Tootfinder

Quantifying and testing dependence to categorical variables
Siegfried H\"ormann, Daniel Strenger-Galvis
https://arxiv.org/abs/2509.10268 https://arxiv…

Quantifying and testing dependence to categorical variables
We suggest a dependence coefficient between a categorical variable and some general variable taking values in a metric space. We derive important theoretical properties and study the large sample behaviour of our suggested estimator. Moreover, we develop an independence test which has an asymptotic $χ^2$-distribution if the variables are independent and prove that this test is consistent against any violation of independence. We discuss some extensions, including a variant of the coefficient f…

Stabilising Lifetime PD Models under Forecast Uncertainty
Vahab Rostampour
https://arxiv.org/abs/2509.10586 https://arxiv.org/pdf/2509.10586

Stabilising Lifetime PD Models under Forecast Uncertainty
Estimating lifetime probabilities of default (PDs) under IFRS~9 and CECL requires projecting point--in--time transition matrices over multiple years. A persistent weakness is that macroeconomic forecast errors compound across horizons, producing unstable and volatile PD term structures. This paper reformulates the problem in a state--space framework and shows that a direct Kalman filter leaves non--vanishing variability. We then introduce an anchored observation model, which incorporates a neut…

Variable Matrix-Weighted Besov Spaces
Dachun Yang, Wen Yuan, Zongze Zeng
https://arxiv.org/abs/2509.07786 https://arxiv.org/pdf/2509.07786

Variable Matrix-Weighted Besov Spaces
In this article, applying matrix ${\mathcal A}_{p(\cdot),\infty}$ weights introduced in our previous work, we introduce the matrix-weighted variable Besov space via the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, Then we show that, defined either by the matrix weight $W$ or the reducing operators ${\mathbb{A}}$ of order $p(\cdot)$ for $W$, the matrix-weighted variable Besov spaces (respectively, the matrix-weighted variable Besov sequence spaces) are …

Exploring the Design Space of Fair Tree Learning Algorithms
Kiara Stempel, Mattia Cerrato, Stefan Kramer
https://arxiv.org/abs/2509.03204 https://arxiv.org…

Exploring the Design Space of Fair Tree Learning Algorithms
Decision trees have been studied extensively in the context of fairness, aiming to maximize prediction performance while ensuring non-discrimination against different groups. Techniques in this space usually focus on imposing constraints at training time, constraining the search space so that solutions which display unacceptable values of relevant metrics are not considered, discarded, or discouraged. If we assume one target variable y and one sensitive attribute s, the design space of tree lea…

Conic optimization techniques yield sufficient conditions for set-completely positive matrix completion under arrowhead specification pattern
Markus Gabl
https://arxiv.org/abs/2509.19348

Conic optimization techniques yield sufficient conditions for set-completely positive matrix completion under arrowhead specification pattern
Matrix completion results deal with the question of when a partially specified symmetric matrix can be completed to a member of certain matrix cones. Results from positive semidefinite matrix completion and completely positive matrix completion have been successfully applied in optimization to greatly reduce the number of variables in conic optimization problems in the space of symmetric matrices. In this text, we go the other direction and show that we can use tools from conic optimization (mo…

Interpretable Kernels
Patrick J. F. Groenen, Michael Greenacre
https://arxiv.org/abs/2508.15932 https://arxiv.org/pdf/2508.15932

Interpretable Kernels
The use of kernels for nonlinear prediction is widespread in machine learning. They have been popularized in support vector machines and used in kernel ridge regression, amongst others. Kernel methods share three aspects. First, instead of the original matrix of predictor variables or features, each observation is mapped into an enlarged feature space. Second, a ridge penalty term is used to shrink the coefficients on the features in the enlarged feature space. Third, the solution is not obtain…

On the Kolmogorov-Feller weak law of large numbers for Frechet mean on non-compact symmetric spaces
Jongmin Lee, Sungkyu Jung
https://arxiv.org/abs/2509.02074 https://

On the Kolmogorov-Feller weak law of large numbers for Frechet mean on non-compact symmetric spaces
We establish the Kolmogorov-Feller weak law of large numbers for Frechet mean on non-compact symmetric spaces under certain regularity conditions. Our results accommodate non-identically distributed random variables and are accompanied by an illustrative example that demonstrates their applicability to the space of symmetric positive-definite matrices.

Rings of non-commutative functions and their fields of fractions
M\'eric L. Augat, Robert T. W. Martin, Eli Shamovich
https://arxiv.org/abs/2509.21270 https://

Rings of non-commutative functions and their fields of fractions
Semi-free ideal rings, or semifirs, were introduced by Paul M. Cohn to study universal localizations in the non-commutative setting. We provide new examples of semifirs consisting of analytic functions in several non-commuting variables. These examples arise canonically in free analysis by completing the free algebra in the topology of ``uniform convergence on operator-space balls'' in the non-commutative universe of tuples of square matrices of any finite size. We show, in particular, that t…

Sums of projections with random coefficients
Leonid Pastur, Alexander Pushnitski
https://arxiv.org/abs/2509.21539 https://arxiv.org/pdf/2509.21539

Sums of projections with random coefficients
We study infinite sums \[ {\mathcal P}_{\varkappa}=\sum_{n=-\infty}^\infty \varkappa_n \langle\cdot, ψ_n\rangleψ_n \] of rank-one projections in a Hilbert space, where $\{ψ_n\}_{n\in\mathbb Z}$ are norm-one vectors, not necessarily orthogonal, and $\{\varkappa_n\}_{n\in\mathbb Z}$ are independent identically distributed positive random variables. Assuming that the Gram matrix $\{\langleψ_n,ψ_m\rangle\}_{n,m\in\mathbb Z}$ defines a bounded operator on $\ell^2(\mathbb Z)$ and that its entrie…

Eigenvalue distribution of some random matrices
Jebalia Mohamed, Ahmed Souabni
https://arxiv.org/abs/2507.15342 https://arxiv.org/pdf…

Eigenvalue distribution of some random matrices
In this paper, we investigate the eigenvalue distribution of a class of kernel random matrices whose $(i,j)$-th entry is $f(X_i,X_j)$ where $f$ is a symmetric function belonging to the Paley-Wiener space $\mathcal{B}_c$ and $(X_i)_{1\leq i \leq N}$ are i.i.d. random variables. We rigorously prove that, with high probability, the eigenvalues of these random matrices are well approximated by those of an underlying estimator. A particularly notable case is when $f=sinc$ , which has been widely…