27,539 research outputs found
Recurrent Pixel Embedding for Instance Grouping
We introduce a differentiable, end-to-end trainable framework for solving
pixel-level grouping problems such as instance segmentation consisting of two
novel components. First, we regress pixels into a hyper-spherical embedding
space so that pixels from the same group have high cosine similarity while
those from different groups have similarity below a specified margin. We
analyze the choice of embedding dimension and margin, relating them to
theoretical results on the problem of distributing points uniformly on the
sphere. Second, to group instances, we utilize a variant of mean-shift
clustering, implemented as a recurrent neural network parameterized by kernel
bandwidth. This recurrent grouping module is differentiable, enjoys convergent
dynamics and probabilistic interpretability. Backpropagating the group-weighted
loss through this module allows learning to focus on only correcting embedding
errors that won't be resolved during subsequent clustering. Our framework,
while conceptually simple and theoretically abundant, is also practically
effective and computationally efficient. We demonstrate substantial
improvements over state-of-the-art instance segmentation for object proposal
generation, as well as demonstrating the benefits of grouping loss for
classification tasks such as boundary detection and semantic segmentation
Toric varieties and spherical embeddings over an arbitrary field
We are interested in two classes of varieties with group action, namely toric
varieties and spherical embeddings. They are classified by combinatorial
objects, called fans in the toric setting, and colored fans in the spherical
setting. We characterize those combinatorial objects corresponding to varieties
defined over an arbitrary field . Then we provide some situations where
toric varieties over are classified by Galois-stable fans, and spherical
embeddings over by Galois-stable colored fans. Moreover, we construct an
example of a smooth toric variety under a 3-dimensional nonsplit torus over
whose fan is Galois-stable but which admits no -form. In the spherical
setting, we offer an example of a spherical homogeneous space over \mr
of rank 2 under the action of SU(2,1) and a smooth embedding of whose fan
is Galois-stable but which admits no \mr-form
Classification of Reductive Monoid Spaces Over an Arbitrary Field
In this semi-expository paper we review the notion of a spherical space. In
particular we present some recent results of Wedhorn on the classification of
spherical spaces over arbitrary fields. As an application, we introduce and
classify reductive monoid spaces over an arbitrary field.Comment: This is the final versio
Demazure roots and spherical varieties: the example of horizontal SL(2)-actions
Let be a connected reductive group, and let be an affine
-spherical variety. We show that the classification of
-actions on normalized by can be reduced to the
description of quasi-affine homogeneous spaces under the action of a
semi-direct product with the following property. The
induced -action is spherical and the complement of the open orbit is either
empty or a -orbit of codimension one. These homogeneous spaces are
parametrized by a subset of the character lattice
of , which we call the set of Demazure roots of . We give a complete
description of the set when is a semi-direct product of and an algebraic torus; we show particularly that can be
obtained explicitly as the intersection of a finite union of polyhedra in
and a sublattice of
. We conjecture that can be described in a similar
combinatorial way for an arbitrary affine spherical variety .Comment: Added Section 4; modified main result, Theorem 5.18 now; other
change
A combinatorial smoothness criterion for spherical varieties
We suggest a combinatorial criterion for the smoothness of an arbitrary
spherical variety using the classification of multiplicity-free spaces,
generalizing an earlier result of Camus for spherical varieties of type .Comment: 14 pages, 2 table
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