arXiv subjects

Polina Barabanshchikova

Publications and source records attributed to Polina Barabanshchikova.

4 recordsLinked to original sources

arXiv2026

Divide-and-Denoise: A Game-Theoretic Method for Fairly Composing Diffusion Models

The abundance of pre-trained diffusion models provides an opportunity for composition. Combining several models, however, runs the risk of one model dominating or models disagreeing with each other. Here, we propose Divide-and-Denoise, a method for coordinating multiple pre-trained diffusion models during sampling. Much like managing a specialized workforce, our method creates a fair but efficient division of labor across models. Central to our method is the notion of an allocation which defines the responsibility of each model to every region of the noisy sample. At every timestep, we then denoise by (i) updating the allocation by solving a fair division game, where we divide the sample into regions that maximize total utility under fairness constraints, and (ii) aligning the models with this allocation, where we guide each model to denoise within its assigned region. This leads to a new composite denoising process that evolves in tandem with a division process. We evaluate Divide-and-Denoise on conditional image generation. Across several quality metrics, including the GenEval benchmark, our method outperforms baselines and resolves common failures including missing objects and mismatched attributes. Experiments show that Divide-and-Denoise utilizes each model's expertise without neglecting any other model.

cs.CV↗

arXiv2025

Tight colorful no-dimensional Tverberg theorem

We study colorful no-dimensional Tverberg-type problems and obtain several optimal results. A colorful no-dimensional Tverberg-type theorem provides a bound on a radius $R$ such that, for any pairwise disjoint $k$-element subsets $Q_1,\dots,Q_n$ of a normed space, there exists a partition of $Q_1\cup\cdots\cup Q_n$ into disjoint transversals $\{P_1,\dots,P_k\}$ for which a ball of radius $R$ intersects the convex hull of each $P_i$ ($1\le i\le k$). Our methods are deterministic and dimension-free, and they are unified by optimizing two functionals: a quadratic \emph{selection} functional whose local maximizers produce a complete system of disjoint transversals, and a convex \emph{intersection} functional that certifies a common point. First, in the Euclidean setting we bound $R$ in terms of the Chebyshev radii (minimal enclosing-ball radii) of the color classes $Q_1,\dots,Q_n$. A key observation is a ``combinatorial'' subadditivity of the squared Chebyshev radius: given sequences $X=(x_1,\dots,x_k)$ and $Y=(y_1,\dots,y_k)$ of points in a Euclidean space, contained in balls of radii $R_X$ and $R_Y$ (not necessarily with the same center), one can reenumerate $Y$ so that the pointwise-sum sequence $Z=(x_1+y_1,\dots,x_k+y_k)$ is contained in a ball of radius $R_Z$ satisfying \[ R_Z^2 \le R_X^2 + R_Y^2 . \] As a corollary, we obtain the best-possible bound \[ R \le \frac{1}{\sqrt{2n}}\sqrt{\frac{k-1}{k}}\, \max_{1\le i\le n} \operatorname{diam}(Q_i). \] Our algorithm returns the desired disjoint transversals in overall time $\mathcal{O}(nk^3)$. Second, we develop a complementary approach based on the inter-color diameter and extend the framework to obtain no-dimensional colorful Tverberg-type results in the hyperbolic setting and in Banach spaces.

math.MG↗

arXiv2023

Intersecting ellipses induced by a max-sum matching

For an even set of points in the plane, choose a max-sum matching, that is, a perfect matching maximizing the sum of Euclidean distances of its edges. For each edge of the max-sum matching, consider the ellipse with foci at the edge's endpoints and eccentricity $\sqrt 3 / 2$. Using an optimization approach, we prove that the convex sets bounded by these ellipses intersect, answering a Tverberg-type question of Andy Fingerhut from 1995.

cs.CG↗

arXiv2023

Intersecting diametral balls induced by a geometric graph II

For a graph whose vertices are points in $\mathbb R^d$, consider the closed balls with diameters induced by its edges. The graph is called a Tverberg graph if these closed balls intersect. A max-sum tree of a finite point set $X \subset \mathbb R^d$ is a tree with vertex set $X$ that maximizes the sum of Euclidean distances of its edges among all trees with vertex set $X$. Similarly, a max-sum matching of an even set $X \subset \mathbb R^d$ is a perfect matching of $X$ maximizing the sum of Euclidean distances between the matched points among all perfect matchings of $X$. We prove that a max-sum tree of any finite point set in $\mathbb R^d$ is a Tverberg graph, which generalizes a recent result of Abu-Affash et al., who established this claim in the plane. Additionally, we provide a new proof of a theorem by Bereg et al., which states that a max-sum matching of any even point set in the plane is a Tverberg graph. Moreover, we proved a slightly stronger version of this theorem.

math.CO↗