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Jackson Walters

Publications and source records attributed to Jackson Walters.

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Fairis: Fairness-Aware Aggregation with Provable Influence Containment against Fairness Poisoning Attacks in Collaborative Machine Learning

Collaborative machine learning among financial institutions must be both group-fair and robust against deliberate adversarial manipulation. Existing fairness-aware aggregation methods remain formally vulnerable to fairness poisoning: a malicious client maximizing group disparity while preserving accuracy evades accuracy-based Byzantine defenses, and in our threat model FairFed's gap-based weighting can be gamed by an adversary who observes the global fairness score. We present Fairis, a server-side reweighting scheme in which each client's update receives the normalized weight $\omega_k = \bar{w}_k / \sum_j \bar{w}_j$ built from the unnormalized score $\bar{w}_k = \eta - \mathcal{F}_k$, with $\mathcal{F}_k \in [0,1]$ the local Equal Opportunity Difference and $\eta > 1$ a security parameter. We prove three properties, Monotone Weight Reduction (MWR), Demographic Participation, and Non-Gamesmanship, extend MWR to colluding minority coalitions, and show that combining MWR with server-side norm clipping bounds the adversary's displacement of the global model by $\omega_0 C$, strictly decreasing in its own reported disparity. Assuming honest score reporting, an assumption this paper does not discharge, Fairis is the only rule evaluated that guarantees every client strictly positive weight while provably reducing an adversary's weight monotonically in its bias; clipped FairFed can reach a lower weight but guarantees nothing and zeroes a client outright on Taiwan Credit. Against an adversary stealthy enough to evade accuracy-based defenses, within 0.04 accuracy of benign, Fairis cuts its weight by 41 to 54% below a size-blind control on Taiwan. On routine non-IID partitions no rule dominates, and a uniform-weighting ablation shows that containment tracks how far the adversary's score separates from the honest mean, providing none when the honest population is already unfair.

cs.CR

Edge-Span Chern Algebras of Graphical Configuration Spaces

Place the vertices of a finite graph at projective points. Each edge defines a span map to $\mathrm{Gr}(2,n)$; the pulled-back Chern classes generate a graded algebra $A_G^{(n)}$. This assignment is a covariant graph functor, so the abstract graded-algebra type is a graph invariant. In ambient dimension four, the Hilbert series is incomparable with the chromatic and Tutte polynomials. On five vertices in dimension three, the $34$ graph classes yield $33$ algebra types, strictly refining both classical polynomials. For every tree $T$, we obtain a closed Hilbert-series formula depending only on $|V(T)|$ and $n$, while $A_T^{(3)}$ determines $T$ up to isomorphism. More precisely, its cubic relation data is a complete tree invariant of polynomial size. Finally, the graphical configuration space embeds as a dense open in the picture variety, and picture spaces with equal additive homology can have nonisomorphic edge-Chern algebras.

math.CO

The Modular DFT of the Symmetric Group

We describe the discrete Fourier transform (DFT) for a cyclic group when $p|N$ by factoring $x^N-1$ over finite fields and constructing the Fourier transform and its inverse using B\'{e}zout's identity for polynomials. For the symmetric group, in the modular case when $p|n!$ we construct the Peirce decomposition using central primitive orthogonal idempotents, yielding a change-of-basis matrix which generalizes the DFT. We compute the unitary DFT for the symmetric group over number fields containing sufficiently many square roots. For $n=3$, we compute the Galois group of the splitting field of the characteristic polynomial. All constructions are implemented in SageMath.

math.RT

Toroidal prefactorization algebras associated to holomorphic fibrations and a relationship to vertex algebras

Let $X$ be a complex manifold, $\pi: E \rightarrow X$ a locally trivial holomorphic fibration with fiber $F$, and $\mathfrak{g}$ a Lie algebra with an invariant symmetric form. We associate to this data a holomorphic prefactorization algebra $\mathcal{F}_{\mathfrak{g}, \pi}$ on $X$ in the formalism of Costello-Gwilliam. When $X=\mathbb{C}$, $\mathfrak{g}$ is simple, and $F$ is a smooth affine variety, we extract from $\mathcal{F}_{\mathfrak{g}, \pi}$ a vertex algebra which is a vacuum module for the universal central extension of the Lie algebra $\mathfrak{g} \otimes H^{0}(F, \mathcal{O})[z,z^{-1}]$. As a special case, when $F$ is an algebraic torus $(\mathbb{C}^{*})^n$, we obtain a vertex algebra naturally associated to an $(n+1)$--toroidal algebra, generalizing the affine vacuum module.

math.QA

Averages of Unlabeled Networks: Geometric Characterization and Asymptotic Behavior

It is becoming increasingly common to see large collections of network data objects -- that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.

math.ST