arXiv subjects

Paul Larsen

Publications and source records attributed to Paul Larsen.

6 recordsLinked to original sources

arXiv2024

Pathological Regularization Regimes in Classification Tasks

In this paper we demonstrate the possibility of a trend reversal in binary classification tasks between the dataset and a classification score obtained from a trained model. This trend reversal occurs for certain choices of the regularization parameter for model training, namely, if the parameter is contained in what we call the pathological regularization regime. For ridge regression, we give necessary and sufficient algebraic conditions on the dataset for the existence of a pathological regularization regime. Moreover, our results provide a data science practitioner with a hands-on tool to avoid hyperparameter choices suffering from trend reversal. We furthermore present numerical results on pathological regularization regimes for logistic regression. Finally, we draw connections to datasets exhibiting Simpson's paradox, providing a natural source of pathological datasets.

stat.ML↗

arXiv2016

Asyptotic Normality for Maximum Likelihood Estimation and Operational Risk

Operational risk models commonly employ maximum likelihood estimation (MLE) to fit loss data to heavy-tailed distributions. Yet several desirable properties of MLE (e.g. asymptotic normality) are generally valid only for large sample-sizes, a situation rarely encountered in operational risk. In this paper, we study how asymptotic normality does--or does not--hold for common severity distributions in operational risk models. We then apply these results to evaluate errors caused by failure of asymptotic normality in constructing confidence intervals around the MLE fitted parameters.

q-fin.RM↗

arXiv2013

Fano schemes of generic intersections and machine learning

We investigate Fano schemes of conditionally generic intersections, i.e. of hypersurfaces in projective space chosen generically up to additional conditions. Via a correspondence between generic properties of algebraic varieties and events in probability spaces that occur with probability one, we use the obtained results on Fano schemes to solve a problem in machine learning.

math.AG↗

arXiv2012

Notes on Fano varieties of complete intersections

Fano varieties are subvarieties of the Grassmannian whose points parametrize linear subspaces contained in a given projective variety. These expository notes give an account of results on Fano varieties of complete intersections, with a view toward an application in machine learning. The prerequisites have been kept to a minimum in order to make these results accessible to a broad audience.

math.AG↗

arXiv2011

Fulton's Conjecture for \bar{M}_{0,7}

Fulton's conjecture for the moduli space of stable pointed rational curves, \bar{M}_{0,n}, claims that a divisor non-negatively intersecting all F-curves is linearly equivalent to an effective sum of boundary divisors. Our main result is a proof of Fulton's conjecture for n=7. A key ingredient in the proof is an \binom{n}{4} dimensional-subspace of the Neron-Severi space of \bar{M}_{0,n}, defined by averages of Keel relations, for which we prove Fulton's conjecture for all n.

math.AG↗

arXiv2011

Permutohedral spaces and the Cox ring of the moduli space of stable pointed rational curves

We study the Cox ring of the moduli space of stable pointed rational curves, \M_{0,n}, via the closely related permutohedral (or Losev-Manin) spaces. Our main result establishes \binom{n}{2} polynomial subrings of the Cox ring, thus giving collections of boundary variables that intersect the ideal of relations trivially. As applications, we give a combinatorial way to partially solve the Riemann-Roch problem for \M_{0,n}, and we show that all relations in degrees of Cox(\M_{0,6}) arising from certain pull-backs from projective spaces are generated by the Plucker relations.

math.AG↗