights. This improves upon the previous best algorithm due to Colbourn et al.
that runs in matrix multiplication time, . For the special case of
unweighted graphs, this improves upon the best previously known running time of
for (Colbourn
et al. '96, Kelner-Madry '09, Madry et al. '15).
The effective resistance metric is essential to our algorithm, as in the work
of Madry et al., but we eschew determinant-based and random walk-based
techniques used by previous algorithms. Instead, our algorithm is based on
Gaussian elimination, and the fact that effective resistance is preserved in
the graph resulting from eliminating a subset of vertices (called a Schur
complement). As part of our algorithm, we show how to compute
-approximate effective resistances for a set of vertex pairs via
approximate Schur complements in time,
without using the Johnson-Lindenstrauss lemma which requires time. We
combine this approximation procedure with an error correction procedure for
handing edges where our estimate isn't sufficiently accurate
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