82 research outputs found
A simplicial homology algorithm for Lipschitz optimisation
The simplicial homology global optimisation (SHGO) algorithm is a general purpose
global optimisation algorithm based on applications of simplicial integral homology and
combinatorial topology. SHGO approximates the homology groups of a complex built on
a hypersurface homeomorphic to a complex on the objective function. This provides both
approximations of locally convex subdomains in the search space through Sperner's lemma
(Sperner, 1928) and a useful visual tool for characterising and e ciently solving higher
dimensional black and grey box optimisation problems. This complex is built up using
sampling points within the feasible search space as vertices. The algorithm is specialised
in nding all the local minima of an objective function with expensive function evaluations
e ciently which is especially suitable to applications such as energy landscape exploration.
SHGO was initially developed as an improvement on the topographical global
optimisation (TGO) method rst proposed by T orn (1986; 1990; 1992). It is proven that
the SHGO algorithm will always outperform TGO on function evaluations if the objective
function is Lipschitz smooth. In this dissertation SHGO is applied to non-convex problems
with linear and box constraints with bounds placed on the variables. Numerical experiments
on linearly constrained test problems show that SHGO gives competitive results
compared to TGO and the recently developed Lc-DISIMPL algorithm (Paulavi cius and
Zilinskas, 2016) as well as the PSwarm and DIRECT-L1 algorithms. Furthermore SHGO
is compared with the TGO, basinhopping (BH) and di erential evolution (DE) global
optimisation algorithms over a large selection of black-box problems with bounds placed
on the variables from the SciPy (Jones, Oliphant, Peterson, et al., 2001{) benchmarking
test suite. A Python implementation of the SHGO and TGO algorithms published under
a MIT license can be found from https://bitbucket.org/upiamcompthermo/shgo/.Dissertation (MEng)--University of Pretoria, 2017.Chemical EngineeringMEngUnrestricte
A simplicial homology algorithm for Lipschitz optimisation
The simplicial homology global optimisation (SHGO) algorithm is a general purpose global optimisation algorithm based on applications of simplicial integral homology and combinatorial topology. SHGO approximates the homology groups of a complex built on a hypersurface homeomorphic to a complex on the objective function. This provides both approximations of locally convex subdomains in the search space through Sperner’s lemma and a useful visual tool for characterising and efficiently solving higher dimensional black and grey box optimisation problems. This complex is built up using sampling points within the feasible search space as vertices. The algorithm is specialised in finding all the local minima of an objective function with expensive function evaluations efficiently which is especially suitable to applications such as energy landscape exploration. SHGO was initially developed as an improvement on the topographical global optimisation (TGO) method. It is proven that the SHGO algorithm will always outperform TGO on function evaluations if the objective function is Lipschitz smooth. In this paper SHGO is applied to non-convex problems with linear and box constraints with bounds placed on the variables. Numerical experiments on linearly constrained test problems show that SHGO gives competitive results compared to TGO and the recently developed Lc-DISIMPL algorithm as well as the PSwarm, LGO and DIRECT-L1 algorithms. Furthermore SHGO is compared with the TGO, basinhopping (BH) and differential evolution (DE) global optimisation algorithms over a large selection of black-box problems with bounds placed on the variables from the SciPy benchmarking test suite. A Python implementation of the SHGO and TGO algorithms published under a MIT license can be found from https://bitbucket.org/upiamcompthermo/shgo/.http://link.springer.com/journal/108982019-10-01hj2018Chemical Engineerin
On the Expressivity of Persistent Homology in Graph Learning
Persistent homology, a technique from computational topology, has recently
shown strong empirical performance in the context of graph classification.
Being able to capture long range graph properties via higher-order topological
features, such as cycles of arbitrary length, in combination with multi-scale
topological descriptors, has improved predictive performance for data sets with
prominent topological structures, such as molecules. At the same time, the
theoretical properties of persistent homology have not been formally assessed
in this context. This paper intends to bridge the gap between computational
topology and graph machine learning by providing a brief introduction to
persistent homology in the context of graphs, as well as a theoretical
discussion and empirical analysis of its expressivity for graph learning tasks
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Topological and geometric inference of data
The overarching problem under consideration is to determine the structure
of the subspace on which a distribution is supported, given
only a finite noisy sample thereof. The special case in
which the subspace is an embedded manifold is given particular
attention owing to its conceptual elegance, and asymptotic bounds are
obtained on the admissible level of noise such that the
manifold can be recovered up to homotopy equivalence.
Attention is turned on how to accomplish this in practice.
Following ideas from topological data analysis, simplicial complexes are used
as discrete analogues of spaces suitable for computation. By utilising
the prior assumption that the data lie on a manifold, topologically
inspired techniques are proposed for refining the simplicial complex
to better approximate this manifold. This is applied to the
problem of nonlinear dimensionality reduction and found to improve accuracy
of reconstructing several synthetic and real-world datasets.
The second chapter focuses on extending this work to the
case where the ambient space is non-Euclidean. The interfaces between
topological data analysis, functional data analysis, and shape analysis
are thoroughly explored. Lipschitz bounds are proved which relate several
metrics on the space of positive semidefinite matrices; they are then
interpreted in the context of topological data analysis. This is
applied to diffusion tensor imaging and phonology.
The final chapter explores the case where the points are
non-uniformly distributed over the embedded subspace. In particular, a method
is proposed to overcome the shortcomings of witness complex construction
when there are large deviations in the density. The theory
of multidimensional persistence is leveraged to provide a succinct setting
in which the structure of the data can be interpreted
as a generalised stratified space.EPSR
Fitting IVIM with Variable Projection and Simplicial Optimization
Fitting multi-exponential models to Diffusion MRI (dMRI) data has always been
challenging due to various underlying complexities. In this work, we introduce
a novel and robust fitting framework for the standard two-compartment IVIM
microstructural model. This framework provides a significant improvement over
the existing methods and helps estimate the associated diffusion and perfusion
parameters of IVIM in an automatic manner. As a part of this work we provide
capabilities to switch between more advanced global optimization methods such
as simplicial homology (SH) and differential evolution (DE). Our experiments
show that the results obtained from this simultaneous fitting procedure
disentangle the model parameters in a reduced subspace. The proposed framework
extends the seminal work originated in the MIX framework, with improved
procedures for multi-stage fitting. This framework has been made available as
an open-source Python implementation and disseminated to the community through
the DIPY project
Bifurcated topological optimization for IVIM
In this work, we shed light on the issue of estimating Intravoxel Incoherent Motion (IVIM)
for diffusion and perfusion estimation by characterizing the objective function using
simplicial homology tools. We provide a robust solution via topological optimization of
this model so that the estimates are more reliable and accurate. Estimating the tissue
microstructure from diffusion MRI is in itself an ill-posed and a non-linear inverse problem.
Using variable projection functional (VarPro) to fit the standard bi-exponential IVIM model
we perform the optimization using simplicial homology based global optimization to
better understand the topology of objective function surface. We theoretically show
how the proposed methodology can recover the model parameters more accurately
and consistently by casting it in a reduced subspace given by VarPro. Additionally
we demonstrate that the IVIM model parameters cannot be accurately reconstructed
using conventional numerical optimization methods due to the presence of infinite
solutions in subspaces. The proposed method helps uncover multiple global minima by
analyzing the local geometry of the model enabling the generation of reliable estimates
of model parameters.The National Institute of Biomedical Imaging And Bioengineering (NIBIB) of the National Institutes of Health (NIH); University of Washington’s Royalty Research Fund; NIH grants; the German Research Foundation (DFG) and a grant from the Alfred P. Sloan Foundation and the Gordon & Betty Moore Foundation to the University of Washington eScience Institute Data Science Environment.http://www.frontiersin.org/Neuroscienceam2022Chemical Engineerin
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