Study shows how non-uniform scaling affects persistence diagrams.
problem Stability of persistence diagrams under non-uniform scaling.
method Explicit bounds on bottleneck distance derived for Euclidean scaling.
result Explicit bounds on the stability of persistence diagrams under non-uniform scaling.
We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zi…
New bounds on self-normalized martingales improve online linear regression performance.
problem Improving regret bounds in online linear regression.
method Characterizing scale-invariant bounds on self-normalized martingales.
result For d=1, O(logT) doubly-uniform regret is possible; for d>1, sublinear doubly-uniform regret is impossible. NUTS mixing time scales as d^(1/4) for Gaussian distributions.
problem Improving the efficiency of the No-U-Turn Sampler (NUTS) for Gaussian distributions.
method Coupling argument leveraging geometric structure of Gaussian concentration, uniformity analysis of NUTS transitions.
result The mixing time of NUTS scales as d^(1/4) for Gaussian distributions, up to logarithmic factors.
Uniformity and proximity are two different ways for defining small scale structures on a set. Coarse structures are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the concept of asymptotic resemblance as a relation between subsets of a set to define a…
The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
problem Finding metrics with prescribed combinatorial curvature on polyhedral surfaces.
method Discrete uniformization theorem, combinatorial α-Yamabe flow, combinatorial α-Calabi flow, edge flipping surgery.
result Longtime existence and convergence of combinatorial α-Yamabe flow and combinatorial α-Calabi flow with surgery.
Uniform scaling limits in AdamW-trained transformers converge to ODEs.
problem Understanding the dynamics of large-depth transformers trained with AdamW.
method Modeling transformer dynamics as an interacting particle system coupled through attention, proving convergence to ODEs.
result The joint dynamics of hidden states and backpropagated variables converge uniformly to an ODE system.
The study compares uniform-price and discriminatory auctions in terms of learning difficulty.
problem Comparing the learning difficulty of uniform-price and discriminatory multi-unit auctions.
method Characterization of learning difficulty through regret minimization in both full-information and bandit feedback settings.
result Regret scales similarly for both auction formats under full-information, but uniform-price auctions can achieve faster learning rates.
We design and mathematically analyze sampling-based algorithms for regularized loss minimization problems that are implementable in popular computational models for large data, in which the access to the data is restricted in some way. Our main result is that if the regularizer's effect does not become negligible as th…
Learning ReLU networks to high uniform accuracy requires exponentially many samples.
problem Achieving high uniform accuracy on ReLU networks for security-critical applications.
method Quantified the number of training samples needed for any algorithm to guarantee uniform accuracy.
result The minimal number of training samples scales exponentially with network depth and input dimension.
Estimates heat equation on shrinking Ricci solitons with uniform bounds.
problem Analyzing heat equation on shrinking Ricci solitons.
method Proved L2 estimate with time-dependent Gaussian weight. result Uniform bounds for heat equation along Ricci flow.
Uniform K-homology theory applied to elliptic operators on manifolds with boundary.
problem Developing a theory to study boundary conditions for elliptic operators on non-compact manifolds.
method Theory of relative uniform K-homology, developing a relative index map.
result Uniform K-homology classes of boundary conditions and their connection to the higher ρ-invariant.
Study uniform rates for estimating Gaussian mixtures without separation assumption.
problem Estimating parameters in two-component Gaussian mixtures without separation.
method Uniform convergence rates derived using minimax lower bounds and careful analysis of polynomial equalities.
result Phase transition in optimal estimation rate based on mixture balance.
New bounds for learning polynomial surrogates with L∞ guarantees.
problem Learning polynomial surrogates for bounded binary functions with L∞ error guarantees. method Characterized minimax sample complexity for two classes of polynomials under subgaussian noise.
result Sample complexity rates differ from noiseless case, scaling as nd+1 for degree d polynomials and ns2 for sparse polynomials. The paper develops time-uniform inference methods for stochastic approximation parameters.
problem Statistical inference for parameters in stochastic approximation problems.
method Analysis of averaged iterates convergence rates and construction of asymptotic confidence sequences.
result Valid asymptotic confidence sequences for parameters in stochastic approximation problems.
Enhances Fourier estimator performance for asynchronous event-data.
problem Improving correlation and covariance estimation on event-data.
method Implement and test NUFFT methods with different averaging kernels.
result Demonstrates improved performance and relationship between averaging scales.
Generatability in metric spaces studied with novel novelty parameters.
problem Understanding generatability in metric spaces with asymmetric novelty parameters.
method Introducing (ε,ε′)-closure dimension to characterize uniform and non-uniform generatability. result Generatability is stable across novelty scales in doubling spaces but can be highly scale-sensitive in general metric spaces.
We consider the rate of volume growth of large Carnot-Carathéodory metric balls on a class of unbounded model hypersurfaces in C2. When the hypersurface has a uniform global structure, we show that a metric ball of radius δ≫1 either has volume on the order of δ3 or δ4. We also give necessary and …
A major challenge for building statistical models in the big data era is that the available data volume far exceeds the computational capability. A common approach for solving this problem is to employ a subsampled dataset that can be handled by available computational resources. In this paper, we propose a general sub…
The paper analyzes LETF option markets using moneyness scaling to find statistical arbitrage opportunities.
problem Statistical discrepancies between levered and unlevered ETF option implied volatility smiles.
method Bootstrap uniform confidence bands, dynamic semiparametric factor model, moneyness scaling, Heston stochastic volatility.
result Trading opportunities exist on LETF market, and a statistical arbitrage strategy generates positive returns.
Transforms uniform learners to work under arbitrary distributions efficiently.
problem Learning under arbitrary distributions from uniform learners.
method Black-box transformation using decision tree decomposition.
result Efficient transformation with runtime scaling with distribution complexity.
Scalable kernel methods for large datasets using Fourier representations and NUFFT.
problem Cubic complexity in kernel methods limits their use on large-scale datasets.
method Fourier representation of kernels combined with NUFFT for O(n log n) complexity.
result Achieves minimax convergence rates and processes up to tens of billions of samples.
Uniform elliptic theory for Dirac operators on orbifold resolutions.
problem Analyzing Dirac operators on orbifold resolutions.
method Viewing orbifolds as conically fibred singular spaces and resolving them by gluing asymptotically conical fibrations.
result Uniform index formula for Dirac operators on orbifold resolutions.
A new model-free subsampling method using uniform designs is proposed.
problem Model-based subsampling methods are often dependent on model assumptions.
method Developed a criterion (GEFD) and a model-free subsampling method based on uniform designs.
result The proposed method outperforms random sampling and is robust under diverse model specifications.
Improved matrix completion for non-uniformly sampled data.
problem Estimating unobserved entries in a matrix with varying sampling probabilities.
method Developed entry-specific bounds for low-rank matrix completion under structured non-uniform sampling.
result Error bounds for each entry match minimax lower bounds under certain conditions.
Concrete distribution properties examined on simplex.
problem Properties of Concrete distribution on simplex.
method Reflection and location-scale transformation of uniform distribution; explicit parameterization to Poincaré half-space.
result Fisher information and information metric are hyperbolic space; Fisher-Rao geodesic distance computed.
Three methods for tuning HMC diagonal scale matrices compared.
problem Improving Hamiltonian Monte Carlo efficiency with diagonal scale matrices.
method Three approaches: ISG, median crossing frequency, and estimated marginal standard deviations.
result ISG method leads to more efficient sampling in many cases.
New algorithm for learning functions with bounds on error and sample complexity.
problem Learning [0,1]-valued functions in a prediction model. method General-purpose algorithm with upper and lower bounds on expected error and sample complexity.
result Improved bounds on sample complexity and agnostic learning conditions.
We present an idea of unifying small scale (topology, proximity spaces, uniform spaces) and large scale (coarse spaces, large scale spaces). It relies on an analog of multilinear forms from Linear Algebra. Each form has a large scale compactification and those include all well-known compactifications: Higson corona, Gr…
A topology on a set X is the same as a projection (i.e. an idempotent linear operator) cl:2X→2X satisfying A⊂cl(A) for all A⊂X. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set X is a dot product ⋅:2X×2X→2Y. Its equivalent form is an or…
We revisit the index leverage effect, that can be decomposed into a volatility effect and a correlation effect. We investigate the latter using a matrix regression analysis, that we call `Principal Regression Analysis' (PRA) and for which we provide some analytical (using Random Matrix Theory) and numerical benchmarks.…
This paper maps the large-scale variation of the Spanish language by employing a corpus based on geographically tagged Twitter messages. Lexical dialects are extracted from an analysis of variants of tens of concepts. The resulting maps show linguistic variation on an unprecedented scale across the globe. We discuss th…
Convolutional Neural Networks (CNN) has become more popular choice for various tasks such as computer vision, speech recognition and natural language processing. Thanks to their large computational capability and throughput, GPUs ,which are not power efficient and therefore does not suit low power systems such as mobil…
New sampling bounds improve uniform coverage verification in machine learning.
problem Conservative bounds in classical coverage analyses at small failure probabilities.
method Variance-based analysis of uniform random sampling on a d-dimensional unit hypercube. result Sample complexity bound with logarithmic dependence on failure probability.
For collapsing sequences of Riemannian manifolds which satisfy a uniform lower Ricci curvature bound it is shown that there is a sequence of scales such that for a set of good base points of large measure the pointed rescaled manifolds subconverge to a product of a Euclidean and a compact space. All Euclidean factors h…
The paper offers error bounds for quantized dynamical models.
problem Accuracy of dynamical models from dependent data sequences.
method Developed uniform error bounds for quantized models and imperfect optimization algorithms.
result Unified bounds for slow and fast rates, scaling with model encoding bits.
Measure-scaling quasi-isometries on graphs have specific scaling groups.
problem Understanding the scaling groups of graphs under quasi-isometries.
method Analyzing measure-scaling quasi-isometries on graphs and their properties.
result The scaling group of a graph is invariant under measure-scaling quasi-isometries.
Existing strategies for finite-armed stochastic bandits mostly depend on a parameter of scale that must be known in advance. Sometimes this is in the form of a bound on the payoffs, or the knowledge of a variance or subgaussian parameter. The notable exceptions are the analysis of Gaussian bandits with unknown mean and…
Energy-efficient sampling for machine learning using magnetic tunnel junctions.
problem Costly and inefficient random sampling in machine learning.
method Energy-efficient algorithm using stochastic magnetic tunnel junctions for uniform Float16 sampling.
result Higher energy efficiency than state-of-the-art algorithms, with a minimum factor of 9721.
Improved sampling accuracy in SG-MCMC methods via non-uniform gradient subsampling.
problem Computational inefficiency and sampling error in stochastic gradient MCMC methods.
method Proposes a non-uniform subsampling scheme to reduce sampling error in EWSG, a variant of SG-MCMC.
result EWSG reduces sampling error compared to uniform subsampling, improving accuracy without sacrificing convergence speed.
Unified bounds for sketched bilinear forms in machine learning and statistics.
problem Uniform bounds on sketched bilinear forms for modern analyses.
method Generic chaining and new techniques for handling suprema over pairs of sets.
result Improved convergence bounds for sketched Federated Learning and bandit algorithms.
Develops statistical confidence sets for multidimensional scaling.
problem Statistical uncertainty in multidimensional scaling of noisy data.
method Formal statistical framework, distributional convergence results, uniform confidence sets, bootstrap procedures.
result Construction of reliable confidence sets for latent configurations in multidimensional scaling.
Study tests uniformity of categorical data against missing-ball alternatives, finding chi-squared test outperforms.
problem Testing uniformity of categorical data against missing-ball alternatives.
method Characterizes minimax risk, uses collisions and chi-squared test, reduces to structured subset of alternatives.
result Minimax test outperforms chi-squared test under least favorable alternative.
We demonstrate that a popular class of nonparametric mutual information (MI) estimators based on k-nearest-neighbor graphs requires number of samples that scales exponentially with the true MI. Consequently, accurate estimation of MI between two strongly dependent variables is possible only for prohibitively large samp…
We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem (P,C) is equivalent to the uniform regularity of the natural family (Px,Cx) of associated boundary value …
New bin-wise scaling methods improve prediction uncertainty calibration for machine learning.
problem Improving prediction uncertainty calibration for machine learning regression.
method Adaptations of Binwise Variance Scaling (BVS) with alternative loss functions and feature-based binning.
result Improved adaptivity and consistency in prediction uncertainty calibration.
A discrete conformality for polyhedral metrics on surfaces is introduced in this paper which generalizes earlier work on the subject. It is shown that each polyhedral metric on a surface is discrete conformal to a constant curvature polyhedral metric which is unique up to scaling. Furthermore, the constant curvature me…
Paper introduces Simplet Frequency Distribution (SFD) for SCs.
problem Frequency analysis of simplets in large SCs.
method Developed SFD vector and uniform sampling-based algorithm.
result Validated theoretical bounds with experiments.