Estimating Degradation of Machine Learning Data Assets

Large-scale adoption of Artificial Intelligence and Machine Learning (AI-ML) models fed by heterogeneous, possibly untrustworthy data sources has spurred interest in estimating degradation of such models due to spurious, adversarial, or low-quality data assets. We propose a quantitative estimate of the severity of classifiers’ training set degradation: an index expressing the deformation of the convex hulls of the classes computed on a held-out dataset generated via an unsupervised technique. We show that our index is computationally light, can be calculated incrementally and complements well existing ML data assets’ quality measures. As an experimentation, we present the computation of our index on a benchmark convolutional image classifier.

METHOD FOR ELIMINATING ANOMALOUS MEASUREMENTS IN ANALYSIS OF THE MULTI-DIMENSIONAL DATABASE IN SOLVING THE DECISION-MAKING PROBLEM

The paper proposes a method for eliminating abnormal measurements (outliers) to improve the quality of multivariate data in statistical studies. Such a problem arises, for example, in the theory of managerial decision-making, since when calculating estimates of the parameters of probability distributions, the presence of anomalous (that is, those that significantly increase the confidence interval) measurements in the sample can distort the results of a statistical study, and, consequently, the main problem. The peculiarity of the proposed method is a combination of statistical and geometric methods, namely: the Gestwirt estimation method, the Tukey procedure, and a modification of the method for constructing the convex hull of a finite set of points in a multidimensional space. A set of multidimensional data is associated with a set of points of a multidimensional space. To find and eliminate outliers, a sequence of nested convex hulls, polytopes, is constructed, each of which is described by the intersection of half-spaces (support facets). A detailed algorithm for finding anomalous measurements is given. Their elimination corresponds to the successive elimination of the boundary points of nested convex hulls. The Gestwirt estimate gives the condition for stopping the operation of the algorithm. The proposed method does not require large computational costs and can be widely used in solving both theoretical and practical problems related to the processing of multidimensional data. The numerical results of the method with the number of data components 4 and 5 are presented.

Normal Polytopes and Ellipsoids

We show that: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary of the polytope if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in $\mathbb{R}^3$ has a unimodular cover, and (3) for every $d\geqslant 5$, there are ellipsoids in $\mathbb{R}^d$, such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (c) answers a question of Bruns, Michałek, and the author.

Angle Sums of Schläfli Orthoschemes

We consider the simplices $$\begin{aligned} K_n^A=\{x\in {\mathbb {R}}^{n+1}:x_1\ge x_2\ge \cdots \ge x_{n+1},x_1-x_{n+1}\le 1,\,x_1+\cdots +x_{n+1}=0\} \end{aligned}$$ K n A = { x ∈ R n + 1 : x 1 ≥ x 2 ≥ ⋯ ≥ x n + 1 , x 1 - x n + 1 ≤ 1 , x 1 + ⋯ + x n + 1 = 0 } and $$\begin{aligned} K_n^B=\{x\in {\mathbb {R}}^n:1\ge x_1\ge x_2\ge \cdots \ge x_n\ge 0\}, \end{aligned}$$ K n B = { x ∈ R n : 1 ≥ x 1 ≥ x 2 ≥ ⋯ ≥ x n ≥ 0 } , which are called the Schläfli orthoschemes of types A and B, respectively. We describe the tangent cones at their j-faces and compute explicitly the sums of the conic intrinsic volumes of these tangent cones at all j-faces of $$K_n^A$$ K n A and $$K_n^B$$ K n B . This setting contains sums of external and internal angles of $$K_n^A$$ K n A and $$K_n^B$$ K n B as special cases. The sums are evaluated in terms of Stirling numbers of both kinds. We generalize these results to finite products of Schläfli orthoschemes of type A and B and, as a probabilistic consequence, derive formulas for the expected number of j-faces of the Minkowski sums of the convex hulls of a finite number of Gaussian random walks and random bridges. Furthermore, we evaluate the analogous angle sums for the tangent cones of Weyl chambers of types A and B and finite products thereof.

Constraining the body mass range of Anzu wyliei using volumetric and extant-scaling methods

The ability to accurately and reliably estimate body mass of extinct taxa is a vital tool for interpreting the physiology and even behavior of long-dead animals. For this reason, paleontologists have developed many possible methods of estimating the body mass of extinct animals, with varying degrees of success. These methods can be divided into two main categories: volumetric mass estimation and extant scaling methods. Each has advantages and disadvantages, which is why, when possible, it is best to perform both, and compare the results to determine what is most plausible within reason. Here we employ volumetric mass estimation (VME) to calculate an approximate body mass for previously described specimens of Anzu wyliei from the Carnegie Museum of Natural History. We also use extant scaling methods to try to obtain a reliable mass estimate for this taxon. In addition, we present the first digital life restoration and convex hull of the dinosaur Anzu wyliei used for mass estimation purposes. We found that the volumetric mass estimation using our digital model was 216-280kg, which falls within the range predicted by extant scaling techniques, while the mass estimate using minimum convex hulls was below the predicted range, between 159-199 kg . The VME method for Anzu wyliei strongly affirms the predictive utility of extant-based scaling. However, volumetric mass estimates are likely more precise because the models are based on comprehensive specimen anatomy rather than regressions of a phylogenetically comprehensive but disparate sample.

Radon Numbers Grow Linearly

Define the k-th Radon number $$r_k$$ r k of a convexity space as the smallest number (if it exists) for which any set of $$r_k$$ r k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that $$r_k$$ r k grows linearly, i.e., $$r_k\le c(r_2)\cdot k$$ r k ≤ c ( r 2 ) · k .