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ORBi

Visualization of AE's Training on Credit Card Transactions with Persistent Homology Charlier, Jérémy Henri J. ; Petit, François ; et al in Proceedings of the International Workshop on Applications of Topological Data Analysis In conjunction with ECML PKDD 2019 (2019, September) Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the ... [more ▼] Auto-encoders are among the most popular neural network architecture for dimension reduction. They are composed of two parts: the encoder which maps the model distribution to a latent manifold and the decoder which maps the latent manifold to a reconstructed distribution. However, auto-encoders are known to provoke chaotically scattered data distribution in the latent manifold resulting in an incomplete reconstructed distribution. Current distance measures fail to detect this problem because they are not able to acknowledge the shape of the data manifolds, i.e. their topological features, and the scale at which the manifolds should be analyzed. We propose Persistent Homology for Wasserstein Auto-Encoders, called PHom-WAE, a new methodology to assess and measure the data distribution of a generative model. PHom-WAE minimizes the Wasserstein distance between the true distribution and the reconstructed distribution and uses persistent homology, the study of the topological features of a space at different spatial resolutions, to compare the nature of the latent manifold and the reconstructed distribution. Our experiments underline the potential of persistent homology for Wasserstein Auto-Encoders in comparison to Variational Auto-Encoders, another type of generative model. The experiments are conducted on a real-world data set particularly challenging for traditional distance measures and auto-encoders. PHom-WAE is the first methodology to propose a topological distance measure, the bottleneck distance, for Wasserstein Auto-Encoders used to compare decoded samples of high quality in the context of credit card transactions. [less ▲] Detailed reference viewed: 67 (4 UL)Ephemeral persistence modules and distance comparison ; Petit, François E-print/Working paper (2019) We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of γ-sheaves. In the case of one ... [more ▼] We provide a definition of ephemeral multi-persistent modules and prove that the quotient of persistent modules by the ephemeral ones is equivalent to the category of γ-sheaves. In the case of one-dimensional persistence, our definition agrees with the usual one showing that the observable category and the category of γ-sheaves are equivalent. We also establish isometry theorems between the category of persistent modules and γ-sheaves both endowed with their interleaving distance. Finally, we compare the interleaving and convolution distances. [less ▲] Detailed reference viewed: 60 (0 UL)Quantization of spectral curves and DQ-modules Petit, François in Journal of Noncommutative Geometry (2018) Given an holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, we prove the existence of an holonomic DQ-module supported by the spectral curve associated to this bundle. Then ... [more ▼] Given an holomorphic Higgs bundle on a compact Riemann surface of genus greater than one, we prove the existence of an holonomic DQ-module supported by the spectral curve associated to this bundle. Then, we relate quantum curves arising in various situations (quantization of spectral curves of Higgs Bundles, quantization of the A-polynomial...) to DQ-modules and show that a quantum curve and the DQ-module canonically associated to it have isomorphic sheaves of solutions. [less ▲] Detailed reference viewed: 116 (4 UL)Holomorphic Frobenius actions for DQ-modules Petit, François E-print/Working paper (2018) Given a complex manifold endowed with a Gm-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the Gm-action is free and proper, then the category ... [more ▼] Given a complex manifold endowed with a Gm-action and a DQ-algebra equipped with a compatible holomorphic Frobenius action (F-action), we prove that if the Gm-action is free and proper, then the category of F-equivariant DQ-modules is equivalent to the category of modules over the sheaf of invariant sections of the DQ-algebra. As an application, we deduce the codimension three conjecture for formal microdifferential modules from the one for DQ-modules on a symplectic manifold. [less ▲] Detailed reference viewed: 46 (3 UL)The codimension-three conjecture for holonomic DQ-modules Petit, François in Selecta Mathematica, New Series (2017) On a complex symplectic manifold, we prove that any holonomic DQ-module endowed with a lattice and defined outside of an analytic subvariety of codimension 3 of its support extends as an holonomic DQ ... [more ▼] On a complex symplectic manifold, we prove that any holonomic DQ-module endowed with a lattice and defined outside of an analytic subvariety of codimension 3 of its support extends as an holonomic DQ-module. This is an analogue for DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. [less ▲] Detailed reference viewed: 79 (3 UL)Tempered subanalytic topology on algebraic varieties Petit, François E-print/Working paper (2017) On a smooth complex algebraic variety, we build the tempered subanalytic and Stein tempered subanalytic sites. We construct the sheaf of holomorphic functions tempered at infinity over these sites and ... [more ▼] On a smooth complex algebraic variety, we build the tempered subanalytic and Stein tempered subanalytic sites. We construct the sheaf of holomorphic functions tempered at infinity over these sites and study their relations with the sheaf of regular functions, proving in particular that these sheaves are isomorphic on Zariski open subsets. We show that these data allow to define the functors of tempered and Stein tempered analytifications. We study the relations between these two functors and the usual analytification functor. We also obtain algebraization results in the non-proper case and flatness results. [less ▲] Detailed reference viewed: 42 (2 UL)Cohomologically enriched categories and DQ-modules Petit, François Presentation (2016, November) Detailed reference viewed: 37 (2 UL)Quantization of spectral curves and DQ-modules Petit, François Presentation (2016, July 28) Detailed reference viewed: 79 (3 UL)Une brêve introduction aux faisceaux pervers Petit, François Presentation (2016, April 13) Detailed reference viewed: 83 (3 UL)Quantization of spectral curves and DQ-modules Petit, François in Oberwolfach Reports (2016) Detailed reference viewed: 75 (0 UL)Quantization of spectral curves and DQ-modules Petit, François Presentation (2016) Detailed reference viewed: 68 (1 UL)The Codimension-Three conjecture for holonomic DQ-modules Petit, François E-print/Working paper (2014) We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module ... [more ▼] We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely beyond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module. [less ▲] Detailed reference viewed: 146 (1 UL)Fourier-Mukai transform in the quantized setting Petit, François in Advances in Mathematics (2014), 256 We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an ... [more ▼] We prove that a coherent DQ-kernel induces an equivalence between the derived categories of DQ-modules with coherent cohomology if and only if the graded commutative kernel associated to it induces an equivalence between the derived categories of coherent sheaves. [less ▲] Detailed reference viewed: 101 (5 UL)The Lefschetz-Lunts formula for deformation quantization modules Petit, François in Mathematische Zeitschrift (2013), 273(3-4), 1119-1138 We adapt to the case of deformation quantization modules a formula of Lunts (Lefschetz fixed point theorems for Fourier–Mukai functors and DG algebras. arXiv:1102.2884. ArXiv e-prints, 2011) who ... [more ▼] We adapt to the case of deformation quantization modules a formula of Lunts (Lefschetz fixed point theorems for Fourier–Mukai functors and DG algebras. arXiv:1102.2884. ArXiv e-prints, 2011) who calculates the trace of a kernel acting on the Hochschild homology of a DQ-algebroid. [less ▲] Detailed reference viewed: 105 (2 UL)A Riemann-Roch Theorem for dg Algebras Petit, François in Bulletin de la Société Mathématique de France (2013), 141(2), 197-223 Given a smooth proper dg algebra A, a perfect dg A-module M and an endomorphism f of M, we define the Hochschild class of the pair (M,f) with values in the Hochschild homology of the algebra A. Our main ... [more ▼] Given a smooth proper dg algebra A, a perfect dg A-module M and an endomorphism f of M, we define the Hochschild class of the pair (M,f) with values in the Hochschild homology of the algebra A. Our main result is a Riemann-Roch type formula involving the convolution of two such Hochschild classes. [less ▲] Detailed reference viewed: 127 (0 UL)DG Affinity of DQ-modules Petit, François in International Mathematics Research Notices (2012), (6), 1414-1438 In this paper, we prove the dg affinity of formal deformation algebroid stacks over complex smooth algebraic varieties. For that purpose, we introduce the triangulated category of formal deformation ... [more ▼] In this paper, we prove the dg affinity of formal deformation algebroid stacks over complex smooth algebraic varieties. For that purpose, we introduce the triangulated category of formal deformation modules which are cohomologically complete and whose associated graded module is quasi-coherent. [less ▲] Detailed reference viewed: 62 (2 UL) |
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