arXiv subjects

Gunnar Carlsson

Publications and source records attributed to Gunnar Carlsson.

At least 37 records · Page 2Linked to original sources

arXiv2020

Asymptotic transfer maps in parametrized K-theory

We define asymptotic transfers in bounded K-theory together with a context where this can be done in great generality. Controlled algebra plays a central role in many advances in geometric topology, including recent work on Novikov, Borel, and Farrell-Jones conjectures. One of the features that appears in various manifestations throughout the subject, starting with the original work of Farrell and Jones, is an asymptotic transfer whose meaning and construction depend on the geometric circumstances. We first develop a general framework that allows us to construct a version of asymptotic transfer maps for any finite aspherical complex. This framework is the equivariant parametrized K-theory with fibred control. We also include several fibrewise excision theorems for its computation and a discussion of where the standard tools break down and which tools replace them.

math.KT↗

arXiv2019

Excision in equivariant fibred G-theory

This paper provides a generalization of excision theorems in controlled algebra in the context of equivariant G-theory with fibred control and families of bounded actions. It also states and proves several characteristic features of this theory such as existence of the fibred assembly and the fibrewise trivialization.

math.KT↗

arXiv2019

Atiyah-Segal Derived Completions for Equivariant Algebraic G-Theory and K-Theory

In the mid 1980s, while working on establishing completion theorems for equivariant Algebraic K- Theory similar to the well-known Atiyah-Segal completion theorem for equivariant topological K-theory, the late Robert Thomason found the strong finiteness conditions that are required in such theorems to be too restrictive. Then he made a conjecture on the existence of a completion theorem in the sense of Atiyah and Segal for equivariant Algebraic G-theory, for actions of linear algebraic groups on schemes that holds without any of the strong finiteness conditions that are required in such theorems proven by him, and also appearing in the original Atiyah-Segal theorem. The main goal of the present paper is to provide a proof of this conjecture in as broad a context as possible, making use of the technique of derived completion, and to consider several of the applications. Our solution is broad enough to allow actions by all linear algebraic groups, irrespective of whether they are connected or not, and acting on any quasi-projective scheme of finite type over a field, irrespective of whether they are regular or projective. This allows us therefore to consider the Equivariant Algebraic G-Theory of large classes of varieties like all Toric varieties (for the action of a torus) and all Spherical varieties (for the action of a reductive group). Restricting to actions by split tori, we are also able to consider actions on Algebraic Spaces. These enable us to obtain a wide range of applications, some of which are briefly sketched and which we plan to explore in detail in the future. A comparison of our results with previously known results, none of which made use of derived completions, shows that without the use of derived completions, one can only obtain results which are indeed very restrictive.

math.AG↗

arXiv2019

Persistent homology of the sum metric

Given finite metric spaces $(X, d_X)$ and $(Y, d_Y)$, we investigate the persistent homology $PH_*(X \times Y)$ of the Cartesian product $X \times Y$ equipped with the sum metric $d_X + d_Y$. Interpreting persistent homology as a module over a polynomial ring, one might expect the usual Künneth short exact sequence to hold. We prove that it holds for $PH_0$ and $PH_1$, and we illustrate with the Hamming cube $\{0,1\}^k$ that it fails for $PH_n,\,\, n \geq 2$. For $n = 2$, the prediction for $PH_2(X \times Y)$ from the expected Künneth short exact sequence has a natural surjection onto $PH_2(X \times Y)$. We compute the nontrivial kernel of this surjection for the splitting of Hamming cubes $\{0,1\}^k = \{0,1\}^{k-1} \times \{0,1\}$. For all $n \geq 0$, the interleaving distance between the prediction for $PH_n(X \times Y)$ and the true persistent homology is bounded above by the minimum of the diameters of $X$ and $Y$. As preliminary results of independent interest, we establish an algebraic Künneth formula for simplicial modules over the ring $κ[\mathbb{R}_+]$ of polynomials with coefficients in a field $κ$ and exponents in $\mathbb{R}_+ = [0,\infty)$, as well as a Künneth formula for the persistent homology of $\mathbb{R}_+$-filtered simplicial sets -- both of these Künneth formulas hold in all homological dimensions $n \geq 0$.

math.AT↗

arXiv2019

Exposition and Interpretation of the Topology of Neural Networks

Convolutional neural networks (CNN's) are powerful and widely used tools. However, their interpretability is far from ideal. One such shortcoming is the difficulty of deducing a network's ability to generalize to unseen data. We use topological data analysis to show that the information encoded in the weights of a CNN can be organized in terms of a topological data model and demonstrate how such information can be interpreted and utilized. We show that the weights of convolutional layers at depths from 1 through 13 learn simple global structures. We also demonstrate the change of the simple structures over the course of training. In particular, we define and analyze the spaces of spatial filters of convolutional layers and show the recurrence, among all networks, depths, and during training, of a simple circle consisting of rotating edges, as well as a less recurring unanticipated complex circle that combines lines, edges, and non-linear patterns. We also demonstrate that topological structure correlates with a network's ability to generalize to unseen data and that topological information can be used to improve a network's performance. We train over a thousand CNN's on MNIST, CIFAR-10, SVHN, and ImageNet.

cs.CV↗

arXiv2019

Equivariant stable homotopy methods in the algebraic K-theory of infinite groups

Equivariant homotopy methods developed over the last 20 years lead to recent breakthroughs in the Borel isomorphism conjectures for Loday assembly maps in K- and L-theories. An important consequence of these algebraic conjectures is the topological rigidity of compact aspherical manifolds. Our goal is to strip the basic idea to the core and follow the evolution over time in order to explain the advantages of the flexible state that exists today. We end with an outline of the proof of the Borel conjecture in algebraic K-theory for groups of finite asymptotic dimension. We also discuss the relation of these methods to the recent work on the Farrell-Jones conjecture.

math.AT↗

arXiv2019

Bounded G-theory with fibred control

We use filtered modules over a Noetherian ring and fibred bounded control on homomorphisms to construct a new kind of controlled algebra with applications in geometric topology. The theory here can be thought of as a "pushout" of the bounded K-theory with fibred control and the controlled G-theory constructed and used by the authors. This paper contains the non-equivariant theory including controlled excision theorems crucial for computations.

math.KT↗

arXiv2018

Topological Approaches to Deep Learning

We perform topological data analysis on the internal states of convolutional deep neural networks to develop an understanding of the computations that they perform. We apply this understanding to modify the computations so as to (a) speed up computations and (b) improve generalization from one data set of digits to another. One byproduct of the analysis is the production of a geometry on new sets of features on data sets of images, and use this observation to develop a methodology for constructing analogues of CNN's for many other geometries, including the graph structures constructed by topological data analysis.

cs.LG↗

arXiv2018

Parametrized Homology via Zigzag Persistence

This paper develops the idea of homology for 1-parameter families of topological spaces. We express parametrized homology as a collection of real intervals with each corresponding to a homological feature supported over that interval or, equivalently, as a persistence diagram. By defining persistence in terms of finite rectangle measures, we classify barcode intervals into four classes. Each of these conveys how the homological features perish at both ends of the interval over which they are defined.

math.AT↗

arXiv2018

K-theory of geometric modules with fibred control

Controlled algebra plays a central role in many recent advances in geometric topology. This paper studies the iteration construction that was present from the very origins of the theory but started being exploited only recently. We develop the general framework for fibred control, prove localization theorems required for fibrewise excision, and then prove several versions of fibrewise excision theorems. However, we also demonstrate how the standard tools break down in the presence of new equivariant phenomena which require advanced localization methods.

math.KT↗

arXiv2018

Fibres of Failure: Classifying errors in predictive processes

We describe Fibres of Failure (FiFa), a method to classify failure modes of predictive processes using the Mapper algorithm from Topological Data Analysis. Our method uses Mapper to build a graph model of input data stratified by prediction error. Groupings found in high-error regions of the Mapper model then provide distinct failure modes of the predictive process. We demonstrate FiFa on misclassifications of MNIST images with added noise, and demonstrate two ways to use the failure mode classification: either to produce a correction layer that adjusts predictions by similarity to the failure modes; or to inspect members of the failure modes to illustrate and investigate what characterizes each failure mode.

cs.CV↗

arXiv2016

Hierarchical Clustering of Asymmetric Networks

This paper considers networks where relationships between nodes are represented by directed dissimilarities. The goal is to study methods that, based on the dissimilarity structure, output hierarchical clusters, i.e., a family of nested partitions indexed by a connectivity parameter. Our construction of hierarchical clustering methods is built around the concept of admissible methods, which are those that abide by the axioms of value - nodes in a network with two nodes are clustered together at the maximum of the two dissimilarities between them - and transformation - when dissimilarities are reduced, the network may become more clustered but not less. Two particular methods, termed reciprocal and nonreciprocal clustering, are shown to provide upper and lower bounds in the space of admissible methods. Furthermore, alternative clustering methodologies and axioms are considered. In particular, modifying the axiom of value such that clustering in two-node networks occurs at the minimum of the two dissimilarities entails the existence of a unique admissible clustering method.

cs.LG↗

arXiv2016

Admissible Hierarchical Clustering Methods and Algorithms for Asymmetric Networks

This paper characterizes hierarchical clustering methods that abide by two previously introduced axioms -- thus, denominated admissible methods -- and proposes tractable algorithms for their implementation. We leverage the fact that, for asymmetric networks, every admissible method must be contained between reciprocal and nonreciprocal clustering, and describe three families of intermediate methods. Grafting methods exchange branches between dendrograms generated by different admissible methods. The convex combination family combines admissible methods through a convex operation in the space of dendrograms, and thirdly, the semi-reciprocal family clusters nodes that are related by strong cyclic influences in the network. Algorithms for the computation of hierarchical clusters generated by reciprocal and nonreciprocal clustering as well as the grafting, convex combination, and semi-reciprocal families are derived using matrix operations in a dioid algebra. Finally, the introduced clustering methods and algorithms are exemplified through their application to a network describing the interrelation between sectors of the United States (U.S.) economy.

cs.LG↗

arXiv2016

Excisive Hierarchical Clustering Methods for Network Data

We introduce two practical properties of hierarchical clustering methods for (possibly asymmetric) network data: excisiveness and linear scale preservation. The latter enforces imperviousness to change in units of measure whereas the former ensures local consistency of the clustering outcome. Algorithmically, excisiveness implies that we can reduce computational complexity by only clustering a data subset of interest while theoretically guaranteeing that the same hierarchical outcome would be observed when clustering the whole dataset. Moreover, we introduce the concept of representability, i.e. a generative model for describing clustering methods through the specification of their action on a collection of networks. We further show that, within a rich set of admissible methods, requiring representability is equivalent to requiring both excisiveness and linear scale preservation. Leveraging this equivalence, we show that all excisive and linear scale preserving methods can be factored into two steps: a transformation of the weights in the input network followed by the application of a canonical clustering method. Furthermore, their factorization can be used to show stability of excisive and linear scale preserving methods in the sense that a bounded perturbation in the input network entails a bounded perturbation in the clustering output.

cs.LG↗

arXiv2016

Symmetric and r-Symmetric Tropical Polynomials and Rational Functions

A tropical polynomial in nr variables divided into blocks of r variables each, is r-symmetric, if it is invariant under the action of Sn that permutes the blocks. For r=1 we call these tropical polynomials symmetric. We can define r-symmetric and symmetric rational functions in a similar manner. In this paper we identify generators for the sets of symmetric tropical polynomials and rational functions. While r-symmetric tropical polynomials are not finitely generated, we show that r-symmetric rational functions are and provide a list of generators.

math.RA↗

arXiv2016

Dimension Independent Matrix Square using MapReduce

We compute the singular values of an $m \times n$ sparse matrix $A$ in a distributed setting, without communication dependence on $m$, which is useful for very large $m$. In particular, we give a simple nonadaptive sampling scheme where the singular values of $A$ are estimated within relative error with constant probability. Our proven bounds focus on the MapReduce framework, which has become the de facto tool for handling such large matrices that cannot be stored or even streamed through a single machine. On the way, we give a general method to compute $A^TA$. We preserve singular values of $A^TA$ with $ε$ relative error with shuffle size $O(n^2/ε^2)$ and reduce-key complexity $O(n/ε^2)$. We further show that if only specific entries of $A^TA$ are required and $A$ has nonnegative entries, then we can reduce the shuffle size to $O(n \log(n) / s)$ and reduce-key complexity to $O(\log(n)/s)$, where $s$ is the minimum cosine similarity for the entries being estimated. All of our bounds are independent of $m$, the larger dimension. We provide open-source implementations in Spark and Scalding, along with experiments in an industrial setting.

cs.DS↗

arXiv2015

Two Countermeasures Against Hardware Trojans Exploiting Non-Zero Aliasing Probability of BIST

The threat of hardware Trojans has been widely recognized by academia, industry, and government agencies. A Trojan can compromise security of a system in spite of cryptographic protection. The damage caused by a Trojan may not be limited to a business or reputation, but could have a severe impact on public safety, national economy, or national security. An extremely stealthy way of implementing hardware Trojans has been presented by Becker et al. at CHES'2012. Their work have shown that it is possible to inject a Trojan in a random number generator compliant with FIPS 140-2 and NIST SP800-90 standards by exploiting non-zero aliasing probability of Logic Built-In-Self-Test (LBIST). In this paper, we present two methods for modifying LBIST to prevent such an attack. The first method makes test patterns dependent on a configurable key which is programed into a chip after the manufacturing stage. The second method uses a remote test management system which can execute LBIST using a different set of test patterns at each test cycle.

cs.CR↗

arXiv2015

On Modules over Infinite Group Rings

Let $R$ be a commutative ring and $Γ$ be an infinite discrete group. The algebraic $K$-theory of the group ring $R[Γ]$ is an important object of computation in geometric topology and number theory. When the group ring is Noetherian there is a companion $G$-theory of $R[Γ]$ which is often easier to compute. However, it is exceptionally rare that the group ring is Noetherian for an infinite group. In this paper, we define a version of $G$-theory for any finitely generated discrete group. This construction is based on the coarse geometry of the group. Therefore it has some expected properties such as independence from the choice of a word metric. We prove that, whenever $R$ is a regular Noetherian ring of finite global homological dimension and $Γ$ has finite asymptotic dimension and a finite model for the classifying space $BΓ$, the natural Cartan map from the $K$-theory of $R[Γ]$ to $G$-theory is an equivalence. On the other hand, our $G$-theory is indeed better suited for computation as we show in a separate paper. Some results and constructions in this paper might be of independent interest as we learn to construct projective resolutions of finite type for certain modules over group rings.

math.KT↗