arXiv subjects

Elena Gal

Publications and source records attributed to Elena Gal.

7 recordsLinked to original sources

arXiv2024

Domain Generalization with Small Data

In this work, we propose to tackle the problem of domain generalization in the context of \textit{insufficient samples}. Instead of extracting latent feature embeddings based on deterministic models, we propose to learn a domain-invariant representation based on the probabilistic framework by mapping each data point into probabilistic embeddings. Specifically, we first extend empirical maximum mean discrepancy (MMD) to a novel probabilistic MMD that can measure the discrepancy between mixture distributions (i.e., source domains) consisting of a series of latent distributions rather than latent points. Moreover, instead of imposing the contrastive semantic alignment (CSA) loss based on pairs of latent points, a novel probabilistic CSA loss encourages positive probabilistic embedding pairs to be closer while pulling other negative ones apart. Benefiting from the learned representation captured by probabilistic models, our proposed method can marriage the measurement on the \textit{distribution over distributions} (i.e., the global perspective alignment) and the distribution-based contrastive semantic alignment (i.e., the local perspective alignment). Extensive experimental results on three challenging medical datasets show the effectiveness of our proposed method in the context of insufficient data compared with state-of-the-art methods.

cs.LG↗

arXiv2023

Unbiased Decisions Reduce Regret: Adversarial Domain Adaptation for the Bank Loan Problem

In many real world settings binary classification decisions are made based on limited data in near real-time, e.g. when assessing a loan application. We focus on a class of these problems that share a common feature: the true label is only observed when a data point is assigned a positive label by the principal, e.g. we only find out whether an applicant defaults if we accepted their loan application. As a consequence, the false rejections become self-reinforcing and cause the labelled training set, that is being continuously updated by the model decisions, to accumulate bias. Prior work mitigates this effect by injecting optimism into the model, however this comes at the cost of increased false acceptance rate. We introduce adversarial optimism (AdOpt) to directly address bias in the training set using adversarial domain adaptation. The goal of AdOpt is to learn an unbiased but informative representation of past data, by reducing the distributional shift between the set of accepted data points and all data points seen thus far. AdOpt significantly exceeds state-of-the-art performance on a set of challenging benchmark problems. Our experiments also provide initial evidence that the introduction of adversarial domain adaptation improves fairness in this setting.

cs.LG↗

arXiv2019

Higher Segal spaces and Lax $\mathbb{A}_\infty$-algebras

The notion of a higher Segal space was introduced by Dyckerhoff and Kapranov as a general framework for studying higher associativity inherent in a wide range of mathematical objects. In the present work we formalize the connection between this notion and the notion of $\mathbb{A}_\infty$-algebra. We introduce the notion of a "$d$-lax $\mathbb{A}_\infty$-algebra object" which generalizes the notion of an $\mathbb{A}_\infty$-algebra object. We describe a construction that assigns to a simplicial object $S_\bullet$ in a category $\mathscr{S}$ a datum of higher associators. We show that this datum defines a $d$-lax $\mathbb{A}_\infty$-algebra object in the category of correspondences in $\mathscr{S}$ precisely when $S_\bullet$ is a $(d+1)$-Segal object. More concretely we prove that for $n\geq d$ the "$n$-dimensional associator" is invertible. The so called "upper" and "lower" $d$-Segal conditions which originally come from the geometry of polytopes appear naturally in our construction as the two conditions which together imply the invertibility of the $d$-dimensional associator. A corollary is that for $d=2$, our construction defines an $\mathbb{A}_\infty$-algebra in the $(\infty,1)$-category of correspondences in $\mathscr{S}$ with the $2$-Segal conditions implying invertibility of all associativity data.

math.AT↗

arXiv2018

Hall categories and KLR categorification

This paper is the first step in the project of categorifying the bialgebra structure on the half of quantum group $U_{q}(\mathfrak{g})$ by using geometry and Hall algebras. We equip the category of D-modules on the moduli stack of objects of the category $Rep_{\mathbb{C}}(Q)$ of representations of a quiver with the structure of an algebra object in the category of stable $\infty$-categories. The data for this construction is provided by an extension of the Waldhausen construction for the category $Rep_{\mathbb{C}}(Q)$. We discuss the connection to the Khovanov-Lauda-Rouquier categorification of half of the quantum group $U_{q}(\mathfrak{g})$ associated to the quiver $Q$ and outline our approach to the categorification of the bialgebra structure.

math.RT↗

arXiv2016

Hopf categories and the categorification of the Heisenberg algebra via graphical calculus

We explore the connection between the notion of Hopf category and the categorification of the infinite dimensional Heisenberg algebra via graphical calculus proposed by M.Khovanov. We show that the existence of a Hopf structure on a semisimple symmetric monoidal abelian category implies existence of a categorical action in the sense of Khovanov and thus leads to a strong categorification of this algebra.

math.RT↗

arXiv2016

A geometric approach to Hall algebras I: Higher Associativity

We construct a geometric system from which the Hall algebra can be recovered. This system inherently satisfies higher associativity conditions and thus leads to a categorification of the Hall algebra. We then suggest how to use this approach to construct categorified representations.

math.RT↗

arXiv2016

Symmetric self-adjoint Hopf categories and a categorical Heisenberg double

Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive self-adjoint Hopf algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the symmetric self-adjoint Hopf structure on the categories of polynomial functors and equivariant polynomial functors. In the case of the category of polynomial functors we obtain categorification of the Fock space representation of the infinite-dimensional Heisenberg algebra.

math.RT↗