Space exploration technology advances exponentially, consistent with Moore's and Wright's laws.
problem Predicting the advancement of space exploration technology.
method Analysis of Moore's and Wright's laws applied to space exploration technology.
result Spacecraft technology advances exponentially, consistent with Moore's and Wright's laws.
Forecasting technological progress is of great interest to engineers, policy makers, and private investors. Several models have been proposed for predicting technological improvement, but how well do these models perform? An early hypothesis made by Theodore Wright in 1936 is that cost decreases as a power law of cumul…
New law predicts first extinction in resampling processes.
problem Intractable extinction times in resampling processes.
method Modeling multinomial updates as independent square-root diffusions.
result Closed-form law for first-extinction time with linear cost.
Following findings by Ormerod and Mounfield, Wright rises the problem whether a power or an exponential law describes the distribution of occurrences of economic recession periods. In order to clarify the controversy a different set of GDP data is hereby examined. The conclusion about a power law distribution of recess…
A new distribution family extends the α-stable distribution with a degree of freedom parameter.
problem Lack of moments in the α-stable distribution. method Wright function framework to combine and extend distribution families.
result Generalized α-stable distribution with valid moments. Paper disproves Wright's periodic map conjecture.
problem Existence of periodic maps on closed hyperbolic surfaces.
method Analyzes automorphisms and curve behavior on surfaces.
result No periodic map can fill every curve with its image.
New proof for higher rank subvarieties in genus three.
problem Classifying higher rank invariant subvarieties in genus three.
method Uses recent techniques developed by Apisa and Wright.
result Short and simplified proof of classification.
Study identifies specific subvarieties in translation surfaces with quadratic field.
problem Characterizing invariant subvarieties in translation surfaces with quadratic field.
method Analyzing algebraically primitive subvarieties in strata of translation surfaces.
result Only specific subvarieties identified: decagon, Weierstrass curves, etc.
The paper generalizes product inequalities for random vectors and their applications.
problem Understanding concentration of measure for products of random vectors.
method Develops expressions for the concentration of functionals of random vectors based on product norms.
result Provides generalized Hanson-Wright inequalities and applications to random matrices.
Constructs subvarieties in translation surface strata using combinatorial input.
problem Creating subvarieties in translation surface strata.
method Combining Hurwitz spaces theory with combinatorial input.
result Constructs Teichmueller curves and orbit closures.
The paper studies neural networks with wide layers and finds a deformed semicircle law.
problem Investigating spectral distributions of neural networks in the ultra-wide regime.
method Analyzes empirical kernel matrices, proves deformed semicircle law, provides nonlinear Hanson-Wright inequality.
result Emergence of a deformed semicircle law in the ultra-wide neural network regime.
In 1992, David Wright proved a remarkable theorem about which contractible open manifolds are covering spaces. He showed that if a one-ended open manifold M has pro-monomorphic fundamental group at infinity which is not pro-trivial and is not stably Z, then M does not cover any manifold (except itself). In the non-mani…
We complete the classification of rank two affine manifolds in the moduli space of translation surfaces in genus three. Combined with a recent result of Mirzakhani and Wright, this completes the classification of higher rank affine manifolds in genus three.
Study disproves conjecture about quadratic differentials.
problem Apisa-Wright conjecture about branched covers of quadratic differentials.
method Provided counterexamples to the conjecture.
result Found infinite family of non-cyclic 1-cylinder surfaces.
Unified analysis simplifies Johnson-Lindenstrauss lemma for data reduction.
problem Efficiently reducing high-dimensional data while preserving geometry.
method Unified analysis of various JL constructions using probabilistic tools.
result First rigorous proof and extension of spherical construction's effectiveness.
The paper analyzes tensor recovery from symmetric rank-one measurements using information theory.
problem Recovering tensors with low symmetric rank from symmetric rank-one measurements.
method Covering numbers argument, Carbery-Wright inequality, orthogonal polynomials, Fano's inequality.
result Near-optimal sample complexity bounds for log-concave distributions.
Study resolves conjecture on overparameterized linear models' generalization.
problem Asymptotic generalization of multiclass classification with overparameterized models.
method Gaussian covariates bi-level model, Hanson-Wright inequality variant.
result Min-norm interpolating classifier can be suboptimal compared to noninterpolating classifiers.
Efficiently estimates covariance for sub-Weibull vectors with sub-Gaussian rate.
problem Outliers in high-dimensional covariance estimation.
method Cross-Fitted Norm-Truncated Estimator for Sub-Weibull distributions.
result Achieves optimal sub-Gaussian rate with O(Nd2) operations. New insights into data geometry reveal manifold structure in grid-cell activity.
problem Understanding the roles of different dimensions in data geometry.
method Generalised Hanson-Wright inequality and random function model analysis.
result Persistence diagrams reveal latent homology and manifold structure.
Classifies components of k-differentials and their orbit closures.
problem Classifying components of strata of k-differentials and their orbit closures.
method Algebraic approach using multiscale compactification.
result Complete classification of components of strata of holomorphic and meromorphic k-differentials.
The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closed orbits, the Prym locus Q(3,-1^3), and H^{odd}(4). Together with work of Matheus-Wright, this implies that there are only f…
Develops a new volatility model for prediction markets.
problem Volatility forecasting in prediction markets differs from standard asset markets.
method Combines Wright-Fisher and Glosten-Milgrom mechanisms to model binary prediction markets.
result Structural model outperforms standard ARCH/GARCH models in volatility forecasting.
Develops a new volatility model for prediction markets.
problem Volatility forecasting in prediction markets differs from standard asset markets.
method Combines Wright-Fisher and Glosten-Milgrom mechanisms to model binary prediction markets.
result Structural model outperforms standard ARCH/GARCH models in volatility forecasting.
The study shows acylindrical hyperbolicity for Artin groups not associated with joins or cones.
problem Proving acylindrical hyperbolicity for Artin groups of infinite type not associated with joins or cones.
method Developing and extending the clique-cube complex and action studies of Charney and Morris-Wright.
result Acylindrical hyperbolicity demonstrated for Artin groups of infinite type associated with graphs that are not cones.
Study shows periodic points of Prym eigenforms in specific genera.
problem Understanding periodic points of Prym eigenforms in translation surfaces.
method Geometric proof using Prym involution and affine automorphism group.
result Fixed points of Prym involution are periodic points of Prym eigenforms.
Geometric flow on curves in S^3 generates YO equations solutions.
problem Modeling short wave-long wave interaction.
method Simple geometric flow on curves in S3. result Constructs transverse curves for YO equations periodic solutions.
The study connects spheres in specific surface curve graphs, proving connectivity and classifying components.
problem Proving connectivity and classifying components of spheres in curve graphs of low and medium complexity surfaces.
method Analyzing specific surfaces Σ2,0,Σ1,3,Σ0,6 and Σ0,5,Σ1,2, proving connectivity and classifying components. result Spheres of any radius are connected in Σ2,0,Σ1,3,Σ0,6, and the union of two consecutive spheres is connected in Σ0,5 and Σ1,2. We introduce a mean-reverting SDE whose solution is naturally defined on the space of correlation matrices. This SDE can be seen as an extension of the well-known Wright-Fisher diffusion. We provide conditions that ensure weak and strong uniqueness of the SDE, and describe its ergodic limit. We also shed light on a use…
Starting from a sequence of independent Wright-Fisher diffusion processes on [0,1], we construct a class of reversible infinite dimensional diffusion processes on $\DD_\infty:= \{{\bf x}\in Let $MbeacompleteRiemnnianmanifoldandμthedistributionofthediffusionprocessgeneratedby\ff 1 2\DD+ZwhereZ$…
Classifies GL(2,R)-invariant subvarieties with zero Lyapunov exponents.
problem Classifying GL(2,R)-invariant subvarieties with specific properties.
method Classification based on homological dimensions and Lyapunov exponents.
result Explicit exceptions list for GL(2,R)-invariant subvarieties with zero Lyapunov exponents.
We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also constr…
Optimizes sub-Gaussian matrices for preserving data distances.
problem Improving the performance of sub-Gaussian matrices in preserving data distances.
method Analyzes sub-Gaussian matrices and their dependence on the sub-Gaussian norm, presenting optimal bounds.
result Optimal dependence on the sub-Gaussian norm for sub-Gaussian matrices as near isometries on sets.
A tutorial on non-asymptotic system identification methods.
problem Identifying system parameters in linear models.
method Covering technique, Hanson-Wright Inequality, method of self-normalized martingales.
result Streamlined proofs of least-squares based estimator performance.
Study linear subvarieties of meromorphic differential strata, proving toric closures and new proofs of theorems.
problem Understanding linear subvarieties in strata of meromorphic differentials.
method Investigate closures in multi-scale compactification, prove restrictions on period coordinates.
result Prove closures are locally toric varieties, generalize cylinder deformation theorem.
We are interested in strong approximations of one-dimensional SDEs which have non-Lipschitz coefficients and which take values in a domain. Under a set of general assumptions we derive an implicit scheme that preserves the domain of the SDEs and is strongly convergent with rate one. Moreover, we show that this general …
Classifies GL(2,R)-invariant subvarieties in complex geometry.
problem Classifying GL(2,R)-invariant subvarieties.
method Classification of natural subvarieties including double covers and Teichmüller space.
result Classification of high rank invariant subvarieties and new examples.
We present the Wright-Fisher Indian buffet process (WF-IBP), a probabilistic model for time-dependent data assumed to have been generated by an unknown number of latent features. This model is suitable as a prior in Bayesian nonparametric feature allocation models in which the features underlying the observed data exhi…
This work explains scaling laws as redundancy laws in deep learning.
problem The mathematical origins of scaling laws in deep learning models remain unclear.
method Kernel regression and analysis of data covariance spectra.
result Scaling laws can be explained as redundancy laws, revealing the learning curve's slope depends on data redundancy.
We consider the following question: Which parameters in the extension of a rational pleating ray across the boundary of $\Cal M$, the Maskit embedding of the Teichmüller space of once punctured tori correspond to a Kleinian group? Using methods of Keen and Series and Wright we prove a local result, stating that on each…
Study on RL on volatility surfaces, proving no free lunch for law-seeking methods.
problem Aligning RL agents with no-arbitrage laws in volatile markets.
method Built a law manifold, defined penalties, and used a Goodhart decomposition.
result No free lunch theorem: Law-seeking RL cannot outperform baselines.
A new scaling law predicts optimal batch size for training models.
problem Finding the optimal batch size for training models efficiently.
method Proposed a three-term scaling law that considers model size, training data, training steps, and batch size.
result The three-term law accurately recovers the optimal batch size and can be robustly fit with fewer training runs.
Artin groups not free of infinity are shown to have finite centers.
problem Characterizing Artin groups with finite centers.
method Reduced clique-cube complexes and actions on them.
result Artin groups not free of infinity have finite centers, and are trivial in many cases.
This work analyzes neural scaling laws using power-law data spectra and derives analytical expressions for generalization error.
problem Understanding how neural network performance scales with key factors like data size and model complexity.
method Statistical mechanics techniques applied to one-pass stochastic gradient descent in a student-teacher framework.
result Derivation of analytical expressions for generalization error under power-law data spectra and identification of conditions for power-law scaling.
An Atlas model is a rank-based system of continuous semimartingales for which the steady-state values of the processes follow a power law, or Pareto distribution. For a power law, the log-log plot of these steady-state values versus rank is a straight line. Zipf's law is a power law for which the slope of this line is …
Large models follow power laws in performance with dataset size or parameters.
problem Understanding neural scaling laws in large language models.
method Joint generative data model and random feature model.
result Modeling and solving the dual limit reveals insights into scaling laws.
The paper studies geometric loci and their invariants in complex dynamics.
problem Analyzing geometric loci and their invariants in complex dynamics.
method Intersection theory and dynamical invariants on the flex and gothic loci.
result Determined the divisor class of the flex locus and various tautological intersection numbers on the gothic locus.
Conservation law for weakly harmonic mappings in high dimensions.
problem Conservation law for harmonic mappings in supercritical dimensions.
method Partial extension of Rivière's conservation law with Lorentz integrability condition.
result Conservation law for weakly harmonic mappings in supercritical dimensions.
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.