IA Scholar Query: Manifold Learning in Quotient Spaces.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgSat, 01 Oct 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Isadore M. Singer (1924–2021) In Memoriam Part 1: Scientific Works
https://scholar.archive.org/work/aejx3oq2lvch5gdpwoqpzzlbqe
Robert Bryant, Jean-Michel Bismut, Jeff Cheeger, Phillip Griffiths, Simon Donaldson, Nigel Hitchin, H Blaine Lawson, Michail Gromov, Adam Marcus, Daniel Spielman, Nikhil Srivastava, Edward Wittenwork_aejx3oq2lvch5gdpwoqpzzlbqeSat, 01 Oct 2022 00:00:00 GMTRemembrances of Ciprian Ilie Foias
https://scholar.archive.org/work/y3ke66suwvcffakbbjadvd6avi
Robert A Becker, Hari Bercovici, Animikh Biswas, Alexey Cheskidov, Peter Constantin, Alp Eden, Art Frazho, Michael Jolly, Igor Kukavica, Carl Pearcy, Ricardo M S Rosa, Jean-Claude Saut, Allen Tannenbaum, Roger Temam, Edriss Titi, Dan Voiculescuwork_y3ke66suwvcffakbbjadvd6aviSat, 01 Oct 2022 00:00:00 GMTModularity of arithmetic special divisors for unitary Shimura varieties (with an appendix by Yujie Xu)
https://scholar.archive.org/work/cxptftd63vepnlzziwsnewq6ne
We construct explicit generating series of arithmetic extensions of Kudla's special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the arithmetic Chow groups. This provides a partial solution to Kudla's modularity problem. The main ingredient in our construction is S.~Zhang's theory of admissible arithmetic divisors. The main ingredient in the proof is an arithmetic mixed Siegel-Weil formula.Congling Qiuwork_cxptftd63vepnlzziwsnewq6neThu, 29 Sep 2022 00:00:00 GMTThe Prime Geodesic Theorem for PSL_2(ℤ[i]) and Spectral Exponential Sums
https://scholar.archive.org/work/uxwfignztjewhnfb36pvc2jaaq
This work addresses the Prime Geodesic Theorem for the Picard manifold ℳ = PSL_2(ℤ[i]) \𝔥^3, which asks for the asymptotic evaluation of a counting function for the closed geodesics on ℳ. Let E_Γ(X) be the error term in the Prime Geodesic Theorem. We establish that E_Γ(X) = O_ε(X^3/2+ε) on average as well as many pointwise bounds. The second moment bound parallels an analogous result for Γ = PSL_2(ℤ) due to Balog et al. and our innovation features the delicate analysis of sums of Kloosterman sums with an explicit manipulation of oscillatory integrals. The proof of the pointwise bounds requires Weyl-strength subconvexity for quadratic Dirichlet L-functions over ℚ(i). Moreover, an asymptotic formula for a spectral exponential sum in the spectral aspect for a cofinite Kleinian group Γ is given. Our numerical experiments visualise in particular that E_Γ(X) obeys a conjectural bound of size O_ϵ(X^1+ε).Ikuya Kanekowork_uxwfignztjewhnfb36pvc2jaaqThu, 29 Sep 2022 00:00:00 GMTThe Geometry of the Bing Involution
https://scholar.archive.org/work/2madzpe3dfd4dmdk4b4efnli7a
In 1952 Bing published a wild (not topologically conjugate to smooth) involution I of the 3-sphere S^3. But exactly how wild is it, analytically? We prove that any involution I^h, topologically conjugate to I, must have a weakly exponential modulus of continuity. Specifically, there exists a constant c > 1.167 such that for a sequence of δs converging to zero, δ > 0, there are points x,y ∈ S^3 with dist(x,y) < δ, yet dist(I^h(x), I^h(y)) > ϵ, where δ^-1 = c^√(ϵ^-1), and dist is the usual Riemannian distance on S^3. In particular, I^h stretches distance much more than a Lipschitz function (δ^-1 = c^'ϵ^-1) or a Hölder function (δ^-1 = c^" (ϵ^-1)^p, 1 < p < ∞). Bing's original construction and known alternatives (see text) for I have a modulus of continuity δ^-1 > c √(2)^ϵ^-1, so the theorem is reasonably tight–we prove the modulus must be at least weakly exponential, whereas the truth may be exponential. Using the same technique we analyze a large class of "ramified" Bing involutions and show, as a scholium, that given any function f: ℝ^+ →ℝ^+, no matter how rapid its growth, we can find a corresponding involution J of the 3-sphere such that any topological conjugate J^h of J must have a modulus of continuity δ^-1(ϵ^-1) growing faster than f (near infinity). There is a literature on inherent differentiability (references in text) but as far as the authors know the subject of inherent modulus of continuity is new. Dedicated to R.H. Bing's life work on the 70th anniversary of his involution.Michael Freedman, Michael Starbirdwork_2madzpe3dfd4dmdk4b4efnli7aThu, 29 Sep 2022 00:00:00 GMTIn Search of Projectively Equivariant Neural Networks
https://scholar.archive.org/work/lf6zpgbtfbcdzmpcvih5mw4wla
Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. In particular, we study the relation of projective and ordinary equivariance and show that for important examples, the problems are in fact equivalent. The rotation group in 3D acts projectively on the projective plane. We experimentally study the practical importance of rotation equivariance when designing networks for filtering 2D-2D correspondences. Fully equivariant models perform poorly, and while a simple addition of invariant features to a strong baseline yields improvements, this seems to not be due to improved equivariance.Georg Bökman, Axel Flinth, Fredrik Kahlwork_lf6zpgbtfbcdzmpcvih5mw4wlaThu, 29 Sep 2022 00:00:00 GMTA JSJ-type decomposition theorem for symplectic fillings
https://scholar.archive.org/work/h7twrdz3ungcdjwqn34ylhifty
We establish a JSJ-type decomposition theorem for splitting exact symplectic fillings of contact 3-manifolds along mixed tori – these are convex tori satisfying a particular geometric condition. As an application, we show that if (M,ξ) is obtained from (S^3,ξ_std) via Legendrian surgery along a knot which has been stabilized both positively and negatively, then (M,ξ) has a unique exact filling.Austin Christian, Michael Menkework_h7twrdz3ungcdjwqn34ylhiftyThu, 29 Sep 2022 00:00:00 GMTOn the Classical Integrability of Root-T T̅ Flows
https://scholar.archive.org/work/peauvhvo5jf6pe7vjagj5uaqwu
The Root-T T̅ flow was recently introduced as a universal and classically marginal deformation of any two-dimensional translation-invariant field theory. The flow commutes with the (irrelevant) T T̅ flow and it can be integrated explicitly for a large class of actions, leading to non-analytic Lagrangians reminiscent of the four-dimensional Modified-Maxwell theory (ModMax). It is not a priori obvious whether the Root-T T̅ flow preserves integrability, like it is the case for the T T̅ flow. In this paper we demonstrate that this is the case for a large class of classical models by explicitly constructing a deformed Lax connection. We discuss the principal chiral model and the non-linear sigma models on symmetric and semi-symmetric spaces, without or with Wess-Zumino term. We also construct Lax connections for the two-parameter families of theories deformed by both Root-T T̅ and T T̅ for all of these models.Riccardo Borsato, Christian Ferko, Alessandro Sfondriniwork_peauvhvo5jf6pe7vjagj5uaqwuWed, 28 Sep 2022 00:00:00 GMTAn Upper Bound on the Critical Volume in a Class of Toric Sasaki-Einstein Manifolds
https://scholar.archive.org/work/6q7yilmjp5artdxakbx6mzs22a
We prove the existence of an upper bound on critical volume of a large class of toric Sasaki-Einstein manifolds with respect to the first Chern class of the resolutions of the Gorenstein singularities in the corresponding toric Calabi-Yau varieties. We examine the canonical metrics obtained by the Delzant construction on these varieties and characterise cases when the bound is attained. We comment on computational tools used in the investigation, in particular Neural Networks and the gradient saliency method.Maksymilian Mankowork_6q7yilmjp5artdxakbx6mzs22aWed, 28 Sep 2022 00:00:00 GMTAn Algebra of Observables for de Sitter Space
https://scholar.archive.org/work/ko2bh23banczjmyjwp2zanuex4
We describe an algebra of observables for a static patch in de Sitter space, with operators gravitationally dressed to the worldline of an observer. The algebra is a von Neumann algebra of Type II_1. There is a natural notion of entropy for a state of such an algebra. There is a maximum entropy state, which corresponds to empty de Sitter space, and the entropy of any semiclassical state of the Type II_1 algebras agrees, up to an additive constant independent of the state, with the expected generalized entropy S_gen=(A/4G_N)+S_out. An arbitrary additive constant is present because of the renormalization that is involved in defining entropy for a Type II_1 algebra.Venkatesa Chandrasekaran, Roberto Longo, Geoff Penington, Edward Wittenwork_ko2bh23banczjmyjwp2zanuex4Wed, 28 Sep 2022 00:00:00 GMTRiemannian foliations and geometric quantization
https://scholar.archive.org/work/hk2fvscuvrbsje2rocli7aeedy
We introduce geometric quantization for constant rank presymplectic structures with Riemannian null foliation and compact leaf closure space. We prove a quantization-commutes-with-reduction theorem in this context. Examples related to symplectic toric quasi-folds, suspensions of isometric actions of discrete groups, and K-contact manifolds are discussed.Yi Lin, Yiannis Loizides, Reyer Sjamaar, Yanli Songwork_hk2fvscuvrbsje2rocli7aeedyWed, 28 Sep 2022 00:00:00 GMTQuantifying Quantum Advantage in Topological Data Analysis
https://scholar.archive.org/work/w3tuawesfnhahpxmkcn4fuy3ua
Lloyd et al. were first to demonstrate the promise of quantum algorithms for computing Betti numbers in persistent homology (a way of characterizing topological features of data sets). Here, we propose, analyze, and optimize an improved quantum algorithm for topological data analysis (TDA) with reduced scaling, including a method for preparing Dicke states based on inequality testing, a more efficient amplitude estimation algorithm using Kaiser windows, and an optimal implementation of eigenvalue projectors based on Chebyshev polynomials. We compile our approach to a fault-tolerant gate set and estimate constant factors in the Toffoli complexity. Relative to the best classical heuristic algorithms, our analysis reveals that super-quadratic quantum speedups are only possible for this problem when targeting a multiplicative error approximation and the Betti number grows asymptotically. Further, we propose a dequantization of the quantum TDA algorithm that shows that having exponentially large dimension and Betti number are necessary, but insufficient conditions, for super-polynomial advantage. We then introduce and analyze specific problem examples for which super-polynomial advantages may be achieved, and argue that quantum circuits with tens of billions of Toffoli gates can solve some seemingly classically intractable instances.Dominic W. Berry, Yuan Su, Casper Gyurik, Robbie King, Joao Basso, Alexander Del Toro Barba, Abhishek Rajput, Nathan Wiebe, Vedran Dunjko, Ryan Babbushwork_w3tuawesfnhahpxmkcn4fuy3uaTue, 27 Sep 2022 00:00:00 GMTChromatic fixed point theory and the Balmer spectrum for extraspecial 2-groups
https://scholar.archive.org/work/mnjaakaywrab3gx6szagmeakg4
In the early 1940's, P.A.Smith showed that if a finite p-group G acts on a finite complex X that is mod p acyclic, then its space of fixed points, X^G, will also be mod p acyclic. In their recent study of the Balmer spectrum of equivariant stable homotopy theory, Balmer and Sanders were led to study chromatic versions of this statement, with the question: given H<G and n, what is the smallest r such that if X^H is acyclic in the (n+r)th Morava K-theory, then X^G must be acyclic in the nth Morava K-theory? Barthel et.al. then answered this when G is abelian, by finding general lower and upper bounds for these 'blue shift' numbers which agree in the abelian case. In our paper, we first prove that these potential chromatic versions of Smith's theorem are equivalent to chromatic versions of a 1952 theorem of E.E.Floyd, which replaces acyclicity by bounds on dimensions of homology, and thus applies to all finite G-spaces. This unlocks new techniques and applications in chromatic fixed point theory. In one direction, we are able to use classic constructions and representation theory to search for blue shift number lower bounds. We give a simple new proof of the known lower bound theorem, and then get the first results about nonabelian 2-groups that don't follow from previously known results. In particular, we are able to determine all blue shift numbers for extraspecial 2-groups. As samples of new applications, we offer a new result about involutions on the 5-dimensional Wu manifold, and a calculation of the mod 2 K-theory of a 100 dimensional real Grassmanian that uses a C_4 chromatic Floyd theorem.Nicholas J. Kuhn, Christopher J.R. Lloydwork_mnjaakaywrab3gx6szagmeakg4Tue, 27 Sep 2022 00:00:00 GMTScalable and Equivariant Spherical CNNs by Discrete-Continuous (DISCO) Convolutions
https://scholar.archive.org/work/6wuikknowfbqvdicbplb5c247a
No existing spherical convolutional neural network (CNN) framework is both computationally scalable and rotationally equivariant. Continuous approaches capture rotational equivariance but are often prohibitively computationally demanding. Discrete approaches offer more favorable computational performance but at the cost of equivariance. We develop a hybrid discrete-continuous (DISCO) group convolution that is simultaneously equivariant and computationally scalable to high-resolution. While our framework can be applied to any compact group, we specialize to the sphere. Our DISCO spherical convolutions not only exhibit SO(3) rotational equivariance but also a form of asymptotic SO(3)/SO(2) rotational equivariance, which is more desirable for many applications (where SO(n) is the special orthogonal group representing rotations in n-dimensions). Through a sparse tensor implementation we achieve linear scaling in number of pixels on the sphere for both computational cost and memory usage. For 4k spherical images we realize a saving of 10^9 in computational cost and 10^4 in memory usage when compared to the most efficient alternative equivariant spherical convolution. We apply the DISCO spherical CNN framework to a number of benchmark dense-prediction problems on the sphere, such as semantic segmentation and depth estimation, on all of which we achieve the state-of-the-art performance.Jeremy Ocampo, Matthew A. Price, Jason D. McEwenwork_6wuikknowfbqvdicbplb5c247aTue, 27 Sep 2022 00:00:00 GMTSelected Topics in Analytic Conformal Bootstrap: A Guided Journey
https://scholar.archive.org/work/hcfgumuob5hmfo55ras6b35rjq
This review aims to offer a pedagogical introduction to the analytic conformal bootstrap program via a journey through selected topics. We review analytic methods which include the large spin perturbation theory, Mellin space methods and the Lorentzian inversion formula. These techniques are applied to a variety of topics ranging from large-N theories, to the epsilon expansion and holographic superconformal correlators, and are demonstrated in a large number of explicit examples.Agnese Bissi, Aninda Sinha, Xinan Zhouwork_hcfgumuob5hmfo55ras6b35rjqTue, 27 Sep 2022 00:00:00 GMTLinear Dimensionality Reduction
https://scholar.archive.org/work/mlko2ih72vc6jk4karnzqtlk7e
These notes are an overview of some classical linear methods in Multivariate Data Analysis. This is an good old domain, well established since the 60's, and refreshed timely as a key step in statistical learning. It can be presented as part of statistical learning, or as dimensionality reduction with a geometric flavor. Both approaches are tightly linked: it is easier to learn patterns from data in low dimensional spaces than in high-dimensional spaces. It is shown how a diversity of methods and tools boil down to a single core methods, PCA with SVD, such that the efforts to optimize codes for analyzing massive data sets can focus on this shared core method, and benefit to all methods. An extension to the study of several arrays is presented (Canonical Analysis).Alain A. Francwork_mlko2ih72vc6jk4karnzqtlk7eTue, 27 Sep 2022 00:00:00 GMTPerverse Microsheaves
https://scholar.archive.org/work/jzkl5gig6namxafthktcbfgbii
An exact complex symplectic manifold carries a sheaf of stable categories, locally equivalent to a microlocalization of a category of constructible sheaves. This sheaf of categories admits a t-structure, whose heart is locally equivalent to a microlocalization of a category of perverse sheaves. The abelian category of local systems on a spin conic complex Lagrangian embeds fully faithfully in the heart. The sheaf of homs between two objects in the heart is itself a perverse sheaf, shifted by half the dimension of the ambient manifold. Analogous results hold for complex contact manifolds. The correspondence between microsheaves and Fukaya categories yields t-structures on Fukaya categories of conic complex symplectic manifolds, with holomorphic Lagrangians in the heart.Laurent Côté, Christopher Kuo, David Nadler, Vivek Shendework_jzkl5gig6namxafthktcbfgbiiMon, 26 Sep 2022 00:00:00 GMTA streamlined quantum algorithm for topological data analysis with exponentially fewer qubits
https://scholar.archive.org/work/bfpxeg4apvglzfhkomsu4xnphe
Topological invariants of a dataset, such as the number of holes that survive from one length scale to another (persistent Betti numbers) can be used to analyse and classify data in machine learning applications. We present an improved quantum algorithm for computing persistent Betti numbers, and provide an end-to-end complexity analysis. Our approach provides large polynomial time improvements, and an exponential space saving, over existing quantum algorithms. Subject to gap dependencies, our algorithm obtains an almost quintic speedup in the number of datapoints over rigorous state-of-the-art classical algorithms for calculating the persistent Betti numbers to constant additive error - the salient task for applications. However, this may be reduced to closer to quadratic when compared against heuristic classical methods and observed scalings. We discuss whether quantum algorithms can achieve an exponential speedup for tasks of practical interest, as claimed previously. We conclude that there is currently no evidence that this is the case.Sam McArdle, András Gilyén, Mario Bertawork_bfpxeg4apvglzfhkomsu4xnpheMon, 26 Sep 2022 00:00:00 GMTGeneralized Permutants and Graph GENEOs
https://scholar.archive.org/work/qusggqp6xrdyhkfx25pavk4cn4
In this paper we establish a bridge between Topological Data Analysis and Geometric Deep Learning, adapting the topological theory of group equivariant non-expansive operators (GENEOs) to act on the space of all graphs weighted on vertices or edges. This is done by showing how the general concept of GENEO can be used to transform graphs and to give information about their structure. This requires the introduction of the new concepts of generalized permutant and generalized permutant measure and the mathematical proof that these concepts allow us to build GENEOs between graphs. An experimental section concludes the paper, illustrating the possible use of our operators to extract information from graphs. This paper is part of a line of research devoted to developing a compositional and geometric theory of GENEOs for Geometric Deep Learning.Faraz Ahmad, Massimo Ferri, Patrizio Frosiniwork_qusggqp6xrdyhkfx25pavk4cn4Sun, 25 Sep 2022 00:00:00 GMTBatalin-Vilkovisky algebra structure on Poisson manifolds with semi-simple modular symmetry
https://scholar.archive.org/work/ggaonkrgxrbs5neqahu2vj7vhy
We study the "twisted" Poincar\'e duality of smooth Poisson manifolds, and show that, if the modular vector field is semi-simple (or say, diagonalizable), then there is a mixed complex associated to the Poisson complex, which, combining with the twisted Poincar\'e duality, gives a Batalin-Vilkovisky algebra structure on the Poisson cohomology. This generalizes the previous results obtained by Xu for unimodular Poisson manifolds. We also show that the Batalin-Vilkovisky algebra structure is preserved under Kontsevich's deformation quantization, and in the case of polynomial algebras it is also preserved by Koszul duality.Xiaojun Chen, Leilei Liu, Sirui Yu, Jieheng Zengwork_ggaonkrgxrbs5neqahu2vj7vhySun, 25 Sep 2022 00:00:00 GMT