simplicial homology
Recently Published Documents


TOTAL DOCUMENTS

41
(FIVE YEARS 14)

H-INDEX

5
(FIVE YEARS 0)

2021 ◽  
Vol 15 ◽  
Author(s):  
Shreyas Fadnavis ◽  
Stefan Endres ◽  
Qiuting Wen ◽  
Yu-Chien Wu ◽  
Hu Cheng ◽  
...  

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.


2021 ◽  
Vol 20 (5s) ◽  
pp. 1-26
Author(s):  
Guihong Li ◽  
Sumit K. Mandal ◽  
Umit Y. Ogras ◽  
Radu Marculescu

Neural architecture search (NAS) is a promising technique to design efficient and high-performance deep neural networks (DNNs). As the performance requirements of ML applications grow continuously, the hardware accelerators start playing a central role in DNN design. This trend makes NAS even more complicated and time-consuming for most real applications. This paper proposes FLASH, a very fast NAS methodology that co-optimizes the DNN accuracy and performance on a real hardware platform. As the main theoretical contribution, we first propose the NN-Degree, an analytical metric to quantify the topological characteristics of DNNs with skip connections (e.g., DenseNets, ResNets, Wide-ResNets, and MobileNets). The newly proposed NN-Degree allows us to do training-free NAS within one second and build an accuracy predictor by training as few as 25 samples out of a vast search space with more than 63 billion configurations. Second, by performing inference on the target hardware, we fine-tune and validate our analytical models to estimate the latency, area, and energy consumption of various DNN architectures while executing standard ML datasets. Third, we construct a hierarchical algorithm based on simplicial homology global optimization (SHGO) to optimize the model-architecture co-design process, while considering the area, latency, and energy consumption of the target hardware. We demonstrate that, compared to the state-of-the-art NAS approaches, our proposed hierarchical SHGO-based algorithm enables more than four orders of magnitude speedup (specifically, the execution time of the proposed algorithm is about 0.1 seconds). Finally, our experimental evaluations show that FLASH is easily transferable to different hardware architectures, thus enabling us to do NAS on a Raspberry Pi-3B processor in less than 3 seconds.


2021 ◽  
Vol 22 (2) ◽  
pp. 223
Author(s):  
P. Christopher Staecker

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images.<br /><br />We introduce a new type of homotopy relation for digitally continuous functions which we call ``strong homotopy.'' Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane.<br /><br />We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of the clique complex, a singular simplicial homology theory by D. W. Lee, a cubical homology theory by Jamil and Ali, and a new kind of cubical homology for digital images with $c_1$-adjacency which is easily computed, and generalizes a construction by Karaca \&amp; Ege. We show that the two simplicial homology theories are isomorphic to each other, but distinct from the two cubical theories.<br /><br />We also show that homotopic maps have the same induced homomorphisms in the cubical homology theory, and strong homotopic maps additionally have the same induced homomorphisms in the simplicial theory.


Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 744
Author(s):  
Andrei Bura ◽  
Qijun He ◽  
Christian Reidys

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.


Author(s):  
Javier Aramayona ◽  
Priyam Patel ◽  
Nicholas G Vlamis

Abstract It is a classical result that pure mapping class groups of connected, orientable surfaces of finite type and genus at least 3 are perfect. In stark contrast, we construct nontrivial homomorphisms from infinite-genus mapping class groups to the integers. Moreover, we compute the first integral cohomology group associated to the pure mapping class group of any connected orientable surface of genus at least 2 in terms of the surface’s simplicial homology. In order to do this, we show that pure mapping class groups of infinite-genus surfaces split as a semi-direct product.


Sign in / Sign up

Export Citation Format

Share Document