arXiv subjects

Daniel T. Soukup

Publications and source records attributed to Daniel T. Soukup.

11 recordsLinked to original sources

arXiv2021

Representative & Fair Synthetic Data

Algorithms learn rules and associations based on the training data that they are exposed to. Yet, the very same data that teaches machines to understand and predict the world, contains societal and historic biases, resulting in biased algorithms with the risk of further amplifying these once put into use for decision support. Synthetic data, on the other hand, emerges with the promise to provide an unlimited amount of representative, realistic training samples, that can be shared further without disclosing the privacy of individual subjects. We present a framework to incorporate fairness constraints into the self-supervised learning process, that allows to then simulate an unlimited amount of representative as well as fair synthetic data. This framework provides a handle to govern and control for privacy as well as for bias within AI at its very source: the training data. We demonstrate the proposed approach by amending an existing generative model architecture and generating a representative as well as fair version of the UCI Adult census data set. While the relationships between attributes are faithfully retained, the gender and racial biases inherent in the original data are controlled for. This is further validated by comparing propensity scores of downstream predictive models that are trained on the original data versus the fair synthetic data. We consider representative & fair synthetic data a promising future building block to teach algorithms not on historic worlds, but rather on the worlds that we strive to live in.

cs.LG↗

arXiv2019

A 0-dimensional, Lindelöf space that is not strongly D

A topological space $X$ is strongly $D$ if for any neighbourhood assignment $\{U_x:x\in X\}$, there is a $D\subseteq X$ such that $\{U_x:x\in D\}$ covers $X$ and $D$ is locally finite in the topology generated by $\{U_x:x\in X\}$. We prove that $\diamondsuit$ implies that there is an $HFC_w$ space in $2^{ω_1}$ (hence 0-dimensional, Hausdorff and hereditarily Lindelöf) which is not strongly $D$. We also show that any $HFC$ space $X$ is dually discrete and if additionally, countable sets have Menger closure then $X$ is a $D$-space.

math.GN↗

arXiv2018

More ZFC inequalities between cardinal invariants

Motivated by recent results and questions of D. Raghavan and S. Shelah, we present ZFC theorems on the bounding and various almost disjointness numbers, as well as on reaping and dominating families on uncountable, regular cardinals. We show that if $κ=λ^+$ for some $λ\geq ω$ and $\mathfrak b(κ)=κ^+$ then $\mathfrak a_e(κ)=\mathfrak a_p(κ)=κ^+$. If, additionally, $2^{<λ}=λ$ then $\mathfrak a_g(κ)=κ^+$ as well. Furthermore, we prove a variety of new bounds for $\mathfrak d(κ)$ in terms of $\mathfrak r(κ)$, including $\mathfrak d(κ)\leq \mathfrak r_σ(κ)\leq \mathrm{cof}([\mathfrak r(κ)]^ω)$, and $\mathfrak d(κ)\leq \mathfrak r(κ)$ whenever $\mathfrak r(κ)<\mathfrak b(κ)^{+κ}$ or $\mathrm{cof}(\mathfrak r(κ))\leq κ$ holds.

math.LO↗

arXiv2017

Cycle reversions and dichromatic number in tournaments

We show that if $D$ is a tournament of arbitrary size then $D$ has finite strong components after reversing a locally finite sequence of cycles. In turn, we prove that any tournament can be covered by two acyclic sets after reversing a locally finite sequence of cycles. This provides a partial solution to a conjecture of S. Thomassé.

math.CO↗

arXiv2017

On spaces with $σ$-closed-discrete dense sets

The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "there is a product of at most $\mathfrak c$ many $e$-separable spaces that fails to be $e$-separable'" is equiconsistent with the existence of a weakly compact cardinal.

math.GN↗

arXiv2017

Balanced independent sets in graphs omitting large cliques

Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such that for any balanced $r$-colouring of the vertices of $G$ one can find an independent set which meets at least $m$ colour classes in a set of size $|V|$. Answering a conjecture of S. Thomassé, we express the exact value of $r(H_n,m)$ (using Ramsey-numbers for finite digraphs), where $H_n$ is Henson's countable universal homogeneous $K_n$-free graph. In turn, we deduce a new partition property of $H_n$ regarding balanced embeddings of bipartite graphs: for any finite bipartite $G$ with bipartition $A,B$, if the vertices of $H_n$ are partitioned into two infinite classes then there is an induced copy of $G$ in $H_n$ such that the images of $A$ and $B$ are contained in different classes.

math.CO↗

arXiv2016

Decompositions of edge-coloured infinite complete graphs into monochromatic paths II

P. Erdős proved that every 2-edge coloured complete graph on the natural numbers can be vertex decomposed into two monochromatic paths of different colour. This result was extended by R. Rado to an arbitrary finite number of colours. We prove that the vertices of every finite-edge coloured infinite complete graph can be partitioned into disjoint monochromatic paths of different colours. This answers a question of R. Rado from 1978.

math.CO↗

arXiv2014

Davies-trees in infinite combinatorics

This short note, prepared for the Logic Colloquium 2014, provides an introduction to Davies-trees and presents new applications in infinite combinatorics. In particular, we give new and simple proofs to the following theorems of P. Komjáth: every $n$-almost disjoint family of sets is essentially disjoint for any $n\in \mathbb N$; $\mathbb R^2$ is the union of $n+2$ clouds if the continuum is at most $\aleph_n$ for any $n\in \mathbb N$; every uncountably chromatic graph contains $n$-connected uncountably chromatic subgraphs for every $n\in \mathbb N$.

math.LO↗

arXiv2014

Partitioning bases of topological spaces

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a T_3 Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first countable, 0-dimensional, Hausdorff space of size continuum and weight ω_1 which admits a point countable base without a partition to two bases. Several related results are proved and the paper finishes with a list of open problems.

math.GN↗

arXiv2012

Comparing weak versions of separability

Our aim is to investigate spaces with sigma-discrete and meager dense sets, as well as selective versions of these properties. We construct numerous examples to point out the differences between these classes while answering questions of Tkachuk [30], Hutchinson [17] and the authors of [8].

math.GN↗

arXiv2011

A counterexample in the theory of $D$-spaces

Assuming $\diamondsuit$, we construct a $T_2$ example of a hereditarily Lindelöf space of size $ω_1$ which is not a $D$-space. The example has the property that all finite powers are also Lindelöf.

math.GN↗