| 1. |
- Agerberg, Jens, et al.
(författare)
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Global and Relative Topological Features from Homological Invariants of Subsampled Datasets
- 2023
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Ingår i: Proceedings of the 2nd Annual Topology, Algebra, and Geometry in Machine Learning, TAG-ML 2023. - : ML Research Press. ; , s. 302-312
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Konferensbidrag (refereegranskat)abstract
- Homology-based invariants can be used to characterize the geometry of datasets and thereby gain some understanding of the processes generating those datasets. In this work we investigate how the geometry of a dataset changes when it is subsampled in various ways. In our framework the dataset serves as a reference object; we then consider different points in the ambient space and endow them with a geometry defined in relation to the reference object, for instance by subsampling the dataset proportionally to the distance between its elements and the point under consideration. We illustrate how this process can be used to extract rich geometrical information, allowing for example to classify points coming from different data distributions.
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| 2. |
- Agerberg, Jens, et al.
(författare)
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Supervised Learning Using Homology Stable Rank Kernels
- 2021
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Ingår i: FRONTIERS IN APPLIED MATHEMATICS AND STATISTICS. - : FRONTIERS MEDIA SA. - 2297-4687. ; 7
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Tidskriftsartikel (refereegranskat)abstract
- Exciting recent developments in Topological Data Analysis have aimed at combining homology-based invariants with Machine Learning. In this article, we use hierarchical stabilization to bridge between persistence and kernel-based methods by introducing the so-called stable rank kernels. A fundamental property of the stable rank kernels is that they depend on metrics to compare persistence modules. We illustrate their use on artificial and real-world datasets and show that by varying the metric we can improve accuracy in classification tasks.
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| 3. |
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| 4. |
- Blomgren, Martin
(författare)
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Fibrations and Idempotent Functors
- 2011
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of two articles. Both articles concern homotopical algebra. In Paper I we study functors indexed by a small category into a model category whose value at each morphism is a weak equivalence. We show that the category of such functors can be understood as a certain mapping space. Specializing to topological spaces, this result is used to reprove a classical theorem that classifies fibrations with a fixed base and homotopy fiber. In Paper II we study augmented idempotent functors, i.e., co-localizations, operating on the category of groups. We relate these functors to cellular coverings of groups and show that a number of properties, such as finiteness, nilpotency etc., are preserved by such functors. Furthermore, we classify the values that such functors can take upon finite simple groups and give an explicit construction of such values.
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| 5. |
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| 6. |
- Blomgren, Mats, et al.
(författare)
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Idempotent transformations of finite groups
- 2013
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Ingår i: Advances in Mathematics. - : Elsevier BV. - 0001-8708 .- 1090-2082. ; 233:1, s. 56-86
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Tidskriftsartikel (refereegranskat)abstract
- We describe the action of idempotent transformations on finite groups. We show that finiteness is preserved by such transformations and enumerate all possible values such transformations can assign to a fixed finite simple group. This is done in terms of the first two homology groups. We prove for example that except special linear groups, such an orbit can have at most 7 elements. We also study the action of monomials of idempotent transformations on finite groups and show for example that orbits of this action are always finite.
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| 7. |
- Blomgren, Martin, et al.
(författare)
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On the classification of fibrations
- 2015
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Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 367:1, s. 519-557
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Tidskriftsartikel (refereegranskat)abstract
- We identify the homotopy type of the moduli of maps with a given homotopy type of the base and the homotopy fiber. A new model for the space of weak equivalences and its classifying space is given.
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| 8. |
- Carannante, Ilaria, et al.
(författare)
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The impact of Parkinson’s disease on striatal network connectivity and cortico-striatal drive : an in-silico study
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Striatum, the input stage of the basal ganglia, is important for sensory-motor integration, initiation and selection of behaviour, as well as reward learning. Striatum receives glutamatergic inputs from mainly cortex and thalamus. In rodents, the striatal projection neurons (SPNs), giving rise to the direct and the indirect pathway (dSPNs and iSPNs, respectively), account for 95% of the neurons and the remaining 5% are GABAergic and cholinergic interneurons. Interneuron axon terminals as well as local dSPN and iSPN axon collaterals form an intricate striatal network. Following chronic dopamine depletion as in Parkinson’s disease (PD), both morphological and electrophysiological striatal neuronal features are altered. Our goal with this \textit{in-silico} study is twofold: a) to predict and quantify how the intrastriatal network connectivity structure becomes altered as a consequence of the morphological changes reported at the single neuron level, and b) to investigate how the effective glutamatergic drive to the SPNs would need to be altered to account for the activity level seen in SPNs during PD. In summary we find that the richness of the connectivity motifs is significantly decreased during PD, while at the same time a substantial enhancement of the effective glutamatergic drive to striatum is present.
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| 9. |
- Chachólski, Wojciech, et al.
(författare)
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Cellular covers of divisible abelian groups
- 2009
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Ingår i: ALPINE PERSPECTIVES ON ALGEBRAIC TOPOLOGY. - 9780821848395 ; , s. 77-97
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Konferensbidrag (refereegranskat)abstract
- We determine all the possible values of the cellular approximation functor c(A) : cell(A)E -> E, where A is an arbitrary group and E is a divisible abelian group.
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| 10. |
- Chacholski, Wojciech, et al.
(författare)
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Cellular generators
- 2004
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Ingår i: Proceedings of the American Mathematical Society. - 0002-9939 .- 1088-6826. ; 132:11, s. 3397-3409
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Tidskriftsartikel (refereegranskat)abstract
- The aim of this paper is twofold. On the one hand, we show that the kernelof the Bousfield periodization functor P-A is cellularly generated by a space B, i.e., we construct a space B such that the smallest closed class C( B) containing B is exactly C( A). On the other hand, we show that the partial order (Spaces, much greater than) is a complete lattice, where B much greater than A if B is an element of C(A). Finally, as a corollary we obtain Bousfield's theorem, which states that (Spaces, >) is a complete lattice, where B > A if B is an element of.
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