Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
3D spheres can't be swept by short curves, complicating geodesic length estimates.
problem Obstructing geodesic length estimates in 3D spheres.
method Constructing specific 3D spheres with controlled diameter and volume.
result Min-max methods for geodesic lengths fail for certain 3D spheres.
Minimum Description Length prevents overfitting in noisy data.
problem Learning from noisy data with overfitting risk.
method Minimum Description Length learning rule with tempered guarantees.
result Tempered agnostic finite sample learning guarantees and asymptotic behavior characterization.
Proves existence of curves with constant curvature in a sphere.
problem Existence of curves with constant geodesic curvature in a Riemannian 2-sphere.
method Develops a min-max scheme for a weighted length functional.
result Proves existence for almost every prescribed curvature.
Let $\mbox{Len}(K)$ be the minimum length of a knot on the cubic lattice (namely the minimum length necessary to construct the knot in the cubic lattice). This paper provides upper bounds for $\mbox{Len}(K)$ of a nontrivial knot K in terms of its crossing number c(K) as follows: $\mbox{Len}(K) \leq \min \left\{ \fr…
Balancing graph summarization and change detection in streaming data.
problem Balancing compression rate in graph summarization and accuracy in change detection.
method Introducing a probabilistic hierarchical latent variable model and optimizing parameters based on the minimum description length principle to balance the trade-off.
result Guaranteed suppression of Type I error probability (false alarms) in change detection.
For a Riemannian metric g on the two-sphere, let ℓmin(g) be the length of the shortest closed geodesic and ℓmax(g) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalities \[ \ell_{\rm min}(g…
Study finds geodesic networks for surfaces with convex boundary.
problem Finding geodesic networks for surfaces with convex boundary.
method Investigates free boundary geodesic networks in surfaces with non-negative sectional curvature and convex boundary.
result Existence of a geodesic network realizing the first width of a surface with non-negative sectional curvature and strictly convex boundary.
New geometric invariant from min-max width of spheres on Riemannian 2-spheres.
problem Understanding the min-max width of spheres associated to distance functions.
method Application of min-max methods to pairs of points on Riemannian 2-spheres.
result The min-max width does not always equal half the length of a simple closed geodesic.
This paper introduces a new method for model selection and more generally hyperparameter selection in machine learning. Minimum description length (MDL) is an established method for model selection, which is however not directly aimed at minimizing generalization error, which is often the primary goal in machine learni…
Study shows LLC correlates with neural network compressibility.
problem Evaluating limits of neural network compression.
method Extended minimum description length principle using singular learning theory.
result Complexity estimates based on LLC are linearly correlated with compressibility.
Equity-Transformer solves NP-hard min-max routing problems efficiently.
problem Min-max routing problems with multiple agents and large-scale applications.
method Sequential planning approach with Transformer and equitable workload distribution inductive biases.
result Significant runtime and cost reductions in min-max mTSP and min-max mPDP tasks.
Study shows Transformers can generalize to varying task lengths.
problem Understanding when and how Transformers can generalize to different input lengths.
method Proposed a unifying framework and introduced the RASP-Generalization Conjecture.
result Transformers tend to length generalize on tasks if solvable by short RASP programs.
We present an asymptotic criterion to determine the optimal number of clusters in k-means. We consider k-means as data compression, and propose to adopt the number of clusters that minimizes the estimated description length after compression. Here we report two types of compression ratio based on two ways to quantify t…
We prove that given a three manifold with an arbitrary metric (M3,g) of positive Ricci curvature, there exists a sweepout of M by surfaces of genus ≤3 and areas bounded by Cvol(M3,g)2/3. We use this result to construct a sweepout of M by 1-cycles of length at most Cvol(M3,g)1/3. The sweepo…
The study quantifies the information needed for causal queries at different levels of Pearl's hierarchy.
problem How much additional information is needed for interventional and counterfactual queries compared to observational queries?
method Formalized via query-class description length, using Kolmogorov complexity of answer oracles induced by SCMs.
result Binary acyclic SCMs show a quadratic gap between observational and interventional descriptions, and a logarithmic gap between interventional and counterfactual descriptions.
New method improves bivariate causal discovery by accurately estimating cause variable complexity.
problem Improper estimation of cause variable complexity in current MDL-based methods.
method Rate-distortion MDL (RDMDL) using information dimension for cause variable complexity estimation.
result RDMDL achieves competitive performance on Tübingen dataset.
New complexity measure ADL connects to classical complexity measures.
problem Deriving generalization bounds for neural networks.
method Exploring ADL's relationship to Covering Numbers and VC Dimension.
result ADL is equivalent to Covering Numbers and VC Dimension for real-valued functions.
Paper proves rigidity of minimal disks in 3-balls with non-negative Ricci curvature.
problem Rigidity of free boundary minimal disks in 3-balls with non-negative Ricci curvature.
method Min-max methods and rigidity statements for half-balls with non-negative Ricci curvature.
result Existence and properties of minimal disks with least area in 3-balls.
Method estimates dataset utility via minimal program length proxy.
problem Determining if data labels are generated by useful subroutines.
method Rissanen Data Analysis (RDA) estimates minimum description length (MDL) as a proxy.
result Method reveals dataset characteristics and utility in various NLP settings.
Improved dynamic regret analysis for strongly convex and smooth functions.
problem Analyzing dynamic regret for online learning algorithms.
method Improved analysis of the Online Multiple Gradient Descent (OMGD) algorithm.
result Achieved a best-of-three-worlds guarantee for dynamic regret.
Critical trajectories in a sphere are found for a specific bending functional.
problem Finding closed trajectories in a sphere for a specific bending functional.
method Existence of infinitely many closed trajectories shown for a given Lagrange multiplier.
result Existence of closed trajectories dependent on a pair of relatively prime natural numbers.
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
Reformulated Markov's conjecture in combinatorial terms.
problem Markov's uniqueness conjecture in integral necklaces.
method Geometric reformulation and combinatorial description.
result Explicitly described set of lengths on modular torus.
Time-invariant linear dynamical system arises in many real-world applications,and its usefulness is widely acknowledged. A practical limitation with this model is that its latent dimension that has a large impact on the model capability needs to be manually specified. It can be demonstrated that a lower-order model cla…
The paper explains how simple methods can converge to optimal solutions in complex neural games.
problem Finding optimal solutions in neural games with non-convex objectives.
method Theoretical framework using hidden convexity and overparameterization, with path-length bounds and PŁ conditions.
result Simple gradient methods can converge to Nash equilibria in non-convex min-max games under certain conditions.
Given a sweepout of a Riemannian 2-sphere which is composed of curves of length less than L, we construct a second sweepout composed of curves of length less than L which are either constant curves or simple curves. This result, and the methods used to prove it, have several consequences; we answer a question of M. Fre…
Neural networks generalize on simple data generated by a programming language.
problem Generalization of neural networks on low complexity data.
method Minimum description length (MDL) approach for feedforward neural networks.
result MDL feedforward neural networks generalize with high probability on simple data.
DL/FBF improves GPSR solutions by selecting compact, generalising expressions.
problem Overfitting and structural bloat in symbolic regression with genetic programming.
method Description length (DL) and fractional Bayes factor (FBF) criteria for selecting compact, generalising expressions.
result DL/FBF post-selection improves test performance compared to AIC/BIC baseline.
PCA (Principal Component Analysis) and its variants areubiquitous techniques for matrix dimension reduction and reduced-dimensionlatent-factor extraction. One significant challenge in using PCA, is thechoice of the number of principal components. The information-theoreticMDL (Minimum Description Length) principle gives…
New algorithm reduces regret in private online learning with optimal gap-dependent rate.
problem Optimal gap-dependent regret rate for private stochastic decision-theoretic online learning.
method Horizon-free pure-DP algorithm with exponential block partitioning and softmax selection.
result Explicit regret bound of 1000⋅(ΔminlogK+εlogK). This paper studies spectral properties of spheres with one equator.
problem Spectral rigidity and flexibility of spheres with one equator.
method Defined marked length spectrum, proved isospectrality, and classified contact forms.
result Marked length spectrum determines the metric up to Z2-symmetry. The Fisher information approximation (FIA) is an implementation of the minimum description length principle for model selection. Unlike information criteria such as AIC or BIC, it has the advantage of taking the functional form of a model into account. Unfortunately, FIA can be misleading in finite samples, resulting i…
Study improves neural network performance in sequential learning for image classification.
problem Improving neural network performance in sequential learning for image classification.
method Evaluation of approaches for computing prequential description lengths, proposing forward-calibration and replay-streams.
result Improved description lengths for image classification datasets, outperforming previous results.
CDL index improves clustering validation for non-convex data.
problem Selecting clustering algorithms and hyperparameters without labeled data.
method CDL uses compactness, centers, and covariances to compute a probabilistic description length bound.
result CDL outperforms conventional CVIs on synthetic and image benchmarks.
In this paper we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold M: 1) the convexity radius of p, $\operatorname{conv}(p)\ge \min\{\frac{1}{2}\operatorname{inj}(p),\operatorname{foc}(B_{\operatorname…
We tackle the problem of penalty selection of regularization on the basis of the minimum description length (MDL) principle. In particular, we consider that the design space of the penalty function is high-dimensional. In this situation, the luckiness-normalized-maximum-likelihood(LNML)-minimization approach is favorab…
Algorithm stabilizes queues in asymmetric systems with unknown service rates.
problem Stabilizing queues in multi-class multi-server systems with unknown service rates.
method Proposes UCB and Thompson Sampling algorithms to stabilize queues while learning service rates.
result Achieves system stability with an average queue length bound of \(O(\min\{N,K\}/ε)\) for large time horizon \(T\).
Paper establishes generalization bounds for representation learning using Minimum Description Length.
problem Designing efficient statistical supervised learning algorithms that generalize well to unseen data.
method Developed a compressibility framework using Minimum Description Length (MDL) to derive upper bounds on generalization error.
result Established the first theoretical generalization bounds for Information Bottleneck type encoders and representation learning.
The Minimum Description Length (MDL) principle states that the optimal model for a given data set is that which compresses it best. Due to practial limitations the model can be restricted to a class such as linear regression models, which we address in this study. As in other formulations such as the LASSO and forward …
Kernel networks' stability edge linked to Fisher Information singularity.
problem Understanding the stability edge in high-capacity kernel Hopfield networks.
method Statistical manifold analysis and Riemannian geometry.
result The Ridge of Optimization corresponds to the Edge of Stability, revealing a dual equilibrium.
A new method avoids overfitting in network reconstruction by using the minimum description length principle.
problem Determining the optimal model complexity in network reconstruction to prevent overfitting.
method Hierarchical Bayesian inference and weight quantization based on the minimum description length principle.
result The method yields increased accuracy in reconstructing both artificial and empirical networks.
We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained f…
Given a surface of infinite topological type, there are several Teichmüller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmüller space and the length-spectrum Teichm…
The paper explores how different network architectures learn logical functions under GOTU, finding that a min-degree-interpolator is learned.
problem Learning logical functions with a focus on generalization on the unseen.
method Study of different network architectures trained by SGD under GOTU.
result For sparse functions and certain network models, a min-degree-interpolator is learned on the unseen.
New methods evaluate data representations by complexity of low-loss predictor learning.
problem Evaluating quality of data representations for downstream tasks.
method Surplus Description Length (SDL) and ε Sample Complexity (εSC) methods.
result Methods measure the information needed to approximate optimal predictor up to specified tolerance.
The closed string field theory minimal-area problem asks for the conformal metric of least area on a Riemann surface with the condition that all non-contractible closed curves have length at least 2π. This is an extremal length problem in conformal geometry as well as a problem in systolic geometry. We consider the ana…
APD method decomposes neural network parameters into simple, faithful components.
problem Understanding the internal mechanisms learned by neural networks.
method Attribution-based Parameter Decomposition (APD) method.
result Demonstrated effectiveness in recovering features, separating computations, and identifying representations.