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Author Description

D. B. A. Epstein is David B. A. Epstein, a British mathematician and Emeritus Professor at the University of Warwick. He is known for his work in geometry and topology, including contributions to geometric group theory and hyperbolic geometry. He co-founded and served as a managing editor of the journal Geometry & Topology. Epstein’s research has helped shape modern approaches to the interface of geometry, group theory, and topology.

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All published works (78)

Dense Steerable Filter CNNs for Exploiting Rotational Symmetry in Histology Images

2020-07-31
Simon Graham , D. B. A. Epstein , Nasir Rajpoot
Histology images are inherently symmetric under rotation, where each orientation is equally as likely to appear. However, this rotational symmetry is not widely utilised as prior knowledge in modern Convolutional … Histology images are inherently symmetric under rotation, where each orientation is equally as likely to appear. However, this rotational symmetry is not widely utilised as prior knowledge in modern Convolutional Neural Networks (CNNs), resulting in data hungry models that learn independent features at each orientation. Allowing CNNs to be rotation-equivariant removes the necessity to learn this set of transformations from the data and instead frees up model capacity, allowing more discriminative features to be learned. This reduction in the number of required parameters also reduces the risk of overfitting. In this paper, we propose Dense Steerable Filter CNNs (DSF-CNNs) that use group convolutions with multiple rotated copies of each filter in a densely connected framework. Each filter is defined as a linear combination of steerable basis filters, enabling exact rotation and decreasing the number of trainable parameters compared to standard filters. We also provide the first in-depth comparison of different rotation-equivariant CNNs for histology image analysis and demonstrate the advantage of encoding rotational symmetry into modern architectures. We show that DSF-CNNs achieve state-of-the-art performance, with significantly fewer parameters, when applied to three different tasks in the area of computational pathology: breast tumour classification, colon gland segmentation and multi-tissue nuclear segmentation.
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Dense Steerable Filter CNNs for Exploiting Rotational Symmetry in Histology Images

2020-04-06
Simon Graham , D. B. A. Epstein , Nasir Rajpoot
Histology images are inherently symmetric under rotation, where each orientation is equally as likely to appear. However, this rotational symmetry is not widely utilised as prior knowledge in modern Convolutional … Histology images are inherently symmetric under rotation, where each orientation is equally as likely to appear. However, this rotational symmetry is not widely utilised as prior knowledge in modern Convolutional Neural Networks (CNNs), resulting in data hungry models that learn independent features at each orientation. Allowing CNNs to be rotation-equivariant removes the necessity to learn this set of transformations from the data and instead frees up model capacity, allowing more discriminative features to be learned. This reduction in the number of required parameters also reduces the risk of overfitting. In this paper, we propose Dense Steerable Filter CNNs (DSF-CNNs) that use group convolutions with multiple rotated copies of each filter in a densely connected framework. Each filter is defined as a linear combination of steerable basis filters, enabling exact rotation and decreasing the number of trainable parameters compared to standard filters. We also provide the first in-depth comparison of different rotation-equivariant CNNs for histology image analysis and demonstrate the advantage of encoding rotational symmetry into modern architectures. We show that DSF-CNNs achieve state-of-the-art performance, with significantly fewer parameters, when applied to three different tasks in the area of computational pathology: breast tumour classification, colon gland segmentation and multi-tissue nuclear segmentation.

Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features

2019-04-04
Talha Qaiser , Yee‐Wah Tsang , Daiki Taniyama +4 more
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Micro-Net: A unified model for segmentation of various objects in microscopy images

2018-12-15
Shan E Ahmed Raza , Linda Cheung , Muhammad Shaban +5 more
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Fast and Accurate Tumor Segmentation of Histology Images using Persistent Homology and Deep Convolutional Features

2018-05-09
Talha Qaiser , Yee‐Wah Tsang , Daiki Taniyama +4 more
Tumor segmentation in whole-slide images of histology slides is an important step towards computer-assisted diagnosis. In this work, we propose a tumor segmentation framework based on the novel concept of … Tumor segmentation in whole-slide images of histology slides is an important step towards computer-assisted diagnosis. In this work, we propose a tumor segmentation framework based on the novel concept of persistent homology profiles (PHPs). For a given image patch, the homology profiles are derived by efficient computation of persistent homology, which is an algebraic tool from homology theory. We propose an efficient way of computing topological persistence of an image, alternative to simplicial homology. The PHPs are devised to distinguish tumor regions from their normal counterparts by modeling the atypical characteristics of tumor nuclei. We propose two variants of our method for tumor segmentation: one that targets speed without compromising accuracy and the other that targets higher accuracy. The fast version is based on the selection of exemplar image patches from a convolution neural network (CNN) and patch classification by quantifying the divergence between the PHPs of exemplars and the input image patch. Detailed comparative evaluation shows that the proposed algorithm is significantly faster than competing algorithms while achieving comparable results. The accurate version combines the PHPs and high-level CNN features and employs a multi-stage ensemble strategy for image patch labeling. Experimental results demonstrate that the combination of PHPs and CNN features outperforms competing algorithms. This study is performed on two independently collected colorectal datasets containing adenoma, adenocarcinoma, signet and healthy cases. Collectively, the accurate tumor segmentation produces the highest average patch-level F1-score, as compared with competing algorithms, on malignant and healthy cases from both the datasets. Overall the proposed framework highlights the utility of persistent homology for histopathology image analysis.

Novel digital tissue phenotypic signatures of distant metastasis in colorectal cancer.

2018-01-23
Korsuk Sirinukunwattana , David Snead , D. B. A. Epstein +5 more
Distant metastasis is the major cause of death in colorectal cancer (CRC). Patients at high risk of developing distant metastasis could benefit from appropriate adjuvant and follow-up treatments if stratified … Distant metastasis is the major cause of death in colorectal cancer (CRC). Patients at high risk of developing distant metastasis could benefit from appropriate adjuvant and follow-up treatments if stratified accurately at an early stage of the disease. Studies have increasingly recognized the role of diverse cellular components within the tumor microenvironment in the development and progression of CRC tumors. In this paper, we show that a new method of automated analysis of digitized images from colorectal cancer tissue slides can provide important estimates of distant metastasis-free survival (DMFS, the time before metastasis is first observed) on the basis of details of the microenvironment. Specifically, we determine what cell types are found in the vicinity of other cell types, and in what numbers, rather than concentrating exclusively on the cancerous cells. We then extract novel tissue phenotypic signatures using statistical measurements about tissue composition. Such signatures can underpin clinical decisions about the advisability of various types of adjuvant therapy.

Maximizing the area of intersection of rectangles

2017-01-01
D. B. A. Epstein , Mike Paterson
This paper attacks the following problem. We are given a large number $N$ of rectangles in the plane, each with horizontal and vertical sides, and also a number $r This paper attacks the following problem. We are given a large number $N$ of rectangles in the plane, each with horizontal and vertical sides, and also a number $r
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Extending homeomorphisms of the circle to quasiconformal homeomorphisms of the disk

2007-03-16
D. B. A. Epstein , Vladimir Marković
We prove that it is not possible to extend, in a homomorphic fashion, each quasisymmetric homeomorphism of the circle to a quasiconformal homeomorphism of the disk. We prove that it is not possible to extend, in a homomorphic fashion, each quasisymmetric homeomorphism of the circle to a quasiconformal homeomorphism of the disk.
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Complex earthquakes and deformations of the unit disk

2006-05-01
D. B. A. Epstein , Albert Marden , Vladimir Marković
We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded … We define deformations of certain geometric objects in hyperbolic 3-space. Such an object starts life as a hyperbolic plane with a measured geometric lamination. Initially the hyperbolic plane is embedded as a standard hyperbolic subspace. Given a complex number t, we obtain a corresponding object in hyperbolic 3-space by earthquaking along the lamination, parametrized by the real part of t, and then bending along the image lamination, parametrized by the complex part of t. In the literature, it is usually assumed that there is a quasifuchsian group that preserves the structure, but this paper is more general and makes no such assumption. Our deformation is holomorphic, as in the λ-lemma, which is a result that underlies the results in this paper. Our deformation is used to produce a new, more natural proof of Sullivan's theorem: that, under standard topological hypotheses, the boundary of the convex hull in hyperbolic 3-space of the complement of an open subset U of the 2-sphere is quasi-conformally equivalent to U, and that, furthermore, the constant of quasiconformality is a universal constant. Our paper presents a precise statement of Sullivan's Theorem. We also generalize much of McMullen's Disk Theorem, describing certain aspects of the parameter space for certain parametrized spaces of 2-dimensional hyperbolic structures.
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CONVEX REGIONS IN THE PLANE AND THEIR DOMES

2006-04-18
D. B. A. Epstein , Albert Marden , Vladimir Marković
We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace … We make a detailed study of the relation of a euclidean convex region $\Omega \subset \mathbb C$ to $\mathrm{Dome} (\Omega)$. The dome is the relative boundary, in the upper halfspace model of hyperbolic space, of the hyperbolic convex hull of the complement of $\Omega$. The first result is to prove that the nearest point retraction $r: \Omega \to \mathrm{Dome} (\Omega)$ is 2-quasiconformal. The second is to establish precise estimates of the distortion of $r$ near $\partial \Omega$.
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Hyperbolic structures

2006-04-13
Richard D. Canary , Albert Marden , D. B. A. Epstein
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(<i>G, X</i>)-structures

2006-04-13
R. D. Canary , A. Marden , D. B. A. Epstein
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Spaces of hyberbolic manifolds

2006-04-13
Richard D. Canary , Albert Marden , D. B. A. Epstein
We shall define a topology on the set of closed subsets of a topological space. We shall thereby derive topologies for both the space of complete hyperbolic manifolds and the … We shall define a topology on the set of closed subsets of a topological space. We shall thereby derive topologies for both the space of complete hyperbolic manifolds and the space of geodesic laminations (see Section I.4.1 (Geodesic Laminations)). This topology was first considered by Chabauty (1950) as a topology on the space of closed subgroups of a locally compact topological group, and later by Harvey (1977) with specific reference to Fuchsian groups. See also Michael, (1951).

Fundamentals of Hyperbolic Manifolds

2006-04-13
Richard D. Canary , D. B. A. Epstein , William P. Thurston +1 more
Presents reissued articles from two classic sources on hyperbolic manifolds. Part I is an exposition of Chapters 8 and 9 of Thurston's pioneering Princeton Notes; there is a new introduction … Presents reissued articles from two classic sources on hyperbolic manifolds. Part I is an exposition of Chapters 8 and 9 of Thurston's pioneering Princeton Notes; there is a new introduction describing recent advances, with an up-to-date bibliography, giving a contemporary context in which the work can be set. Part II expounds the theory of convex hull boundaries and their bending laminations. A new appendix describes recent work. Part III is Thurston's famous paper that presents the notion of earthquakes in hyperbolic geometry and proves the earthquake theorem. The final part introduces the theory of measures on the limit set, drawing attention to related ergodic theory and the exponent of convergence. The book will be welcomed by graduate students and professional mathematicians who want a rigorous introduction to some basic tools essential for the modern theory of hyperbolic manifolds.

THE LINEARITY OF THE CONJUGACY PROBLEM IN WORD-HYPERBOLIC GROUPS

2006-04-01
D. B. A. Epstein , Derek F. Holt
The main result proved in this paper is that the conjugacy problem in word-hyperbolic groups is solvable in linear time. This is using a standard RAM model of computation, in … The main result proved in this paper is that the conjugacy problem in word-hyperbolic groups is solvable in linear time. This is using a standard RAM model of computation, in which basic arithmetical operations on integers are assumed to take place in constant time. The constants involved in the linear time solution are all computable explicitly. We also give a proof of the result of Mike Shapiro that in a word-hyperbolic group a word in the generators can be transformed into short-lex normal form in linear time. This is used in the proof of our main theorem, but is a significant theoretical result of independent interest, which deserves to be in the literature. Previously the best known result was a quadratic estimate.

Fundamentals of hyperbolic geometry : selected expositions

2006-01-01
Richard D. Canary , D. B. A. Epstein , Albert Marden
Preface 2005 Preface Part I. Notes on Notes of Thurston R. D. Canary, D. B. A. Epstein and P. Green Part II. Convex Hulls in Hyperbolic Space, a Theorem of … Preface 2005 Preface Part I. Notes on Notes of Thurston R. D. Canary, D. B. A. Epstein and P. Green Part II. Convex Hulls in Hyperbolic Space, a Theorem of Sullivan, and Measured Pleated Surfaces D. B. A. Epstein and A. Marden Part III. Earthquakes in Two-Dimensional Hyperbolic Geometry William P. Thurston Part IV. Lectures on Measures on Limit Sets of Kleinian Groups S. J. Patterson.

Fundamentals of Hyperbolic Manifolds: CONVEX HULLS IN HYPERBOLIC SPACE, A THEOREM OF SULLIVAN, AND MEASURED PLEATED SURFACES

2006-01-01
Richard D. Canary , D. B. A. Epstein , P.L. Green

The logarithmic spiral: a counterexample to the K = 2 conjecture

2005-03-01
D. B. A. Epstein , Vladimir Marković
Given a nonempty compact connected subset X subset of S-2 with complement a simply-connected open subset Omega subset of S-2, let Dome (Omega) be the boundary of the hyperbolic convex … Given a nonempty compact connected subset X subset of S-2 with complement a simply-connected open subset Omega subset of S-2, let Dome (Omega) be the boundary of the hyperbolic convex hull in H-3 of X. We show that if X is a certain logarithmic spiral, then we obtain a counterexample to the conjecture of Thurston and Sullivan that there is a 2-quasiconformal homeomorphism Omega -> Dome (Omega) which extends to the identity map on their common boundary in S-2. This leads to related counterexamples when the boundary is real analytic, or a finite union of intervals (straight intervals, if we take S-2 = C boolean OR {infinity}). We also show how this counterexample enables us to construct a related counterexample which is a domain of discontinuity of a torsion-free quasifuchsian group with compact quotient. Another result is that the average long range bending of the convex hull boundary associated to a certain logarithmic spiral is approximately .98 pi/2, which is substantially larger than that of any previously known example.

Quasiconformal homeomorphisms and the convex hull boundary

2004-01-01
D. B. A. Epstein , Albert Marden , Vladimir Marković
We investigate the relationship between an open simply-connected region Ω ⊂ S 2 and the boundary Y of the hyperbolic convex hull in H 3 of S 2 \ Ω.A … We investigate the relationship between an open simply-connected region Ω ⊂ S 2 and the boundary Y of the hyperbolic convex hull in H 3 of S 2 \ Ω.A counterexample is given to Thurston's conjecture that these spaces are related by a 2-quasiconformal homeomorphism which extends to the identity map on their common boundary, in the case when the homeomorphism is required to respect any group of Möbius transformations which preserves Ω.We show that the best possible universal lipschitz constant for the nearest point retraction r : Ω → Y is 2. We find explicit universal constants 0 < c 2 < c 1 , such that no pleating map which bends more than c 1 in some interval of unit length is an embedding, and such that any pleating map which bends less than c 2 in each interval of unit length is embedded.We show that every K-quasiconformal homeomorphism D 2 → D 2 is a (K, a(K))-quasi-isometry, where a(K) is an explicitly computed function.The multiplicative constant is best possible and the additive constant a(K) is best possible for some values of K.
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Complex angle scaling

2003-11-10
D. B. A. Epstein , Albert Marden , Vladimir Marković
We introduce the method of complex angle scaling to study deformations of hyperbolic structure. We show how the method leads to a construction of counterexamples to the equivariant K = … We introduce the method of complex angle scaling to study deformations of hyperbolic structure. We show how the method leads to a construction of counterexamples to the equivariant K = 2 conjecture for hyperbolic convex hulls.

COMPUTATION IN WORD-HYPERBOLIC GROUPS

2001-08-01
D. B. A. Epstein , Derek F. Holt
We describe two practical algorithms for computing with word-hyperbolic groups, both of which we have implemented. The first is a method for estimating the maximum width, if it exists, of … We describe two practical algorithms for computing with word-hyperbolic groups, both of which we have implemented. The first is a method for estimating the maximum width, if it exists, of geodesic bigons in the Cayley graph of a finitely presented group G. Our procedure will terminate if and only this maximum width exists, and it has been proved by Papasoglu that this is the case if and only if G is word-hyperbolic. So the algorithm amounts to a method of verifying the property of word-hyperbolicity of G. The aim of the second algorithm is to compute the thinness constant for geodesic triangles in the Cayley graph of G. This seems to be a much more difficult problem, but our implementation does succeed with straightforward examples. Both algorithms involve substantial computations with finite state automata.
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KNUTH–BENDIX FOR GROUPS WITH INFINITELY MANY RULES

2000-10-01
D. B. A. Epstein , Paul J. Sanders
We introduce a new class of groups with solvable word problem, namely groups specified by a confluent set of short-lex-reducing Knuth–Bendix rules which form a regular language. This simultaneously generalizes … We introduce a new class of groups with solvable word problem, namely groups specified by a confluent set of short-lex-reducing Knuth–Bendix rules which form a regular language. This simultaneously generalizes short-lex-automatic groups and groups with a finite confluent set of short-lex-reducing rules. We describe a computer program which looks for such a set of rules in an arbitrary finitely presented group. Our main theorem is that our computer program finds the set of rules, if it exists, given enough time and space. (This is an optimistic description of our result — for the more pessimistic details, see the body of the paper.) The set of rules is embodied in a finite state automaton in two variables. A central feature of our program is an operation, which we call welding, used to combine existing rules with new rules as they are found. Welding can be defined on arbitrary finite state automata, and we investigate this operation in abstract, proving that it can be considered as a process which takes as input one regular language and outputs another regular language. In our programs we need to convert several nondeterministic finite state automata to deterministic versions accepting the same language. We show how to improve somewhat on the standard subset construction, due to special features in our case. We axiomatize these special features, in the hope that these improvements can be used in other applications. The Knuth–Bendix process normally spends most of its time in reduction, so its efficiency depends on doing reduction quickly. Standard data structures for doing this can become very large, ultimately limiting the set of presentations of groups which can be so analyzed. We are able to give a method for rapid reduction using our much smaller two variable automaton, encoding the (usually infinite) regular language of rules found so far. Time taken for reduction in a given group is a small constant times the time taken for reduction in the best schemes known (see [5]), which is not too bad since we are reducing with respect to an infinite set of rules, whereas known schemes use a finite set of rules. We hope that the method described here might lead to the computation of automatic structures in groups for which this is currently infeasible. Some proofs have been omitted from this paper in the interests of brevity. Full details are provided in [4].
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Efficient computation in word-hyperbolic groups

2000-06-15
D. B. A. Epstein , Derek F. Holt
. We describe briefly some practical procedures for computing the various constants associated with a word-hyperbolic group, and report on the performance of their implementations in the KBMAG package on … . We describe briefly some practical procedures for computing the various constants associated with a word-hyperbolic group, and report on the performance of their implementations in the KBMAG package on a number of examples. More complete technical details will be published elsewhere.

THREE-DIMENSIONAL GEOMETRY AND TOPOLOGY, VOLUME 1 (Princeton Mathematical Series 35)

2000-01-01
D. B. A. Epstein
By William P. Thurston (edited by Silvio Levy): 311 pp., £29.95/US$39.50, isbn 0 691 08304 5 (Princeton University Press, 1997). By William P. Thurston (edited by Silvio Levy): 311 pp., £29.95/US$39.50, isbn 0 691 08304 5 (Princeton University Press, 1997).

Knuth-Bendix for groups with infinitely many rules

2000-01-01
D. B. A. Epstein , Paul J. Sanders
It is shown how to use a small finite state automaton in two variables in order to carry out the Knuth-Bendix process for rewriting words in a group in shortlex … It is shown how to use a small finite state automaton in two variables in order to carry out the Knuth-Bendix process for rewriting words in a group in shortlex order. The two-variable automaton can be used to store an infinite set of rules and to carry out fast reduction of arbitrary words using this infinite set. We introduce a new operation, which we call welding, which applies to an arbitrary finite state automaton. We show how to improve on the standard subset construction to determinize a non-deterministic automaton under special conditions which hold in our situation.
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Automatic Groups and Knuth-Bendix with Infinitely Many Rules

1998-01-01
D. B. A. Epstein , Paul J. Sanders
It is shown how to use a small finite state automaton in two variables in order to carry out part of the Knuth--Bendix process for rewriting words in a group. … It is shown how to use a small finite state automaton in two variables in order to carry out part of the Knuth--Bendix process for rewriting words in a group. The main objective is to provide a substitute for the most space-demanding module of the existing software which attempts to find a shortlex-automatic structure for a group. The two-variable automaton can be used to store an infinite set of rules and to carry out fast reduction of arbitrary words using this infinite set. We introduce a new operation, which we call welding, which applies to an arbitrary finite state automaton. In our context this operation is vital. We point out a small potential improvement in the subset algorithm for making a non-deterministic automaton deterministic.
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Computation in word-hyperbolic groups

1998-01-01
D. B. A. Epstein , Derek F. Holt
We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient … We describe a procedure which verifies that a group given by generators and relators is word-hyperbolic. This procedure always works with a group which is word-hyperbolic, provided there is sufficient memory and time devoted to the problem. If the group is not word-hyperbolic, the procedure continues indefinitely. We also describe a procedure which computes the thinness of geodesic triangles in the Cayley graph of a word-hyperbolic group. Again this procedure is bound to work, given sufficient memory and time.
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The second bounded cohomology of word-hyperbolic groups

1997-11-01
D. B. A. Epstein , Koji Fujiwara

Growth Functions and Automatic Groups

1996-01-01
D. B. A. Epstein , Anthony R. lano-Fletcher , Uri Zwick
In this paper we study growth functions of automatic and hyperbolic groups. In addition to standard growth functions, we also want to count the number of finite graphs isomorphic to … In this paper we study growth functions of automatic and hyperbolic groups. In addition to standard growth functions, we also want to count the number of finite graphs isomorphic to a given finite graph in the ball of radius n around the identity element in the Cayley graph. This topic was introduced to us by K. Saito [1991]. We report on fast methods to compute the growth function once we know the automatic structure. We prove that for a geodesic automatic structure, the growth function for any fixed finite connected graph is a rational function. For a word-hyperbolic group, we show that one can choose the denominator of the rational function independently of the finite graph.
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On Frobenius groups of finite Morley rank II

1994-08-11
D. B. A. Epstein , Ali Nesin
Abstract This is a continuation of the previous article by Nesin in this volume. The reader is invited to read the introduction of that article. We shall adopt the same … Abstract This is a continuation of the previous article by Nesin in this volume. The reader is invited to read the introduction of that article. We shall adopt the same notation throughout. In this article, we will first show that if T &amp;lt; B is a Frobenius group of finite Morley rank where Tis finite, then Bis split as U × T (Theorem 3.1). Then we will deal with the minimal counterexamples to the following two conjectures.

Word Processing in Groups

1992-11-02
D. B. A. Epstein
From the Publisher: This study in combinatorial group theory introduces the concept of automatic groups. It contains a succinct introduction to the theory of regular languages, a discussion of related … From the Publisher: This study in combinatorial group theory introduces the concept of automatic groups. It contains a succinct introduction to the theory of regular languages, a discussion of related topics in combinatorial group theory, and the connections between automatic groups and geometry which motivated the development of this new theory. It is of interest to mathematicians and computer scientists and includes open problems that will dominate the research for years to come.

The use of Knuth-Bendix methods to solve the wordproblem in automatic groups

1991-10-01
D. B. A. Epstein , Derek F. Holt , Sarah Rees

Euclidean decompositions of noncompact hyperbolic manifolds

1988-01-01
D. B. A. Epstein , Robert Penner
In this paper, we introduce a method for dividing up a noncompact hyperbolic manifold of finite volume into canonical Euclidean pieces.The construction first arose in the setting of surfaces (see … In this paper, we introduce a method for dividing up a noncompact hyperbolic manifold of finite volume into canonical Euclidean pieces.The construction first arose in the setting of surfaces (see [7]), and in this case one gets a canonical cell decomposition of the surface and a canonical Euclidean structure.(The Euclidean structure, of course, is not complete.)The conformal structure underlying this Euclidean structure does not agree with the underlying hyperbolic structure, but the two conformal structures are probably not too distant (cf.Sullivan's theorem [5] for an analogous result).This investigation arose from an attempt to understand the coordinates and cell decomposition of Teichmϋller space due to Harer and Mumford [6] and independently to Thurston.Such coordinates and cell decompositions are also provided in [3] and [7]; in the latter, the action of the mapping class group on the coordinates is considered.We would like to thank J. Harer for the inspiration of his work and for several helpful remarks.Our method is to work in Minkowski space and to represent a cusp by a point on the light-cone.The orbit of this point turns out to be discrete (even though the action of the group on the light-cone is ergodic), and we take the convex hull of the orbit.The boundary of this convex hull is decomposed into affine pieces, and one should think of the convex hull boundary as a kind of piecewise linear approximation to the upper sheet of the hyperboloid in Minkowski space.Each piece has a natural Euclidean structure.The suggestion that this might be possible first arose in a conversation between the authors and Lee Mosher.We thank Mosher for his contribution to this crucial idea.Comments by Brian Bowditch have also been helpful on a number of occasions.As a final credit, we wish to thank Bill Thurston.
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Analytical and geometric aspects of hyperbolic space : Warwick and Durham 1984

1987-01-01
D. B. A. Epstein , dimensional Topology
This work and its companion volume form the collected papers from two symposia held at Durham and Warwick in 1984. Volume I contains an expository account by David Epstein and … This work and its companion volume form the collected papers from two symposia held at Durham and Warwick in 1984. Volume I contains an expository account by David Epstein and his students of certain parts of Thurston's famous mimeographed notes. This is preceded by a clear and comprehensive account by S. J. Patterson of his fundamental work on measures on limit sets of Kleinian groups.

Commutators ofC ∞-diffeomorphisms. Appendix to “A Curious Remark Concerning the Geometric Transfer Map” by John N. Mather

1984-12-01
D. B. A. Epstein

Measures on Sigma-Compact Manifolds and their Equivalence Under Homeomorphisms

1983-02-01
Ricardo Berlanga , D. B. A. Epstein

Pointwise Periodic Homeomorphisms

1981-05-01
D. B. A. Epstein

Prime Ends

1981-05-01
D. B. A. Epstein

Transformation Groups and Natural Bundles

1979-03-01
D. B. A. Epstein , William P. Thurston

A Counterexample to the Periodic Orbit Conjecture in Codimension 3

1978-11-01
D. B. A. Epstein , Elmar Vogt

Foliations by Geodesic Circles

1978-01-01
D. B. A. Epstein

Leaves Without Holonomy

1977-12-01
D. B. A. Epstein , Kenneth C. Millett , David Tischler
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Stability of compact foliations

1977-01-01
D. B. A. Epstein , Harold Rosenberg

A topology for the space of foliations

1977-01-01
D. B. A. Epstein

Foliations with all leaves compact

1976-01-01
D. B. A. Epstein
The notion of the "volume" of a leaf in a foliated space is defined. If L is a compact leaf, then any leaf entering a small neighbourhood of L either … The notion of the "volume" of a leaf in a foliated space is defined. If L is a compact leaf, then any leaf entering a small neighbourhood of L either has a very large volume, or a volume which is approximatively an integral multiple of the volume of L. If all leaves are compact there are three related objects to study. Firstly the topology of the quotient space obtained by identifying each leaf to a point ; secondly the holonomy of a leaf ; and thirdly whether the leaves have a locally bounded volume. We prove various implications relating these concepts and we also give some counterexamples. We give a proof of the result, published by Ehresmann without proof, that in a foliated manifold, if a compact leaf has arbitrarily small neighbourhoods, which are saturated by the leaves, then its holonomy is finite.
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Natural tensors on Riemannian manifolds

1975-01-01
D. B. A. Epstein
Suppose that for each differentiate manifold M (without boundary) and each C°° metric g, we are given a C°° tensor field t {Mig) satisfying the following naturality axiom:If φ: (M, … Suppose that for each differentiate manifold M (without boundary) and each C°° metric g, we are given a C°° tensor field t {Mig) satisfying the following naturality axiom:If φ: (M, g) -> (TV, h) is an isometry of M onto an open subset of N, thenIn these circumstances we say that / is a natural tensor.Our objective is to elucidate the nature of natural tensors.It will emerge that the situation is too complicated for there to be any hope of a complete classification.Therefore we try to find additional conditions which can be imposed on a natural tensor which will imply that it lies in a good class of natural tensors.Our main results are as follows.In § 5 we classify all natural tensors which depend in a polynomial way of the oo-jet of g.In § 6 we show that it is sufficient for the dependence on the oo-jet to be a differentiable dependence (we demand C°°-dependence in Theorem 6.2, but the proof obviously goes through with less differentiability), if in addition the tensor is homogeneous (Definition 5.1): these two conditions imply polynomial dependence.In Theorem 7.3 we prove a special result where only homogeneity is assumed and nothing whatever concerning the dependence on the oo-jet of g.The fact that this is not trivial is shown by the existence of an example of a natural tensor depending only on the 4-jet of g, but with the dependence not even continuous (Theorem 4.1).We observe in passing that the space of germs of C°° Riemannian manifolds of dimension 2 can be parametrized in terms of the orbits of a linear 0(2)-action on an infinite dimensional vector space (Corollary 2.4).Finally we show that there is a unique "natural" connection V for Riemannian manifolds-namely the Levi-Civita connection.In other words the fact that V is torsion free and preserves the metric follows from the naturality.(For a precise statement, see Theorem 5.6.)This works was stimulated by G. Lusztig when he asked whether the Levi-Civita connection was the unique natural connection.This was during lectures on the work of Atiyah, Bott and Patodi [1] whose treatment of Gilkey's theorem has heavily influenced this paper.In fact Gilkey's theorem deals with the problem of classifying natural g-forms.Here we relax the condition that
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Foliations with all leaves compact

1975-01-01
D. B. A. Epstein
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A 3-manifold which is not a product

1973-11-01
D. B. A. Epstein , G. P. Scott
The purpose of this note is to construct a non-compact 3-manifold M with the following properties: (i) M is orientable and irreducible (ii) ∂ M is connected and compact and … The purpose of this note is to construct a non-compact 3-manifold M with the following properties: (i) M is orientable and irreducible (ii) ∂ M is connected and compact and its inclusion in M is a homotopy equivalence (iii) M is not homeomorphic to ∂ M × R + . (Throughout this paper, all manifolds will be PL and all embeddings will be PL and locally flat.)

THE TOPOLOGY OF CLASSICAL GROUPS AND RELATED TOPICS

1973-07-01
D. B. A. Epstein
By S. Y. Husseini: pp. viii, 128. £4.75; $11.40. (Gordon and Breach, Science Publishers, Inc., New York, 1970.) By S. Y. Husseini: pp. viii, 128. £4.75; $11.40. (Gordon and Breach, Science Publishers, Inc., New York, 1970.)

Periodic Flows on Three-Manifolds

1972-01-01
D. B. A. Epstein

Common Coauthors

Coauthor Papers Together
Albert Marden 7
Nasir Rajpoot 6
Richard D. Canary 5
Derek F. Holt 5
Vladimir Marković 5
Simon Graham 3
Yee‐Wah Tsang 3
Paul J. Sanders 3
William P. Thurston 2
Talha Qaiser 2
Daiki Taniyama 2
Kazuaki Nakane 2
Naoya Sakamoto 2
Imaad Mujeeb 1
Vladimir Marković 1
R. D. Canary 1
Martin Kneser 1
Anthony R. lano-Fletcher 1
Mike Paterson 1
Robert Penner 1
David Tischler 1
Sarah Rees 1
Ali Nesin 1
R. L. E. Schwarzenberger 1
David Snead 1
A. Marden 1
P.L. Green 1
S. J. Patterson 1
Harold Rosenberg 1
G. P. Scott 1
Koji Fujiwara 1
Linda Cheung 1
Heiner Zieschang 1
Stella Pelengaris 1
dimensional Topology 1
Uri Zwick 1
Zia Aftab 1
Muhammad Shaban 1
Elmar Vogt 1
Ian A. Cree 1
Michael Khan 1
Ricardo Berlanga 1
Korsuk Sirinukunwattana 1
Kenneth C. Millett 1
Shan E Ahmed Raza 1

Commonly Cited References

Gland segmentation in colon histology images: The glas challenge contest

2016-09-10
Korsuk Sirinukunwattana , Josien P. W. Pluim , Hao Chen +7 more
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The use of Knuth-Bendix methods to solve the wordproblem in automatic groups

1991-10-01
D. B. A. Epstein , Derek F. Holt , Sarah Rees

Foliations with all leaves compact

1976-01-01
D. B. A. Epstein
The notion of the "volume" of a leaf in a foliated space is defined. If L is a compact leaf, then any leaf entering a small neighbourhood of L either … The notion of the "volume" of a leaf in a foliated space is defined. If L is a compact leaf, then any leaf entering a small neighbourhood of L either has a very large volume, or a volume which is approximatively an integral multiple of the volume of L. If all leaves are compact there are three related objects to study. Firstly the topology of the quotient space obtained by identifying each leaf to a point ; secondly the holonomy of a leaf ; and thirdly whether the leaves have a locally bounded volume. We prove various implications relating these concepts and we also give some counterexamples. We give a proof of the result, published by Ehresmann without proof, that in a foliated manifold, if a compact leaf has arbitrarily small neighbourhoods, which are saturated by the leaves, then its holonomy is finite.
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Quasiconformal homeomorphisms and the convex hull boundary

2004-01-01
D. B. A. Epstein , Albert Marden , Vladimir Marković
We investigate the relationship between an open simply-connected region Ω ⊂ S 2 and the boundary Y of the hyperbolic convex hull in H 3 of S 2 \ Ω.A … We investigate the relationship between an open simply-connected region Ω ⊂ S 2 and the boundary Y of the hyperbolic convex hull in H 3 of S 2 \ Ω.A counterexample is given to Thurston's conjecture that these spaces are related by a 2-quasiconformal homeomorphism which extends to the identity map on their common boundary, in the case when the homeomorphism is required to respect any group of Möbius transformations which preserves Ω.We show that the best possible universal lipschitz constant for the nearest point retraction r : Ω → Y is 2. We find explicit universal constants 0 < c 2 < c 1 , such that no pleating map which bends more than c 1 in some interval of unit length is an embedding, and such that any pleating map which bends less than c 2 in each interval of unit length is embedded.We show that every K-quasiconformal homeomorphism D 2 → D 2 is a (K, a(K))-quasi-isometry, where a(K) is an explicitly computed function.The multiplicative constant is best possible and the additive constant a(K) is best possible for some values of K.
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Word Processing in Groups

1992-11-02
D. B. A. Epstein
From the Publisher: This study in combinatorial group theory introduces the concept of automatic groups. It contains a succinct introduction to the theory of regular languages, a discussion of related … From the Publisher: This study in combinatorial group theory introduces the concept of automatic groups. It contains a succinct introduction to the theory of regular languages, a discussion of related topics in combinatorial group theory, and the connections between automatic groups and geometry which motivated the development of this new theory. It is of interest to mathematicians and computer scientists and includes open problems that will dominate the research for years to come.

A counterexample to the periodic orbit conjecture

1976-12-01
Dennis Sullivan
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Deep Residual Learning for Image Recognition

2016-06-01
Kaiming He , Xiangyu Zhang , Shaoqing Ren +1 more
Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly … Deeper neural networks are more difficult to train. We present a residual learning framework to ease the training of networks that are substantially deeper than those used previously. We explicitly reformulate the layers as learning residual functions with reference to the layer inputs, instead of learning unreferenced functions. We provide comprehensive empirical evidence showing that these residual networks are easier to optimize, and can gain accuracy from considerably increased depth. On the ImageNet dataset we evaluate residual nets with a depth of up to 152 layers - 8× deeper than VGG nets [40] but still having lower complexity. An ensemble of these residual nets achieves 3.57% error on the ImageNet test set. This result won the 1st place on the ILSVRC 2015 classification task. We also present analysis on CIFAR-10 with 100 and 1000 layers. The depth of representations is of central importance for many visual recognition tasks. Solely due to our extremely deep representations, we obtain a 28% relative improvement on the COCO object detection dataset. Deep residual nets are foundations of our submissions to ILSVRC & COCO 2015 competitions1, where we also won the 1st places on the tasks of ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation.
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Lectures on Quasiconformal Mappings

2006-07-14
Lars V. Ahlfors
The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces Editors' notes The Additional Chapters: A supplement to Ahlfors's lectures … The Ahlfors Lectures: Acknowledgments Differentiable quasiconformal mappings The general definition Extremal geometric properties Boundary correspondence The mapping theorem Teichmuller spaces Editors' notes The Additional Chapters: A supplement to Ahlfors's lectures Complex dynamics and quasiconformal mappings Hyperbolic structures on three-manifolds that fiber over the circle.
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Sur les Groupes Hyperboliques d’après Mikhael Gromov

1990-01-01
Étienne Ghys , Pierre de la Harpe

Quantifying the effects of data augmentation and stain color normalization in convolutional neural networks for computational pathology

2019-08-21
David Tellez , Geert Litjens , Péter Bándi +4 more
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Roto-Translation Covariant Convolutional Networks for Medical Image Analysis

2018-01-01
Erik J. Bekkers , Maxime W. Lafarge , Mitko Veta +3 more
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RotDCF: Decomposition of Convolutional Filters for Rotation-Equivariant Deep Networks

2018-01-01
Xiuyuan Cheng , Qiang Qiu , A.R. Calderbank +1 more
Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision … Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group geometry simultaneously, namely a rotation-equivariant CNN with decomposed convolutional filters (RotDCF). This decomposition facilitates computing the joint convolution, which is proved to be necessary for the group equivariance. It significantly reduces the model size and computational complexity while preserving performance, and truncation of the bases expansion serves implicitly to regularize the filters. On datasets involving in-plane and out-of-plane object rotations, RotDCF deep features demonstrate greater robustness and interpretability than regular CNNs. The stability of the equivariant representation to input variations is also proved theoretically under generic assumptions on the filters in the decomposed form. The RotDCF framework can be extended to groups other than rotations, providing a general approach which achieves both group equivariance and representation stability at a reduced model size.
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What can topology tell us about the neural code?

2016-09-27
Carina Curto
Neuroscience is undergoing a period of rapid experimental progress and expansion. New mathematical tools, previously unknown in the neuroscience community, are now being used to tackle fundamental questions and analyze … Neuroscience is undergoing a period of rapid experimental progress and expansion. New mathematical tools, previously unknown in the neuroscience community, are now being used to tackle fundamental questions and analyze emerging data sets. Consistent with this trend, the last decade has seen an uptick in the use of topological ideas and methods in neuroscience. In this paper I will survey recent applications of topology in neuroscience, and explain why topology is an especially natural tool for understanding neural codes.
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Persistent homology transform for modeling shapes and surfaces

2014-12-01
Katharine Turner , Sayan Mukherjee , Douglas Boyer
We introduce a statistic, the persistent homology transform (PHT), to model surfaces in R3 and shapes in R2. This statistic is a collection of persistence diagrams—multiscale topological summaries used extensively … We introduce a statistic, the persistent homology transform (PHT), to model surfaces in R3 and shapes in R2. This statistic is a collection of persistence diagrams—multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We provide a constructive proof that the map from the space of simplicial complexes in R3 into the space spanned by this statistic is injective. This implies that we can use it to determine a metric on the space of piecewise linear shapes. Stability results justify that we can approximate this metric using finitely many persistence diagrams. We illustrate the utility of this statistic on simulated and real data.
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Dynamic Routing Between Capsules

2017-01-01
Sara Sabour , Nicholas Frosst , Geoffrey E. Hinton
A capsule is a group of neurons whose activity vector represents the instantiation parameters of a specific type of entity such as an object or an object part. We use … A capsule is a group of neurons whose activity vector represents the instantiation parameters of a specific type of entity such as an object or an object part. We use the length of the activity vector to represent the probability that the entity exists and its orientation to represent the instantiation parameters. Active capsules at one level make predictions, via transformation matrices, for the instantiation parameters of higher-level capsules. When multiple predictions agree, a higher level capsule becomes active. We show that a discrimininatively trained, multi-layer capsule system achieves state-of-the-art performance on MNIST and is considerably better than a convolutional net at recognizing highly overlapping digits. To achieve these results we use an iterative routing-by-agreement mechanism: A lower-level capsule prefers to send its output to higher level capsules whose activity vectors have a big scalar product with the prediction coming from the lower-level capsule.
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General E(2)-Equivariant Steerable CNNs

2019-01-01
Maurice Weiler , Gabriele Cesa
The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been … The big empirical success of group equivariant networks has led in recent years to the sprouting of a great variety of equivariant network architectures. A particular focus has thereby been on rotation and reflection equivariant CNNs for planar images. Here we give a general description of E(2)-equivariant convolutions in the framework of Steerable CNNs. The theory of Steerable CNNs thereby yields constraints on the convolution kernels which depend on group representations describing the transformation laws of feature spaces. We show that these constraints for arbitrary group representations can be reduced to constraints under irreducible representations. A general solution of the kernel space constraint is given for arbitrary representations of the Euclidean group E(2) and its subgroups. We implement a wide range of previously proposed and entirely new equivariant network architectures and extensively compare their performances. E(2)-steerable convolutions are further shown to yield remarkable gains on CIFAR-10, CIFAR-100 and STL-10 when used as drop in replacement for non-equivariant convolutions.
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Rotation Equivariant CNNs for Digital Pathology

2018-01-01
Bastiaan S. Veeling , Jasper Linmans , Jim Winkens +2 more
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Substantial boundary points for plane domains and Gardiner's conjecture

2000-01-01
Nikola Lakic
The local dilatation Hp at a boundary point of a quasiconformal mapping on a plane domain of arbitrary connectivity is defined and it is shown that there is always a … The local dilatation Hp at a boundary point of a quasiconformal mapping on a plane domain of arbitrary connectivity is defined and it is shown that there is always a substantial point p, such that Hp = H, where H is the boundary dilatation. Infinitesimal local boundary dilatation is also defined and it is shown that the sets of infinitesimally substantial and substantial boundary points coincide.

Periodic Flows on Three-Manifolds

1972-01-01
D. B. A. Epstein

Die erste Cohomologiegruppe von Überlagerungen und Homotopie-Eigenschaften dreidimensionaler Mannigfaltigkeiten

1949-12-01
Bonny Specker

SRN: Side-Output Residual Network for Object Symmetry Detection in the Wild

2017-07-01
Wei Ke , Jie Chen , Jianbin Jiao +2 more
In this paper, we establish a baseline for object symmetry detection in complex backgrounds by presenting a new benchmark and an end-to-end deep learning approach, opening up a promising direction … In this paper, we establish a baseline for object symmetry detection in complex backgrounds by presenting a new benchmark and an end-to-end deep learning approach, opening up a promising direction for symmetry detection in the wild. The new benchmark, named Sym-PASCAL, spans challenges including object diversity, multi-objects, part-invisibility, and various complex backgrounds that are far beyond those in existing datasets. The proposed symmetry detection approach, named Side-output Residual Network (SRN), leverages output Residual Units (RUs) to fit the errors between the object symmetry ground-truth and the outputs of RUs. By stacking RUs in a deep-to-shallow manner, SRN exploits the flow of errors among multiple scales to ease the problems of fitting complex outputs with limited layers, suppressing the complex backgrounds, and effectively matching object symmetry of different scales. Experimental results validate both the benchmark and its challenging aspects related to real-world images, and the state-of-the-art performance of our symmetry detection approach. The benchmark and the code for SRN are publicly available at https://github.com/KevinKecc/SRN.
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MILD-Net: Minimal information loss dilated network for gland instance segmentation in colon histology images

2018-12-20
Simon Graham , Hao Chen , Jevgenij Gamper +5 more
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Extremal Quasiconformal Mappings of the Disk

2002-01-01
Edgar Reich

Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift

2015-02-11
Sergey Ioffe , Christian Szegedy
Training Deep Neural Networks is complicated by the fact that the distribution of each layer's inputs changes during training, as the parameters of the previous layers change. This slows down … Training Deep Neural Networks is complicated by the fact that the distribution of each layer's inputs changes during training, as the parameters of the previous layers change. This slows down the training by requiring lower learning rates and careful parameter initialization, and makes it notoriously hard to train models with saturating nonlinearities. We refer to this phenomenon as internal covariate shift, and address the problem by normalizing layer inputs. Our method draws its strength from making normalization a part of the model architecture and performing the normalization for each training mini-batch. Batch Normalization allows us to use much higher learning rates and be less careful about initialization. It also acts as a regularizer, in some cases eliminating the need for Dropout. Applied to a state-of-the-art image classification model, Batch Normalization achieves the same accuracy with 14 times fewer training steps, and beats the original model by a significant margin. Using an ensemble of batch-normalized networks, we improve upon the best published result on ImageNet classification: reaching 4.9% top-5 validation error (and 4.8% test error), exceeding the accuracy of human raters.

Simple Word Problems in Universal Algebras

1983-01-01
Donald E. Knuth , PETER B. BENDIX

Oriented Response Networks

2017-07-01
Yanzhao Zhou , Qixiang Ye , Qiang Qiu +1 more
Deep Convolution Neural Networks (DCNNs) are capable of learning unprecedentedly effective image representations. However, their ability in handling significant local and global image rotations remains limited. In this paper, we … Deep Convolution Neural Networks (DCNNs) are capable of learning unprecedentedly effective image representations. However, their ability in handling significant local and global image rotations remains limited. In this paper, we propose Active Rotating Filters (ARFs) that actively rotate during convolution and produce feature maps with location and orientation explicitly encoded. An ARF acts as a virtual filter bank containing the filter itself and its multiple unmaterialised rotated versions. During back-propagation, an ARF is collectively updated using errors from all its rotated versions. DCNNs using ARFs, referred to as Oriented Response Networks (ORNs), can produce within-class rotation-invariant deep features while maintaining inter-class discrimination for classification tasks. The oriented response produced by ORNs can also be used for image and object orientation estimation tasks. Over multiple state-of-the-art DCNN architectures, such as VGG, ResNet, and STN, we consistently observe that replacing regular filters with the proposed ARFs leads to significant reduction in the number of network parameters and improvement in classification performance. We report the best results on several commonly used benchmarks.
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Why do deep convolutional networks generalize so poorly to small image transformations

2018-05-30
Aharon Azulay , Yair Weiss
Convolutional Neural Networks (CNNs) are commonly assumed to be invariant to small image transformations: either because of the convolutional architecture or because they were trained using data augmentation. Recently, several … Convolutional Neural Networks (CNNs) are commonly assumed to be invariant to small image transformations: either because of the convolutional architecture or because they were trained using data augmentation. Recently, several authors have shown that this is not the case: small translations or rescalings of the input image can drastically change the network's prediction. In this paper, we quantify this phenomena and ask why neither the convolutional architecture nor data augmentation are sufficient to achieve the desired invariance. Specifically, we show that the convolutional architecture does not give invariance since architectures ignore the classical sampling theorem, and data augmentation does not give invariance because the CNNs learn to be invariant to transformations only for images that are very similar to typical images from the training set. We discuss two possible solutions to this problem: (1) antialiasing the intermediate representations and (2) increasing data augmentation and show that they provide only a partial solution at best. Taken together, our results indicate that the problem of insuring invariance to small image transformations in neural networks while preserving high accuracy remains unsolved.

Loopy belief propagation for approximate inference: an empirical study

1999-07-30
Kevin P. Murphy , Yair Weiss , Michael I. Jordan
Recently, researchers have demonstrated that belief -- the use of Pearl's polytree algorithm in a Bayesian network with loops -- can perform well in the context of error-correcting codes. The … Recently, researchers have demonstrated that belief -- the use of Pearl's polytree algorithm in a Bayesian network with loops -- can perform well in the context of error-correcting codes. The most dramatic instance of this is the near Shannon-limit performance of Turbo Codes -- codes whose decoding algorithm is equivalent to loopy belief propagation in a chain-structured Bayesian network. In this paper we ask: is there something special about the error-correcting code context, or does loopy propagation work as an approximate inference scheme in a more general setting? We compare the marginals computed using loopy propagation to the exact ones in four Bayesian network architectures, including two real-world networks: ALARM and QMR. We find that the loopy beliefs often converge and when they do, they give a good approximation to the correct marginals. However, on the QMR network, the loopy beliefs oscillated and had no obvious relationship to the correct posteriors. We present some initial investigations into the cause of these oscillations, and show that some simple methods of preventing them lead to the wrong results.
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DCAN: Deep Contour-Aware Networks for Accurate Gland Segmentation

2016-06-01
Hao Chen , Xiaojuan Qi , Lequan Yu +1 more
The morphology of glands has been used routinely by pathologists to assess the malignancy degree of adenocarcinomas. Accurate segmentation of glands from histology images is a crucial step to obtain … The morphology of glands has been used routinely by pathologists to assess the malignancy degree of adenocarcinomas. Accurate segmentation of glands from histology images is a crucial step to obtain reliable morphological statistics for quantitative diagnosis. In this paper, we proposed an efficient deep contour-aware network (DCAN) to solve this challenging problem under a unified multi-task learning framework. In the proposed network, multi-level contextual features from the hierarchical architecture are explored with auxiliary supervision for accurate gland segmentation. When incorporated with multi-task regularization during the training, the discriminative capability of intermediate features can be further improved. Moreover, our network can not only output accurate probability maps of glands, but also depict clear contours simultaneously for separating clustered objects, which further boosts the gland segmentation performance. This unified framework can be efficient when applied to large-scale histopathological data without resorting to additional steps to generate contours based on low-level cues for post-separating. Our method won the 2015 MICCAI Gland Segmentation Challenge out of 13 competitive teams, surpassing all the other methods by a significant margin.
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SegNet: A Deep Convolutional Encoder-Decoder Architecture for Image Segmentation

2017-01-02
Vijay Badrinarayanan , A. C. Kendall , Roberto Cipolla
We present a novel and practical deep fully convolutional neural network architecture for semantic pixel-wise segmentation termed SegNet. This core trainable segmentation engine consists of an encoder network, a corresponding … We present a novel and practical deep fully convolutional neural network architecture for semantic pixel-wise segmentation termed SegNet. This core trainable segmentation engine consists of an encoder network, a corresponding decoder network followed by a pixel-wise classification layer. The architecture of the encoder network is topologically identical to the 13 convolutional layers in the VGG16 network [1] . The role of the decoder network is to map the low resolution encoder feature maps to full input resolution feature maps for pixel-wise classification. The novelty of SegNet lies is in the manner in which the decoder upsamples its lower resolution input feature map(s). Specifically, the decoder uses pooling indices computed in the max-pooling step of the corresponding encoder to perform non-linear upsampling. This eliminates the need for learning to upsample. The upsampled maps are sparse and are then convolved with trainable filters to produce dense feature maps. We compare our proposed architecture with the widely adopted FCN [2] and also with the well known DeepLab-LargeFOV [3] , DeconvNet [4] architectures. This comparison reveals the memory versus accuracy trade-off involved in achieving good segmentation performance. SegNet was primarily motivated by scene understanding applications. Hence, it is designed to be efficient both in terms of memory and computational time during inference. It is also significantly smaller in the number of trainable parameters than other competing architectures and can be trained end-to-end using stochastic gradient descent. We also performed a controlled benchmark of SegNet and other architectures on both road scenes and SUN RGB-D indoor scene segmentation tasks. These quantitative assessments show that SegNet provides good performance with competitive inference time and most efficient inference memory-wise as compared to other architectures. We also provide a Caffe implementation of SegNet and a web demo at http://mi.eng.cam.ac.uk/projects/segnet/.
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Fully convolutional networks for semantic segmentation

2015-06-01
Jonathan Long , Evan Shelhamer , Trevor Darrell
Convolutional networks are powerful visual models that yield hierarchies of features. We show that convolutional networks by themselves, trained end-to-end, pixels-to-pixels, exceed the state-of-the-art in semantic segmentation. Our key insight … Convolutional networks are powerful visual models that yield hierarchies of features. We show that convolutional networks by themselves, trained end-to-end, pixels-to-pixels, exceed the state-of-the-art in semantic segmentation. Our key insight is to build "fully convolutional" networks that take input of arbitrary size and produce correspondingly-sized output with efficient inference and learning. We define and detail the space of fully convolutional networks, explain their application to spatially dense prediction tasks, and draw connections to prior models. We adapt contemporary classification networks (AlexNet [20], the VGG net [31], and GoogLeNet [32]) into fully convolutional networks and transfer their learned representations by fine-tuning [3] to the segmentation task. We then define a skip architecture that combines semantic information from a deep, coarse layer with appearance information from a shallow, fine layer to produce accurate and detailed segmentations. Our fully convolutional network achieves state-of-the-art segmentation of PASCAL VOC (20% relative improvement to 62.2% mean IU on 2012), NYUDv2, and SIFT Flow, while inference takes less than one fifth of a second for a typical image.
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Barcodes: The persistent topology of data

2007-10-26
Robert Ghrist
This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool … This article surveys recent work of Carlsson and collaborators on applications of computational algebraic topology to problems of feature detection and shape recognition in high-dimensional data. The primary mathematical tool considered is a homology theory for point-cloud data sets—
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Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)

2015-11-23
Djork-Arné Clevert , Thomas Unterthiner , Sepp Hochreiter
We introduce the exponential linear (ELU) which speeds up learning in deep neural networks and leads to higher classification accuracies. Like rectified linear units (ReLUs), leaky ReLUs (LReLUs) and parametrized … We introduce the exponential linear (ELU) which speeds up learning in deep neural networks and leads to higher classification accuracies. Like rectified linear units (ReLUs), leaky ReLUs (LReLUs) and parametrized ReLUs (PReLUs), ELUs alleviate the vanishing gradient problem via the identity for positive values. However, ELUs have improved learning characteristics compared to the units with other activation functions. In contrast to ReLUs, ELUs have negative values which allows them to push mean unit activations closer to zero like batch normalization but with lower computational complexity. Mean shifts toward zero speed up learning by bringing the normal gradient closer to the unit natural gradient because of a reduced bias shift effect. While LReLUs and PReLUs have negative values, too, they do not ensure a noise-robust deactivation state. ELUs saturate to a negative value with smaller inputs and thereby decrease the forward propagated variation and information. Therefore, ELUs code the degree of presence of particular phenomena in the input, while they do not quantitatively model the degree of their absence. In experiments, ELUs lead not only to faster learning, but also to significantly better generalization performance than ReLUs and LReLUs on networks with more than 5 layers. On CIFAR-100 ELUs networks significantly outperform ReLU networks with batch normalization while batch normalization does not improve ELU networks. ELU networks are among the top 10 reported CIFAR-10 results and yield the best published result on CIFAR-100, without resorting to multi-view evaluation or model averaging. On ImageNet, ELU networks considerably speed up learning compared to a ReLU network with the same architecture, obtaining less than 10% classification error for a single crop, single model network.
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Persistent homology: an introduction and a new text representation for natural language processing

2013-08-03
Xiaojin Zhu
Persistent homology is a mathematical tool from topological data analysis. It performs multi-scale analysis on a set of points and identifies clusters, holes, and voids therein. These latter topological structures … Persistent homology is a mathematical tool from topological data analysis. It performs multi-scale analysis on a set of points and identifies clusters, holes, and voids therein. These latter topological structures complement standard feature representations, making persistent homology an attractive feature extractor for artificial intelligence. Research on persistent homology for AI is in its infancy, and is currently hindered by two issues: the lack of an accessible introduction to AI researchers, and the paucity of applications. In response, the first part of this paper presents a tutorial on persistent homology specifically aimed at a broader audience without sacrificing mathematical rigor. The second part contains one of the first applications of persistent homology to natural language processing. Specifically, our Similarity Filtration with Time Skeleton (SIFTS) algorithm identifies holes that can be interpreted as semantic tie-backs in a text document, providing a new document structure representation. We illustrate our algorithm on documents ranging from nursery rhymes to novels, and on a corpus with child and adolescent writings.

Average bending of convex pleated planes in hyperbolic three-space

1998-04-24
Martin Bridgeman

Functors between tensored categories

1966-09-01
D. B. A. Epstein

The boundary correspondence under quasiconformal mappings

1956-01-01
Arne Beurling , Lars V. Ahlfors
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Travaux de thurston sur les groupes quasi-fuchsiens et les variétés hyperboliques de dimension 3 fibrées sur s1

2006-11-24
Dennis Sullivan

Persistent Homology for Path Planning in Uncertain Environments

2015-04-03
Subhrajit Bhattacharya , Robert Ghrist , Vijay Kumar
We address the fundamental problem of goal-directed path planning in an uncertain environment represented as a probability (of occupancy) map. Most methods generally use a threshold to reduce the grayscale … We address the fundamental problem of goal-directed path planning in an uncertain environment represented as a probability (of occupancy) map. Most methods generally use a threshold to reduce the grayscale map to a binary map before applying off-the-shelf techniques to find the best path. This raises the somewhat ill-posed question, what is the right (optimal) value to threshold the map? We instead suggest a persistent homology approach to the problem-a topological approach in which we seek the homology class of trajectories that is most persistent for the given probability map. In other words, we want the class of trajectories that is free of obstacles over the largest range of threshold values. In order to make this problem tractable, we use homology in ℤ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> coefficients (instead of the standard ℤ coefficients), and describe how graph search-based algorithms can be used to find trajectories in different homology classes. Our simulation results demonstrate the efficiency and practical applicability of the algorithm proposed in this paper.paper.
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On the Groups H(Π, n), I

1953-07-01
Samuel Eilenberg , Saunders Mac Lane

On Dehn's Lemma and the Asphericity of Knots

1957-07-01
C. D. Papakyriakopoulos
The present paper contains a 'roof of Dehn's lemma and an analogous result that we call the sphere theorem, from which other theorems follow.' DEHN'S LEMMA. Let M be a … The present paper contains a 'roof of Dehn's lemma and an analogous result that we call the sphere theorem, from which other theorems follow.' DEHN'S LEMMA. Let M be a 3-manifold, compact or not, with boundary which may be empty, and in M let D be a 2-cell with self-intersections (singularities), having as boundary the simple closed polygonal curve C and such that there exists a closed neighborhood of C in D which is an annulus (i.e. no point of C is singular). Then there exists a 2-cell Do with boundary C, semi-linearly imbedded in M. SPHERE THEOREM. Let M be an orientable 3-manifold, compact or not, with boundary which may be empty, such that 7r2(M) # 0, and which can be semi-linearly2 imbedded in a 3-manifold N, having the following property: the commutator quotient group of any non-trivial (but not necessarily proper) finitely generated subgroup of 7r,(N) has an element of infinite order (n.b. in particular this holds if 7r,(N) = 1). Then there exists a 2-sphere S semi-linearly imbedded in M, such that3 S X 0 in M. Dehn's lemma was included in a 1910 paper of M. Dehn [4] p. 147, but in 1928 H. Kneser [13] p. 260, observed that Dehn's proof contained a serious gap. In 1935 and 1938 appeared two papers by I. Johansson [11], [12], on Dehn's lemma. In the second one, p. 659, he proves that, if Dehn's lemma holds for all orientable 3-manifolds, it then holds for all non-orientable ones. We now prove in this paper that Dehn's lemma holds for all orientable 3-manifolds. Our proof makes use also of I. Johansson's first paper. As far as the sphere theorem is concerned we have to remark that, to the best knowledge of this author, the first one to attempt a theorem of this kind was H. Kneser in 1928, [13] p. 257; however his proof does not seem to be conclusive. In 1937 S. Eilenberg [5] p. 242, Remark 1, observed a relation between the nonvanishing of the second homotopy group and the existence of a non-contractible 2-sphere. Finally in 1939 J. H. C. Whitehead [25] p. 161, posed a problem which stimulated the author to prove the sphere theorem, stated above. We emphasize that, if 7r,(N) is a free group4 then the hypotheses of the sphere theorem are fulfilled, according to the following NIELSEN-SCHREIER THEOREM. Every subgroup of a free group is itself a free group.5
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TI-POOLING: Transformation-Invariant Pooling for Feature Learning in Convolutional Neural Networks

2016-06-01
Dmitry Laptev , Nikolay Savinov , Joachim M. Buhmann +1 more
In this paper we present a deep neural network topology that incorporates a simple to implement transformationinvariant pooling operator (TI-POOLING). This operator is able to efficiently handle prior knowledge on … In this paper we present a deep neural network topology that incorporates a simple to implement transformationinvariant pooling operator (TI-POOLING). This operator is able to efficiently handle prior knowledge on nuisance variations in the data, such as rotation or scale changes. Most current methods usually make use of dataset augmentation to address this issue, but this requires larger number of model parameters and more training data, and results in significantly increased training time and larger chance of under-or overfitting. The main reason for these drawbacks is that that the learned model needs to capture adequate features for all the possible transformations of the input. On the other hand, we formulate features in convolutional neural networks to be transformation-invariant. We achieve that using parallel siamese architectures for the considered transformation set and applying the TI-POOLING operator on their outputs before the fully-connected layers. We show that this topology internally finds the most optimal "canonical" instance of the input image for training and therefore limits the redundancy in learned features. This more efficient use of training data results in better performance on popular benchmark datasets with smaller number of parameters when comparing to standard convolutional neural networks with dataset augmentation and to other baselines.
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Rotation Equivariant Vector Field Networks

2017-10-01
Diego Marcos , Michele Volpi , Nikos Komodakis +1 more
In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input image. If this relationship is explicitly encoded, instead of treated … In many computer vision tasks, we expect a particular behavior of the output with respect to rotations of the input image. If this relationship is explicitly encoded, instead of treated as any other variation, the complexity of the problem is decreased, leading to a reduction in the size of the required model. In this paper, we propose the Rotation Equivariant Vector Field Networks (RotEqNet), a Convolutional Neural Network (CNN) architecture encoding rotation equivariance, invariance and covariance. Each convolutional filter is applied at multiple orientations and returns a vector field representing magnitude and angle of the highest scoring orientation at every spatial location. We develop a modified convolution operator relying on this representation to obtain deep architectures. We test RotEqNet on several problems requiring different responses with respect to the inputs' rotation: image classification, biomedical image segmentation, orientation estimation and patch matching. In all cases, we show that RotEqNet offers extremely compact models in terms of number of parameters and provides results in line to those of networks orders of magnitude larger.
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Harmonic Networks: Deep Translation and Rotation Equivariance

2017-07-01
Daniel E. Worrall , Stephan J. Garbin , Daniyar Turmukhambetov +1 more
Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature … Translating or rotating an input image should not affect the results of many computer vision tasks. Convolutional neural networks (CNNs) are already translation equivariant: input image translations produce proportionate feature map translations. This is not the case for rotations. Global rotation equivariance is typically sought through data augmentation, but patch-wise equivariance is more difficult. We present Harmonic Networks or H-Nets, a CNN exhibiting equivariance to patch-wise translation and 360-rotation. We achieve this by replacing regular CNN filters with circular harmonics, returning a maximal response and orientation for every receptive field patch. H-Nets use a rich, parameter-efficient and fixed computational complexity representation, and we show that deep feature maps within the network encode complicated rotational invariants. We demonstrate that our layers are general enough to be used in conjunction with the latest architectures and techniques, such as deep supervision and batch normalization. We also achieve state-of-the-art classification on rotated-MNIST, and competitive results on other benchmark challenges.
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<scp>HER</scp>2 challenge contest: a detailed assessment of automated <scp>HER</scp>2 scoring algorithms in whole slide images of breast cancer tissues

2017-08-03
Talha Qaiser , Abhik Mukherjee , Chaitanya Reddy PB +7 more
Evaluating expression of the human epidermal growth factor receptor 2 (HER2) by visual examination of immunohistochemistry (IHC) on invasive breast cancer (BCa) is a key part of the diagnostic assessment … Evaluating expression of the human epidermal growth factor receptor 2 (HER2) by visual examination of immunohistochemistry (IHC) on invasive breast cancer (BCa) is a key part of the diagnostic assessment of BCa due to its recognized importance as a predictive and prognostic marker in clinical practice. However, visual scoring of HER2 is subjective, and consequently prone to interobserver variability. Given the prognostic and therapeutic implications of HER2 scoring, a more objective method is required. In this paper, we report on a recent automated HER2 scoring contest, held in conjunction with the annual PathSoc meeting held in Nottingham in June 2016, aimed at systematically comparing and advancing the state-of-the-art artificial intelligence (AI)-based automated methods for HER2 scoring.The contest data set comprised digitized whole slide images (WSI) of sections from 86 cases of invasive breast carcinoma stained with both haematoxylin and eosin (H&E) and IHC for HER2. The contesting algorithms predicted scores of the IHC slides automatically for an unseen subset of the data set and the predicted scores were compared with the 'ground truth' (a consensus score from at least two experts). We also report on a simple 'Man versus Machine' contest for the scoring of HER2 and show that the automated methods could beat the pathology experts on this contest data set.This paper presents a benchmark for comparing the performance of automated algorithms for scoring of HER2. It also demonstrates the enormous potential of automated algorithms in assisting the pathologist with objective IHC scoring.
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Micro-Net: A unified model for segmentation of various objects in microscopy images

2018-12-15
Shan E Ahmed Raza , Linda Cheung , Muhammad Shaban +5 more
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Sample Efficient Semantic Segmentation using Rotation Equivariant Convolutional Networks

2018-01-01
Jasper Linmans , Jim Winkens , Bastiaan S. Veeling +2 more
We propose a semantic segmentation model that exploits rotation and reflection symmetries. We demonstrate significant gains in sample efficiency due to increased weight sharing, as well as improvements in robustness … We propose a semantic segmentation model that exploits rotation and reflection symmetries. We demonstrate significant gains in sample efficiency due to increased weight sharing, as well as improvements in robustness to symmetry transformations. The group equivariant CNN framework is extended for segmentation by introducing a new equivariant (G-&gt;Z2)-convolution that transforms feature maps on a group to planar feature maps. Also, equivariant transposed convolution is formulated for up-sampling in an encoder-decoder network. To demonstrate improvements in sample efficiency we evaluate on multiple data regimes of a rotation-equivariant segmentation task: cancer metastases detection in histopathology images. We further show the effectiveness of exploiting more symmetries by varying the size of the group.
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Leveraging Unlabeled Whole-Slide-Images for Mitosis Detection

2018-01-01
Saad Ullah Akram , Talha Qaiser , Simon Graham +3 more
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Holomorphic families of injections

1986-01-01
Lipman Bers , H. L. Royden
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Hover-Net: Simultaneous segmentation and classification of nuclei in multi-tissue histology images

2019-09-18
Simon Graham , Quoc Dang Vu , Shan E Ahmed Raza +4 more
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