12 research outputs found
Embeddings and immersions of tropical curves
We construct immersions of trivalent abstract tropical curves in the
Euclidean plane and embeddings of all abstract tropical curves in higher
dimensional Euclidean space. Since not all curves have an embedding in the
plane, we define the tropical crossing number of an abstract tropical curve to
be the minimum number of self-intersections, counted with multiplicity, over
all its immersions in the plane. We show that the tropical crossing number is
at most quadratic in the number of edges and this bound is sharp. For curves of
genus up to two, we systematically compute the crossing number. Finally, we use
our immersed tropical curves to construct totally faithful nodal algebraic
curves via lifting results of Mikhalkin and Shustin.Comment: 23 pages, 14 figures, final submitted versio
Asynchronous Algorithmic Alignment with Cocycles
State-of-the-art neural algorithmic reasoners make use of message passing in
graph neural networks (GNNs). But typical GNNs blur the distinction between the
definition and invocation of the message function, forcing a node to send
messages to its neighbours at every layer, synchronously. When applying GNNs to
learn to execute dynamic programming algorithms, however, on most steps only a
handful of the nodes would have meaningful updates to send. One, hence, runs
the risk of inefficiencies by sending too much irrelevant data across the graph
-- with many intermediate GNN steps having to learn identity functions. In this
work, we explicitly separate the concepts of node state update and message
function invocation. With this separation, we obtain a mathematical formulation
that allows us to reason about asynchronous computation in both algorithms and
neural networks
A Generalist Neural Algorithmic Learner
The cornerstone of neural algorithmic reasoning is the ability to solve
algorithmic tasks, especially in a way that generalises out of distribution.
While recent years have seen a surge in methodological improvements in this
area, they mostly focused on building specialist models. Specialist models are
capable of learning to neurally execute either only one algorithm or a
collection of algorithms with identical control-flow backbone. Here, instead,
we focus on constructing a generalist neural algorithmic learner -- a single
graph neural network processor capable of learning to execute a wide range of
algorithms, such as sorting, searching, dynamic programming, path-finding and
geometry. We leverage the CLRS benchmark to empirically show that, much like
recent successes in the domain of perception, generalist algorithmic learners
can be built by "incorporating" knowledge. That is, it is possible to
effectively learn algorithms in a multi-task manner, so long as we can learn to
execute them well in a single-task regime. Motivated by this, we present a
series of improvements to the input representation, training regime and
processor architecture over CLRS, improving average single-task performance by
over 20% from prior art. We then conduct a thorough ablation of multi-task
learners leveraging these improvements. Our results demonstrate a generalist
learner that effectively incorporates knowledge captured by specialist models.Comment: 20 pages, 10 figure
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Quantales and Hyperstructures
We present a theory of lattice-enriched semirings, called \emph{quantic semirings}, which generalize both quantales and powersets of hyperrings. Using these structures, we show how to recover the spectrum of a Krasner hyperring (and in particular, a commutative ring with unity) via universal constructions, and generalize the spectrum to a new class of hyperstructures, \emph{hypersemirings}. (These include hyperstructures currently studied under the name ``semihyperrings'', but we have weakened the distributivity axioms.)Much of the work consists of background material on closure systems, suplattices, quantales, and hyperoperations, some of which is new. In particular, we define the category of covered semigroups, show their close relationship with quantales, and construct their spectra by exploiting the construction of a universal quotient frame by Rosenthal.We extend these results to hypersemigroups, demonstrating various folkloric correspondences between hyperstructures and lattice-enriched structures on the powerset. Building on this, we proceed to define quantic semirings, and show that they are the lattice-enriched counterparts of hypersemirings. To a quantic semiring, we show how to define a universal quotient quantale, which we call the \emph{quantic spectrum}, and using this, we show how to obtain the spectrum of a hypersemiring as a topological space in a canonical fashion.Finally, we we conclude with some applications of the theory to the ordered blueprints of Lorscheid
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Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge
Gliomas are the most common primary brain malignancies, with different
degrees of aggressiveness, variable prognosis and various heterogeneous
histologic sub-regions, i.e., peritumoral edematous/invaded tissue, necrotic
core, active and non-enhancing core. This intrinsic heterogeneity is also
portrayed in their radio-phenotype, as their sub-regions are depicted by
varying intensity profiles disseminated across multi-parametric magnetic
resonance imaging (mpMRI) scans, reflecting varying biological properties.
Their heterogeneous shape, extent, and location are some of the factors that
make these tumors difficult to resect, and in some cases inoperable. The amount
of resected tumor is a factor also considered in longitudinal scans, when
evaluating the apparent tumor for potential diagnosis of progression.
Furthermore, there is mounting evidence that accurate segmentation of the
various tumor sub-regions can offer the basis for quantitative image analysis
towards prediction of patient overall survival. This study assesses the
state-of-the-art machine learning (ML) methods used for brain tumor image
analysis in mpMRI scans, during the last seven instances of the International
Brain Tumor Segmentation (BraTS) challenge, i.e., 2012-2018. Specifically, we
focus on i) evaluating segmentations of the various glioma sub-regions in
pre-operative mpMRI scans, ii) assessing potential tumor progression by virtue
of longitudinal growth of tumor sub-regions, beyond use of the RECIST/RANO
criteria, and iii) predicting the overall survival from pre-operative mpMRI
scans of patients that underwent gross total resection. Finally, we investigate
the challenge of identifying the best ML algorithms for each of these tasks,
considering that apart from being diverse on each instance of the challenge,
the multi-institutional mpMRI BraTS dataset has also been a continuously
evolving/growing dataset
Identifying the Best Machine Learning Algorithms for Brain Tumor Segmentation, Progression Assessment, and Overall Survival Prediction in the BRATS Challenge
Gliomas are the most common primary brain malignancies, with different
degrees of aggressiveness, variable prognosis and various heterogeneous
histologic sub-regions, i.e., peritumoral edematous/invaded tissue, necrotic
core, active and non-enhancing core. This intrinsic heterogeneity is also
portrayed in their radio-phenotype, as their sub-regions are depicted by
varying intensity profiles disseminated across multi-parametric magnetic
resonance imaging (mpMRI) scans, reflecting varying biological properties.
Their heterogeneous shape, extent, and location are some of the factors that
make these tumors difficult to resect, and in some cases inoperable. The amount
of resected tumor is a factor also considered in longitudinal scans, when
evaluating the apparent tumor for potential diagnosis of progression.
Furthermore, there is mounting evidence that accurate segmentation of the
various tumor sub-regions can offer the basis for quantitative image analysis
towards prediction of patient overall survival. This study assesses the
state-of-the-art machine learning (ML) methods used for brain tumor image
analysis in mpMRI scans, during the last seven instances of the International
Brain Tumor Segmentation (BraTS) challenge, i.e., 2012-2018. Specifically, we
focus on i) evaluating segmentations of the various glioma sub-regions in
pre-operative mpMRI scans, ii) assessing potential tumor progression by virtue
of longitudinal growth of tumor sub-regions, beyond use of the RECIST/RANO
criteria, and iii) predicting the overall survival from pre-operative mpMRI
scans of patients that underwent gross total resection. Finally, we investigate
the challenge of identifying the best ML algorithms for each of these tasks,
considering that apart from being diverse on each instance of the challenge,
the multi-institutional mpMRI BraTS dataset has also been a continuously
evolving/growing dataset