Uniformizing maps and period maps are topologically tame, leading to algebraicity of Hodge loci.
problem Understanding the algebraicity of Hodge loci in arithmetic quotients.
method Proving topological tameness of uniformizing maps and period maps, applying Peterzil-Starchenko's o-minimal GAGA theorem.
result The Hodge locus of (S,V) is a countable union of algebraic subvarieties of S. Proves GAGA-style result for toric vector bundles.
problem None explicitly stated in the abstract.
method Algebraic construction of Frölicher approximating vector bundle.
result Proves GAGA-style result for toric vector bundles.
Study reveals GAGA phenomenon in Poisson cohomology for plane structures with isolated singularities.
problem Understanding Poisson cohomology for plane structures with isolated singularities.
method Determined Gerstenhaber algebra structure over Poisson cohomology groups.
result GAGA type phenomenon observed in Poisson cohomology.
GAGA learns a warped metric for geometry-aware data generation and interpolation.
problem Challenges in generating data with meaningful geometry in high-dimensional datasets.
method Combines manifold learning with generative modeling to learn a warped Riemannian metric.
result GAGA improves trajectory inference by 30% in single-cell population-level data.
GAGA accelerates 3D molecular generation by replacing long trajectories with Gaussian approximations.
problem High computational cost of long generative trajectories in 3D molecular generation.
method GAGA identifies a characteristic step where molecular data becomes sufficiently Gaussian, replacing the trajectory with a Gaussian approximation.
result Significant improvement in both generation quality and computational efficiency.
Develops o-minimal de Rham cohomology for smooth manifolds.
problem Cohomology of smooth manifolds in o-minimal settings.
method Defines o-minimal de Rham cohomology for smooth manifolds in an o-minimal expansion of the real field.
result Establishes properties of o-minimal cohomology groups and invariance under diffeomorphisms.
We prove the existence of Verdier stratifications for sets definable in any o-minimal structure on (R, +, .). It is also shown that the Verdier condition (w) implies the Whitney condition (b) in o-minimal structures on (R, +, .). As a consequence the Whitney Stratification Theorem holds. The existence of (wf)-stratific…
Proposes GAGA algorithm for automatic hyperparameter learning in signal recovery.
problem Difficulty in selecting hyperparameters in traditional signal recovery methods.
method Global Adaptive Generative Adjustment (GAGA) algorithm for automatic hyperparameter learning and signal estimate.
result Consistency of model selection and signal estimate output.
FC-GAGA forecasts traffic using a novel gating mechanism.
problem Forecasting multivariate time-series, especially with graph relationships.
method Learnable fully connected hard graph gating mechanism for fully connected time-series forecasting.
result Competitive or better performance than existing algorithms without graph knowledge.
Defines Schwartz and tempered functions on o-minimal manifolds.
problem Defining Schwartz and tempered functions on non-polynomially bounded o-minimal manifolds.
method Defining Schwartz and tempered functions on manifolds definable in polynomially bounded o-minimal structures, and showing classical properties hold.
result The theory of Schwartz and tempered functions can be constructed on manifolds definable in polynomially bounded o-minimal structures but not on non-polynomially bounded ones.
Paper studies planar extensions in o-minimal structures.
problem Establishing conditions for definable homeomorphic extensions.
method Combinatorial conditions involving cyclic orders and orientations.
result Necessary and sufficient conditions for extensions.
Smooth maps in o-minimal structures are mostly transverse.
problem Transversality of smooth definable maps in o-minimal structures.
method Definable smooth version of Thom transversality theorem, proving nowhere density of non-transverse maps, and a definable version of Trotman's theorem.
result Non-transverse maps are nowhere dense in the definable smooth topology.
We expose some ideas from mathematical logics, i.e. the background of the theory of o-minimal structures, and demonstrate how they lead to the notion of a tame integral of motion and some extensions and clarifications of previous results on obstructions to integrability of geodesic flows.
Arguments on PL,(=piecewise linear) topology work over any ordered field in the same way as over the real field, and those on differential topology do over a real closed field R in an o-minimal structure that expands (R,<,0,1,+,cdot). One of the most fundamental properties of definable sets is that a compact definable …
New findings show modern neural networks have finite sample complexity in o-minimal structures.
problem Understanding the learnability of modern neural networks in a broad context.
method Analyzing feedforward neural networks definable in o-minimal structures.
result Modern neural networks, including MLPs, CNNs, GNNs, and transformers, have finite sample complexity in the agnostic PAC setting.
We prove that a theorem of Pawlucki, showing that Whitney regularity for a subanalytic set with a smooth singular locus of codimension one implies the set is a finite union of differentiable manifolds with boundary, applies to definable sets in polynomially bounded o-minimal structures. We give a refined version of Paw…
The paper extends a link criterion for Lipschitz normal embeddings to definable sets in o-minimal structures.
problem Characterizing Lipschitz normal embeddings of definable sets.
method Extending a known result about subanalytic germs to definable germs in any o-minimal structure.
result The link criterion holds for definable germs in o-minimal structures, but is not sufficient for all homomorphisms.
Proves embedding theorem for definable manifolds.
problem Embedding abstract-definable Cp manifolds into Euclidean space. method Proves Whitney embedding theorem for definable manifolds.
result Abstract-definable Cp manifolds are Cp embedded into RN. We give a geometric proof of existence of Whitney stratifications of definable sets in o-minimal structures.
Proves algebraicity of Hodge loci in arithmetic quotients.
problem Proving algebraicity of Hodge loci in arithmetic quotients.
method Real semi-algebraic structure and o-minimal theory.
result Hodge locus is a countable union of algebraic subvarieties.
Gradient flows of neural networks converge to optimal values or diverge, with thresholds and asymptotic behaviors.
problem Understanding the convergence and divergence of gradient flows in neural networks.
method Analysis of gradient flows on loss landscapes of neural networks using o-minimal structures.
result Gradient flows either converge to optimal values or diverge to infinity, with thresholds and asymptotic behaviors.
The paper studies coherent sheaves on subvarieties of Hopf manifolds.
problem Understanding coherent sheaves on subvarieties of Hopf manifolds.
method Proves a version of GAGA theorem, shows natural algebraic structure, and uses quotient and embedding properties.
result Any reflexive coherent sheaf on M is filtrable. Deep learning models viewed through tame geometry for convergence guarantees.
problem Understanding convergence guarantees in deep learning models.
method Introducing tame geometry concepts and tools for nonsmooth nonconvex settings.
result Illustrates tame geometry as a natural framework for AI systems, especially deep learning.
Defines smoothness of definable sets in o-minimal structures.
problem Characterizing smoothness of definable sets in o-minimal structures.
method Characterizes smoothness using tangent cones and metric properties.
result Equivalence of several conditions for C1 smoothness of definable sets. We clarify measurability assumptions in the agnostic PAC learning theorem.
problem Measurability assumptions in the Fundamental Theorem of Statistical Learning.
method Measure-theoretic scrutiny of existing proofs to extract minimal assumptions.
result Sound statement and detailed proof of the Fundamental Theorem in the agnostic setting.
We present a short complete proof of the existence of the normal cycle of a compact subanalytic set. The approach is inspired by some old ides of Joseph Fu, uses Morse theoretic techniques and o-minimal topology.
The paper defines a stratification for Lie groupoids in a tame topology context.
problem Presenting a tame topology counterpart to canonical stratification of Lie groupoids.
method Using Shiota's isotopy lemma and approximation theorem, the paper defines a canonical Whitney stratification of definable Lie groupoids into invariant strata.
result A canonical Whitney stratification of the Lie groupoid into definable strata invariant under the groupoid action.
In [5] I solved the Thom's conjecture that a proper Thom map is triangulable. In this paper I drop the properness condition in the semialgebraic case and, moreover, in the definable case in an o-minimal structure.
Paper proves generalizations of Bernstein's theorem in higher dimensions.
problem Proving theorems about sets in higher-dimensional spaces.
method Analyzes properties of sets with monotonicity formula and uses geometric and analytic techniques.
result Generalizations of Bernstein's theorem in higher dimensions.
Introduces Grassmann Distance Complexity to measure algebraic set nearest point problems.
problem Measuring complexity of finding nearest points in Grassmannian space.
method Uses Lipschitz critical point theory and o-minimal geometry.
result Establishes fundamental properties of GDC, including bounds and finiteness conditions.
The paper links set cuspidality to function regularity and flatness.
problem Linking set cuspidality to function regularity and flatness.
method Analyzes arc-smooth functions and their properties on various sets.
result Establishes a precise link between set cuspidality and function regularity.
In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or semi-Pfaffian cases, which depend only on the combinatoric data representing the objects…
Deep neural networks have almost linear sample complexity.
problem Sample complexity of deep neural networks.
method o-minimal expansion of the real field to bound sample complexity.
result Almost linear bound on sample complexity of neural networks.
The paper explains how continuous language models can produce discrete, interpretable meanings.
problem Semantic collapse in continuous systems of large language models.
method Formalizing large language models as Continuous State Machines (CSMs) and analyzing the associated transfer operator.
result The leading eigenfunctions of the transfer operator induce a finite number of invariant meaning basins, explaining how continuous computation can produce discrete, interpretable semantics.
We give several versions of local and global inverse mapping theorem for tame non necessarily smooth, mappings. Here tame mapping means a mapping which is subanalytic or, more generally, definable in some o-minimal structure. Our sufficient conditions are formulated in terms of various properties (convexity, positivity…
Let R be an o-minimal expansion of the real field. We introduce a class of Hausdorff limits, the T-infinity limits over R, that do not in general fall under the scope of Marker and Steinhorn's definability-of-types theorem. We prove that if R admits analytic cell decomposition, then every T-infinity limit over R is def…
In this paper we investigate how germs of real functions can change under deformation. In particular we look at deformations of germs of isolated singularities from R_n to R_k (n >= k) and the relation with there natural stratification in some tame categorie (algebraic, analytic, semi-algebraic, subanalytic, o-minimal …
We relate the Lipschitz-Killing measures of a definable set X⊂Rn in an o-minimal structure to the volumes of generic polar images. For smooth submanifolds of Rn, such results were established by Langevin and Shifrin.Then we give infinitesimal versions of these results. As a corollary, we…
Deep networks converge in direction, with implications for predictions and margins.
problem Understanding convergence and alignment in deep learning networks.
method Developed a theory of unbounded nonsmooth Kurdyka-Łojasiewicz inequalities for functions definable in an o-minimal structure.
result Network weights, predictions, training errors, and margin distribution converge in direction and align with gradient flow.
Stochastic subgradient descent avoids critical points in definable functions.
problem Finding local minima in definable functions.
method Stochastic subgradient descent with density-like perturbation.
result SGD converges to a local minimum in definable functions.
Euler calculus is based on integrating simple functions with respect to the Euler characteristic. This paper makes the case for extending Euler calculus to continuous integrands by integrating with respect to (Gaussian) curvature. This requires a metric but is nevertheless defined within any O-minimal theory. It satisf…
A coordinate cone in R^n is an intersection of some coordinate hyperplanes and open coordinate half-spaces. A semi-monotone set is a defnable in an o-minimal structure over the reals, open bounded subset of R^n such that its intersection with any translation of any coordinate cone is connected. This can be viewed as a …
Let X⊂Rn be a connected locally closed definable set in an o-minimal structure. We prove that the following three statements are equivalent: (i) X is a C1 manifold, (ii) the tangent cone and the paratangent cone of X coincide at every point in X, (iii) for every x∈X, the tangent cone of…
Smooth approximations proved for triangulable sets.
problem Universal approximation of continuous maps to triangulable sets.
method New approximation techniques for weakly Cr triangulable sets. result Triangulable sets are Cr-approximation targets. New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
Let S(R) be an o-minimal structure over R, T⊂Rk1+k2+ℓ a closed definable set, and $$ \displaylines{π_1: \R^{k_1+k_2+\ell}\to \R^{k_1 + k_2}, π_2: \R^{k_1+k_2+\ell}\to \R^{\ell}, \ π_3: \R^{k_1 + k_2} \to \R^{k_2}} $$ the projection maps. For any collection ${\mathcal A} = \{A_1,...,A…
Consider a transitive action of a Lie group G on a (real analytic) manifold M of dimension m, and two (embedded) submanifolds A and B in M of sufficiently large class and of dimension k and l, respectively. We prove that, for a generic σ∈G, the intersection σ(A)∩B is transversal, whence a su…
Let R be an o-minimal expansion of the real field, and let L(R) be the language consisting of all nested Rolle leaves over R. We call a set nested subpfaffian over R if it is the projection of a boolean combination of definable sets and nested Rolle leaves over R. Assuming that R admits analytic cell decomposition, we …