arXiv research

Paper shows how to embed Möbius bands with many twists and small aspect ratios.

This paper finds bounds on the ribbonlength of knots and links with up to 9 crossings.

The infimal Heegaard gradient of a compact 3-manifold was defined and studied by Marc Lackenby in an approach toward the well-known virtually Haken conjecture. As instructive examples, we consider Seifert fibered 3-manifolds, and show that a Seifert fibered 3-manifold has zero infimal Heegaard gradient if and only if i…

The paper introduces a new FOR framework using Huber and ε-insensitive losses.

Improved bound for optimal Moebius band aspect ratio.

Proves bounds on ribbonlength for various knot types.

New principle controls graph-informed adversarial discrepancies.

The study shows that inscribed rectangles in smooth curves cover at least one third of all possible aspect ratios.

New equivalences found between subsampling and ridge regularization methods.

Two trees in the boundary of outer space are said to be \emph{primitive-equivalent} whenever their translation length functions are equal in restriction to the set of primitive elements of $F_N$. We give an explicit description of this equivalence relation, showing in particular that it is nontrivial. This question is …

We establish linear regret bounds for convex smooth losses using Fenchel-Young losses.

Many manifold embedding algorithms fail apparently when the data manifold has a large aspect ratio (such as a long, thin strip). Here, we formulate success and failure in terms of finding a smooth embedding, showing also that the problem is pervasive and more complex than previously recognized. Mathematically, success …

Study on the ribbonlength of knots and links, improving upper bounds.

We show that any finitely generated non-elementary Kleinian group has a co-final family of finite index normal subgroups with respect to which it has Property $τ$. As a consequence, any closed hyperbolic 3-manifold has a co-final family of finite index normal subgroups for which the infimal Heegaard gradient is positiv…

Optimizes kernel density ratios for better predictions and information measures.

Optimal risk sharing found for heterogeneous risk attitudes using distortion risk measures.

We present an elementary analysis of the dynamical aspects of the GDP / government surplus multiplier with relevance to the assessment of a country's debt repayment policy. We show the (at first) counter intuitive result that in order to reduce the Debt/GDP ratio, countries with high Debt to GDP should go into further …

Multicomponent bilayer structures arise as the ubiquitous plasma membrane in cellular biology and as blends of amphiphilic copolymers used in electrolyte membranes, drug delivery, and emulsion stabilization within the context of synthetic chemistry. We develop the multicomponent functionalized Cahn-Hilliard (mFCH) free…

Gradient descent dynamics in nonconvex models explained with universality.

Importance weighting is a general way to adjust Monte Carlo integration to account for draws from the wrong distribution, but the resulting estimate can be highly variable when the importance ratios have a heavy right tail. This routinely occurs when there are aspects of the target distribution that are not well captur…

Noise increases the Rashomon ratio, leading simpler models to perform similarly to complex ones.

Deep neural networks solve optimal risk sharing problems.

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spe…

We present a reinforcement learning approach for detecting objects within an image. Our approach performs a step-wise deformation of a bounding box with the goal of tightly framing the object. It uses a hierarchical tree-like representation of predefined region candidates, which the agent can zoom in on. This reduces t…

Study optimal ridge regularization for out-of-distribution prediction.

Paper introduces lexical ratio to measure portfolio diversification.

In this second part of a series of papers on the long-time behavior of Ricci flows with surgery, we establish a bound on the evolution of the infimal area of simplicial complexes inside a 3-manifold under the Ricci flow. This estimate generalizes an area estimate of Hamilton, which we will recall in the first part of t…

It had been believed in the conventional practice that the risk of a bank going bankrupt is lessened in a straightforward manner by transferring the risk of loan defaults. But the failure of American International Group in 2008 posed a more complex aspect of financial contagion. This study presents an extension of the …

New algorithm for MDS with quasi-polynomial dependency on aspect ratio.

Optimizes energy of mappings from complex projective spaces.

This paper uses GSNR to explain why neural networks generalize well.

The paper analyzes the benefit-cost ratio for feature selection in machine learning.

Study ridge ensembles in proportional feature-to-sample size regime, proving risk equivalence and GCV consistency.

Framework mitigates risk non-monotonicity in high-dimensional predictions.

The book explains deep learning theory and how networks learn nontrivial representations.

Upper bound on 3-manifold volumes from surface homeomorphisms.

Estimates Gaussian location model with ridge regularization, comparing variational and spectral methods.

New algorithm for safer machine learning with different testing and training distributions.

Aggregation defenses improve deep learning models' robustness against data poisoning attacks.

We establish general versions of a variety of results for quasiconvex, lower-semicontinuous, and law-invariant functionals. Our results extend well-known results from the literature to a large class of spaces of random variables. We sometimes obtain sharper versions, even for the well-studied case of bounded random var…

The aims of this study are twofold. First, we consider an optimal risk allocation problem with non-convex preferences. By establishing an infimal representation for distortion risk measures, we give some necessary and sufficient conditions for the existence of optimal and asymptotic optimal allocations. We will show th…

This article considers algorithmic and statistical aspects of linear regression when the correspondence between the covariates and the responses is unknown. First, a fully polynomial-time approximation scheme is given for the natural least squares optimization problem in any constant dimension. Next, in an average-case…

New approach for algorithms that learn predictors to improve performance.

The purpose of this work is to develop and study a distributed strategy for Pareto optimization of an aggregate cost consisting of regularized risks. Each risk is modeled as the expectation of some loss function with unknown probability distribution while the regularizers are assumed deterministic, but are not required…

In this paper, we propose a novel learning method for image classification called Between-Class learning (BC learning). We generate between-class images by mixing two images belonging to different classes with a random ratio. We then input the mixed image to the model and train the model to output the mixing ratio. BC …

Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.

PolyModel theory and iTransformer improve hedge fund portfolio construction.

Gaussian process is a very promising novel technology that has been applied to both the regression problem and the classification problem. While for the regression problem it yields simple exact solutions, this is not the case for the classification problem, because we encounter intractable integrals. In this paper we …