Simon Dohn | Searcharxiv

arXiv2024

Improved Hardness Results for the Clearing Problem in Financial Networks with Credit Default Swaps

We study computational problems in financial networks of banks connected by debt contracts and credit default swaps (CDSs). A main problem is to determine \emph{clearing} payments, for instance right after some banks have been exposed to a financial shock. Previous works have shown the $\varepsilon$-approximate version of the problem to be $\mathrm{PPAD}$-complete and the exact problem $\mathrm{FIXP}$-complete. We show that $\mathrm{PPAD}$-hardness hold when $\varepsilon \approx 0.101$, improving the previously best bound significantly. Due to the fact that the clearing problem typically does not have a unique solution, or that it may not have a solution at all in the presence of default costs, several natural decision problems are also of great interest. We show two such problems to be $\exists\mathbb{R}$-complete, complementing previous $\mathrm{NP}$-hardness results for the approximate setting.

cs.GT↗

arXiv2023

Relating tSNE and UMAP to Classical Dimensionality Reduction

It has become standard to use gradient-based dimensionality reduction (DR) methods like tSNE and UMAP when explaining what AI models have learned. This makes sense: these methods are fast, robust, and have an uncanny ability to find semantic patterns in high-dimensional data without supervision. Despite this, gradient-based DR methods lack the most important quality that an explainability method should possess: themselves being explainable. That is, given a UMAP output, it is currently unclear what one can say about the corresponding input. We work towards closing this question by relating UMAP to classical DR techniques. Specifically, we show that one can fully recover methods like PCA, MDS, and ISOMAP in the modern DR paradigm: by applying attractions and repulsions onto a randomly initialized dataset. We also show that, with a small change, Locally Linear Embeddings (LLE) can indistinguishably reproduce UMAP outputs. This implies that the UMAP effective objective is minimized by this modified version of LLE (and vice versa). Given this, we discuss what must be true of UMAP emebddings and present avenues for future work.

cs.LG↗