3 research outputs found
Combining Method of Alternating Projections and Augmented Lagrangian for Task Constrained Trajectory Optimization
Motion planning for manipulators under task space constraints is difficult as
it constrains the joint configurations to always lie on an implicitly defined
manifold. It is possible to view task constrained motion planning as an
optimization problem with non-linear equality constraints which can be solved
by general non-linear optimization techniques. In this paper, we present a
novel custom optimizer which exploits the underlying structure present in many
task constraints.
At the core of our approach are some simple reformulations, which when
coupled with the \emph{method of alternating projection}, leads to an efficient
convex optimization based routine for computing a feasible solution to the task
constraints. We subsequently build on this result and use the concept of
Augmented Lagrangian to guide the feasible solutions towards those which also
minimize the user defined cost function. We show that the proposed optimizer is
fully distributive and thus, can be easily parallelized. We validate our
formulation on some common robotic benchmark problems. In particular, we show
that the proposed optimizer achieves cyclic motion in the joint space
corresponding to a similar nature trajectory in the task space. Furthermore, as
a baseline, we compare the proposed optimizer with an off-the-shelf non-linear
solver provide in open source package SciPy. We show that for similar task
constraint residuals and smoothness cost, it can be upto more than three times
faster than the SciPy alternative.Comment: 8 page
Are Girls Neko or Sh\=ojo? Cross-Lingual Alignment of Non-Isomorphic Embeddings with Iterative Normalization
Cross-lingual word embeddings (CLWE) underlie many multilingual natural
language processing systems, often through orthogonal transformations of
pre-trained monolingual embeddings. However, orthogonal mapping only works on
language pairs whose embeddings are naturally isomorphic. For non-isomorphic
pairs, our method (Iterative Normalization) transforms monolingual embeddings
to make orthogonal alignment easier by simultaneously enforcing that (1)
individual word vectors are unit length, and (2) each language's average vector
is zero. Iterative Normalization consistently improves word translation
accuracy of three CLWE methods, with the largest improvement observed on
English-Japanese (from 2% to 44% test accuracy).Comment: ACL 201
Dropping Symmetry for Fast Symmetric Nonnegative Matrix Factorization
Symmetric nonnegative matrix factorization (NMF), a special but important
class of the general NMF, is demonstrated to be useful for data analysis and in
particular for various clustering tasks. Unfortunately, designing fast
algorithms for Symmetric NMF is not as easy as for the nonsymmetric
counterpart, the latter admitting the splitting property that allows efficient
alternating-type algorithms. To overcome this issue, we transfer the symmetric
NMF to a nonsymmetric one, then we can adopt the idea from the state-of-the-art
algorithms for nonsymmetric NMF to design fast algorithms solving symmetric
NMF. We rigorously establish that solving nonsymmetric reformulation returns a
solution for symmetric NMF and then apply fast alternating based algorithms for
the corresponding reformulated problem. Furthermore, we show these fast
algorithms admit strong convergence guarantee in the sense that the generated
sequence is convergent at least at a sublinear rate and it converges globally
to a critical point of the symmetric NMF. We conduct experiments on both
synthetic data and image clustering to support our result.Comment: Accepted in NIPS 201