arXiv subjects

Daniel Rademacher

Publications and source records attributed to Daniel Rademacher.

8 recordsLinked to original sources

arXiv2026

Frequency Domain Bootstrap for Functional Time Series

A frequency domain bootstrap procedure for functional time series is proposed and applied to the class of spectral mean operators. The procedure works by first generating independent pseudo periodogram operators across the positive Fourier frequencies using an estimator of the spectral density operator involved. Functional replicates of the spectral mean operators of interest are then generated. Through an additive, projection-based decomposition of the bootstrapped spectral mean operator, its leading $m$-dimensional part is properly complemented to also capture the relevant fourth order characteristics of the process. The complementation is achieved by means of a resampling procedure based on convolved periodogram operators of subsamples. The resulting bootstrap spectral mean operator consistently estimates the entire second order as well as the $m$-dimensional fourth order structure of the distribution of spectral mean operators. By allowing for the decomposition parameter $m$ to increase to infinity as the sample size increases to infinity, consistency in estimating the entire fourth order structure of the process also is achieved. The asymptotic theory developed investigates properties of the procedure for fixed and for increasing $m$ and establishes validity of the frequency domain bootstrap proposal under rather weak conditions on the underlying functional process class.

math.ST↗

arXiv2026

Structure of Cayley Codes

Cayley codes, introduced by Kaufman and Wigderson, are linear codes constructed from a Cayley graph and a smaller linear code. We explore general properties of the class of Cayley codes for finite groups. In particular we give a reduction to Cayley codes for connected Cayley graphs that maintains code properties such as rate, minimum distance and symmetry. Also, for a given Cayley code, we identify a family of symmetric Cayley codes, each associated with a normal edge-transitive Cayley graph, such that the given Cayley code embeds into the direct sum of the symmetric Cayley codes. We analyse several families of examples, in particular studying the behaviour of the Cayley code construction under forming direct products and cartesian products of Cayley graphs, and we pose a number of open questions.

math.CO↗

arXiv2024

Constructive Recognition of Special Linear Groups

We introduce a new constructive recognition algorithm for finite special linear groups in their natural representation. Given a group $G$ generated by a set of $d\times d$ matrices over a finite field $\mathbb{F}_q$, known to be isomorphic to the special linear group $\mathrm{SL}(d,q)$, the algorithm computes a special generating set $S$ for $G$. These generators enable efficient computations with the input group, including solving the word problem. Implemented in the computer algebra system GAP, our algorithm outperforms existing state-of-the-art algorithms by a significant margin. A detailed complexity analysis of the algorithm will be presented in an upcoming publication.

math.GR↗

arXiv2024

Two-sample tests for relevant differences in persistence diagrams

We study two-sample tests for relevant differences in persistence diagrams obtained from $L^p$-$m$-approximable data $(\mathcal{X}_t)_t$ and $(\mathcal{Y}_t)_t$. To this end, we compare variance estimates w.r.t.\ the Wasserstein metrics on the space of persistence diagrams. In detail, we consider two test procedures. The first compares the Fr{\'e}chet variances of the two samples based on estimators for the Fr{\'e}chet mean of the observed persistence diagrams $PD(\mathcal{X}_i)$ ($1\le i\le m$), resp., $PD(\mathcal{Y}_j)$ ($1\le j\le n$) of a given feature dimension. We use classical functional central limit theorems to establish consistency of the testing procedure. The second procedure relies on a comparison of the so-called independent copy variances of the respective samples. Technically, this leads to functional central limit theorems for U-statistics built on $L^p$-$m$-approximable sample data.

math.ST↗

arXiv2023

On the stability of filtration functions for dependent data with applications to break detection

In this paper, we study the stability of commonly used filtration functions in topological data analysis under small perturbations of the underlying nonrandom point cloud. Relying on these stability results, we then develop a test procedure to detect and determine structural breaks in a sequence of topological data objects obtained from weakly dependent data. The proposed method applies for instance to statistics of persistence diagrams of $\mathbb{R}^d$-valued Bernoulli shift systems under the \v{C}ech or Vietoris-Rips filtration.

math.ST↗

arXiv2022

Statistical inference for intrinsic wavelet estimators of SPD matrices in a log-Euclidean manifold

In this paper we treat statistical inference for an intrinsic wavelet estimator of curves of symmetric positive definite (SPD) matrices in a log-Euclidean manifold. This estimator preserves positive-definiteness and enjoys permutation-equivariance, which is particularly relevant for covariance matrices. Our second-generation wavelet estimator is based on average-interpolation and allows the same powerful properties, including fast algorithms, known from nonparametric curve estimation with wavelets in standard Euclidean set-ups. The core of our work is the proposition of confidence sets for our high-level wavelet estimator in a non-Euclidean geometry. We derive asymptotic normality of this estimator, including explicit expressions of its asymptotic variance. This opens the door for constructing asymptotic confidence regions which we compare with our proposed bootstrap scheme for inference. Detailed numerical simulations confirm the appropriateness of our suggested inference schemes.

stat.ME↗

arXiv2021

Testing Equality of Spectral Density Operators for Functional Processes

The problem of comparing the entire second order structure of two functional processes is considered and a $L^2$-type statistic for testing equality of the corresponding spectral density operators is investigated. The test statistic evaluates, over all frequencies, the Hilbert-Schmidt distance between the two estimated spectral density operators. Under certain assumptions, the limiting distribution under the null hypothesis is derived. A novel frequency domain bootstrap method is introduced, which leads to a more accurate approximation of the distribution of the test statistic under the null than the large sample Gaussian approximation derived. Under quite general conditions, asymptotic validity of the bootstrap procedure is established for estimating the distribution of the test statistic under the null. Furthermore, consistency of the bootstrap-based test under the alternative is proved. Numerical simulations show that, even for small samples, the bootstrap-based test has a very good size and power behavior. An application to a bivariate real-life functional time series illustrates the methodology proposed.

math.ST↗

arXiv2013

Showcasing straight-line programs with memory via matrix Bruhat decomposition

We suggest that straight-line programs designed for algebraic computations should be accompanied by a comprehensive complexity analysis that takes into account both the number of fundamental algebraic operations needed, as well as memory requirements arising during evaluation. We introduce an approach for formalising this idea and, as illustration, construct and analyse straight-line programs for the Bruhat decomposition of $d\times d$ matrices with determinant $1$ over a finite field of order $q$ that have length $O(d^2\log(q))$ and require storing only $O(\log(q))$ matrices during evaluation.

cs.DS↗