Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2205.03852

Randomized geometric tools for anomaly detection in stock markets

Cyril Bachelard·Apostolos Chalkis·Vissarion Fisikopoulos·Elias Tsigaridas
View original source ↗Read source document ↗Export metadata ↓

Abstract

We propose novel randomized geometric tools to detect low-volatility anomalies in stock markets; a principal problem in financial economics. Our modeling of the (detection) problem results in sampling and estimating the (relative) volume of geodesically non-convex and non-connected spherical patches that arise by intersecting a non-standard simplex with a sphere. To sample, we introduce two novel Markov Chain Monte Carlo (MCMC) algorithms that exploit the geometry of the problem and employ state-of-the-art continuous geometric random walks (such as Billiard walk and Hit-and-Run) adapted on spherical patches. To our knowledge, this is the first geometric formulation and MCMC-based analysis of the volatility puzzle in stock markets. We have implemented our algorithms in C++ (along with an R interface) and we illustrate the power of our approach by performing extensive experiments on real data. Our analyses provide accurate detection and new insights into the distribution of portfolios' performance characteristics. Moreover, we use our tools to show that classical methods for low-volatility anomaly detection in finance form bad proxies that could lead to misleading or inaccurate results.

Explore related subjects

cs.CGecon.GNq-fin.ECstat.APstat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Cyril Bachelard, Apostolos Chalkis, Vissarion Fisikopoulos, Elias Tsigaridas. 2022-05-15. Randomized geometric tools for anomaly detection in stock markets. https://arxiv.org/abs/2205.03852

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Unlabeled Multi-Robot Motion Planning with Improved Separation Trade-offs

We study unlabeled MRMP for unit-disk robots in a polygonal environment. Although the problem is hard in general, polynomial-time solutions exist under appropriate separation assumptions on start and target positions. Banyassady et al.(SoCG'22) guarantee feasibility in simple polygons under start--start and target--target distances of at least $4$, and start--target distances of at least $3$, but without optimality guarantees. Solovey et al.(RSS'15) provide a near-optimal solution in general polygonal domains, under stricter conditions: start/target positions must have pairwise distance at least $4$, and at least $\sqrt{5}\approx2.236$ from obstacles. This raises the question of whether polynomial-time algorithms can be obtained in even more densely packed environments. In this paper we present a generalized algorithm that achieve different tradeoffs on the robots-separation $ρ$ and obstacles-separation $ω$, all significantly improving upon the state of the art. Specifically, we obtain polynomial-time constant-approximation algorithms to minimize the total path length when (i) $ρ=2\frac{2}{3}$ and $ω=1\frac{2}{3}$, or (ii) $ρ\approx3.291$ and $ω\approx1.354$. These solutions are weakly-monotone; we also provide a monotone solution requiring $ω=\approx1.614$ and $ρ=4$. We prove that monotone plans may not exist when $ω<1.614$, and weakly-monotone plans may not exist when $ω<1.354$. We then present tradeoffs between the separation bounds and the approximation factor, specifically achieving an (almost) optimal bound of $ρ=2$ at the cost of a linear approximation factor and requiring $ω=2$. This applies also for the labeled variant of MRMP, in which case we show a tight bound on $ω$. Finally, we show that without any robots-separation assumption, obstacles-separation of at least $1.5$ may be necessary for a solution to exist.

cs.CG↗
arXiv2026

Low-Dimensional Embeddings for Gaussian Kernels on Manifolds

The Gaussian kernel is a widely used similarity measure underlying kernel methods such as kernel PCA and spectral clustering, but computing Gaussian kernel distances for many pairs of points can be expensive. Using Random Fourier Features (RFF), Chen and Phillips [ALT 2017] showed that for points in a $d$-dimensional Euclidean ball in ${\mathbb R}^N$, $t=Ω((d/\varepsilon^2)\log(dR/\varepsilon))$ features suffice to preserve all pairwise Gaussian kernel distances within a $(1\pm\varepsilon)$ factor with high probability. We establish a uniform relative-error embedding theorem for the more general setting of an arbitrary positive-reach submanifold $\mathcal M\subset{\mathbb R}^N$ of intrinsic dimension $d$. We show that $t=O((d/\varepsilon^2)\log(\operatorname{vol}(\mathcal M)^2N^{2d}/(\operatorname{vol}(B_1^d(0))^2\operatorname{rch}(\mathcal M)^{2d}\varepsilon^{2d+1}δ)))$, or approximately $O((d^2/\varepsilon^2)(\log N+\log(1/(\varepsilonδ))))$, RFFs suffice, with probability $1-δ$, to preserve the Gaussian kernel distance between every pair of manifold points up to relative error $\varepsilon$. Thus the bound depends only logarithmically on the ambient dimension and on manifold parameters such as volume and reach, while retaining the $1/\varepsilon^2$ Euclidean rate. We also prove a topological consequence: under the same RFF embedding, persistent homology is preserved in the sense that weighted Cech and Rips filtrations built from Gaussian kernel power distance are $(1\pm\varepsilon_\star)$-interleaved, where $\varepsilon_\star$ accounts for both distance distortion and kernel-weight approximation.

cs.CG↗
arXiv2026

Perfectly Guarding Straits: Exact Algorithms for Weak Visibility Polygons

The Art Gallery Problem (AGP) asks for the fewest guards that see all of a simple polygon. It is $\exists\mathbb{R}$-complete, hence NP-hard. We show that for a particular class of polygons, confining guards to a single edge makes AGP exactly and efficiently solvable. We call this the Strait Guarding Problem (SGP). Its input is a weak visibility polygon (WVP): a simple polygon where every point is seen from some point of one fixed edge, the base. SGP places the fewest guards on the base that jointly see the whole polygon. First, a structural fact: guards on the base edge that cover the boundary already cover the entire interior, turning a two-dimensional covering problem into a one-dimensional one. Our main result is the Witness-Guard Algorithm, which solves SGP exactly in $O((n + \mathrm{OPT} \cdot ρ)(\log n + \log \mathrm{OPT}))$ time, where $ρ$ is the number of reflex vertices in the WVP and OPT is the minimum number of guards. It is output-sensitive and certifies optimality by a witness set of size OPT derived from its output. We also study the guarding-the-vertex version and prove a tight $Θ(n \log n)$ bound, with the lower bound following from Sorting. As a corollary of SGP, we obtain two results for altitude terrain guarding (ATG), a special case that SGP generalizes. We give a linear-time perfect-guarding algorithm, improving the previous $O(n^2 \log n)$ bound of Daescu, Friedrichs, Malik, Polishchuk and Schmidt. We also resolve their problem on the minimum guarding altitude, in $O(nk + k^2 \log k)$ time, improving on the $O(k^2 λ_{k-1}(n) \log n)$ bound of Kang, Kim and Ahn.

cs.CG↗