This work simplifies proximal mapping for low-rank norms.
problem Efficient computation of proximal mappings for low-rank inducing norms.
method Reduces proximal mapping to nested binary search, solving simpler problems analytically.
result Simplified computation of proximal mappings for various norms.
Paper analyzes structured matrix recovery using generalized Dantzig selector.
problem Structured matrix recovery for applications like recommender systems and computer vision.
method Non-asymptotic analysis of generalized Dantzig selector for estimation of generally structured matrices.
result Estimation error can be expressed in terms of geometric measures of suitable sets.
Study analyzes perturbations in singular subspaces under random noise.
problem Understanding singular vector and subspace changes in signal-plus-noise models.
method Generalized Davis-Kahan-Wedin theorem for any unitarily invariant norm, considering ℓ∞ and ℓ2,∞ bounds. result Fine-grained insights into singular vector and subspace perturbations, including ℓ∞ and ℓ2,∞ bounds. The hermitian analog of Aleksandrov's area measures of convex bodies is investigated. A characterization of those area measures which arise as the first variation of unitarily invariant valuations is established. General smooth area measures are shown to form a module over smooth valuations and the module of unitarily …
When data is sampled from an unknown subspace, principal component analysis (PCA) provides an effective way to estimate the subspace and hence reduce the dimension of the data. At the heart of PCA is the Eckart-Young-Mirsky theorem, which characterizes the best rank k approximation of a matrix. In this paper, we prove …
We obtain new general results on the structure of the space of translation invariant continuous valuations on convex sets (a version of the hard Lefschetz theorem). Using these and our previous results we obtain explicit characterization of unitarily invariant translation invariant continuous valuations. It implies new…
Convexity proven for sums of angles of unitary paths.
problem Proving convexity of sums of eigenvalues of unitary matrices.
method Analyzing paths of unitary matrices and their angles, using operator norms.
result Sum of first m angles of unitary path is convex.
Researchers classify and decompose valuations on convex functions.
problem Classifying valuations on convex functions.
method Geometric decomposition of valuations, using properties of special subspaces and Monge-Ampère-type operators.
result Valuations decompose into subspaces defined by vanishing properties.
Groups of importance in group theory have flexible stability properties.
problem Stability and flexibility of groups in geometric and combinatorial group theory.
method Establishing Kirchberg's Local Lifting Property and Lubotzky--Shalom's Property FD for specific groups.
result Groups like 3-manifold groups, limit groups, and certain one-relator groups are very flexibly stable. We obtain upper bounds for the eigenvalues of the Schrödinger operator L=Δg+q depending on integral quantities of the potential q and a conformal invariant called the min-conformal volume. Moreover, when the Schrödinger operator L is positive, integral quantities of q which appear in upper bounds, can be repla…
Given a positive and unitarily invariant Lagrangian L defined in the algebra of Hermitian matrices, and a fixed interval [a,b]⊂R, we study the action defined in the Lie group of n×n unitary matrices U(n) by S(α)=∫abL(α˙(t))dt, where α:[a,b]→U(n) is a …
Paper analyzes singular subspace estimation in noisy matrix models.
problem Estimating low-rank signals in noisy matrix data.
method Asymptotic distributional theory, extreme value theory, saddle point approximation, random matrix theory.
result Plug-in test statistic based on two-to-infinity norm has higher power for detecting structured alternatives.
Abstract: Proves Tutte's sequence connection to complex space forms.
problem Relating algebra of isometry invariant valuations to combinatorics.
method Proves Fu's power series conjecture.
result Fu's power series conjecture is proven, linking algebra to combinatorics.
Muons and random optimizers perform similarly, challenging geometric optimization theory.
problem Empirical success of Muon optimizer challenges geometric optimization theory.
method Introducing Freon and Kaon optimizers, demonstrating performance without precise geometric structure.
result Performance of optimizers is controlled by alignment and descent potential, not geometric structure.
New algebra structure for curvature measures in complex space forms.
problem Understanding curvature measures in complex space forms.
method Explicitly describing the algebra structure of dual unitarily invariant curvature measures.
result Characterization of invariant valuations on complex space forms.
We prove that Nelson's massless scalar field model is infrared divergent in three dimensions. In particular, the Nelson Hamiltonian and the Hamiltonian obtained from Euclidean quantization are not unitarily equivalent. In contrast, for dimensions higher than three the Nelson Hamiltonian has a unique ground state in Foc…
Let p: M -> B be a family of compact manifolds equipped with a unitarily flat vector bundle F -> M. We generalize Igusa's higher Franz-Reidemeister torsion τ(M/B;F) to the case that the fibre-wise cohomology H^*(M/B;F) -> B carries a parallel metric. If moreover M admits a fibre-wise Morse function, we compute the diff…
The paper compares two torsion invariants in complex vector bundles.
problem Comparing two torsion invariants in complex vector bundles.
method Constructing Bismut-Lott analytic torsion classes and showing they coincide with Igusa-Klein torsions.
result Bismut-Lott analytic torsion classes coincide with Igusa-Klein torsions for trivial flat line bundles.
Study on constant curvature immersions of surfaces into flag manifolds.
problem Investigate constant curvature immersions of Riemann surfaces into flag manifolds.
method Investigate pseudoholomorphic maps and invariant metrics on flag manifolds.
result Unitarily equivalent primitive immersions of the two-sphere into full flag manifolds have constant curvature under all invariant metrics.
New method shows unitarity in quantization for toric manifolds.
problem Unitarity in quantization commutes with reduction for toric manifolds.
method Generalized coherent state transform (gCST) and geodesic rays of toric Kähler polarizations.
result Quantization commutes unitarily with reduction for the new mixed polarization.
New non-semisimple Ising anyons enable robust universal quantum computation.
problem Limitation of semisimple theories in universal topological quantum computation.
method Developed non-semisimple Ising anyon model with new anyon types indexed by α. result Robust universality of braiding persists over an open interval of α. The paper studies norms and invariants for free-by-cyclic groups.
problem Investigating norms and invariants for a specific class of groups.
method Analyzes universal L2-torsion invariants and their Newton polytopes. result Establishes inequalities between different norms and invariants.
The paper proves a duality theorem for conjugation-invariant norms and quasimorphisms.
problem Understanding conjugation-invariant norms and their relationship to quasimorphisms.
method Develops a Bavard-type duality theorem for conjugation-invariant norms and subset-controlled quasimorphisms.
result Proves a duality theorem between conjugation-invariant norms and subset-controlled quasimorphisms.
Study reflection symmetry and APS boundary conditions on a warped cylinder.
problem Analyzing reflection symmetry and APS boundary conditions for twisted Dirac operators on a finite warped cylinder.
method Examined reflection symmetry and APS boundary conditions for twisted Dirac operators on a finite warped cylinder, considering both fixed and varying holonomy.
result Reflection symmetry lifts to a unitary symmetry under specific conditions, and the spectral flow admits an RO(O(2))-valued decomposition for fixed holonomy.
Survey of Thurston norm properties and connections to 3-manifold invariants.
problem Understanding Thurston norm in 3-manifolds.
method Review and analysis of existing literature.
result Relationships between Thurston norm and various topological invariants.
Surveying hermitian integral geometry, the paper describes new kinematic formulas for complex space forms.
problem Understanding curvature measures and valuations on complex spaces.
method Analyzing valuations and curvature measures on complex space forms, deriving kinematic formulas.
result New local kinematic formulas for hermitian geometry, containing more information than global formulas.
The paper connects Turaev-Viro invariants and Gromov norm of 3-manifolds.
problem Relating Turaev-Viro invariants and Gromov norm of 3-manifolds.
method Combines TQFT techniques, geometric decomposition theory, and analytical estimates of 6j-symbols. result Established a relation between the asymptotics of Turaev-Viro invariants and Gromov norm of 3-manifolds.
Invariants for 3-manifolds with toral boundaries, related by sutured decompositions.
problem Invariants for 3-manifolds with toral boundaries and non-degenerate Thurston norm.
method Constructing an invariant called guts and proving its invariance under sutured decompositions.
result The guts of different homology classes are related by sutured decompositions.
Analytic torsion equals Ruelle zeta function value for certain manifolds.
problem Equality of analytic torsion and dynamical zeta function values.
method Proof using Ruelle dynamical zeta functions and acyclic unitarily flat vector bundles.
result Solves Fried's conjecture on equality of analytic torsion and zeta function values.
Lower bound for L^2-norm of Hermitian scalar curvature derived from Futaki invariant.
problem Finding lower bounds for the L^2-norm of Hermitian scalar curvature.
method Using the symplectic Futaki invariant as an asymptotic obstruction to the existence of constant Hermitian scalar curvature almost-Kähler metrics.
result Deduced a lower bound for the L^2-norm of the Hermitian scalar curvature.
Let Γ1 and Γ2 be Bieberbach groups contained in the full isometry group G of Rn. We prove that if the compact flat manifolds Γ1\Rn and Γ2\Rn are strongly isospectral then the Bieberbach groups Γ1 and Γ2 are representation equivalent, that is, the rig…
SAM improves generalization in overparameterized models, but its behavior in tensorized models is less understood.
problem Understanding the implicit regularization of SAM in tensorized models.
method Scale-invariance analysis and gradient flow analysis to derive Norm Deviation as a measure of core norm imbalance, and propose Deviation-Aware Scaling (DAS).
result DAS achieves competitive or improved performance over SAM, while offering reduced computational overhead.
The paper proves norms on braid group commutator subgroup are unbounded.
problem Understanding norms on the commutator subgroup of infinite braid groups.
method Constructed stably unbounded norms and showed equivalence to biinvariant word norms.
result Found norms on commutator subgroup are equivalent to biinvariant word norms and are stably unbounded.
New Thurston norm defined for a specific type of groups using L2-invariants.
problem Measuring splitting complexity of integral characters in coherent right-angled Artin groups.
method Defining splitting complexity via L2-Euler characteristic and using Friedl--Lück's L2-polytope. result A Thurston-type semi-norm defined for measuring splitting complexity of integral characters.
For a 3-manifold M, McMullen derived from the Alexander polynomial of M a norm on H^1(M, R) called the Alexander norm. He showed that the Thurston norm on H^1(M, R), which measures the complexity of a dual surface, is an upper bound for the Alexander norm. He asked if these two norms were equal on all of H^1(M,R) when …
This paper shows excessive invariance in adversarial robust models can make them more vulnerable to certain types of attacks.
problem Excessive invariance in adversarial robust models can make them more vulnerable to certain types of attacks.
method Analytical constructions and empirical studies of vision classifiers with state-of-the-art robustness to perturbation-based adversaries constrained by an ℓp norm. result Robustness to perturbation-based adversarial examples does not guarantee general robustness and can increase vulnerability to invariance-based adversarial examples.
Scale-invariant algorithms for unconstrained online learning.
problem Designing online algorithms invariant to arbitrary linear transformations of input vectors.
method Exploiting scale invariance symmetry, developing algorithms for coordinate-wise and general invariance.
result Achieved optimal regret bound for coordinate-wise invariance, and almost achieved it for general invariance with logarithmic overhead.
Lie groups with bi-invariant distance are products of abelian and compact groups.
problem Characterizing Lie groups with bi-invariant distances.
method Analyzing the structure of Lie groups and introducing a Finsler norm.
result The sectional curvature of bi-invariant distances is non-negative and vanishes only for abelian subalgebras.
Analytic torsion equals dynamical zeta function for certain bundles.
problem Equalities between analytic torsion and dynamical zeta functions.
method Analytic torsion and Ruelle dynamical zeta function for admissible twists.
result Generalization of previous results to admissible twists.
We define an infinite sequence of new invariants, delta_n, of a group G that measure the size of the successive quotients of the derived series of G. In the case that G is the fundamental group of a 3-manifold, we obtain new 3-manifold invariants. These invariants are closely related to the topology of the 3-manifold. …
In a previous paper, the second author defined integer-valued functions delta_n on the first cohomology of a 3-manifold, generalizing McMullen's Alexander norm. It was shown that these functions give lower bounds on the Thurston norm. In this paper, we reformulate these invariants in terms of Reidemeister torsion over …
In this note we observe that the no two of the three invariants defined for contact structures by Etnyre and Ozbagci -- that is, the support genus, binding number and support norm -- determine the third.
Geodesic distance vanishes for critical Sobolev norms on diffeomorphism groups.
problem Analyzing geodesic distance in diffeomorphism groups for critical Sobolev norms.
method Combining techniques from [JM19] and [BHP18]
result Geodesic distance vanishes for Ws,n/s norms when s∈(0,1) and sp≤n. New capacity measure based on Fisher-Rao norm for neural networks.
problem Understanding the complexity and capacity of neural networks.
method Introducing Fisher-Rao norm and studying its invariance properties.
result The Fisher-Rao norm serves as an umbrella for existing norm-based complexity measures.
New adaptive signal denoising method mimics oracle with better statistical properties.
problem Adaptive discrete-time signal denoising with linear oracle structure.
method Minimizes the ℓ2-norm of the estimation residual, proving oracle inequalities for ℓ2-loss. result Improved statistical properties over ℓ∞-fit estimators, especially in ℓ2- and pointwise losses. A polynomial invariant for veering triangulations helps in understanding 3-manifold fibers.
problem Understanding the fibers of 3-manifolds using veering triangulations.
method Introducing a polynomial invariant Vτ associated to veering triangulations and using flow graphs. result The invariant Vτ recovers the Teichmüller polynomial for fibered faces and determines cones in homology. Extends risk measure theory to general Orlicz spaces.
problem Applying risk measure theory to non-standard spaces.
method Generalizes results from bounded random variables to general Orlicz spaces, proving new characterizations and extensions.
result Characterizations and extensions of the Fatou property and Kusuoka representation in Orlicz spaces.
Riemannian cubics are critical points for the L2 norm of acceleration of curves in Riemannian manifolds M. In the present paper the L∞ norm replaces the L2 norm, and a less direct argument is used to derive necessary conditions analogous to those for Riemannian cubics. The necessary conditions are exami…