We present a graded-geometric approach to modular classes of Lie algebroids and their generalizations, introducing in this setting an idea of relative modular class of a Dirac structure for a certain type of Courant algebroids, called projectable. This novel approach puts several concepts related to Poisson geometry an…
Equivariant localization techniques give a rigorous interpretation of the Witten genus as an integral over the double loop space. This provides a geometric explanation for its modularity properties. It also reveals an interplay between the geometry of double loop spaces and complex analytic elliptic cohomology. In part…
New method uses hyperspherical geometry to improve community detection.
problem Improving community detection methods in network analysis.
method Mapping networks to points on a hypersphere, then projecting to clustering vectors.
result Modularity maximization is equivalent to minimizing angular distance on the hypersphere.
Regularity properties of intrinsic objects for a large class of Stein Manifolds, namely of Monge-Ampère exhaustions and Kobayashi distance, is interpreted in terms of modular data. The results lead to a construction of an infinite dimensional family of convex domains with squared Kobayashi distance of prescribed regula…
The {\em Wiman-Edge pencil} is the universal family $\Cs/\mathcal B$ of projective, genus 6, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The goal of this paper is to prove that the monodromy of $\Cs/\mathcal B$ is commensurable with a Hilbert modular group; in particular is …
Modular deep learning framework using pairwise labels without backpropagation.
problem Efficiently training deep neural networks with limited supervision.
method Stacked linear models in feature spaces, provably optimal modular learning framework.
result High accuracy (94.88%) achieved with minimal labeled examples (1 per class).
Paper studies Wiman-Edge pencil and Wiman curve, providing uniformizations and modular interpretations.
problem Understanding the geometry and uniformization of the Wiman-Edge pencil and Wiman curve.
method Explicit uniformizations of the Wiman-Edge pencil and Wiman curve as quotients of the hyperbolic plane and arithmetic quotients.
result Explicit uniformizations and modular interpretations of the Wiman-Edge pencil and Wiman curve.
We derive formulae which lend themselves to TQFT interpretations of the Milnor torsion, the Lescop invariant, the Casson invariant, and the Casson-Morita cocyle of a 3-manifold, and, furthermore, relate them to the Reshetikhin-Turaev theory.
VTIRT speeds up IRT inference for dynamic learner proficiency.
problem Expensive and slow inference algorithms for dynamic IRT models.
method Variational Temporal IRT (VTIRT) for fast, accurate inference.
result Orders of magnitude speedup in inference runtime with accurate results.
Flexible framework for deep distributional regression models.
problem Learning conditional distributions from semi-structured data.
method Combines additive regression models with deep networks using TensorFlow.
result State-of-the-art predictive performance with interpretability.
Study of knot complements yields quantum modularity insights.
problem Understanding quantum invariants of knot complements.
method Large-N analysis of q-series invariants, counts of holomorphic curves. result Closed-form expressions for a-deformed FK for (2,2p+1)-torus knots. Interpreting the prediction mechanism of complex models is currently one of the most important tasks in the machine learning field, especially with layered neural networks, which have achieved high predictive performance with various practical data sets. To reveal the global structure of a trained neural network in an …
In this paper we define the p-adic framed braid group F∞,n, arising as the inverse limit of the modular framed braids and we give topological generators for F∞,n. We also give geometric interpretations for the p-adic framed braids. We then construct a p-adic Yokonuma-Hec…
The f-invariant is a higher version of the e-invariant that takes values in the divided congruences between modular forms; it can be formulated as an elliptic genus of manifolds with corners of codimension two. In this thesis, we develop a geometrical interpretation of the f-invariant in terms of index theory, thereby …
Transformers learn to solve modular arithmetic tasks by in-context learning and skill composition.
problem Understanding how large language models generalize to unseen tasks in modular arithmetic.
method Pre-training on a set of modular arithmetic tasks and evaluating out-of-distribution performance.
result Transformers require two transformer blocks for out-of-distribution generalization, and deeper models exhibit transient out-of-distribution performance.
Fractal Flow enhances normalizing flows with interpretable latent space and hierarchical modeling.
problem High-dimensional density estimation and generative modeling challenges.
method Integrates topic modeling (LDA) and fractal strategy into normalizing flows.
result Achieves latent clustering, controllable generation, and superior estimation accuracy.
Scaling model capacity has been vital in the success of deep learning. For a typical network, necessary compute resources and training time grow dramatically with model size. Conditional computation is a promising way to increase the number of parameters with a relatively small increase in resources. We propose a train…
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…
MOCA uses modular attention to estimate causal effects from complex data.
problem Estimating causal effects from observational data with complex, non-linear, and high-dimensional treatment and outcome mechanisms.
method MOCA is a transformer-based framework that separates treatment and outcome modeling through modular design and one-way attention mechanism, with cutting-feedback to prevent outcome influence on treatment representations.
result MOCA outperforms classical estimators and machine learning approaches across various simulated and real-world scenarios.
Study interprets neural network generalization and memorization on corrupted data.
problem Understanding when a neural network has memorized corrupted data versus learned the underlying rule.
method Analyzes multi-layer perceptrons and Transformers on modular arithmetic tasks with corrupted labels.
result Regularization methods can force networks to ignore corrupted data, improving accuracy on uncorrupted data.
A new robust prefix-tuning framework improves model robustness against adversarial attacks.
problem Lack of robustness in prefix-tuning for adversarial attacks.
method Leveraging layerwise activations of pretrained models for additional prefix finetuning during the test phase.
result Framework substantially improves robustness over strong baselines while maintaining comparable accuracy on clean texts.
We explain how neural networks learn to solve modular addition tasks.
problem How two-layer neural networks learn to solve modular addition tasks.
method Formalized a diversification condition during training, proving it allows the network to approximate the correct logic for modular addition.
result Neural networks can robustly identify the correct sum through phase symmetry and frequency diversification.
The paper studies how neural policies can be interpreted using decision trees.
problem Understanding how machine learning controllers make decisions in complex environments.
method The approach involves disentangled representation using decision trees to interpret neural policies.
result The paper shows that disentanglement of learned neural dynamics improves interpretability.
New GCNs solve graph embedding problems efficiently and interpretably.
problem Graph embedding for scalable and interpretable machine learning.
method Proposed two GCNs: CAFE-GCN and sphere-GCN, based on constrained optimization.
result Both GCNs yield good approximations of dominant eigenvectors and perform dimensionality reduction.
We extend the theory of the universal eta-invariant to the case of relative bordism groups of manifolds with boundaries. This allows the construction of secondary descendants of the universal eta-invariant. We obtain an interpretation of Laures' f-invariant as an example of this general construction. As an aside we imp…
SX-GeoTree improves spatially coherent explanations in geospatial regression trees.
problem Capturing spatial dependence and producing robust explanations in tabular prediction models.
method Integrates three objectives: impurity reduction, spatial residual control, and explanation robustness via modularity maximization on a consensus similarity network.
result Improves residual spatial evenness and doubles attribution consensus (modularity: Fujian 0.19 vs 0.09; Seattle 0.10 vs 0.05).
Study of q-rationals and their geometric properties, including deformed Farey triangulation and Springborn operations.
problem Geometry of q-rationals and their properties. method Construction and analysis of deformed Farey triangulation and deformed modular surface; definition and study of Springborn operations.
result Derivation of a formula for the q-deformed midpoint and new q-deformation of Markov numbers. We find and propose an explanation for a large variety of modularity-related symmetries in problems of 3-manifold topology and physics of 3d N=2 theories where such structures a priori are not manifest. These modular structures include: mock modular forms, SL(2,Z) Weil representations, quantum mo…
Flexible spatial models improve predictive performance over nonstationary alternatives.
problem Improving predictive performance in nonstationary spatial modeling.
method Introduces a modular parametric covariance function that extends nonstationary spatial models.
result The proposed covariance function outperforms nonparametric methods in predictive performance.
Researchers found the global topology of the Eisenstein-Picard modular surface.
problem Understanding the global topology of the Eisenstein-Picard modular surface.
method Quotient space of the complex hyperbolic plane by the modular group.
result Determined the global topology of the Eisenstein-Picard modular surface as a 4-orbifold.
Study abelian varieties' Weil-Petersson metric asymptotics.
problem Asymptotic behavior of Weil-Petersson metric on abelian varieties.
method Linking asymptotic with multi-scale collapsing limits of parametrized flat tori.
result Refined description of Weil-Petersson metric on abelian varieties.
Study modular surfaces in Lorentz-Minkowski 3-space, classifying and analyzing their curvature and applications.
problem Understanding the curvature properties of modular surfaces in Lorentz-Minkowski space.
method Analyzing the sign of Gaussian and mean curvature, classifying surfaces, and applying to conformal field theories.
result Complete classification of zero Gaussian curvature modular surfaces and non-existence of non-planar maximal modular surfaces.
BL learns interpretable optimization structures from data.
problem Learning interpretable optimization structures from data.
method BL parameterizes a compositional utility function from intrinsically interpretable modular blocks.
result BL supports architectures from single to hierarchical compositions, modeling hierarchical optimization structures.
Modular neural networks generalize better with less data.
problem Theoretical and practical understanding of how modularity improves neural network generalization.
method Theoretical analysis of sample complexity, development of a novel learning rule.
result Modular networks require fewer samples to generalize compared to nonmodular networks, especially in high-dimensional tasks.
Study modular forms over Γ^0(2) and anomaly cancellation formulas.
problem Anomaly cancellation formulas for modular forms over Γ^0(2).
method Study and analysis of modular forms over Γ^0(2).
result Anomaly cancellation formulas derived for modular forms over Γ^0(2).
We propose a novel statistical model for sparse networks with overlapping community structure. The model is based on representing the graph as an exchangeable point process, and naturally generalizes existing probabilistic models with overlapping block-structure to the sparse regime. Our construction builds on vectors …
Our aim is to introduce and advocate non-Σ (non-symmetric) modular operads. While ordinary modular operads were inspired by the structure of the moduli space of stable complex curves, non-Σ modular operads model surfaces with open strings outputs. An immediate application of our theory is a short proof that the mod…
Neural networks learn modular arithmetic but not all, extending known solutions to generalize.
problem Neural networks struggle with modular arithmetic, especially for polynomials.
method Developed analytical solutions for MLP networks to learn modular addition and multiplication, then combined these solutions to generalize on arbitrary modular polynomials.
result Neural networks can learn and generalize solutions to modular polynomials, supporting the hypothesis that some polynomials are learnable.
Fuchsian groups with a modular embedding have the richest arithmetic properties among non-arithmetic Fuchsian groups. But they are very rare, all known examples being related either to triangle groups or to Teichmueller curves. In Part I of this paper we study the arithmetic properties of the modular embedding and deve…
Improved algorithm for modular links provides upper volume bounds.
problem Understanding the geometry of modular links and Lorenz links.
method Bunch algorithm to study modular links and provide upper volume bounds.
result First upper volume bound independent of word exponents and quadratic in braid index.
Geodesics on modular surface yield arithmetic 3-manifolds.
problem Understanding arithmetic properties of modular surfaces.
method Constructing geodesics and analyzing their lifts.
result Complements of canonical lifts are arithmetic 3-manifolds.
We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold M with its Nambu tensor Λ as the modular class of the tangent Lie algebroid TM with Nambu structure Λ. We show that many known properties of th…
New modular forms for anomaly cancellation formulas on any dimensional manifolds.
problem Constructing new modular forms for anomaly cancellation formulas.
method Using E8 bundles, constructing modular forms on any dimensional manifolds. result Derived new anomaly cancellation formulas and applications.
Using the Weil-Brezin-Zak transform of solid state physics, we describe line bundles over elliptic curves in terms of Weyl operators. We then discuss the connection with finitely-generated projective modules over the algebra Aθ of the noncommutative torus. We show that such Aθ-modules have a natural interpretatio…
Quantum modularity proved for SU(2) TQFT signature on genus 2 surfaces.
problem Proving quantum modularity of SU(2) TQFT signature for genus 2 surfaces.
method Using quantum modularity of generalized Dedekind sums associated with modular forms and trigonometric sum expressions.
result Quantum modularity of SU(2) TQFT signature on genus 2 surfaces proved.
Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that e…
Countable modular groups found on surfaces with infinite type.
problem Finding modular groups of infinite type surfaces.
method Proving countable modular groups for orientable infinite type surfaces.
result Every orientable infinite type surface has a countable modular group.
Quantum modularity proven for specific theta series.
problem Proving quantum modularity for partial theta series with periodic coefficients.
method Explicit proof using Kontsevich-Zagier series and colored Jones polynomials.
result Kontsevich-Zagier series is a weight 3/2 quantum modular form.