We consider a class of learning problems regularized by a structured sparsity-inducing norm defined as the sum of l_2- or l_infinity-norms over groups of variables. Whereas much effort has been put in developing fast optimization techniques when the groups are disjoint or embedded in a hierarchy, we address here the ca…
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48 results for “L-infinity norm”
Unified framework for various adversarial attacks on deep networks.
problem Vulnerability of deep neural networks to adversarial attacks.
method ADMM (Alternating Direction Method of Multipliers) for generating adversarial examples.
result ADMM-based attacks achieve highest success rates and minimal distortion.
Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those coming from Lie algebroids. Given a Lie algebroid, we show that there is an L-inf…
The procedure "Lie group --> Lie algebra" has a generalization "simplicial manifold --> L_infinity algebra", or yet better, "presheaf on the category of surjective submersions --> L_infinity algebra". We describe this generalization, together with its higher-order extensions.
L-infinity algebra actionsmath.DG
We define the notion of action of an L-infinity algebra on a graded manifold , and show that such an action corresponds to a homological vector field on of a specific form. This generalizes the correspondence between Lie algebra actions on manifolds and transformation Lie algebroids. In particula…
New neural network design resists small -norm adversarial perturbations.
problem Vulnerability of neural networks to small -norm adversarial perturbations.
method Designing -dist neurons and constructing -dist nets, proving their 1-Lipschitz property and expressive power.
result Certified robustness of -dist nets with state-of-the-art performance on various datasets.
This study extends verifiable learning to boosted tree ensembles, enabling efficient security verification.
problem Efficiently verifying the robustness of boosted tree ensembles against norm-based attackers.
method Formal verification of robustness for large-spread boosted tree ensembles, considering -norm and pseudo-polynomial time for -norm verification.
result Polynomial time verification for -norm attackers, NP-hard for other norms, and pseudo-polynomial time for -norm verification.
Study shows closed Bach-flat manifolds with positive scalar curvature are locally spherical.
problem Characterizing closed Bach-flat manifolds with positive scalar curvature.
method Applied a different method to show local sphericality compared to previous complete non-compact cases.
result Closed Bach-flat manifolds with positive scalar curvature are locally spherical.
We develop further the approach to derived differential geometry introduced in Costello's work on the Witten genus. In particular, we introduce several new examples of L-infinity spaces, discuss vector bundles and shifted symplectic structures on L-infinity spaces, and examine in some detail the example of derived loop…
Adaptive sampling improves convex function learning.
problem Learning convex functions in the norm.
method Function-specific complexity measure for adaptive sampling.
result Adaptive sampling nearly achieves optimal error rate.
Multisymplectic geometry admits an operation that has no counterpart in symplectic geometry, namely, taking the product of two multisymplectic manifolds endowed with the wedge product of the multisymplectic forms. We show that there is an L-infinity-embedding of the L-infinity-algebra of observables of the individual f…
New approach to Lagrangian field theories using pro-finite structures and L-infinity algebras.
problem Formulating Lagrangian field theories with locality constraints.
method Using pro-finite structures and L-infinity algebras to define local observables and a pre-multisymplectic form.
result Definition of L-infinity algebra of local observables based on Lagrangian cohomology.
Establishes higher T-duality for super M-branes.
problem Generalizing T-duality for super p-branes.
method Super L-infinity-algebraic T-duality for super WZW-terms.
result Spherical T-duality of super M5-branes.
The Deligne groupoid is a functor from nilpotent differential graded Lie algebras concentrated in positive degrees to groupoids; in the special case of Lie algebras over a field of characteristic zero, it gives the associated simply connected Lie group. We generalize the Deligne groupoid to a functor gamma from L-infin…
This paper upgrades Khovanov homology to an L-infinity module structure.
problem Exploring Khovanov homology with L-infinity algebra structures.
method Developed an L-infinity algebra structure on sl2(∧) and showed annular Khovanov homology is an L-infinity module over it.
result The annular Khovanov homology of a link L is an L-infinity module over sl2(∧) up to quasi-isomorphism.
Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group.…
Homotopy moment mapsmath.DG
Associated to any manifold equipped with a closed form of degree >1 is an `L-infinity algebra of observables' which acts as a higher/homotopy analog of the Poisson algebra of functions on a symplectic manifold. In order to study Lie group actions on these manifolds, we introduce a theory of homotopy moment maps. Such a…
New algebra structure for Legendrian knots preserves contact homology invariants.
problem Constructing an algebra for Legendrian knots.
method Combining rational Symplectic Field Theory and combinatorial methods.
result Invariant Poisson algebra of Legendrian links under isotopy.
Generalizes Lie bialgebroids to supermanifolds with homotopy Poisson structures.
problem Relating Lie bialgebroids to homotopy Poisson structures on supermanifolds.
method Introduces L-infinity bialgebroids and higher Koszul brackets to connect these structures.
result Shows that has an L-infinity bialgebroid structure for homotopy Poisson structures.
Constructs structure on symplectic cohomology.
problem None explicitly stated; focuses on construction.
method Constructs structure on symplectic cohomology.
result Symplectic cohomology gains an structure.
In the first chapter, we give a precise and general description of gerbes valued in arbitrary crossed module and over an arbitrary differential stack. We do it using only Lie groupoids, hence ordinary differential geometry, by considering differential stacks as being Lie groupoids up to Morita equivalence. We prove the…
Low-Rank Covariance Completion for Graph Quilting with Applications to Functional Connectivitystat.ME
New method estimates neuronal connectivity from partially observed data.
problem Estimating neuronal connectivity from partially observed data.
method Two-step approach: low-rank covariance completion followed by graph structure estimation.
result Graph selection consistency demonstrated for one approach.
Inverse function theorem and homotopy description for L-infinity bundles.
problem Inverse function theorem and homotopy description for L-infinity bundles.
method Local sections composed of elementary morphisms.
result Simple description of homotopy category of L-infinity bundles.
Explains how pre-symplectic structures can be changed.
problem Understanding how pre-symplectic structures can be deformed.
method Uses Dirac geometry to explain the geometric origin of -algebra controlling deformations.
result Discovers the geometric origin of the -algebra controlling deformations of pre-symplectic structures.
New bounds for learning polynomial surrogates with guarantees.
problem Learning polynomial surrogates for bounded binary functions with error guarantees.
method Characterized minimax sample complexity for two classes of polynomials under subgaussian noise.
result Sample complexity rates differ from noiseless case, scaling as for degree polynomials and for sparse polynomials.
Study on deformations of pre-symplectic structures using an L-infinity algebra.
problem Deformation theory of pre-symplectic structures.
method Parametrization of deformations using Koszul L-infinity algebra.
result A quotient of the Koszul L-infinity algebra is isomorphic to the L-infinity algebra controlling foliations.
Improved training boosts certified robustness of L-infinity distance nets.
problem Certified robustness of L-infinity distance nets is not as strong as conventional networks.
method Improved training process combining scaled cross-entropy and clipped hinge loss with a decaying mixing coefficient.
result Certified accuracy of L-infinity distance nets improved from 33.30% to 40.06% on CIFAR-10.
An -algebra is built on symplectic manifold homology.
problem No specific problem stated; focuses on construction of algebra.
method Construction of an -algebra on symplectic manifold homology.
result The constructed -algebra naturally projects to a Lie algebra extension.
The work of Oh and Park ([OP]) on the deformation problem of coisotropic submanifolds opened the possibility of studying a large and interesting class of foliations with some explicit geometric tools. These tools assemble into the structure of an L-infinity algebra on the shifted foliation complex (Ω^*[1](\fol), d_\fol…
Homotopy operators help describe structures in equivariant deformation problems.
problem Equivariant deformation problems in algebraic structures.
method Use homotopy operators for an -algebra associated with the problem.
result Smooth parametrization of the space of structures around a given one.
New method for flux quantization on phase space stacks.
problem Defining and constructing flux-quantized phase space stacks.
method Observation of flux densities and characterization of Cauchy data.
result Flux-quantized phase space stacks have classifying spaces with rational Whitehead L-infinity algebra.
Homotopy momentum map extends Noether's theorem in general relativity.
problem Extending Noether's theorem to spacetime vector fields.
method Using homotopy momentum map and -algebras.
result Extension of conserved currents to spacetime vector fields.
Paper certifies neural network control policies against persistent adversarial perturbations.
problem Neural networks' fragility to adversarial perturbations in control systems.
method Combining neural network certification tools with robust control theory.
result Certifies neural network policies in a control loop under l-infinity norm bounded adversarial perturbations.
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise i…
New liftings derived from Chern-Simons classes for coherent sheaves.
problem Liftings of semiregularity maps for coherent sheaves on complex manifolds.
method Introducing Chern-Simons classes for curved DG-pairs and proving canonical liftings.
result Canonical liftings of Buchweitz-Flenner semiregularity maps.
Split Courant algebroids linked to special algebra structures.
problem Understanding the structure of split Courant algebroids.
method Established a correspondence with multiplicative curved -algebras.
result Split Courant algebroids correspond to multiplicative curved -algebras.
We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields on a dg-manifold with homological vector field admits a structure of L-infinity algebra with the Lie derivative as unary …
Paper constructs observables using multisymplectic geometry and algebraic methods.
problem Building observables in multisymplectic geometry.
method Uses -algebras, Gerstenhaber algebras, BV-modules, and constraint triples.
result Reconstructs and explains recent geometric results.
New method makes deep nets more robust to attacks without increasing training time.
problem Gradient obfuscation makes models vulnerable to stronger attacks.
method Local linearization regularizer to penalize non-linearity in loss surface.
result Models trained with regularizer achieve 47% adversarial accuracy on ImageNet.
Homotopy equivalence between formalities with different covariant derivatives.
problem Formality of Dolgushev depends on covariant derivative choice.
method Proved homotopy equivalence of -morphisms twisted by gauge equivalent elements.
result Globalized formalities with different covariant derivatives are homotopic.
We define and study the degeneration property for BV-infinity algebras and show that it implies that the underlying L-infinity algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity Δ(e^x)=e^x(Δ(x)+[x,x]/2) which holds in all BV algebras. As an application we show that the high…
Study shows equivalence in foliations and pre-symplectic forms aligns with gauge equivalence.
problem Deformation theory of foliations and pre-symplectic forms.
method Proved geometric equivalence agrees with algebraic gauge equivalence using -algebras.
result Gauge equivalences for foliations and pre-symplectic structures are consistent.
Study rational homotopy types of embedding spaces of manifolds.
problem Understanding the rational homotopy types of embedding spaces of manifolds.
method Express rational homotopy types through combinatorially defined L-infinity algebras of diagrams.
result Expressed the rational homotopy type of connected components of embedding spaces.
We give a generalization of the notion of a Cartan-Ehresmann connection from Lie algebras to L-infinity algebras and use it to study the obstruction theory of lifts through higher String-like extensions of Lie algebras. We find (generalized) Chern-Simons and BF-theory functionals this way and describe aspects of their …
Reviewing Q-manifolds, modular classes, and applications.
problem Obtaining invariant volumes on Q-manifolds.
method Exploring Q-manifolds, modular classes, and applying to specific examples.
result Applications to -algebroids and higher Poisson manifolds.
Study of manifolds with special holonomy using Frölicher-Nijenhuis bracket.
problem Understanding manifolds with special holonomy.
method Use of Frölicher-Nijenhuis bracket to define cohomologies and -algebras.
result Definition and computation of Frölicher-Nijenhuis cohomology.
Homotopy equivalence of cotangent bundles' function algebras is shown.
problem Understanding homotopy equivalence in cotangent bundles and their function algebras.
method Using shifted Poisson algebras and homotopy equivalence of bundles.
result Homotopy equivalent bundles have equivalent Poisson algebras.
The paper studies deformations of Lagrangian submanifolds using algebraic tools.
problem Deformation theory of Lagrangian submanifolds in symplectic geometry.
method Graded versions of the Darboux Theorem and Weinstein's Lagrangian tubular neighbourhood Theorem, attaching an -algebra to each submanifold.
result Controls the deformation theory of Lagrangian -submanifolds using an -algebra.