Outer automorphisms of free products are represented by CTs.
problem Representing outer automorphisms of free products by relative train track maps.
method Developed theory of attracting laminations and CTs for free products.
result Outer automorphisms of free products satisfy an index inequality.
A new neural network model ONCF improves collaborative filtering by explicitly modeling embedding correlations.
problem Improving collaborative filtering for better recommendation quality.
method ONCF uses an outer product to model embedding correlations and a convolutional layer to learn high-order correlations.
result ONCF outperforms existing models in implicit feedback data experiments.
Outer automorphism groups of Coxeter groups are trivial except for small ranks.
problem Triviality of outer automorphism groups of Coxeter groups.
method Using Guirardel-Levitt outer space for free products, proving triviality and cyclic order for specific ranks.
result Outer automorphism groups are trivial except for small ranks.
Paper shows neural network derivatives have outer product structure.
problem Understanding and utilizing higher-order information in neural networks.
method Analyzes feedforward, recurrent, and convolutional neural networks to identify derivative structure.
result Derivatives of feedforward and recurrent networks have an outer product structure, while convolutional networks do not.
We show that the horoboundary of outer space for the Lipschitz metric is a quotient of Culler and Morgan's classical boundary, two trees being identified whenever their translation length functions are homothetic in restriction to the set of primitive elements of FN. We identify the set of Busemann points with the s…
The paper shows symmetries of a geometric space for Coxeter groups.
problem Understanding symmetries in the Outer space of a Coxeter group.
method Analyzing the geometric rigidity of the universal Coxeter group of rank n.
result For n ≥ 4, the symmetries of the spine of the outer space are only the outer automorphisms.
Study embeddings of free group products into automorphism groups.
problem Embeddings of direct products of free groups into automorphism groups.
method Proved existence of canonical fixed points in the boundary of Outer space.
result Complete description of embeddings of direct products of free groups.
The study classifies subgroups of outer automorphisms of free products.
problem Classifying subgroups of outer automorphisms of free products.
method Geometric tool: boundaries of relative factor graphs and equivalence classes of arational trees.
result Every finitely generated subgroup either contains a relatively fully irreducible automorphism or virtually preserves a conjugacy class.
New method uses outer product manifolds to simplify neural networks.
problem Exponential inefficiency of hierarchical neural networks.
method Reparametrization invariant Riemannian metrics and tangent subspace computation.
result Significant improvement in network performance after early training.
Outer automorphism group of hyperbolic groups is HHG under certain conditions.
problem Characterizing the outer automorphism group of hyperbolic groups.
method Proving finite-index subgroups are central extensions of orbifold mapping class groups with bounded Euler class.
result Outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group.
Paper improves SDR estimation speed and conditions.
problem Improving sufficient dimension reduction for multi-index models.
method Estimating expected smoothed gradient outer product.
result Achieves fast parametric convergence rate of Cd⋅n−1/2. In this paper we develop the metric theory for the outer space of a free product of groups. This generalizes the theory of the outer space of a free group, and includes its relative versions. The outer space of a free product is made of G-trees with possibly non-trivial vertex stabilisers. The strategies are the same…
The paper shows that for Coxeter groups, the commensurator of outer automorphisms is rigid.
problem The rigidity of commensurator of outer automorphisms of Coxeter groups.
method Study of the abstract commensurator of the outer automorphism group of a universal Coxeter group.
result For n≥5, the natural map is an isomorphism and every isomorphism between finite index subgroups is conjugation. The study examines how automorphism growth rates of a group can be deduced from its simpler decompositions.
problem Determine automorphism growth rates of a group from its simpler decompositions.
method Analyze group decompositions into simpler pieces (direct products, free products, graph of groups) and deduce growth rates.
result Information about automorphism growth rates of a group can be deduced from its simpler decompositions.
We prove central limit theorems for the random walks on either the mapping class group of a closed, connected, orientable, hyperbolic surface, or on Out(FN), each time under a finite second moment condition on the measure (either with respect to the Teichmüller metric, or with respect to the Lipschitz metric …
Study on automorphism groups of Coxeter groups, proving they are not CAT(0).
problem Determining the curvature of automorphism groups of Coxeter groups.
method Combinatorial and geometric analysis of automorphism groups of universal right-angled Coxeter groups.
result Proved that the natural model space for outer automorphism groups is not CAT(0).
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a fr…
Let G=G1∗⋯∗Gk∗F be a countable group which splits as a free product, where all groups Gi are freely indecomposable and not isomorphic to Z, and F is a finitely generated free group. If for all i∈{1,…,k}, both Gi and its outer automorphism group Out(Gi) satisfy t…
Given a real matrix A with n columns, the problem is to approximate the Gram product AA^T by c << n weighted outer products of columns of A. Necessary and sufficient conditions for the exact computation of AA^T (in exact arithmetic) from c >= rank(A) columns depend on the right singular vector matrix of A. For a Monte-…
We associate a contractible ``outer space'' to any free product of groups G=G_1*...*G_q. It equals Culler-Vogtmann space when G is free, McCullough-Miller space when no G_i is Z. Our proof of contractibility (given when G is not free) is based on Skora's idea of deforming morphisms between trees. Using the action of Ou…
In this paper we study obstructions to presentability by products for finitely generated groups. Along the way we develop both the concept of acentral subgroups, and the relations between presentability by products on the one hand, and certain geometric and measure or orbit equivalence invariants of groups on the other…
Outer billiards defined on geodesics surfaces in 3D space forms.
problem Defining and analyzing outer billiards on geodesics in 3D space forms.
method Defined an outer billiard map on the space of oriented geodesics, showing diffeomorphism and symplectomorphism properties.
result Outer billiard map is a diffeomorphism and symplectomorphism under certain conditions.
We study aspherical manifolds that do not support Anosov diffeomorphisms. Weakening conditions of Gogolev and Lafont, we show that the product of an infranilmanifold with finitely many aspherical manifolds whose fundamental groups have trivial center and finite outer automorphism group does not support Anosov diffeomor…
Let G be a countable group that splits as a free product of groups of the form G=G1∗⋯∗Gk∗FN, where FN is a finitely generated free group. We identify the closure of the outer space PO(G,{G1,…,Gk}) for the axes topology with the space of projective minimal, \emph{very small} …
Calibration in 16D disproves Federer's product question.
problem Proving Federer's product question in 16D.
method Adapted Cayley calibration in R^16, constructing a specific calibration Φ.
result Product of two orthogonally supported calibrations is not always a calibration.
AGOP mechanism explains deep neural collapse in neural networks.
problem Explaining the rigid structure of data representations in deep neural networks.
method Introducing AGOP and Deep RFM to demonstrate DNC.
result AGOP mechanism causes deep neural collapse in neural networks.
We define analogues of the graphs of free splittings, of cyclic splittings, and of maximally-cyclic splittings of FN for free products of groups, and show their hyperbolicity. Given a countable group G which splits as G=G1∗⋯∗Gk∗F, where F denotes a finitely generated free group, we identify th…
The paper proves rigidity results for manifolds with scalar curvature and boundary.
problem Rigidity of manifolds with scalar curvature and boundary.
method Analysis of Obata's equation and application of marginally outer trapped surfaces.
result Several rigidity results for manifolds with scalar curvature and boundary.
Recursive Feature Machines show grokking in modular arithmetic without neural networks.
problem Grokking in modular arithmetic tasks.
method Recursive Feature Machines (RFM) with Average Gradient Outer Product (AGOP).
result RFM and neural networks learn block-circulant features to solve modular arithmetic.
In a recent work, Galloway [9] proved a local foliation theorem by MOTSs for a 3-dimensional initial data set (M,g,K) with mean curvature τ≤0 in a 4-dimensional spacetime (M,g) when (under suitable assumptions) M has a stable spherical MOTS Σ which achieves an upper bound for the area. H…
We prove that for k≥5 there does not exist a continuous map ∂CV(Fk)→PCurr(Fk) that is either Out(Fk)-equivariant or Out(Fk)-anti-equivariant. Here ∂CV(Fk) is the "length-function" boundary of Culler-Vogtmann's Outer space CV(Fk), and PCurr(Fk) is the space of pr…
Graphical notation simplifies tensor operations and decompositions.
problem Complex tensor operations are difficult to understand and represent.
method Introduces graphical notation to represent tensor operations.
result Simplified representation of tensor operations and decompositions.
The thesis shows how automorphisms of hyperbolic groups can be represented by train track maps.
problem Representing automorphisms of hyperbolic groups using train track maps.
method Using graphs of groups and Bestvina-Handel's irreducible train track maps, the thesis constructs relative train track maps.
result Outer automorphisms of finitely-generated word hyperbolic groups satisfy a dynamical trichotomy.
The paper proves rigidity for cocycles from higher rank lattices to Out(FN).
problem Proving rigidity for cocycles from higher rank lattices to Out(FN).
method Geometric tool: barycenter map.
result Every Borel cocycle is cohomologous to a cocycle with finite image.
Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.
problem Finiteness properties and contractibility of symmetric automorphisms of RAAGs.
method Definition of symmetric automorphism group, construction of symmetric Outer space, proof of contractibility.
result Finiteness properties and contractibility results for symmetric automorphisms of RAAGs.
New insights into optimizing Local SGD's outer optimizer for faster convergence.
problem Understanding the impact of outer optimizer and its hyperparameters in Local SGD.
method Analyzing convergence guarantees with new outer learning rates and momentum.
result Tuning the outer learning rate can improve convergence and handle inner learning rate ill-tuning.
This work is the first step towards a description of the Gromov boundary of the free factor graph of a free product, with applications to subgroup classification for outer automorphisms. We extend the theory of algebraic laminations dual to trees, as developed by Coulbois, Hilion, Lustig and Reynolds, to the context of…
Criterion for subgroup separability in outer automorphism groups.
problem Subgroup separability in outer automorphism groups.
method Criterion for separability of subgroups.
result Strengthening and generalizing a previous result on mapping class groups.
Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.
problem Understanding the structure of trees in Outer space.
method Associate simplices to R-trees and estimate their dimensions. result Estimates the dimensions of maximal simplices for both rational and irrational trees.
We give estimates on the length of paths defined in the sphere model of outer space using a surgery process, and show that they make definite progress in some sense when they remain in some thick part of outer space. To do so, we relate the Lipschitz metric on outer space to a notion of intersection numbers.
Random trees found in Outer space boundary.
problem Understanding trees in the boundary of Outer space.
method Proving properties of harmonic measure on random walks.
result Typical tree is trivalent and nongeometric.
Study improves curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
problem Curvature estimate for stable marginally outer trapped hypersurfaces with a free boundary.
method Iteration argument based on uniform area bound.
result Improved curvature estimate for stable marginally outer trapped hypersurfaces.
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.
KoPA approximates matrices using Kronecker products for better flexibility.
problem Matrix approximation and denoising with Kronecker product decomposition.
method Approximate a matrix as a sum of Kronecker products of smaller matrices using extended information criteria for configuration selection.
result KoPA selects the true configuration with high probability under suitable conditions.
We show that any vacuum initial data set containing a marginally outer trapped surface S and satisfying a "no KIDs" condition can be perturbed near S so that S becomes strictly outer trapped in the new vacuum initial data set. This, together with the results in [9], gives a precise sense in which generic initial data c…
Paper identifies how neural networks learn features.
problem Understanding how neural networks automatically select features.
method Deep Neural Feature Ansatz, implementing average gradient outer product.
result Recursive Feature Machines achieve state-of-the-art performance.
This paper aims at achieving a "good" estimator for the gradient of a function on a high-dimensional space. Often such functions are not sensitive in all coordinates and the gradient of the function is almost sparse. We propose a method for gradient estimation that combines ideas from Spall's Simultaneous Perturbation …
Unfolding paths in Outer space accumulate on a simplex, not converge.
problem Understanding accumulation points in Outer space.
method Constructing an unfolding path in Outer space.
result Unfolding paths accumulate on a 1-simplex, not converge.